![Christoph Aistleitner
Metric number theory, lacunary series and
systems of dilated functions
Abstract: By a classical result of Weyl, for any increasing sequence (𝑛𝑘)𝑘≥1 of integers
the sequence of fractional parts ({𝑛𝑘𝑥})𝑘≥1 is uniformly distributed modulo 1 for al-
most all 𝑥 ∈ [0, 1]. Except for a few special cases, e.g. when 𝑛𝑘 = 𝑘, 𝑘 ≥ 1, the excep-
tionalsetcannotbedescribed explicitly. Theexactasymptotic order of thediscrepancy
of ({𝑛𝑘𝑥})𝑘≥1 for almost all 𝑥 is only known in a few special cases, for example when
(𝑛𝑘)𝑘≥1 is a (Hadamard) lacunary sequence, that is when 𝑛𝑘+1/𝑛𝑘 ≥ 𝑞 > 1, 𝑘 ≥ 1.
In this case of quickly increasing (𝑛𝑘)𝑘≥1 the system ({𝑛𝑘𝑥})𝑘≥1 (or, more generally,
(𝑓(𝑛𝑘𝑥))𝑘≥1 for a 1-periodic function 𝑓) shows many asymptotic properties which are
typical for the behavior of systems of independent random variables. Precise results
depend on a fascinating interplay between analytic, probabilistic and number-theo-
retic phenomena.
Without any growth conditions on (𝑛𝑘)𝑘≥1 the situation becomes much more compli-
cated, and the system (𝑓(𝑛𝑘𝑥))𝑘≥1 will typically fail to satisfy probabilistic limit the-
orems. An important problem which remains is to study the almost everywhere con-
vergence of series ∑∞
𝑘=1 𝑐𝑘𝑓(𝑘𝑥), which is closely related to finding upper bounds for
maximal 𝐿2
-norms of the form
1
∫
0
( max
1≤𝑀≤𝑁
𝑀
∑
𝑘=1
𝑐𝑘𝑓(𝑘𝑥)
)
2
𝑑𝑥 .
The most striking example of this connection is the equivalence of the Carleson con-
vergence theorem and the Carleson–Hunt inequality for maximal partial sums of
Fourier series. For general functions 𝑓 this is a very difficult problem, which is related
to finding upper bounds for certain sums involving greatest common divisors.
Keywords: uniform distribution theory, discrepancy theory, metric number theory, la-
cunary series, systems of dilated functions
Mathematics Subject Classification 2010: 11J83, 11K38, 42A55, 60F15, 11A05, 42A20
||
Christoph Aistleitner: Department of Applied Mathematics, School of Mathematics and Statistics,
University of New South Wales, Sydney NSW 2052, Australia, e-mail: aistleitner@math.tugraz.at
The author is supported by a Schrödinger scholarship of the Austrian Research Foundation (FWF).](https://image.slidesharecdn.com/25565190-250514212713-9e6dff5a/85/Uniform-Distribution-And-Quasimonte-Carlo-Methods-Discrepancy-Integration-And-Applications-15-320.jpg)
![2 | Christoph Aistleitner
1 Uniform distribution modulo 1
A sequence of real numbers from the unit interval (𝑥𝑘)𝑘≥1 is called uniformly dis-
tributed modulo 1 (u.d. mod 1) if for all 0 ≤ 𝑎 < 𝑏 ≤ 1 the asymptotic relation
lim
𝑁→∞
1
𝑁
𝑁
∑
𝑘=1
1[𝑎,𝑏)(𝑥𝑘) = 𝑏 − 𝑎 (1.1)
holds. Roughly speaking, a sequence is u.d. mod 1 if asymptotically every interval
[𝑎, 𝑏) ⊂ [0, 1] receives its fair share of points, which is proportional to its length. In
an informal way, uniformly distributed sequences are often considered as sequences
showing random behavior; this is justified by the Glivenko–Cantelli theorem, which
asserts that for a sequence (𝑈𝑘)𝑘≥1 of independent, uniformly [0, 1]-distributed ran-
dom variables we have
lim
𝑁→∞
1
𝑁
𝑁
∑
𝑘=1
1[𝑎,𝑏)(𝑈𝑘) = 𝑏 − 𝑎 almost surely
for all [𝑎, 𝑏) ⊂ [0, 1]. Thus a deterministic sequence (𝑥𝑘)𝑘≥1 which is u.d. mod 1 can
be seen as a typical realization of a random (uniformly [0, 1]-distributed) sequence.
The theory of uniform distribution was boosted by Weyl’s [43] seminal paper of
1916, which contains the celebrated Weyl criterion for uniform distribution of a se-
quence: a sequence (𝑥𝑘)𝑘≥1 is u.d. mod 1 if and only if for all integers ℎ ̸
= 0
lim
𝑁→∞
1
𝑁
𝑁
∑
𝑘=1
𝑒2𝜋𝑖ℎ𝑥𝑘
= 0 . (1.2)
This criterion can be used to give an easy proof for the fact that the sequence ({𝑘𝑥})𝑘≥1
is u.d. mod 1 if and only if 𝑥 ̸
∈ ℚ (here {⋅} stands for the fractional part function;
usually this sequence is called 𝑛𝛼 rather than 𝑘𝑥, but for the sake of consistency of
the notation with later parts of this article we will denote it by 𝑘𝑥). In fact, assume
that 𝑥 ̸
∈ ℚ; then, using the well-known formula for the geometric series, we have
1
𝑁
𝑁
∑
𝑘=1
𝑒2𝜋𝑖ℎ𝑘𝑥
=
1
𝑁
𝑒2𝜋𝑖ℎ(𝑁+1)𝑥
− 𝑒2𝜋𝑖ℎ𝑥
𝑒2𝜋𝑖ℎ𝑥 − 1
→ 0 as 𝑁 → ∞ ,
where we used the fact that 𝑒2𝜋𝑖ℎ𝑥
− 1 ̸
= 0 for ℎ ̸
= 0 and 𝑥 ̸
∈ ℚ. It is easy to see that
for 𝑥 ∈ ℚ the sequence ({𝑘𝑥})𝑘≥1 is not u.d. mod 1; thus the problem of deciding for
which 𝑥 ∈ [0, 1] the parametric sequence (𝑘𝑥)𝑘≥1 is u.d. mod 1 is completely solved.
Weyl’s paper also contained a general result for parametric sequences of the form
({𝑛𝑘𝑥})𝑘≥1, where (𝑛𝑘)𝑘≥1 is a sequence of distinct positive integers and 𝑥 is a real
number from [0, 1]: for almost all 𝑥 (in the sense of Lebesgue measure) the sequence
({𝑛𝑘𝑥})𝑘≥1 is u.d. mod 1. Accordingly, the general case ({𝑛𝑘𝑥})𝑘≥1 resembles the prop-
erties of the case ({𝑘𝑥})𝑘≥1 insofar as in both cases the exceptional set is of measure](https://image.slidesharecdn.com/25565190-250514212713-9e6dff5a/85/Uniform-Distribution-And-Quasimonte-Carlo-Methods-Discrepancy-Integration-And-Applications-16-320.jpg)
![Metric number theory, lacunary series and systems of dilated functions | 3
zero; however, while in the latter case the exceptional set can be explicitly determined,
it is generally very difficult to decide whether for a given sequence (𝑛𝑘)𝑘≥1 and a given
parameter 𝑥 the sequence ({𝑛𝑘𝑥})𝑘≥1 is u.d. mod 1 or not (see also Section 2).
In the following paragraph, we want to prove Weyl’s result that ({𝑛𝑘𝑥})𝑘≥1 is u.d.
mod 1 for almost all 𝑥. Throughout this article, we will repeatedly use methods from
probability theory; this makes perfect sense, since the unit interval, equipped with
Borel sets and Lebesgue measure 𝜆, is a probability space (that is, a measure space
(𝛺, F, ℙ) for which ℙ(𝛺) = 1). We will use the Rademacher–Menshov inequality,
which states that for a real orthonormal system 𝜙1(𝑥), . . . , 𝜙𝑁(𝑥) and for real coeffi-
cients 𝛼1, . . . , 𝛼𝑁 we have
1
∫
0
max
1≤𝑀≤𝑁
(
𝑀
∑
𝑘=1
𝛼𝑘𝜙𝑘)
2
𝑑𝑥 ≤ (log2 𝑁 + 2)2
𝑁
∑
𝑘=1
𝛼2
𝑘 (1.3)
(this inequality has been obtained independently by Rademacher [39] and Men-
shov [36]; it can be proved quite easily using a dyadic splitting method, see e.g. [34]).
Note that an equivalent formulation of the Weyl criterion (1.2) is
lim
𝑁→∞
1
𝑁
𝑁
∑
𝑘=1
cos 2𝜋ℎ𝑥𝑘 = 0 and lim
𝑁→∞
1
𝑁
𝑁
∑
𝑘=1
sin 2𝜋ℎ𝑥𝑘 = 0
for all integers ℎ ̸
= 0. For integers 𝑚 ≥ 1 and ℎ ̸
= 0 we set
𝑆𝑚,ℎ = {𝑥 ∈ [0, 1]: max
1≤𝑀≤2𝑚
𝑀
∑
𝑘=1
cos 2𝜋ℎ𝑛𝑘𝑥
> 2𝑚/2
𝑚2
} .
Then by (1.3), by the orthogonality of the trigonometric system and by the fact that by
assumption the numbers (𝑛𝑘)𝑘≥1 are distinct we have
1
∫
0
max
1≤𝑀≤2𝑚
(
𝑀
∑
𝑘=1
cos 2𝜋ℎ𝑛𝑘𝑥)
2
𝑑𝑥 ≪ 2𝑚
(log 2𝑚
)2
≪ 2𝑚
𝑚2
(1.4)
(where “≪” is the Vinogradov symbol). Chebyshev’s inequality states that for any
square-integrable function 𝑓 on [0, 1] we have that for any 𝑡 > 0
𝜆(𝑥 ∈ [0, 1]: |𝑓(𝑥)| ≥ 𝑡) ≤
1
𝑡2
1
∫
0
𝑓(𝑥)2
𝑑𝑥 . (1.5)
Applying this inequality, by (1.4) we have
𝜆(𝑆𝑚,ℎ) ≪
1
𝑚2
.](https://image.slidesharecdn.com/25565190-250514212713-9e6dff5a/85/Uniform-Distribution-And-Quasimonte-Carlo-Methods-Discrepancy-Integration-And-Applications-17-320.jpg)
![4 | Christoph Aistleitner
Thus
∞
∑
𝑚=1
𝜆(𝑆𝑚,ℎ) < ∞ ,
which by the first Borel–Cantelli lemma implies that with probability one (with re-
spect to the Lebesgue measure on [0, 1]) only finitely many events 𝑆𝑚,ℎ occur. Thus for
almost all 𝑥 ∈ [0, 1] there exists an 𝑚0 = 𝑚0(𝑥) such that
max
1≤𝑀≤2𝑚
𝑀
∑
𝑘=1
cos 2𝜋ℎ𝑛𝑘𝑥
≤ 2𝑚/2
𝑚2
for all 𝑚 ≥ 𝑚0; consequently, there also exists an 𝑁0 = 𝑁0(𝑥) such that
𝑁
∑
𝑘=1
cos 2𝜋ℎ𝑛𝑘𝑥
≤ 2𝑁1/2
(log2 𝑁)2
for 𝑁 ≥ 𝑁0, which implies
lim
𝑁→∞
1
𝑁
𝑁
∑
𝑘=1
cos 2𝜋ℎ𝑛𝑘𝑥 = 0 .
The same argument applies if we replace the function cos by sin. Consequently,
({𝑛𝑘𝑥})𝑘≥1 is u.d. mod 1 for almost all 𝑥 ∈ [0, 1].
2 Metric number theory
One of the aims of metric number theory is to describe properties which are typical
for (real) numbers, where “typical” means that the exceptional set of numbers not
possessing this property is small; in our case, we consider a property to be “typical” if
it holds for almost all numbers in the sense of Lebesgue measure (but of course there
are also other possibilities of deciding what a “typical” property is, for example by
means of the Hausdorff dimension).
An early result from metric number theory is due to Borel: he proved that almost
all numbers are normal with respect to a given base 𝑏 ≥ 2 (𝑏 being an integer). Here,
a real number 𝑥 ∈ [0, 1] is called “normal” if in its base-𝑏 expansion
𝑥 =
∞
∑
𝑖=1
𝑟𝑖𝑏−𝑖
each digit 0, 1, . . . , 𝑏 − 1 appears asymptotically with frequency 𝑏−1
, each block of 2
digits appears asymptotically withfrequency 𝑏−2
, and, generally, eachblock of 𝑑digits
appears with asymptotic frequency 𝑏−𝑑
. Formally, this can be written as
lim
𝑁→∞
1
𝑁
𝑁
∑
𝑘=1
1[𝑎𝑏−𝑑,(𝑎+1)𝑏−𝑑)({𝑏𝑘−1
𝑥}) = 𝑏−𝑑
(1.6)](https://image.slidesharecdn.com/25565190-250514212713-9e6dff5a/85/Uniform-Distribution-And-Quasimonte-Carlo-Methods-Discrepancy-Integration-And-Applications-18-320.jpg)
![Metric number theory, lacunary series and systems of dilated functions | 5
for all integers 𝑑 ≥ 1 and all integers 𝑎 ∈ {0, . . . , 𝑏𝑑
− 1}. Historically, Borel’s result is
the first appearance of what we call the strong law of large numbers, today. To see this,
we choose 𝑏 = 2 for simplicity, and let for 𝑥 ∈ [0, 1] the function 𝑟𝑘(𝑥) be defined as
the 𝑘-th digit (after the decimal point) of the binary expansion of 𝑥. Then it is an easy
exercise to check that the functions (𝑟𝑘(𝑥))𝑘≥1, interpreted as random variables over
the probability space ([0, 1], B([0, 1]), 𝜆), form a sequence of independent, identically
distributed (i.i.d.) random variables (remember that by definition, a random variable
is just a measurable function). Thus Borel’s theorem, which in the special case 𝑑 = 1
(that is, for single digits) states that
lim
𝑁→∞
1
𝑁
𝑁
∑
𝑘=1
𝑟𝑘(𝑥) =
1
2
=
1
∫
0
𝑟1(𝑥) 𝑑𝑥 = 𝔼𝜆(𝑟1) a.e.
is the strong law of large numbers for i.i.d. fair Bernoulli-distributed random variables
(the functions (𝑟𝑘(𝑥))𝑘≥1 are called Rademacher functions; the fact that they form a
system of i.i.d. random variables was first observed by Steinhaus in the 1920s). We
note, by the way, that Borel’s theorem can also be interpreted as an early appearance
of the pointwise ergodic theorem (for the transformation 𝑇𝑥 = {𝑏𝑥}).
Written in the form (1.6) (which is not Borel’s original notation) it is quite obvious
that there is a connection between normal numbers and the criterion for uniform dis-
tribution modulo 1 in (1.1). Surprisingly, this connection was not noted (or, at least, not
rigorously proved) before 1949, when Wall [42] showed that a number 𝑥 is normal in a
base 𝑏 if and only if the sequence ({𝑏𝑘
𝑥})𝑘≥1 is u.d. mod 1. Thus, Borel’s theorem can
also be seen as a special case of Weyl’s metric theorem on the uniform distribution of
({𝑛𝑘𝑥})𝑘≥1 for a.e. 𝑥.
Now we know that almost all numbers are normal (which was not so difficult to
establish); on the other hand, constructing a normal number is rather difficult, and
checking whether a given number is normal or not is (usually) absolutely infeasible.
Most constructions of normal numbers are based on the principle of concatenating
blocks of digits generated by “simple” functions; for example, Champernowne’s num-
ber (in base 10)
0. 1 2 3 4 5 6 7 8 9 10 11 12 . . .
is obtained by concatenating the decimal expansions of the positive integers in con-
secutive order, the Copeland–Erdős number (again in base 10) is obtained by concate-
nating the decimal expansions of the primes
0. 2 3 5 7 11 13 17 19 23 29 . . . ,
and there are several other constructions of this type (for example concatenating the
values of polynomials [37] or other entire functions [35]). As mentioned before, check-
ing whether a given number is normal or not is extremely difficult, and it is unknown
whether constants such as √2, 𝑒, 𝜋 are normal. It is conjectured that all algebraic ir-
rationals are normal, but no example or counterexample is known. For more details](https://image.slidesharecdn.com/25565190-250514212713-9e6dff5a/85/Uniform-Distribution-And-Quasimonte-Carlo-Methods-Discrepancy-Integration-And-Applications-19-320.jpg)
![6 | Christoph Aistleitner
on this problem, see [6]. A closely-related problem concerns sequences of the form
({𝑥𝑘
})𝑘≥1. By a result of Koksma [30] this sequence is u.d. mod 1 for almost all 𝑥 > 1;
however, not a single explicit value of 𝑥 for which this is the case, is known. The
sequence ({(3/2)𝑘
})𝑘≥1 has attracted particular attention, but it is not even known
whether lim sup𝑘→∞{(3/2)𝑘
} − lim inf𝑘→∞{(3/2)𝑘
} ≥ 1/2 (Vijayaraghavan’s conjec-
ture of 1940). For more information concerning this problem see [17, 18].
The most important open problem in metric number theory is probably the Duffin–
Schaeffer conjecture in metric Diophantine approximation. For a nonnegative function
𝜓: ℕ → ℝ, let 𝑊(𝜓) denote the set of real numbers 𝑥 ∈ [0, 1] for which the in-
equality |𝑛𝑥 − 𝑎| < 𝜓(𝑛) has infinitely many coprime solutions (𝑎, 𝑛). It is an easy
application of the first Borel–Cantelli lemma to prove that 𝜆(𝑊(𝜓)) = 0 if
∞
∑
𝑛=1
𝜓(𝑛)𝜑(𝑛)
𝑛
< ∞ (1.7)
(here 𝜑 denotes the Euler totient function); that means, divergence of the sum in (1.7)
is a necessary condition for 𝜆(𝑊(𝜓)) = 1. The Duffin–Schaeffer conjecture, proposed
by R. J. Duffin and A. C. Schaeffer [19] in 1941, asserts that divergence of the sum in
(1.7) is also sufficient to have 𝜆(𝑊(𝜓)) = 1. Several special cases of the conjecture
have been established (see for example [25]), but a complete solution of the problem
seems to be far out of reach.
More information on the problems discussed in this section can be found in the
books of Bugeaud [13] and Harman [26].
3 Discrepancy
The notion of the discrepancy of a sequence has been introduced as a measure of the
quality of the u.d. mod 1 of a sequence. For a finite sequence (𝑥1, . . . , 𝑥𝑁) of points in
the unit interval, the discrepancy 𝐷𝑁 and the star-discrepancy 𝐷∗
𝑁 are defined as
𝐷𝑁(𝑥1, . . . , 𝑥𝑁) = sup
0≤𝑎<𝑏≤1
∑𝑁
𝑘=1 1[𝑎,𝑏)(𝑥𝑘)
𝑁
− (𝑏 − 𝑎)
and
𝐷∗
𝑁(𝑥1, . . . , 𝑥𝑁) = sup
0≤𝑎≤1
∑𝑁
𝑘=1 1[0,𝑎)(𝑥𝑘)
𝑁
− 𝑎
.
It is easy to see that these two discrepancies are equivalent in the sense that at all
times 𝐷∗
𝑁 ≤ 𝐷𝑁 ≤ 2𝐷∗
𝑁, and that an infinite sequence (𝑥𝑘)𝑘≥1 is u.d. mod 1 if and
only if 𝐷∗
𝑁(𝑥1, . . . , 𝑥𝑁) → 0 as 𝑁 → ∞. An important inequality to estimate the
discrepancy of a sequence is the Erdős–Turán inequality, which (in one out of many](https://image.slidesharecdn.com/25565190-250514212713-9e6dff5a/85/Uniform-Distribution-And-Quasimonte-Carlo-Methods-Discrepancy-Integration-And-Applications-20-320.jpg)
![Metric number theory, lacunary series and systems of dilated functions | 7
possible formulations) states that for any positive integer 𝐻
𝐷∗
𝑁(𝑥1, . . . , 𝑥𝑁) ≤
3
𝐻
+ 3
𝐻
∑
ℎ=1
1
ℎ
1
𝑁
𝑁
∑
𝑘=1
𝑒2𝜋𝑖ℎ𝑥𝑘
. (1.8)
We will use this inequality in Section 5 to obtain an upper bound for the discrepancy
of ({𝑛𝑘𝑥})𝑘≥1 for almost all 𝑥, by this means establishing a quantitative version of the
theorem of Weyl mentioned in Section 1. Another important inequality concerning the
discrepancy of sequences of points is Koksma’s inequality, which states that for any
function 𝑓 which has bounded variation Var(𝑓) in the unit interval the estimate
1
𝑁
𝑁
∑
𝑘=1
𝑓(𝑥𝑘) −
1
∫
0
𝑓(𝑥) 𝑑𝑥
≤ Var(𝑓) ⋅ 𝐷∗
𝑁(𝑥1, . . . , 𝑥𝑁) (1.9)
holds. The notions of u.d. mod 1 and discrepancy can be generalized in a natural way
to the multidimensional setting, as can be the Erdős–Turán inequality and Koksma’s
inequality (then called the Koksma–Hlawka inequality). The multidimensional ver-
sion of (1.9) forms the foundation of the so-called Quasi-Monte Carlo method, which
is based on the observation that sequences having small discrepancy can be used for
numerical integration.
By a result of Schmidt [41] there exists a positive constant 𝑐 such that for any infi-
nite sequence (𝑥𝑘)𝑘≥1 of points in the unit interval the inequality
𝐷∗
𝑁(𝑥1, . . . , 𝑥𝑁) > 𝑐
log 𝑁
𝑁
holds for infinitely many 𝑁. On the other hand, there exist several constructions of
sequences satisfying 𝐷∗
𝑁(𝑥1, . . . , 𝑥𝑁) = O((log 𝑁)𝑁−1
) as 𝑁 → ∞, so in the one-
dimensional case the problem of the optimal asymptotic order of the discrepancy is
solved. On the contrary, determining the optimal asymptotic order of the discrepancy
in the multidimensional case turned out to be a very difficult problem, which is still
open (see [11] for a survey).
More information on discrepancy theory and the Quasi-Monte Carlo method can
be found in the books of Dick and Pillichshammer [15] and Drmota and Tichy [16].
4 Lacunary series
The word lacunary originates from the Latin lacuna (ditch, gap), which is the diminu-
tive form of lacus (lake). Accordingly, a lacunary Fourier series is a series which has
“gaps” in the sense that it is composed of trigonometric functions whose frequencies
are far apart from each other. A classicalgap condition is the Hadamard gap condition,
requiring that
𝑛𝑘+1
𝑛𝑘
≥ 𝑞 > 1 , 𝑘 ≥ 1 ; (1.10)](https://image.slidesharecdn.com/25565190-250514212713-9e6dff5a/85/Uniform-Distribution-And-Quasimonte-Carlo-Methods-Discrepancy-Integration-And-Applications-21-320.jpg)
![8 | Christoph Aistleitner
thus a (Hadamard) lacunary Fourier series is of the form
∞
∑
𝑘=1
(𝑎𝑘 cos 2𝜋𝑛𝑘𝑥 + 𝑏𝑘 sin 2𝜋𝑛𝑘𝑥) (1.11)
for (𝑛𝑘)𝑘≥1 satisfying (1.10). By a classical heuristics, lacunary sequences resemble
many properties which are typical for sequences of independent random variables.
For example, by Kolmogorov’s three series theorem, a sequence of centered and uni-
formly bounded independent random variables (𝑋𝑘)𝑘≥1 is almost surely convergent if
and only if the variances satisfy
∞
∑
𝑘=1
𝕍(𝑋𝑘) < ∞, (1.12)
and by a counterpart for lacunary series, also due to Kolmogorov, the series (1.11) is
almost convergent everywhere if and only if
∞
∑
𝑘=1
(𝑎2
𝑘 + 𝑏2
𝑘 ) < ∞ . (1.13)
Note here that the variance of the function 𝑎𝑘 cos 2𝜋𝑛𝑘𝑥+𝑏𝑘 sin 2𝜋𝑛𝑘𝑥, considered as
a random variable over the probability space ([0, 1], B([0, 1]), 𝜆), is simply given by
1
∫
0
(𝑎𝑘 cos 2𝜋𝑛𝑘𝑥 + 𝑏𝑘 sin 2𝜋𝑛𝑘𝑥)
2
𝑑𝑥 = 𝑎2
𝑘 + 𝑏2
𝑘 .
Thus the almost sure (a.s.) convergence behavior of series of independent random
variables and of lacunary trigonometric series are in perfect accordance. Many sim-
ilar results of the same type exist: for example, by a classical result of Salem and Zyg-
mund [40], under the gap condition (1.10) we have
𝜆 (𝑥 ∈ [0, 1]:
𝑁
∑
𝑘=1
cos 2𝜋𝑛𝑘𝑥 < 𝑡√𝑁/2) → 𝛷(𝑡) ,
where𝛷(𝑡)denotes thestandard normaldistributionfunction. In other words, the sys-
tem (cos 2𝜋𝑛𝑘𝑥)𝑘≥1 satisfies the central limit theorem. By a result of Erdős and Gál [21],
the same system also satisfies the law of the iterated logarithm (LIL), that is
lim sup
𝑁→∞
∑
𝑁
𝑘=1 cos 2𝜋𝑛𝑘𝑥
√2𝑁 log log 𝑁
=
1
√2
a.e.
The situation gets significantly more complicated if we consider the more general
sequence (𝑓(𝑛𝑘𝑥))𝑘≥1 for a (in some sense) “nice” function 𝑓 satisfying
𝑓(𝑥 + 1) = 𝑓(𝑥) ,
1
∫
0
𝑓(𝑥) 𝑑𝑥 = 0 , (1.14)](https://image.slidesharecdn.com/25565190-250514212713-9e6dff5a/85/Uniform-Distribution-And-Quasimonte-Carlo-Methods-Discrepancy-Integration-And-Applications-22-320.jpg)
![Metric number theory, lacunary series and systems of dilated functions | 9
instead of (cos 2𝜋𝑛𝑘𝑥)𝑘≥1. A striking result for this general setting is a theorem of
Philipp [38], who confirmed the so-called Erdős–Gál conjecture by proving that un-
der (1.10) we have
1
4√2
≤ lim sup
𝑁→∞
𝑁𝐷∗
𝑁({𝑛1𝑥}, . . . , {𝑛𝑁𝑥})
√2𝑁 log log 𝑁
≤ 𝐶𝑞 a.e.; (1.15)
this is a counterpart of the Chung–Smirnov LIL for the Kolmogorov–Smirnov statistic
in probability theory. As a consequence of (1.15) and Koksma’s inequality (1.9) we have
lim sup
𝑁→∞
∑
𝑁
𝑘=1 𝑓(𝑛𝑘𝑥)
√2𝑁 log log 𝑁
≤ 𝐶𝑓,𝑞 a.e. (1.16)
Calculating the precise value of the lim sup in (1.15) and (1.16) is a very difficult prob-
lem, and depends on number-theoretic properties of (𝑛𝑘)𝑘≥1 and Fourier-analytic
properties of 𝑓 (or of the indicator functions in the case of (1.15)) in a very delicate way.
In the case of 𝑛𝑘 = 𝜃𝑘
for an integer 𝜃 the problem has been solved by Fukuyama [23];
he proved that almost everywhere
lim sup
𝑁→∞
𝑁𝐷∗
𝑁({𝜃𝑥}, . . . , {𝜃𝑁
𝑥})
√2𝑁 log log 𝑁
=
{
{
{
{
{
{
{
√42
9
if 𝜃 = 2,
√(𝜃+1)𝜃(𝜃−2)
2√(𝜃−1)3
if 𝜃 ≥ 4 is even,
√𝜃+1
2√𝜃−1
if 𝜃 ≥ 3 is odd.
In view of the results mentioned in Section 2, Fukuyama’s theorem establishes the
typical asymptotic order of the discrepancy of normal numbers. In a sense, the se-
quence (𝜃𝑘
)𝑘≥1 is a pathological example of a lacunary sequence, exhibiting an ex-
tremely strong relation between its consecutive terms. For a lacunary sequence for
which no such strong arithmetic relations exist, the LIL is satisfied in the form
lim sup
𝑁→∞
𝑁𝐷∗
𝑁({𝑛1𝑥}, . . . , {𝑛𝑁𝑥})
√2𝑁 log log 𝑁
=
1
2
a.e. ,
which is in perfect accordance (including the value of the constant of the right-hand
side) with the Chung–Smirnov LIL for i.i.d. random variables (see [1] for details). For
more information on lacunary sequences in the context of metric discrepancy theory
and probabilistic limit theorems, see the survey paper [3].
Lacunary functions are well known in analysis for several other interesting prop-
erties, apart from their resemblance of the behavior of systems of independent random
variables. For example, Weierstrass’ celebrated example of a nowhere differentiable
function is defined by means of a lacunary trigonometric series. It should be noted
that the notion of lacunary series does not only include lacunary trigonometric se-
ries, but also other series such as for example lacunary Taylor series. For a survey,
see [29].](https://image.slidesharecdn.com/25565190-250514212713-9e6dff5a/85/Uniform-Distribution-And-Quasimonte-Carlo-Methods-Discrepancy-Integration-And-Applications-23-320.jpg)
![10 | Christoph Aistleitner
5 Almost everywhere convergence
The Kolmogorov three series theorem gives a full characterization of the a.s. conver-
gence behavior of sums of independent random variables. In general, the a.s. conver-
gence condition comprises of three conditions about the convergence or divergence of
certain series (which explains the name three series theorem), but in the case of cen-
tered, uniformly-bounded random variables the criterion reduces to the simple condi-
tion (1.12).
As noted in the previous section, there exists an analogue of the three series
theorem for the case of Hadamard lacunary trigonometric series; however, surpris-
ingly, the requirement of considering a Fourier series which contains only frequencies
along an exponentially growing subsequence canbe entirely dropped. This is the cele-
brated Carleson’s theorem [14], which is considered as one of the major achievements
of Fourier analysis in twentieth century mathematics: a Fourier series
∞
∑
𝑘=1
(𝑎𝑘 cos 2𝜋𝑘𝑥 + 𝑏𝑘 sin 2𝜋𝑘𝑥)
is almost everywhere (a.e.) convergent, provided (1.13) holds. Moreover, for any func-
tion 𝑓 ∈ 𝐿2
([0, 1]) and
𝑓(𝑥) ∼
∞
∑
𝑘=1
(𝑎𝑘 cos 2𝜋𝑘𝑥 + 𝑏𝑘 sin 2𝜋𝑘𝑥) ,
setting
𝑠𝑁(𝑓; 𝑥) =
𝑁
∑
𝑘=1
(𝑎𝑘 cos 2𝜋𝑘𝑥 + 𝑏𝑘 sin 2𝜋𝑘𝑥)
we have
𝑠𝑁(𝑓; 𝑥) → 𝑓(𝑥) as 𝑁 → ∞ for a.e. 𝑥 .
Carleson’s theorem has a breathtaking consequence: there exists an absolute con-
stant 𝑐 such that for any function 𝑓 in 𝐿2
([0, 1]) we have, writing ‖⋅‖ for the 𝐿2
([0, 1])
norm,
sup
𝑁≥1
𝑠𝑁(𝑓; 𝑥)
≤ 𝑐‖𝑓‖ . (1.17)
Carleson’s theorem has been extended to the case 𝑓 ∈ 𝐿𝑝
([0, 1]), 𝑝 > 1, by Hunt [28].
It is a very deep result, and although alternative proofs have been given by Feffer-
man [22] and Lacey and Thiele [33], no “easy” proof exists. For a comprehensive
treatment of the subject, see the monograph of Arias de Reyna [5] and the survey
paper [32].
As an application of Carleson’s theorem, we will show how it can be used to obtain
a quantitative version of the results on a.e. uniform distribution of Weyl mentioned in](https://image.slidesharecdn.com/25565190-250514212713-9e6dff5a/85/Uniform-Distribution-And-Quasimonte-Carlo-Methods-Discrepancy-Integration-And-Applications-24-320.jpg)
![Metric number theory, lacunary series and systems of dilated functions | 11
Section 1. This argument is due to Baker [7] and leads to the upper bound
𝐷∗
𝑁({𝑛1𝑥}, . . . , {𝑛𝑁𝑥}) ≪
(log 𝑁)3/2+𝜀
√𝑁
a.e. (1.18)
for any strictly-increasing sequence of positive integers (𝑛𝑘)𝑘≥1 and any 𝜀 > 0. To
outline the similarity between this proof and the one given in Section 1, we will use a
real version of the Erdős–Turán inequality (1.8).
Let 𝜀 > 0. For integers 𝑚 ≥ 1 we set
𝑆𝑚 = {𝑥 ∈ [0, 1]: max
1≤𝑀≤2𝑚
∑
1≤ℎ≤2𝑚/2
1
ℎ
𝑀
∑
𝑘=1
cos 2𝜋ℎ𝑛𝑘𝑥
> 2𝑚/2
𝑚3/2+𝜀
} .
Note that by (1.17) for any ℎ we have
max
1≤𝑀≤2𝑚
𝑀
∑
𝑘=1
cos 2𝜋ℎ𝑛𝑘𝑥
≤ 𝑐
2𝑚
∑
𝑘=1
cos 2𝜋ℎ𝑛𝑘𝑥
≤ 𝑐2𝑚/2
(1.19)
for an absolute constant 𝑐. Thus by Minkowski’s inequality we have
max
1≤𝑀≤2𝑚
∑
1≤ℎ≤2𝑚/2
1
ℎ
𝑀
∑
𝑘=1
cos 2𝜋ℎ𝑛𝑘𝑥
≤ ∑
1≤ℎ≤2𝑚/2
1
ℎ
max
1≤𝑀≤2𝑚
𝑀
∑
𝑘=1
cos 2𝜋ℎ𝑛𝑘𝑥
≪ 𝑐𝑚2𝑚/2
.
Consequently, by Chebyshev’s inequality (1.5) we have
𝜆(𝑆𝑚) ≪
1
𝑚1+2𝜀
.
Thus
∞
∑
𝑚=1
𝜆(𝑆𝑚) < ∞ ,
which by the first Borel–Cantelli lemma means that with probability one only finitely
many events 𝑆𝑚 occur. Thus for almost all 𝑥 ∈ [0, 1] there exists an 𝑚0 = 𝑚0(𝑥) such
that
max
1≤𝑀≤2𝑚
∑
1≤ℎ≤2𝑚/2
1
ℎ
𝑀
∑
𝑘=1
cos 2𝜋ℎ𝑛𝑘𝑥
≤ 2𝑚/2
𝑚3/2+𝜀
for all 𝑚 ≥ 𝑚0; consequently, there also exists an 𝑁0 = 𝑁0(𝑥) such that
∑
1≤ℎ≤√𝑁
1
ℎ
𝑁
∑
𝑘=1
cos 2𝜋ℎ𝑛𝑘𝑥
≤ 2𝑁1/2
(log2 𝑁)3/2+𝜀](https://image.slidesharecdn.com/25565190-250514212713-9e6dff5a/85/Uniform-Distribution-And-Quasimonte-Carlo-Methods-Discrepancy-Integration-And-Applications-25-320.jpg)
![12 | Christoph Aistleitner
for 𝑁 ≥ 𝑁0. The same result holds if we replace the function cos by sin, and using
(1.8) (split into a real and imaginary part) we get (1.18). Carleson’s inequality (1.17) in
the form (1.19) plays a key role in this proof, and if it is replaced by the Rademacher–
Menshov inequality, which gives an additional logarithmic factor as in (1.4), one can
only obtain (1.18) with the exponent 3/2 + 𝜀 replaced by 5/2 + 𝜀.
The optimal exponent of the logarithmic term in (1.18) is an important open prob-
lem in metric discrepancy theory. Note that by Koksma’s inequality (1.9) as a conse-
quence of (1.18) we get
𝑁
∑
𝑘=1
𝑓(𝑛𝑘𝑥)
≪ √𝑁(log 𝑁)3/2+𝜀
a.e. (1.20)
for any function 𝑓satisfying (1.14) which has bounded variation on [0, 1]. On the other
hand, Berkes and Philipp [8] constructed a sequence (𝑛𝑘)𝑘≥1 for which
lim sup
𝑁→∞
∑
𝑁
𝑘=1 cos 2𝜋𝑛𝑘𝑥
√𝑁 log 𝑁
= ∞ a.e. ,
which again by Koksma’s inequality implies that the exponent of the logarithmic term
in (1.18) can in general not be reduced below 1/2. In the following section we will see
that 1/2 is in fact the optimal exponent, at least if we consider only a single function 𝑓
(as in (1.20)) and not the discrepancy 𝐷∗
𝑁.
For more information on the a.e. convergence of sums of dilated functions, the in-
terested reader is referred to the comprehensive survey article of Berkes and Weber [9].
6 Sums involving greatest common divisors
In the context of counting lattice points in right-angled triangles, around 1920 Hardy
and Littlewood investigated the problem of finding good upper bounds for the asymp-
totic order of
𝑁
∑
𝑘=1
({𝑘𝑥} − 1/2) as 𝑁 → ∞ . (1.21)
The 𝐿2
([0, 1])-norm of (1.21) can be calculated using the formula
1
∫
0
({𝑚𝑥} − 1/2) ({𝑛𝑥} − 1/2) 𝑑𝑥 =
1
12
(gcd(𝑚, 𝑛))2
𝑚𝑛
(1.22)
for integers 𝑚, 𝑛(first stated by Franel and proved by Landau in 1924). The generalized
problem of estimating
𝑁
∑
𝑘,𝑙=1
(gcd(𝑛𝑘, 𝑛𝑙))2
𝑛𝑘𝑛𝑙
(1.23)](https://image.slidesharecdn.com/25565190-250514212713-9e6dff5a/85/Uniform-Distribution-And-Quasimonte-Carlo-Methods-Discrepancy-Integration-And-Applications-26-320.jpg)
![Metric number theory, lacunary series and systems of dilated functions | 13
for an arbitrary sequence of distinct positive integers 𝑛1, . . . , 𝑛𝑁 was posed as a prize
problem by the Scientific Society in Amsterdam in 1947 (following a suggestion of
Erdős), and solved by Gál [24] in 1949. He proved that there exists an absolute con-
stant 𝑐 such that (1.23) is bounded by 𝑐𝑁(log log 𝑁)2
, and that this upper bound is
asymptotically optimal. Koksma [31] observed that as a consequence for any centered,
periodically-extended indicator function 𝑓(𝑥) = 1(𝑎,𝑏)({𝑥}) − (𝑏 − 𝑎) (and in fact even
for any 1-periodic function 𝑓 having mean zero and bounded variation on [0,1]) the
estimate
1
∫
0
(
𝑁
∑
𝑘=1
𝑓(𝑛𝑘𝑥))
2
𝑑𝑥 ≪ 𝑁(log log 𝑁)2
(1.24)
holds. This follows from a generalized version of (1.22), which we will deduce in the
next few lines. Assume that 𝑓 satisfies (1.14) and is of bounded variation on [0, 1]. Let
𝑓(𝑥) ∼
∞
∑
𝑗=1
𝑎𝑗 cos 2𝜋𝑗𝑥
denote the Fourier series of 𝑓 (for simplicity we assume that it is a pure cosine-series;
the general case works in exactly the same way). Then
|𝑎𝑗| ≪ 𝑗−1
(1.25)
(see [44, p. 48]; this estimate can be easily proved using the fact that any function
of bounded variation can be written as the sum of two bounded and monotone func-
tions). Thus for integers 𝑚, 𝑛 we have, by (1.25) and the orthogonality of the trigono-
metric system, and writing 𝛿(⋅, ⋅) for the Kronecker function,
1
∫
0
𝑓(𝑚𝑥)𝑓(𝑛𝑥) 𝑑𝑥 =
∞
∑
𝑗1,𝑗2=1
𝑎𝑗1
𝑎𝑗2
𝛿(𝑚𝑗1, 𝑛𝑗2)
≪
∞
∑
𝑗1,𝑗2=1
1
𝑗1𝑗2
𝛿(𝑚𝑗1, 𝑛𝑗2) . (1.26)
Now 𝑚𝑗1 = 𝑛𝑗2 is only possible if 𝑗1 = 𝑗𝑛/ gcd(𝑚, 𝑛) and 𝑗2 = 𝑗𝑚/ gcd(𝑚, 𝑛) for some
integer 𝑗 ≥ 1. Consequently, (1.26) is at most
∞
∑
𝑗=1
gcd(𝑚, 𝑛)
𝑗𝑛
gcd(𝑚, 𝑛)
𝑗𝑚
≪
gcd(𝑚, 𝑛)2
𝑚𝑛
,
which implies, together with the aforementioned result of Gál, that (1.24) holds. Sums
involving common divisors similar to (1.23) were studied by Dyer and Harman [20] in
the context of metric Diophantine approximation. They investigated
max
𝑛1<⋅⋅⋅<𝑛𝑁
𝑁
∑
𝑘,𝑙=1
(gcd(𝑛𝑘, 𝑛𝑙))2𝛼
(𝑛𝑘𝑛𝑙)𝛼
, 𝛼 ∈ [1/2, 1) , (1.27)](https://image.slidesharecdn.com/25565190-250514212713-9e6dff5a/85/Uniform-Distribution-And-Quasimonte-Carlo-Methods-Discrepancy-Integration-And-Applications-27-320.jpg)
![14 | Christoph Aistleitner
and, amongst other results, proved for the particularly interesting case 𝛼 = 1/2 the
upper bound
max
𝑛1<⋅⋅⋅<𝑛𝑁
𝑁
∑
𝑘,𝑙=1
gcd(𝑛𝑘, 𝑛𝑙)
√𝑛𝑘𝑛𝑙
≪ 𝑁 exp (
5 log 𝑁
log log 𝑁
) .
Recently, Aistleitner, Berkes and Seip [4] obtained upper bounds for (1.27) which are
essentially optimal. They proved that
max
𝑛1<⋅⋅⋅<𝑛𝑁
𝑁
∑
𝑘,𝑙=1
(gcd(𝑛𝑘, 𝑛𝑙))2𝛼
(𝑛𝑘𝑛𝑙)𝛼
≤ 𝐶𝜀𝑁 exp ((1 + 𝜀)𝑔(𝛼, 𝑁)) ,
where for 1/2 < 𝛼 < 1 we have
𝑔(𝛼, 𝑁) =
( 8
1−𝛼
+ 16⋅2−𝛼
√2𝛼−1
) (log 𝑁)1−𝛼
(log log 𝑁)𝛼
+
(log 𝑁)(1−𝛼)/2
1 − 𝛼
, (1.28)
for 𝛼 = 1/2 we have
𝑔(1/2, 𝑁) = 25√log 𝑁√log log 𝑁 , (1.29)
and where 𝐶𝜀 is a constant only depending on 𝜀 > 0. Here the asymptotic order of
𝑔(𝛼, 𝑁) in (1.28) is optimal, and the asymptotic order of 𝑔(1/2, 𝑁) in (1.29) can per-
haps be reduced from √log 𝑁√log log 𝑁 to √log 𝑁/√log log 𝑁 (but not below). As
an application, Aistleitner, Berkes and Seip improved the exponent of the logarithmic
term in (1.20) to 1/2 + 𝜀, which is optimal (up to the 𝜀). Another application of such
GCD sums is concerning the a.e. convergence of series ∑∞
𝑘=1 𝑐𝑘𝑓(𝑛𝑘𝑥) for functions 𝑓
of bounded variation or being Hölder-continuous (see [4]; cf. also [2]) or of so-called
Davenport series (see [12]). There is also a close connection with certain properties of
the Riemann zeta function, which requires further investigation (cf. [27]).
As a consequence of (1.28) (and using a trick to modify the argument around (1.26)
in such a way to get a generalized GCD sum of the form (1.27) instead of (1.23)) one can
show that for any 𝑓 of bounded variation satisfying (1.14) and for any strictly increas-
ing sequence (𝑛𝑘)𝑘≥1 of positive integers, the Carleson-type inequality
max
1≤𝑀≤𝑁
𝑀
∑
𝑘=1
𝑐𝑘𝑓(𝑛𝑘𝑥)
≤ 𝑐(log log 𝑁)4
𝑁
∑
𝑘=1
𝑐2
𝑘
holds, which, using an argument similar to the one used to prove (1.18) in the previous
section, leads to
𝑁
∑
𝑘=1
𝑓(𝑛𝑘𝑥)
≪ √𝑁(log 𝑁)1/2+𝜀
a.e. (1.30)
As mentioned at the end of the previous section this means that the problem con-
cerning the optimal (a.e.) asymptotic order of the sum ∑ 𝑓(𝑛𝑘𝑥) is solved; how-
ever, the more difficult case of the precise a.e. asymptotic order of the discrepancy
𝐷∗
𝑁({𝑛1𝑥}, . . . , {𝑛𝑁𝑥}) remains open.](https://image.slidesharecdn.com/25565190-250514212713-9e6dff5a/85/Uniform-Distribution-And-Quasimonte-Carlo-Methods-Discrepancy-Integration-And-Applications-28-320.jpg)
![Metric number theory, lacunary series and systems of dilated functions | 15
Concluding remark: There exists a close connection between discrepancy theory
and harmonic analysis, which we have for example observed in Weyl’s criterion (1.2),
in the Erdős–Turán inequality (1.8) and in the Carleson convergence theorem in Sec-
tion 5. This connection goes far beyond the material contained in this article, and is
comprehensively presented in a survey article of Dmitriy Bilyk in the present volume
(see [10]).
References
[1] Christoph Aistleitner, On the law of the iterated logarithm for the discrepancy of lacunary se-
quences, Trans. Amer. Math. Soc. 362 (2010), 5967–5982.
[2] Christoph Aistleitner, Convergence of ∑ 𝑐𝑘𝑓(𝑘𝑥) and the Lip 𝛼 class, Proc. Amer. Math. Soc.
140 (2012), 3893–3903.
[3] Christoph Aistleitner and Istvan Berkes, Probability and metric discrepancy theory, Stoch. Dyn.
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University Press, Cambridge, 1988, Reprint of the 1979 edition.](https://image.slidesharecdn.com/25565190-250514212713-9e6dff5a/85/Uniform-Distribution-And-Quasimonte-Carlo-Methods-Discrepancy-Integration-And-Applications-30-320.jpg)
![József Beck
Strong uniformity
Abstract: We attempt to develop a new chapter of the theory of Uniform Distribution;
we call it strong uniformity. Strong uniformity, in a nutshell, means that we combine
Lebesguemeasurewiththeclassicaltheory of Uniform Distribution, basically founded
by H. Weyl in his famous paper from 1916 [12], which is built around nice test sets, such
as axis-parallel rectangles and boxes. We prove the continuous version of the well-
known Khinchin’s conjecture (see [7]) in every dimension 𝑑 ≥ 2 (the discrete version
turned out to be false – it was disproved by Marstrand [10]). We consider an arbitrarily
complicated but fixed measurable test set 𝑆 in the 𝑑-dimensional unit cube, and study
the uniformity of a typical member of some natural families of curves, such as all torus
lines or billiard paths starting from the origin, with respect to 𝑆. In the 2-dimensional
case we have the very surprising superuniformity of the typical torus lines and billiard
paths. In dimensions ≥ 3 we still have strong uniformity, but not superuniformity.
However, in dimension 3 we have the even more striking super-duper uniformity for 2-
dimensional rays (replacing the torus lines). Finally, we briefly indicate how to exhibit
superuniform motions on every “reasonable” plane region (e.g. the circular disk) and
on every “reasonable” closed surface (sphere, torus and so on).
Keywords: Discrepancy of curves, time discrepancy of motions, strong uniformity, su-
peruniformity
Mathematics Subject Classification 2010: 11K38
||
József Beck: Mathematics Department, Busch Campus, Hill Center, Rutgers University, New
Brunswick, NJ 08903, USA, e-mail: jbeck@math.rutgers.edu
1 Introduction
In 1923, Khinchin [7] made the following conjecture: given a Lebesgue measurable set
𝑆 ⊂ [0, 1], the sequence 𝛼, 2𝛼, 3𝛼, . . . modulo 1 is uniformly distributed with respect
to 𝑆 for almost every 𝛼. Formally, the conjecture states that
lim
𝑛→∞
1
𝑛
∑
1≤𝑘≤𝑛:
{𝑘𝛼}∈𝑆
1 = meas(𝑆) for almost every 𝛼 . (2.1)
Here 0 ≤ {𝑥} < 1 denotes the fractional part of a real number 𝑥, and meas stands for
the 1-dimensional Lebesgue measure.
Khinchin’s conjecture was motivated by the classical equidistribution theorem
that (2.1) holds for every irrational 𝛼 if 𝑆 = [𝑎, 𝑏) is an arbitrary subinterval of [0, 1).](https://image.slidesharecdn.com/25565190-250514212713-9e6dff5a/85/Uniform-Distribution-And-Quasimonte-Carlo-Methods-Discrepancy-Integration-And-Applications-31-320.jpg)
![18 | József Beck
Formally,
lim
𝑛→∞
1
𝑛
∑
1≤𝑘≤𝑛:
𝑎≤{𝑘𝛼}<𝑏
1 = 𝑏 − 𝑎 (2.2)
holds for every 0 < 𝑎 < 𝑏 < 1 and every irrational 𝛼.
We obtain a second positive result if the sequence 𝛼, 2𝛼, 3𝛼, . . . is replaced by the
translated copy 𝛽+𝛼, 𝛽+2𝛼, 𝛽+3𝛼, . . . , translated by a typical real number 𝛽, that is,
if we start from a typical 𝛽 instead of 0. More precisely, for every Lebesgue measurable
set 𝑆 ⊂ [0, 1] and for every irrational 𝛼, we have
lim
𝑛→∞
1
𝑛
∑
1≤𝑘≤𝑛:
{𝛽+𝑘𝛼}∈𝑆
1 = meas(𝑆) for almost every 𝛽. (2.3)
Note that (2.3) is a special case of Birkhoff’s well-known individual ergodic theo-
rem. Also there is a simple way of deriving (2.3) directly from (2.2).
A third positive result was proved by Raikov [11]: (2.1) holds if the linear sequence
𝑘𝛼 is replaced by the exponential sequence 2𝑘
𝛼, or in general, by any sequence 𝑞𝑘
𝛼,
where 𝑞 is a fixed integer greater than one (𝑘 = 1, 2, 3, . . . ).
Khinchin’s conjecture remained among the most famous open problems in the
subject of Uniform Distribution for several decades. The likely reason why the conjec-
ture resisted every attack is that people were convinced about its truth, and wanted to
prove a positive result. However, despite the three positive results mentioned above
(see (2.2), (2.3) and Raikov’s theorem), conjecture (2.1) turned out to be false. In 1970,
Marstrand [10] proved that there exists an open set 𝑆 ⊂ [0, 1] with meas(𝑆) < 1 such
that
lim sup
𝑛→∞
1
𝑛
∑
1≤𝑘≤𝑛:
{𝑘𝛼}∈𝑆
1 = 1 for every 𝛼
The fact that open sets are the simplest in the Borel hierarchy makes Marstrand’s neg-
ative result even more surprising.
Marstrand’s result demonstrates that Khinchin was too optimistic. In this paper
we show how to “save” Khinchin’s conjecture in the continuous case, i.e. by switching
from a sequence (arithmetic progression) to the continuous torus line. Of course, we
have to increase the dimension of the underlying set: we replace the unit torus [0, 1)
with the 2-dimensional unit torus [0, 1)2
= 𝐼2
, and replace the arithmetic progres-
sion 𝛼, 2𝛼, 3𝛼, . . . starting from 0 with the straight line (𝑡 cos 𝜃, 𝑡 sin 𝜃), 𝑡 ≥ 0 start-
ing from the origin (0, 0) with a fixed angle 𝜃. The continuous version of the classical
equidistribution theorem (Kronecker–Weyl) says that the torus line (𝑡 cos 𝜃, 𝑡 sin 𝜃),
𝑡 ≥ 0 modulo 1 is uniformly distributed with respect to every axis-parallel rectangle
in the unit square [0, 1]2
= 𝐼2
if and only if the slope tan 𝜃 is irrational.
The continuous analog of Khinchin’s conjecture goes as follows: What happens if
one replaces the rectangle with an arbitrary Lebesgue measurable test set 𝑆 ⊂ [0, 1]2
?](https://image.slidesharecdn.com/25565190-250514212713-9e6dff5a/85/Uniform-Distribution-And-Quasimonte-Carlo-Methods-Discrepancy-Integration-And-Applications-32-320.jpg)
![Strong uniformity | 19
Can one prove uniformity with respect to 𝑆 for almost every angle 𝜃? In other words,
can one prove “strong uniformity” in the continuous case? Here, “strong uniformity”
refers to the fact that Lebesgue measurable sets form the largest class of sets for which
we can define measure (without measure we cannot define uniformity).
The answer is yes; our Theorem 1.1 below proves strong uniformity in the contin-
uous case. We actually prove much more: we prove superuniformity in the sense that
the error term turns out to be shockingly small: we prove a polylogarithmic error term.
More precisely, let 𝑆 ⊂ [0, 1]2
= 𝐼2
be an arbitrary Lebesgue measurable set in the
unit square, and assume that 0 < area(𝑆) < 1, where area denotes the 2-dimensional
Lebesgue measure. Let 𝜃 ∈ [0, 2𝜋)be an arbitraryangle, and consider the straight line
(𝑡 cos 𝜃, 𝑡 sin 𝜃), 𝑡 ≥ 0 starting from the origin (0, 0) with angle 𝜃. Let 𝑇𝑆(𝜃) denote the
time the line (𝑡 cos 𝜃, 𝑡 sin 𝜃) modulo 1 spends in the given set 𝑆 as 0 ≤ 𝑡 ≤ 𝑇 (line
modulo 1 = torus line). Formally,
𝑇𝑆(𝜃) = meas {𝑡 ∈ [0, 𝑇]: ({𝑡 cos 𝜃}, {𝑡 sin 𝜃}) ∈ 𝑆} .
Note that it may happen that the set {𝑡 ∈ [0, 𝑇]: ({𝑡 cos 𝜃}, {𝑡 sin 𝜃}) ∈ 𝑆} is not
measurable for some particular angle 𝜃 ∈ [0, 2𝜋), and so 𝑇𝑆(𝜃) is not defined. But
this technical nuisance happens only for a negligible set of angles: 𝑇𝑆(𝜃) is certainly
well-defined for almost every 𝜃 ∈ [0, 2𝜋). This follows from some general results in
Lebesgueintegration. Wearereferringto Fubini’s theorem and whatwecallthechange
of variables in multiple integrals, applied in the special case of polar coordinates. (The
relevance of polar coordinates is clear from the fact that the lines (𝑡 cos 𝜃, 𝑡 sin 𝜃) are
passing through the origin.)
Uniformity of the torus line (𝑡 cos 𝜃, 𝑡 sin 𝜃) modulo 1 in 𝑆 means that
lim
𝑇→∞
𝑇𝑆(𝜃) − area(𝑆)𝑇
𝑇
= 0 . (2.4)
Our first result says that we can replace the factor of 𝑇 in the denominator of (2.4) with
the much smaller (log 𝑇)3+𝜀
for almost every 𝜃. In this paper log 𝑥 and log2 𝑥 denote,
respectively, the natural and the binary logarithm of 𝑥.
Theorem 1.1. Let 𝑆 ⊂ [0, 1)2
be an arbitrary Lebesgue measurable set in the unit square
with 0 < area(𝑆) < 1. Then for every 𝜀 > 0,
lim
𝑇→∞
𝑇𝑆(𝜃) − area(𝑆)𝑇
(log 𝑇)3+𝜀
= 0
for almost every angle 𝜃.
We can of course rewrite Theorem 1.1 in the equivalent form
𝑇𝑆(𝜃) = area(𝑆)𝑇 + 𝑜 ((log 𝑇)3+𝜀
) .
Notice that the polylogarithmic error term is shockingly small compared to the linear
main term area(𝑆)𝑇. This is why we call Theorem 1.1 a superuniformity result.](https://image.slidesharecdn.com/25565190-250514212713-9e6dff5a/85/Uniform-Distribution-And-Quasimonte-Carlo-Methods-Discrepancy-Integration-And-Applications-33-320.jpg)
![20 | József Beck
What makes the “continuous” Theorem 1.1 particularly interesting is the sharp
contrast with the “discrete” version (Khinchin’s conjecture), where there is no unifor-
mity at all.
We actually prove a more general version of Theorem 1.1, extending the case of
the 0–1 valued characteristic function 𝜒𝑆 of an arbitrary measurable subset 𝑆 of the
unit square to an arbitrary real-valued square-integrable function 𝑓 ∈ 𝐿2 defined on
the unit torus [0, 1)2
(one could also extend it to complex-valued functions 𝑓 ∈ 𝐿2).
This generalization (from sets to functions) will be needed in some later applications,
where we will study superuniform motions inside finite regions more general than the
rectangle, and will also study superuniform motions on closed surfaces; see Section 3.
Of course, a function defined on the unit torus (in any dimension) means that the
function is periodic with period one in each variable. (Warning: for convenience I use
𝐼𝑑
for both the unit torus [0, 1)𝑑
and the unit cube [0, 1]𝑑
; I hope this minor ambiguity
does not confuse the reader.)
Theorem 1.2. Let 𝑓 ∈ 𝐿2 be an arbitrary real-valued Lebesgue square-integrable func-
tion on the unit torus [0, 1)2
= 𝐼2
. Then for every 𝜀 > 0,
lim
𝑇→∞
∫
𝑇
0
𝑓(𝑡 cos 𝜃, 𝑡 sin 𝜃) 𝑑𝑡 − 𝑇 ∫𝐼2 𝑓(y) 𝑑y
(log 𝑇)3+𝜀
= 0
for almost every angle 𝜃.
Notice that Theorem 1.2 implies Theorem 1.1 in the special case 𝑓 = 𝜒𝑆. We derive
Theorem 1.2 from the following quantitative result.
Theorem 1.3. Let 𝑓 ∈ 𝐿2 be an arbitrary real-valued Lebesgue square-integrable func-
tion on the unit torus [0, 1)2
= 𝐼2
. Then for every 𝜂 > 0 there is a subset A = A(𝑓; 𝜂) of
the interval [0, 2𝜋) such that
1
2𝜋
meas(A) ≥ 1 − 𝜂 ,
and for every 𝜃 ∈ A and 𝑇 ≥ 8,
𝑇
∫
0
𝑓(𝑡 cos 𝜃, 𝑡 sin 𝜃) 𝑑𝑡 − 𝑇 ∫
𝐼2
𝑓(y) 𝑑y
≤
8
∫
0
|𝑓(𝑡 cos 𝜃, 𝑡 sin 𝜃)| 𝑑𝑡
+
12 ⋅ 104
𝜎0(𝑓)
𝜂
(log2 𝑇 + 1)3
(log(log2 𝑇 + 1))2
+ 12𝜎0(𝑓) ,](https://image.slidesharecdn.com/25565190-250514212713-9e6dff5a/85/Uniform-Distribution-And-Quasimonte-Carlo-Methods-Discrepancy-Integration-And-Applications-34-320.jpg)
![Strong uniformity | 21
where
𝜎2
0(𝑓) = ∫
𝐼2
(𝑓(y) − ∫
𝐼2
𝑓(z) 𝑑z)
2
𝑑y
is the “variance” of 𝑓.
To derive Theorem 1.2, choose 𝜂 = 2−𝑛
, 𝑛 = 1, 2, 3, . . . , and consider the union set
A =
∞
⋃
𝑛=1
A(𝑓; 2−𝑛
) .
Then meas(A) = 2𝜋; and by Theorem 1.3, for every 𝜀 > 0 and every 𝜃 ∈ A,
lim
𝑇→∞
∫
𝑇
0
𝑓(𝑡 cos 𝜃, 𝑡 sin 𝜃) 𝑑𝑡 − 𝑇 ∫𝐼2 𝑓(y) 𝑑y
(log 𝑇)3+𝜀
= 0 ,
proving Theorem 1.2.
It is natural to ask what happens in higher dimensions, where we replace the unit
square with the unit cube [0, 1]𝑑
, 𝑑 ≥ 3. Again we study the “strong” uniformity of
typical torus lines starting from the origin. Here, “strong” means that the test set is an
arbitrary measurable set.
Let 𝑆 ⊂ [0, 1]𝑑
= 𝐼𝑑
be an arbitrary Lebesgue measurable set in the unit cube
of dimension 𝑑 ≥ 3, and assume that 0 < volume(𝑆) < 1, where volume denotes
the 𝑑-dimensional Lebesgue measure. Let e ∈ S𝑑−1
be an arbitrary unit vector in the
𝑑-dimensional Euclidean space ℝ𝑑
; S𝑑−1
denotes the unit sphere in ℝ𝑑
. Consider the
straight line 𝑡e, 𝑡 ≥ 0 starting from the origin 0 ∈ ℝ𝑑
. Let 𝑇𝑆(e) denote the time the
line 𝑡e modulo 1 spends in the given set 𝑆 as 0 ≤ 𝑡 ≤ 𝑇 (line modulo 1=torus line).
Uniformity of the torus line 𝑡e modulo 1 in 𝑆 means that
lim
𝑇→∞
𝑇𝑆(e) − volume(𝑆)𝑇
𝑇
= 0 . (2.5)
In the 3-dimensional case we can replace the factor of 𝑇 in the denominator of (2.5)
with the substantially smaller 𝑇1/4
(log 𝑇)3+𝜀
for almost every direction e ∈ S2
in the
3-space. In the 𝑑-dimensional case with 𝑑 ≥ 4 we can replace the factor of 𝑇 in the
denominator of (2.5) with 𝑇
1
2
− 1
2(𝑑−1) (log 𝑇)3+𝜀
for almost every direction e ∈ S𝑑−1
in the
𝑑-space ℝ𝑑
.
Theorem 1.4. (a) Let 𝑆 ⊂ [0, 1)3
be an arbitrary Lebesgue measurable set in the unit
cube with 0 < volume(𝑆) < 1. Then for every 𝜀 > 0,
lim
𝑇→∞
𝑇𝑆(e) − volume(𝑆)𝑇
𝑇1/4(log 𝑇)3+𝜀
= 0 (2.6)
for almost every direction e ∈ S2
in the 3-space.](https://image.slidesharecdn.com/25565190-250514212713-9e6dff5a/85/Uniform-Distribution-And-Quasimonte-Carlo-Methods-Discrepancy-Integration-And-Applications-35-320.jpg)
![Strong uniformity | 23
(b) In the 𝑑-dimensional case with 𝑑 ≥ 4, we have the perfect analog of (2.8) where the
factor of 𝑇1/4
in (2.8) is replaced by 𝑐0(𝑑)𝑇
1
2
− 1
2(𝑑−1) , and 𝑐0(𝑑) is a positive absolute
constant that depends only on the dimension.
If we make the choice 𝑑 = 2 in Theorem 1.4 (or 1.5 or 1.6) then we obtain Theorem 1.1
(or 1.2 or 1.3). Perhaps the reader is wondering, why did we formulate two separate
theorems if the first one is the special case of the second one in the 2-dimensional
case. Well, the answer is that we wanted to emphasize superuniformity – meaning
polylogarithmic error term – which is not the case in dimensions 𝑑 ≥ 3. This follows
from the next result, which states that Theorem 1.4 (and 1.5 and 1.6) is best possible
apart from polylogarithmic factor: the exponent 1
2
− 1
2(𝑑−1)
of 𝑇 in the error term of
Theorem 1.4 (and 1.5 and 1.6) is best possible.
Note that in some problems it is natural to study very high dimensions; e.g. in
statistical mechanics the phase space is 6𝑁-dimensional, where 𝑁 is the number of
gas molecules in a container. Since 𝑁 is typically in the range of the Avogadro num-
ber (close to 1024
), it is natural to ask what happens if the dimension 𝑑 is much larger
than the time parameter 𝑇. Theorem 1.6 (b) is about the general case of arbitrary di-
mension 𝑑, but it does not help, because of the unspecified constant factor 𝑐0(𝑑) in
the upper bound for the discrepancy. What we need is an upper bound on the discrep-
ancy that does not depend on the dimension. I briefly mention such a dimension-free
result.
For simplicity I consider the special case 𝑓 = 𝜒𝑆, where 𝑆 is an arbitrary mea-
surable test set in the 𝑑-dimensional unit cube 𝐼𝑑
= [0, 1)𝑑
, and write 0 < 𝑝 =
volume(𝑆) < 1 and 𝑞 = 1 − 𝑝. Notice that the diameter of the 𝑑-dimensional unit
cube [0, 1)𝑑
is √𝑑, and this explains why it is natural to modify the time-counting
function (e ∈ S𝑑−1
)
𝑇𝑆(e) = meas {𝑡 ∈ [0, 𝑇]: 𝑡e ∈ 𝑆 modulo 1} ,
and to replace it with
𝑇𝑆(𝑑; e) = meas {𝑡 ∈ [0, 𝑇]: 𝑡√𝑑e ∈ 𝑆 modulo 1} ,
which can be interpreted as speeding up the linear motion by a factor of √𝑑. (In this
paper we focus on low dimensions such as 𝑑 = 2, 3, 4; then this switch is rather neg-
ligible.)
Here comes our “very high dimension vs. short time” result. Assume that 𝑇 >
log2 𝑑
𝑝𝑞
, then
( ∫
e∈S𝑑−1
(𝑇𝑆(𝑑; e) − 𝑝𝑇)
2
𝑑𝑆𝐴(e))
1/2
< 40√𝑝𝑞𝑇 (1 + 2
log2 𝑇
𝑑1/4
) , (2.9)
where 𝑑𝑆𝐴(e) denotes the integration with respect to the normalized surface area on
the sphere e ∈ S𝑑−1
, i.e. 𝑆𝐴(S𝑑−1
) = 1.](https://image.slidesharecdn.com/25565190-250514212713-9e6dff5a/85/Uniform-Distribution-And-Quasimonte-Carlo-Methods-Discrepancy-Integration-And-Applications-37-320.jpg)
![24 | József Beck
In view of the weak condition 𝑇 >
log2 𝑑
𝑝𝑞
, the dimension 𝑑 can be much larger
than the time parameter 𝑇, and still the quadratic average of the discrepancy (see the
left-hand side of (2.9)) is in the square-root range √𝑇 (see the right-hand side of (2.9));
notice that the order of magnitude √𝑇 represents a high level of uniformity with re-
spect to the given test set 𝑆. Combining (2.9) with Chebyshev’s inequality, we conclude
that for the majority of the directions e ∈ S𝑑−1
, the torus line segment 𝑡e(mod 1),
0 ≤ 𝑡 ≤ 𝑇 is very uniform with respect to the test set 𝑆. (Note that Theorem 1.6 (b) does
not say anything interesting in the case where the dimension 𝑑 is larger than the time
parameter 𝑇).
Similarly to Theorems 1.1 and 1.4, (2.9) is also “complexity-free”, i.e. the upper
bound in (2.9) does not depend on the complexity (“ugliness”) of the test set 𝑆.
We give a more detailed discussion of (2.9) in another paper, which is devoted
to the applications of dimension-free and complexity-free strong uniformity in non-
equilibrium statistical mechanics.
Let us return to Theorems 1.1 and 1.4: they are about the torus line, which is the
simplest curve on the torus. We can define a simple motion on the torus by assuming
that a particle moves on the torus line with unit speed. Theorem 1.7 below is a general
result about the limitations of the time discrepancy of a motion of a particle in the unit
torus [0, 1)𝑑
, 𝑑 ≥ 3.
Let
𝛤 = {x(𝑡) = ({𝑥1(𝑡)}, . . . , {𝑥𝑑(𝑡)}): 0 ≤ 𝑡 ≤ 𝑇} (2.10)
be an arbitrary (continuous) parametrized curve on the 𝑑-dimensional unit torus
[0, 1)𝑑
, 𝑑 ≥ 2 with arc-length 𝑇; here each coordinate 𝑥𝑗(𝑡), 1 ≤ 𝑗 ≤ 𝑑 is a contin-
uous function of 𝑡, and {𝑥} denotes, as usual, the fractional part of a real number 𝑥.
Note that the parametrized curve 𝛤 in (2.10) represents the motion of a particle on
the torus, and we constantly use this interpretation below; we refer to 𝑇 as the “total
traveling time”. Let 𝑆 ⊂ [0, 1)𝑑
be an arbitrary measurable subset, and let vol(𝑆)
denote the volume, i.e. the 𝑑-dimensional Lebesgue measure (of course vol = area
for 𝑑 = 2). Let 𝑇𝑆(𝛤) denote the time the particle spends in the given set 𝑆; formally,
𝑇𝑆(𝛤) = {0 ≤ 𝑡 ≤ 𝑇: x(𝑡) ∈ 𝑆} .
We call 𝑇𝑆(𝛤) the actual time, and compare it to the expected time, which – assuming
perfect uniformity – is proportional to the volume: expected time = 𝑇 ⋅ vol(𝑆). The
differenceof theactualtimeand theexpected timeis called time discrepancy; formally,
time discrepancy = D𝑆(𝛤) = 𝑇𝑆(𝛤) − 𝑇 ⋅ vol(𝑆) .
Short Detour
Note that the concept of time discrepancy is not new; in some sense it goes back to
Weyl; see also Section 1.9 in the book of Kuipers–Niederreiter [9] and Section 2.3 in the
book of Drmota and Tichy [3], which discuss the branch called “continuous uniform](https://image.slidesharecdn.com/25565190-250514212713-9e6dff5a/85/Uniform-Distribution-And-Quasimonte-Carlo-Methods-Discrepancy-Integration-And-Applications-38-320.jpg)
![Strong uniformity | 25
distribution mod 1”, studying arbitrary continuous parametrized curves. My main re-
sults (Theorems 1.1 and 1.4)are about the simplest curves (e.g. lines) with the most nat-
ural arc-length parametrization. If we allow arbitrary (continuous) parametrizations,
then we can have, at first sight, very surprising results such as the time discrepancy
can tend to zero arbitrarily fast; see e.g. Theorem 2.89 in [3]. But, unfortunately, this
result is almost trivial: the proof is just a few lines, it takes advantage of the extreme
generality of the definition, which allows using arbitrary parametrizations of a curve.
Of course we cannot expect a similar result for the natural arc-length parametrization.
The proofs of Theorems 1.1 and 1.4 (about the natural arc-length parametrization) are
not simple; in fact, they are very long and complicated (see [2]).
Just because Theorem 2.89 in [3] is almost trivial, it does not mean at all that the
results in “continuous uniform distribution mod 1” are all simple. For example, The-
orems 2.93 and 2.96 in [3] are very interesting deep results, but because they are not
too close to our subject of strong uniformity, I stop this short detour about “continu-
ous uniform distribution mod 1”. (Note that “continuous” discrepancy has been also
studied in much more general situations, e.g. in compact Riemann manifolds, but that
would lead us very far from our main topic.)
To prove a nontrivial result, we need an extra assumption. We assume that “𝑇 =
total traveling time = arc-length”, which is equivalent to the requirement that the av-
erage speed is one.
Theorem 1.7. For every integer 𝑑 ≥ 3 and real 𝑇 > 1, there exists an integer 𝑚 =
𝑚(𝑑, 𝑇) ≥ 2 such that we can construct 𝑚 measurable subsets 𝑆1, . . . , 𝑆𝑚 of the unit
torus [0, 1)𝑑
with the following property: given any parametrized curve
𝛤 = {x(𝑡) = ({𝑥1(𝑡)}, . . . , {𝑥𝑑(𝑡)}): 0 ≤ 𝑡 ≤ 𝑇}
of arc-length 𝑇 on the torus [0, 1)𝑑
(i.e. the average speed is one)
D𝑆𝑗
(𝛤)
> 𝑐1(𝑑)𝑇
1
2
− 1
2(𝑑−1)
holds for at least two thirds of the 𝑚 subsets 𝑆1, . . . , 𝑆𝑚. Here 𝑐1(𝑑) > 0 is a constant
depending only on the dimension 𝑑 ≥ 3. In particular, 𝑐1(3) = 1/500 is a good choice
for 𝑑 = 3.
Theorem 1.7 implies, via a standard average argument, that Theorem 1.6 is best pos-
sible apart from a polylogarithmic factor of 𝑇. The explanation goes as follows. First
note that every torus line in Theorem 1.6 is determined by its direction e ∈ S𝑑−1
, and
using the ((𝑑 − 1)-dimensional) surface area on the unit sphere S𝑑−1
, it is meaningful
to talk about the “majority of torus lines”, or more precisely, about “1 − 𝜀 part of all
torus lines passing through the origin”. Now assume that, for some 𝑑 ≥ 3 and 𝑇 > 1,
there exists a continuous family of parametrized curves
{𝛤𝜔 : 𝜔 ∈ 𝛺} (2.11)](https://image.slidesharecdn.com/25565190-250514212713-9e6dff5a/85/Uniform-Distribution-And-Quasimonte-Carlo-Methods-Discrepancy-Integration-And-Applications-39-320.jpg)
![Strong uniformity | 27
Note that the torus line does not represent a continuous motion inside a square
(or a cube): when it hits the boundary, it comes back at the opposite side of the bound-
ary – a big jump. There is a well-known way to switch from a torus line to a billiard
path (see unfolding below). Using this, it is easy to extend the results of Section 1 about
the torus line in a square (or a cube) to the billiard path in a square (or a cube). The ad-
vantage is that the billiard path is a piecewise smooth, continuous curve; it represents
a continuous motion. As a byproduct, we obtain that – roughly speaking – the set of
billiard paths represents the “most strongly uniform family of curves” in [0, 1)𝑑
, 𝑑 ≥ 2
(we can of course extend it to rectangles and axis-parallel boxes). Roughly speaking,
this means that we ignore polylogarithmic factors.
For simplicity we focus on the 2-dimensional case; we assume that the under-
lying rectangle is the unit square [0, 1]2
. Since every rectangle can be mapped to a
square by a linear transformation of the plane, this is not a real restriction. Let x(𝑡) =
(𝑥1(𝑡), 𝑥2(𝑡)), 0 < 𝑡 < ∞ represent an infinite billiard path with starting point x(0) =
(𝑥1(0), 𝑥2(0)) = s ∈ [0, 1]2
and initial direction (=angle) 𝜃 ∈ [0, 2𝜋).
The intuitively plausible concept of “billiard path” precisely means that a point-
mass (representing a tiny billiard ball) moves freely along a straight line inside the
unit square with unit speed until it hits the boundary (i.e. one of the four sides of the
square).
The reflection off the boundary is elastic, meaning the familiar law of reflection:
the angle of incidence equals the angle of reflection. After the reflection, the point-
billiard continues its linear motion with the new velocity (but of course the speed re-
mains the same) until it hits the boundary again, and so forth (we ignore friction, air
resistance, etc.). The initial condition, i.e. the starting point s ∈ [0, 1]2
of the billiard
path and the initial direction 𝜃, uniquely determine an infinite piecewise linear bil-
liard path x(𝑡) = (𝑥1(𝑡), 𝑥2(𝑡)), 0 < 𝑡 < ∞ in the unit square. The law of reflection
implies that there are at most four different directions along the billiard path (the ini-
tial direction is preserved modulo 𝜋/2, which is one-fourth of the whole angle 2𝜋; the
same holds for any rectangle). Because of the unit speed, arc-length and time are the
same.
Formally, a billiard path in the unit square [0, 1]2
has the form
x(𝑡) = (𝑥1(𝑡), 𝑥2(𝑡)) , 0 < 𝑡 < ∞ with 𝑥𝑗(𝑡) = 2
(𝑠𝑗 + 𝑡𝛽𝑗)/2
, 𝑗 = 1, 2 ,
where e = (𝛽1, 𝛽2) is a unit vector, and ‖𝑦‖ denotes the distance of a real number 𝑦
from the nearest integer. Here s = (𝑠1, 𝑠2) is the starting point, and e is the initial
direction. So 𝜃 = arctan
𝛽2
𝛽1
(“inverse tangent”) is the initial angle.
An alternative, more symmetric way is to replace [0, 1]2
with [−1
2
, 1
2
]2
; then
x(𝑡) = (𝑥1(𝑡), 𝑥2(𝑡)) , 0 < 𝑡 < ∞ with 𝑥𝑗(𝑡) = ⟨𝑠𝑗 + 𝑡𝛽𝑗⟩ , 𝑗 = 1, 2 ,
where
⟨𝑦⟩ = ‖𝑦‖ if 𝑦 ≥ 0 , and − ‖ − 𝑦‖ if 𝑦 < 0 .](https://image.slidesharecdn.com/25565190-250514212713-9e6dff5a/85/Uniform-Distribution-And-Quasimonte-Carlo-Methods-Discrepancy-Integration-And-Applications-41-320.jpg)
![Strong uniformity | 29
In the general case, the test set is upgraded to a function 𝑓 ∈ 𝐿2(𝐼2
), i.e. 𝑓(𝑥, 𝑦)
is a periodic real-valued Lebesgue square-integrable function with period one in both
variables. In the case of the picture below, 𝑓 means the 0–1 valued characteristic func-
tion of the union 𝑆1 ∪ 𝑆2 ∪ 𝑆3 ∪ 𝑆4 of the four half-size reflected copies of the given
subset 𝑆 ⊂ 𝐼2
= [0, 1)2
.
S1
S S2
S4
=
S3
A billiard pathintheunitcube[0, 1]𝑑
, 𝑑 ≥ 3canbedefined similarly. Theproblem
of uniformity of a billiard path in the unit cube [0, 1)𝑑
with respect to a given test set
𝑆 is equivalent – via unfolding – to the problem of uniformity of the corresponding
torus-line in the 2 × ⋅ ⋅ ⋅ × 2 cube, where each one of the 2𝑑
unit cubes contains a
reflected copy of the given test set 𝑆.
As far as I know, the first appearance of the geometric trick of unfolding is in a
paper of D. König and A. Szücs from 1913 (see [8]), and it became widely known after
Hardy and Wright included it in their well-known book on number theory [4]. The con-
tinuous form of the equidistribution theorem (Kronecker–Weyl) implies the following
elegant property of the torus line in the unit square: if the slope of the initial direction
is rational, then the torus line is periodic, and if the slope of the initial direction is ir-
rational, then the torus line is dense (so far this is Kronecker’s theorem). And what is
more, for an irrational slope the torus line is uniformly distributed in the unit square,
meaning that, for any axis-parallel subrectangle 𝑆 = [𝑎, 𝑏] × [𝑐, 𝑑] ⊂ [0, 1]2
,
lim
𝑇→∞
1
𝑇
measure {𝑡 ∈ [0, 𝑇]: x(𝑡) ∈ 𝑆} = area(𝑆) = (𝑏 − 𝑎)(𝑑 − 𝑐) ,
where x(𝑡), 0 < 𝑡 < ∞ denotes the torus line in the unit square, parametrized with
the arc-length. (This upgrading of Kronecker’s theorem is due to H. Weyl.)](https://image.slidesharecdn.com/25565190-250514212713-9e6dff5a/85/Uniform-Distribution-And-Quasimonte-Carlo-Methods-Discrepancy-Integration-And-Applications-43-320.jpg)
![30 | József Beck
König and Szücs used the trick of unfolding, combined with the Kronecker–Weyl
theorem, to prove the following analog property of the billiard path in the unit square:
if the slope of the initial direction is rational, then the billiard path is periodic, and if
the slope of the initial direction is irrational, then the billiard path is uniformly dis-
tributed in the unit square. This settles the 2-dimensional case. There is an analog
result in every dimension 𝑑 ≥ 3.
To talk about a “typical” billiard path, we need a measure on the set of all initial
conditions of the billiard paths. Since the initialcondition is the pair of a starting point
s ∈ [0, 1]2
and an initial direction (angle) 𝜃 ∈ [0, 2𝜋), the corresponding measure is
simply the (Cartesian) product of the 2-dimensional Lebesgue measure on the unit
square and the normalized 1-dimensional Lebesgue measure. This way the term “1 −
𝜀 part of all billiard paths” in Theorem 2.1 below will become perfectly precise. The
following result was proved in my first paper about superuniformity.
Theorem 2.1 (Beck [1]). Let 𝑆 be an arbitrary Lebesgue measurable subset of the unit
square [0, 1)2
with 2-dimensional Lebesgue measure area(𝑆) > 0, and let 𝑇 > 100 be
an arbitrarily large but fixed real number. Let x(𝑡) = (𝑥1(𝑡), 𝑥2(𝑡)), 0 ≤ 𝑡 ≤ 𝑇 be a
general billiard path of length 𝑇 (length=time) in the unit square (the starting point is
not necessarily the origin). Let 𝑇𝑆 denote the time the billiard path spends in subset 𝑆:
𝑇𝑆 = measure {𝑡 ∈ [0, 𝑇]: x(𝑡) ∈ 𝑆} .
Let 0 < 𝜂 < 1/2 be arbitrary. Then for the (1 − 𝜂)-part of all billiard paths of length 𝑇 in
the square, the curve-discrepancy |𝑇𝑆 − 𝑇 ⋅ area(𝑆)| is estimated from above as follows:
|𝑇𝑆 − 𝑇 ⋅ area(𝑆)| <
20
𝜂
√area(𝑆)(1 − area(𝑆)) ⋅ √log 𝑇 ⋅ log log 𝑇 . (2.14)
Moreover, Theorem 2.1 remains true if “billiard path” is replaced with “torus line”.
This is how the proof goes: we prove Theorem 2.1 for torus lines only – the case of bil-
liard paths is a corollary via unfolding. This is true in general: if we have a theorem
about torus lines (e.g. Theorem 1.1 or Theorem 1.4), the use of unfolding automatically
extends it to billiard paths. To formulate the billiard versions of Theorem 1.3 and The-
orem 1.6, we use the notation x𝜃(𝑡) and xe(𝑡) to denote, respectively, the billiard path
starting from the origin with initial angle 𝜃 ∈ [0, 2𝜋) in the unit square [0, 1]2
and
with initial direction e ∈ S𝑑−1
in the unit cube [0, 1]𝑑
, 𝑑 ≥ 3.
Theorem 2.2. (a) Let 𝑓 ∈ 𝐿2 be an arbitrary real-valued Lebesgue square integrable
function on the unit square [0, 1]2
= 𝐼2
. Then for every 𝜂 > 0 there is a subset
A = A(𝑓; 𝜂) of the interval [0, 2𝜋) such that
1
2𝜋
meas(A) ≥ 1 − 𝜂 ,](https://image.slidesharecdn.com/25565190-250514212713-9e6dff5a/85/Uniform-Distribution-And-Quasimonte-Carlo-Methods-Discrepancy-Integration-And-Applications-44-320.jpg)
![Strong uniformity | 31
and for every 𝜃 ∈ A and 𝑇 ≥ 8,
𝑇
∫
0
𝑓(x𝜃(𝑡)) 𝑑𝑡 − 𝑇 ∫
𝐼2
𝑓(y) 𝑑y
≤
8
∫
0
|𝑓(x𝜃(𝑡))| 𝑑𝑡
+
12 ⋅ 104
𝜎0(𝑓)
𝜂
(log2 𝑇 + 1)3
(log(log2 𝑇 + 1))2
+ 12𝜎0(𝑓) ,
where
𝜎2
0(𝑓) = ∫
𝐼2
(𝑓(y) − ∫
𝐼2
𝑓(z) 𝑑z)
2
𝑑y .
(b) Let 𝑓 ∈ 𝐿2 be an arbitrary real-valued Lebesgue square integrable function on the
unit cube [0, 1]3
= 𝐼3
. Then for every 𝜂 > 0 there is a subset A = A(𝑓; 𝜂) of the
unit sphere S2
such that
1
4𝜋
SurfaceArea(A) ≥ 1 − 𝜂 ,
and for every direction e ∈ A and 𝑇 ≥ 8,
𝑇
∫
0
𝑓(xe(𝑡)) 𝑑𝑡 − 𝑇 ∫
𝐼3
𝑓(y) 𝑑y
≤
8
∫
0
|𝑓(xe(𝑡))| 𝑑𝑡
+
12 ⋅ 105
𝜎0(𝑓)
𝜂
𝑇1/4
(log2 𝑇 + 1)3
(log(log2 𝑇 + 1))2
+ 12𝜎0(𝑓) , (2.15)
where again
𝜎2
0(𝑓) = ∫
𝐼3
(𝑓(y) − ∫
𝐼3
𝑓(z) 𝑑z)
2
𝑑y .
(c) In the case of the 𝑑-dimensional billiard with 𝑑 ≥ 4, we have the perfect analog of
(2.15) where the factor of 𝑇1/4
in (2.15) is replaced by 𝑐0(𝑑)𝑇
1
2
− 1
2(𝑑−1) , and 𝑐0(𝑑) is a
positive absolute constant that depends only on the dimension.](https://image.slidesharecdn.com/25565190-250514212713-9e6dff5a/85/Uniform-Distribution-And-Quasimonte-Carlo-Methods-Discrepancy-Integration-And-Applications-45-320.jpg)
![32 | József Beck
Let us return to Theorem 2.1: it was my first result proving superuniformity, i.e. su-
peruniformity of curves. Comparing it to Theorem 1.3 we see two major weaknesses:
(1) Theorem 2.1 works for a fixed 𝑇 instead of all 𝑇’s simultaneously, (2) the starting
point is an unspecified “typical” point instead of a specified point such as the origin.
The advantage of Theorem 2.1 is that the error term is substantially smaller:
(log 𝑇)
1
2
+𝜀
instead of (log 𝑇)
3+𝜀
.
Notice that the complexity of the test set 𝑆 does not appear in the explicit form of the
error term – this is true for both Theorem 1.3 and Theorem 2.1.
It is very surprising that, given an arbitrarily complicated measurable subset 𝑆 ⊂
[0, 1)2
(and measurable sets can be very, very complicated!), the curve-discrepancy
|𝑇𝑆 − 𝑇 ⋅ area(𝑆)| is at most roughly square-root logarithmic for the majority of the
billiard paths; see (2.14). Since square-root logarithmic is “almost” constant, the “ug-
liness” (=complexity) of 𝑆 plays a negligible role in (2.14). Indeed, (2.14) is nearly sharp
inthefollowingstrongsense. Evenif Theorem 2.1is restricted to thenicest possibletest
sets – say, the family of axes-parallel subsquares – then (at least) constant size curve-
discrepancy 𝑂(1) in (2.14) is still unavoidable, meaning that we cannot have 𝑜(1) in-
stead; we explain it below. This shows that the complexity of the test set 𝐴 ⊂ [0, 1)2
in
Theorem 2.1 is basically irrelevant: the curve-discrepancy is between roughly √log 𝑇
and 𝑂(1), i.e. it is almost the same independently of the complexity of the test set
𝑆 ⊂ [0, 1)2
.
We now quickly prove the (almost trivial) fact that for the class of axes-parallel
subsquares – as test sets – we must have (at least) constant curve-discrepancy 𝑂(1)
in (2.14). Consider the two subsquares, 𝑆1 = [0, 1/3]2
and 𝑆2 = [2/3, 1]2
; the distance
between them is √2/3. Let x(𝑡)be an arbitrarycontinuous curve in the unit square; we
always assume that the arc-length of every segment x(𝑡), 𝑇1 < 𝑡 < 𝑇2 is exactly 𝑇2 −𝑇1
(meaning: 𝑡 is the time and a point-mass moves along the curve with unit speed). For
any real number 𝜏 > 0 write
𝑆𝑖(𝜏) = measure {𝑡 ∈ [0, 𝜏]: x(𝑡) ∈ 𝑆𝑖} , 𝑖 = 1, 2 ,
where𝑆𝑖, 𝑖 = 1, 2arethetwo subsquares mentioned above. Weshowthatthefollowing
four curve-discrepancies:
|𝑆𝑖(𝑇) − 𝑇 ⋅ area(𝑆𝑖)| , |𝑆𝑖(𝑇 + 𝑐) − (𝑇 + 𝑐) ⋅ area(𝑆𝑖)| , 𝑖 = 1, 2 , (2.16)
where 𝑐 = √2/3 is the distance between the two given subsquares 𝑆1 and 𝑆2 (com-
puted for the same curve!), cannot be all 𝑜(1). Indeed, the middle segment x(𝑡), 𝑇 <
𝑡 < 𝑇 + 𝑐 of the curve cannot visit both subsquares (because the arc-length is exactly
the distance between 𝑆1 and 𝑆2); consequently, at least one of the four curve-discrep-
ancies in (2.16) must be
≥
1
2
𝑐 ⋅ area(𝑆𝑖) =
1
2
⋅
√2
3
⋅
1
9
=
√2
54
.](https://image.slidesharecdn.com/25565190-250514212713-9e6dff5a/85/Uniform-Distribution-And-Quasimonte-Carlo-Methods-Discrepancy-Integration-And-Applications-46-320.jpg)
![34 | József Beck
𝑆 ⊂ [0, 1]2
be an arbitrary measurable test set in the unit square of side length one
meter. We apply Theorem 2.3 with 𝑓 = 𝜒𝑆, 𝑇 = 1027
and 𝜂 = 1/5. Then for at least
80% of the initial conditions, if the point-billiard travels 𝑇 = 1027
meters (which is
more than the diameter of the observable universe!), then the total distance traveled
inside 𝑆 differs from the expected value 𝑇 ⋅ area(𝑆) by less than
30𝜎0(𝑓)
𝜂
√log 𝑇 ≤
30 ⋅ 1/2
1/5
√log 1027 < 630 meters .
Assume the point-billiard represents a gas molecule. A typical gas molecule moves
with speed 103
m/s, and to cover the total distance 1027
meters, it takes 1024
seconds,
which is a million times more than the estimated age of the universe.
On the other hand, it takes less than one second for the gas molecule to cover 630
meters (=discrepancy). One second versus a million times the age of the universe is a
strikingcontrast; itrepresents anastonishingprecision, especially thatthecomplexity
of the given test set 𝑆 is irrelevant.
Theorem 2.3 provides a continuous family of curves of length 𝑇 in a square –
all billiard paths of length 𝑇 – that has strong discrepancy 𝑂(√log 𝑇). Here, strong
means that we allow arbitrary measurable test sets, and want uniformity for the ma-
jority of the curves. The analog problem in one less dimension – namely, the strong
discrepancy of the most uniform continuous family of 𝑁-element point sets in the
unit interval – exhibits a much worse quantitative behavior: we have √𝑁 instead of
√log 𝑇, which is a super-exponential jump! The fact that discrepancy of size √𝑁 is
inevitable follows from the following result, which is basically a discrete analog of
Theorem 1.7.
Theorem 2.5. For every integer 𝑁 ≥ 1 there exists another integer 𝑚 = 𝑚(𝑁) such that
we can construct 𝑚 Lebesgue measurable subsets 𝑆1, . . . , 𝑆𝑚 of the unit interval [0, 1)
with the following property: given any 𝑁-element subset X = {𝑥1, 𝑥2, . . . , 𝑥𝑁} in the
unit interval [0, 1], for at least two thirds of the 𝑚 subsets 𝑆1, . . . , 𝑆𝑚,
|X ∩ 𝑆𝑗| − 𝑁 ⋅ meas(𝑆𝑗)
>
√2
20
√𝑁 .
Theorem 2.5 implies, via a standard average argument, that for continuous families of
𝑁-element point sets it is inevitable to have strong discrepancy of size √𝑁. Indeed,
it follows from (basically) repeating the argument after Theorem 1.7. Assume that, for
some 𝑁 > 1, there exists a continuous family of 𝑁-element point sets
{X𝜔 : 𝜔 ∈ 𝛺} (2.17)
on the unit interval [0, 1) such that there is a probability measure 𝜇 on the index-set 𝛺
(i.e. 𝜇(𝛺) = 1, so it is meaningful to talk about a (1−𝜀)-part), and the family of 𝑁-sets
in (2.17) is strongly uniform in the following quantitative sense: Given any measurable](https://image.slidesharecdn.com/25565190-250514212713-9e6dff5a/85/Uniform-Distribution-And-Quasimonte-Carlo-Methods-Discrepancy-Integration-And-Applications-48-320.jpg)
![Strong uniformity | 35
subset 𝑆 ⊂ [0, 1),
D𝑆(X𝜔)
<
√2
20
√𝑁 (2.18)
holds for at least two-thirds of the 𝑁-sets X𝜔, 𝜔 ∈ 𝛺 in the sense of the 𝜇-measure.
We show that this contradicts Theorem 2.5. Indeed, we apply (2.18) for the 𝑚 =
𝑚(𝑁) ≥ 2 sets 𝑆𝑗, 1 ≤ 𝑗 ≤ 𝑚 whose existence is guaranteed by Theorem 2.5. Thus
for every 1 ≤ 𝑗 ≤ 𝑚 there exists a (measurable) subset 𝛺𝑗 of the index-set such that
𝜇(𝛺𝑗) ≥ 2/3, and
D𝑆𝑗
(X𝜔)
<
√2
20
√𝑁
holds for all 𝜔 ∈ 𝛺𝑗. The fact
1
𝑚
𝑚
∑
𝑗=1
𝜇(𝛺𝑗) ≥ 2/3
implies that there must exist an index 𝜔0 ∈ 𝛺 which is contained by at least 2𝑚/3
sets 𝛺𝑗. In other words, there is an 𝑁-set X𝜔0
such that (2.18))
D𝑆𝑗
(X𝜔0
)
<
√2
20
√𝑁 (2.19)
holds for at least two-thirds of the 𝑚 sets 𝑆1, . . . , 𝑆𝑚. But (2.19) clearly contradicts The-
orem 2.5, and this contradiction proves that an error of size 𝑜(√𝑁) is impossible.
On the other hand, the family of arithmetic progressions 𝛽 + 𝑘𝛼, 0 ≤ 𝑘 < 𝑁
modulo 1 shows that the error √𝑁 is best possible, that is, Theorem 2.5 is sharp. This
follows from the next two statements.
Proposition 2.6. Let 𝑆 ⊂ [0, 1] be an arbitrary Lebesgue measurable subset of the unit
interval, and let 𝑁 ≥ 1 be an arbitrarily large integer. Then for more than 75% of the
pairs (𝛼, 𝛽) ∈ [0, 1]2
,
∑
0≤𝑘≤𝑁−1:
𝑘𝛼+𝛽∈𝑆(mod 1)
1 − 𝑁 ⋅ meas(𝑆)
≤ √𝑁 ,
where meas stands for the 1-dimensional Lebesgue measure.
Proposition 2.7. Let 𝑆𝑗 ⊂ [0, 1], 𝑗 = 1, 2, . . . , 𝑚 be an arbitrary finite sequence of
Lebesgue measurable subsets of the unit interval, and let 𝑁 ≥ 1 be an arbitrarily large
integer. Then there is a pair (𝛼0, 𝛽0) ∈ [0, 1]2
such that
∑
0≤𝑘≤𝑁−1:
𝑘𝛼0+𝛽0∈𝑆𝑗(mod 1)
1 − 𝑁 ⋅ meas(𝑆𝑗)
≤ √𝑁
holds for more than 75% of the sets 𝑆𝑗, 𝑗 = 1, 2, . . . , 𝑚.](https://image.slidesharecdn.com/25565190-250514212713-9e6dff5a/85/Uniform-Distribution-And-Quasimonte-Carlo-Methods-Discrepancy-Integration-And-Applications-49-320.jpg)
![38 | József Beck
We choose a test set 𝑆: it is an arbitrary Lebesgue measurable subset of the unit
torus [0, 1)3
with 3-dimensional Lebesgue measure 0 < vol(𝑆) < 1.
Let
𝑆(𝑇1, 𝑇2; b1, b2; s) = 𝛤(𝑇1, 𝑇2; b1, b2; s) ∩ 𝑆
denote the intersection of the torus-parallelogram (2.22) with the given subset 𝑆; the
area of the intersection equals
area(𝑆(𝑇1, 𝑇2; b1, b2; s))
= |b1 × b2| ⋅ area {(𝑡1, 𝑡2) ∈ [0, 𝑇1] × [0, 𝑇2]: x(𝑡1, 𝑡2) ∈ 𝑆(mod 1)} , (2.23)
where b1 × b2 denotes the usual cross product of 3-dimensional vectors.
In view of (2.23) it is natural to call
|b1 × b2|𝑇1𝑇2 ⋅ vol(𝑆)
the expectation: in the case of perfect uniformity the actual intersection area of the
2 − 𝑑-ray area(𝑆(𝑇1, 𝑇2; b1, b2; s)) equals the expectation.
Theorem 2.9. Let 𝑆 be an arbitrary measurable subset of the unit cube [0, 1)3
with 3-
dimensional Lebesgue measure 0 < vol(𝑆) < 1. Let 0 < 𝜂 < 1/2 be arbitrary. Then for
a (1 − 𝜂)-part of all directions (b1, b2) ∈ S2
× S2
,
discrepancy = | area(𝑆(𝑇1, 𝑇2; b1, b2; s)) − |b1 × b2|𝑇1𝑇2 ⋅ vol(𝑆)|
<
1000 log2
(1/𝜂) + 200
𝜂2
⋅ √vol(𝑆)(1 − vol(𝑆)) , (2.24)
for all values of the size parameters 𝑇1, 𝑇2 and for all starting points s ∈ ℝ3
.
Remarks 2.10. (1) An interesting special case is where one of the size parameters
is fixed, and the other one tends to infinity. For example, let 𝑇2 = 10−3
and
𝑇1 = 𝑇 → ∞. Then the 2 − 𝑑-ray in (2.22) represents a “long-and-narrow flat
laser beam” traveling inside a cube, reflecting on the walls (this “billiard beam”
is reduced to the “torus beam” via unfolding). Equation (2.24) describes the super-
duper uniformity of the narrow flat laser beam: the error term remains uniformly
bounded for all 0 < 𝑇 < ∞.
(2) As usual, the “ugliness” (= complexity) of 𝑆 plays absolutely no role here; see
(2.24). Given an arbitrarily complicated subset 𝑆 ⊂ [0, 1)3
, the typical surface-dis-
crepancy (see (2.24)) is less than an absolute constant. (Note that I did not make a
serious effort to find the best values of the constants in the last line of (2.24).) The
constant upper bound is sharp: even if Theorem 2.9 is restricted to the simplest
subsets, say, to the family of axes-parallel subcubes, then (at least) constant sur-
face-discrepancy 𝑂(1) in (2.24) is still unavoidable – I explain it in (3) below. This
means that the complexity of the test set 𝑆 ⊂ [0, 1)3
in Theorem 2.9 is completely
irrelevant.](https://image.slidesharecdn.com/25565190-250514212713-9e6dff5a/85/Uniform-Distribution-And-Quasimonte-Carlo-Methods-Discrepancy-Integration-And-Applications-52-320.jpg)
![Strong uniformity | 39
(3) I outline the almost trivial fact why even for the narrow class of axes-parallel sub-
cubes we must have (at least) constant surface-discrepancy 𝑂(1) in (2.24). The
following argument is a straightforward adaptation of the argument in part (3)
of the Remarks after Theorem 2.1. Consider the two subcubes 𝑆1 = [0, 1/3]3
and
𝑆2 = [2/3, 1]3
; the distance between them is √3/3. We show that the following
four surface-discrepancies cannot be all very small like 𝑜(1): let s = 0 and con-
sider
| area(𝑆𝑖(𝑇1, 𝑇2; b1, b2; 0)) − 𝑇1𝑇2|b1 × b2| ⋅ vol(𝑆𝑖)
where 𝑖 = 1, 2 and (𝑇1, 𝑇2) runs through the pair (𝑇, 10−3
) and (𝑇 + 𝑐, 10−3
); here
𝑐 > 0 is a sufficiently small positive constant. Indeed, the difference set
[𝑇, 𝑇 + 𝑐] × [0, 10−3
] (2.25)
is a “small” rectangle, and in terms of the torus parallelogram (2.22), the equation
(𝑡1, 𝑡2) ∈ rectangle (2.25) gives a “small” torus parallelogram of diameter less
than √3/3, i.e. the diameter is less than the distance between the two subcubes 𝑆1
and 𝑆2, if 𝑐 > 0 is a small enough absolute constant. Therefore, because of the
small diameter, this “small” constant size torus parallelogram (see (2.25)) cannot
intersect both 𝑆1 and 𝑆2, which yields a constant size surface-discrepancy. This
proves that in Theorem 2.9 we cannot expect that the surface-discrepancy tends
to zero as the product 𝑇1𝑇2 tends to infinity, not even for the simplest families of
subsets, such as the family of all axes-parallel subcubes.
(4) In Theorem 2.9 we test the uniformity of a 𝑘-dimensional ray with respect to a
(measurable) subset 𝑆 ⊂ [0, 1]𝑑
, where 𝑘 = 2 and 𝑑 = 3 (Theorem 1.1 corresponds
to the case 𝑘 = 1and 𝑑 = 2). Consider now the general case with arbitrary 1 ≤ 𝑘 <
𝑑: 𝑘-dimensional ray in the 𝑑-dimensional unit torus [0, 1)𝑑
. By a straightforward
adaptation of our proof technique, one can show that the corresponding upper
bound for the discrepancy is 𝑂(1) if 𝑘 > 𝑑/2, and it is
𝑇
1
2
− 𝑘
2(𝑑−𝑘)
+𝜀
(2.26)
if 𝑘 ≤ 𝑑/2, where 𝑇 denotes the total 𝑘-dimensional surface area (arc-length for
𝑘 = 1). Moreover, the exponent
1
2
− 𝑘
2(𝑑−𝑘)
in (2.26) is sharp: it cannot be replaced
by any smaller constant
1
2
− 𝑘
2(𝑑−𝑘)
− 𝜀.
What is going on here? To give a guiding intuition, let 𝑆 ⊂ [0, 1)𝑑
be an arbitrary
Lebesgue measurable subset, and consider the Fourier series of the characteristic
function 𝜒𝑆:
𝜒𝑆(u) = ∑
r∈ℤ𝑑
𝑎r𝑒2𝜋ir⋅u
with 𝑎r = ∫
𝑆
𝑒−2𝜋ir⋅y
𝑑y ,
where 𝑎0 = vol(𝑆)(𝑑-dimensional Lebesgue measure), and by Parseval’s formula,
∑
r∈ℤ𝑑0
|𝑎r|2
= vol(𝑆) − vol
2
(𝑆) . (2.27)](https://image.slidesharecdn.com/25565190-250514212713-9e6dff5a/85/Uniform-Distribution-And-Quasimonte-Carlo-Methods-Discrepancy-Integration-And-Applications-53-320.jpg)
![40 | József Beck
Assume we want to test the uniformity of a 𝑘-dimensional ray with respect to the
given 𝑆 ⊂ [0, 1]𝑑
. The following sum plays a critical role:
∑
r∈ℤ𝑑0
|𝑎r|
|r|𝑘
. (2.28)
Using the Cauchy–Schwarz inequality in (2.28), we obtain
∑
r∈ℤ𝑑0
|𝑎r|
|r|𝑘
≤ ( ∑
r∈ℤ𝑑0
|𝑎r|2
)
1/2
( ∑
r∈ℤ𝑑0
|r|−2𝑘
)
1/2
= √vol(𝑆)(1 − vol(𝑆)) ( ∑
r∈ℤ𝑑0
|r|−2𝑘
)
1/2
,
where we used (2.27).
Simple lattice point counting shows that the infinite sum
∑
r∈ℤ𝑑0
|r|−2𝑘
(2.29)
is convergent if 2𝑘 > 𝑑. This is when we have “super-duper uniformity” (like
Theorem 2.9).
If 2𝑘 = 𝑑, then (2.29) has a very slow, logarithmic divergence. This is when we still
have superuniformity (like Theorems 1.1 and 2.3).
(5) As we explained in (4) above, “super-duper uniformity” is very common in higher
dimensions. On the other hand, there is no similar “super-duper uniformity” in
the discrete case. We reduce the dimension by 2: the 2 − 𝑑-ray becomes a point
sequence, and the unit cube is reduced to the unit interval. The discrete analog of
a ray is the 𝑛𝛼 sequence
𝛼, 2𝛼, 3𝛼, . . . , 𝑛𝛼, . . . (mod 1).
Consider the simplest possible test set in the unit interval: let 𝐼 = [𝑎, 𝑏) ⊂ [0, 1)
be a subinterval. It is easy to see that for all 𝑁 ≥ 1,
∑
1≤𝑛≤𝑁:
𝑛𝛼∈𝐼(mod 1)
1 − 𝑁(𝑏 − 𝑎)
≤ 1
holds for every subinterval 𝐼 of the unit torus that has length 𝛼 > 0. In general,
sup
𝑁≥1
∑
1≤𝑛≤𝑁:
𝑛𝛼∈𝐼(mod 1)
1 − 𝑁(𝑏 − 𝑎)
< ∞ (2.30)
holds for every subinterval 𝐼 of the unit torus that has length 𝑘𝛼 (mod 1) for some
integer 𝑘. A well-known theorem of Kesten [6] proves the converse of (2.30): if](https://image.slidesharecdn.com/25565190-250514212713-9e6dff5a/85/Uniform-Distribution-And-Quasimonte-Carlo-Methods-Discrepancy-Integration-And-Applications-54-320.jpg)
Uniform Distribution And Quasimonte Carlo Methods Discrepancy Integration And Applications
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- 5. Peter Kritzer, Harald Niederreiter, Friedrich Pillichshammer, Arne Winterhof (Eds.) Uniform Distribution and Quasi-Monte Carlo Methods
- 6. Radon Series on Computational and Applied Mathematics | Managing Editor Ulrich Langer, Linz, Austria Editorial Board Hansjörg Albrecher, Lausanne, Switzerland Heinz W. Engl, Linz/Vienna, Austria Ronald H. W. Hoppe, Houston, Texas, USA Karl Kunisch, Linz/Graz, Austria Harald Niederreiter, Linz, Austria Volume 15
- 7. Uniform Distribution and Quasi-Monte Carlo Methods | Discrepancy, Integration and Applications Edited by Peter Kritzer Harald Niederreiter Friedrich Pillichshammer Arne Winterhof
- 8. Mathematics Subject Classification 2010 11K, 65D, 91G, 11N, 42A ISBN 978-3-11-031789-3 e-ISBN 978-3-11-031793-0 ISSN 1865-3707 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available in the Internet at http://dnb.dnb.de. © 2014 Walter de Gruyter GmbH, Berlin/Boston Typesetting: le-tex publishing services GmbH, Leipzig Printing and binding: CPI books GmbH, Leck ♾Printed on acid-free paper Printed in Germany www.degruyter.com
- 9. Preface This book is based on several invited talks of the workshop “Uniform Distribution and Quasi-Monte Carlo Methods” which was part of the RICAM Special Semester on “Ap- plications of Algebra and Number Theory”. The workshop took place in Linz, Austria, on October 14–18, 2013. The workshop and the papers contained in this book focus on number theoretic point constructions, uniform distribution theory and quasi-Monte Carlo methods. As deterministic versions of the Monte Carlo method, quasi-Monte Carlo rules enjoy in- creasing popularity, with many fruitful applications in mathematical practice, as for example in finance, computer graphics and biology. These rules require nodes with good uniform distribution properties, and number theoretic constructions are known to be excellent candidates for such point sets. The goal of this book is to give an overview of recent developments in uniform distribution theory, quasi-Monte Carlo methods, and their applications, presented by leading experts in these vivid fields of research. The collection of surveys can be grouped into papers covering theoretical aspects of uniform distribution, and articles discussing quasi-Monte Carlo methods and their applications. We shortly summa- rize the topics of this volume, ranging from the more theoretical papers to the more applied ones. Chapter “Metric number theory, lacunary series and systems of dilated functions” deals with special aspects of metric number theory and lacunary series related to uni- form distribution. Chapter “Strong uniformity” introduces a new concept of uniform distribution linked with the Lebesgue measure, which is called strong uniformity. Chapter “Discrepancy theory and harmonic analysis” highlights relations of discrep- ancy theory to harmonic analysis and provides insights into very recent results ex- ploiting these methods. Chapters “Explicit constructions of point sets and sequences with low discrepancy” and “On Atanassov’s methods for discrepancy bounds of low-discrepancy sequences” deal with the discrepancy of well known types of uni- formly distributed point sets, namely higher order digital nets and classical digital (𝑡, 𝑠)-sequences, respectively. Chapter “Subsequences of automatic sequences and uniform distribution” discusses the distribution of subsequences of automatic se- quences. Chapter “The hybrid spectral test: a unifying concept” presents the hybrid spectral test as a measure for the uniformity of sequences, and Chapter “Tractability of multivariate analytic problems” covers new results on the tractability of multivari- ate analytic problems. Chapter “Discrepancy estimates for sequences: new results and open problems” provides an overview on recent results on discrepancy estimates for concrete sequences in the unit-cube and states a list of open problems. Chap- ter “A short introduction to quasi-Monte Carlo option pricing” discusses applications to finance, and finally, Chapter “The construction of good lattice rules and polyno- mial lattice rules” deals with the challenge of effectively constructing point sets for numerical integration.
- 10. vi | Preface All chapters were reviewed by renowned experts in this field. We wish to thank the anonymous referees for their precious help. We also would like to thank Annette Weihs, Mersiha Džihanić and Wolfgang Forsthuber for administrative support and all the speakers of the workshop who con- tributed excellent talks and made the workshop a great success: Christoph Aistleitner, József Beck, Dmitriy Bilyk, Johann Brauchart, Ronald Cools, Josef Dick, Michael Dr- mota, Henri Faure, Michael Gnewuch, Peter Grabner, Peter Hellekalek, Christian Irrge- her, Frances Kuo, Gerhard Larcher, Pierre L’Ecuyer, Christian Mauduit, Dirk Nuyens, Ian Sloan, Alev Topuzoğlu, Aljoša Volčič and Henryk Woźniakowski. More details on the RICAM special semester “Applications of Algebra and Number Theory” can be found at the webpage http://www.ricam.oeaw.ac.at/specsem/specsem2013/. We also thank the Johann Radon Institute for Computational and Applied Mathemat- ics (RICAM) of the Austrian Academy of Sciences for financial support. We hope that this book will be a useful source for many people who study or apply quasi-Monte Carlo methods. Peter Kritzer Linz, Harald Niederreiter January 2014 Friedrich Pillichshammer Arne Winterhof
- 11. Contents Preface | v Christoph Aistleitner Metric number theory, lacunary series and systems of dilated functions | 1 1 Uniform distribution modulo 1 | 2 2 Metric number theory | 4 3 Discrepancy | 6 4 Lacunary series | 7 5 Almost everywhere convergence | 10 6 Sums involving greatest common divisors | 12 József Beck Strong uniformity | 17 1 Introduction | 17 2 Superuniformity and super-duper uniformity | 26 2.1 Superuniformity of the typical billiard paths | 26 2.2 Super-duper uniformity of the 2-dimensional ray | 37 3 Superuniform motions | 41 3.1 Billiards in other shapes | 41 3.2 Superuniformity of the geodesics on an equifacial tetrahedron surface | 42 Dmitriy Bilyk Discrepancy theory and harmonic analysis | 45 1 Introduction | 45 2 Exponential sums | 46 3 Fourier analysis methods | 49 3.1 Rotated rectangles | 49 3.2 The lower bound for circles | 51 3.3 Further remarks | 53 4 Dyadic harmonic analysis: discrepancy function estimates | 54 4.1 𝐿𝑝 -discrepancy, 1 < 𝑝 < ∞ | 55 4.2 The 𝐿∞ discrepancy estimates | 56 4.3 The other endpoint, 𝐿1 | 58
- 12. viii | Contents Josef Dick and Friedrich Pillichshammer Explicit constructions of point sets and sequences with low discrepancy | 63 1 Introduction | 63 2 Lower bounds | 65 3 Upper bounds | 67 4 Digital nets and sequences | 69 5 Walsh series expansion of the discrepancy function | 71 6 The construction of finite point sets according to Chen and Skriganov | 77 7 The construction of infinite sequences according to Dick and Pillichshammer | 79 8 Extensions to the L𝑞 discrepancy | 82 9 Extensions to Orlicz norms of the discrepancy function | 83 Michael Drmota Subsequences of automatic sequences and uniform distribution | 87 1 Introduction | 87 2 Automatic sequences | 90 3 Subsequences along the sequence ⌊𝑛𝑐 ⌋ | 93 4 Polynomial subsequences | 95 5 Subsequences along the primes | 98 Henri Faure On Atanassov’s methods for discrepancy bounds of low-discrepancy sequences | 105 1 Introduction | 105 2 Atanassov’s methods for Halton sequences | 107 2.1 Review of Halton sequences | 107 2.2 Review of previous bounds for the discrepancy of Halton sequences | 108 2.3 Atanassov’s methods applied to Halton sequences | 108 2.4 Scrambling Halton sequences with matrices | 113 3 Atanassov’s method for (𝑡, 𝑠)-sequences | 118 3.1 Review of (𝑡, 𝑠)-sequences | 118 3.2 Review of bounds for the discrepancy of (𝑡, 𝑠)-sequences | 119 3.3 Atanassov’s method applied to (𝑡, 𝑠)-sequences | 119 3.4 The special case of even bases for (𝑡, 𝑠)-sequences | 121 4 Atanassov’s methods for generalized Niederreiter sequences and (𝑡, e, 𝑠)- sequences | 124
- 13. Contents | ix Peter Hellekalek The hybrid spectral test: a unifying concept | 127 1 Introduction | 127 2 Adding digit vectors | 129 3 Notation | 132 4 The hybrid spectral test | 134 5 Examples | 137 5.1 Example I: Integration lattices | 137 5.2 Example II: Extreme and star discrepancy | 140 Peter Kritzer, Friedrich Pillichshammer, and Henryk Woźniakowski Tractability of multivariate analytic problems | 147 1 Introduction | 147 2 Tractability | 149 3 A weighted Korobov space of analytic functions | 154 4 Integration in 𝐻(𝐾𝑠,a,b) | 156 5 𝐿2-approximation in 𝐻(𝐾𝑠,a,b) | 162 6 Conclusion and outlook | 169 Gerhard Larcher Discrepancy estimates for sequences: new results and open problems | 171 1 Introduction | 171 2 Metrical and average type discrepancy estimates for digital point sets and sequences and for good lattice point sets | 174 3 Discrepancy estimates for and applications of hybrid sequences | 181 4 Miscellaneous problems | 185 Gunther Leobacher A short introduction to quasi-Monte Carlo option pricing | 191 1 Overview | 191 2 Foundations of financial mathematics | 192 2.1 Bonds, stocks and derivatives | 192 2.2 Arbitrage and the no-arbitrage principle | 194 2.3 The Black–Scholes model | 196 2.4 SDE models | 197 2.5 Lévy models | 199 2.6 Examples | 200 3 MC and QMC simulation | 201 3.1 Nonuniform random number generation | 201 3.2 Generation of Brownian paths | 208 3.3 Generation of Lévy paths | 214
- 14. x | Contents 3.4 Multilevel (quasi-)Monte Carlo | 216 3.5 Examples | 218 Dirk Nuyens The construction of good lattice rules and polynomial lattice rules | 223 1 Lattice rules and polynomial lattice rules | 223 1.1 Lattice rules | 224 1.2 Polynomial lattice rules | 225 2 The worst-case error | 227 2.1 Koksma–Hlawka error bound | 227 2.2 Lattice rules | 229 2.3 Polynomial lattice rules | 232 3 Weighted worst-case errors | 236 4 Some standard spaces | 238 4.1 Lattice rules and Fourier spaces | 238 4.2 Randomly-shifted lattice rules and the unanchored Sobolev space | 239 4.3 Tent-transformed lattice rules and the cosine space | 241 4.4 Polynomial lattice rules and Walsh spaces | 243 5 Component-by-component constructions | 245 5.1 Component-by-component construction | 245 5.2 Fast component-by-component construction | 249 6 Conclusion | 252 Index | 257
- 15. Christoph Aistleitner Metric number theory, lacunary series and systems of dilated functions Abstract: By a classical result of Weyl, for any increasing sequence (𝑛𝑘)𝑘≥1 of integers the sequence of fractional parts ({𝑛𝑘𝑥})𝑘≥1 is uniformly distributed modulo 1 for al- most all 𝑥 ∈ [0, 1]. Except for a few special cases, e.g. when 𝑛𝑘 = 𝑘, 𝑘 ≥ 1, the excep- tionalsetcannotbedescribed explicitly. Theexactasymptotic order of thediscrepancy of ({𝑛𝑘𝑥})𝑘≥1 for almost all 𝑥 is only known in a few special cases, for example when (𝑛𝑘)𝑘≥1 is a (Hadamard) lacunary sequence, that is when 𝑛𝑘+1/𝑛𝑘 ≥ 𝑞 > 1, 𝑘 ≥ 1. In this case of quickly increasing (𝑛𝑘)𝑘≥1 the system ({𝑛𝑘𝑥})𝑘≥1 (or, more generally, (𝑓(𝑛𝑘𝑥))𝑘≥1 for a 1-periodic function 𝑓) shows many asymptotic properties which are typical for the behavior of systems of independent random variables. Precise results depend on a fascinating interplay between analytic, probabilistic and number-theo- retic phenomena. Without any growth conditions on (𝑛𝑘)𝑘≥1 the situation becomes much more compli- cated, and the system (𝑓(𝑛𝑘𝑥))𝑘≥1 will typically fail to satisfy probabilistic limit the- orems. An important problem which remains is to study the almost everywhere con- vergence of series ∑∞ 𝑘=1 𝑐𝑘𝑓(𝑘𝑥), which is closely related to finding upper bounds for maximal 𝐿2 -norms of the form 1 ∫ 0 ( max 1≤𝑀≤𝑁 𝑀 ∑ 𝑘=1 𝑐𝑘𝑓(𝑘𝑥) ) 2 𝑑𝑥 . The most striking example of this connection is the equivalence of the Carleson con- vergence theorem and the Carleson–Hunt inequality for maximal partial sums of Fourier series. For general functions 𝑓 this is a very difficult problem, which is related to finding upper bounds for certain sums involving greatest common divisors. Keywords: uniform distribution theory, discrepancy theory, metric number theory, la- cunary series, systems of dilated functions Mathematics Subject Classification 2010: 11J83, 11K38, 42A55, 60F15, 11A05, 42A20 || Christoph Aistleitner: Department of Applied Mathematics, School of Mathematics and Statistics, University of New South Wales, Sydney NSW 2052, Australia, e-mail: aistleitner@math.tugraz.at The author is supported by a Schrödinger scholarship of the Austrian Research Foundation (FWF).
- 16. 2 | Christoph Aistleitner 1 Uniform distribution modulo 1 A sequence of real numbers from the unit interval (𝑥𝑘)𝑘≥1 is called uniformly dis- tributed modulo 1 (u.d. mod 1) if for all 0 ≤ 𝑎 < 𝑏 ≤ 1 the asymptotic relation lim 𝑁→∞ 1 𝑁 𝑁 ∑ 𝑘=1 1[𝑎,𝑏)(𝑥𝑘) = 𝑏 − 𝑎 (1.1) holds. Roughly speaking, a sequence is u.d. mod 1 if asymptotically every interval [𝑎, 𝑏) ⊂ [0, 1] receives its fair share of points, which is proportional to its length. In an informal way, uniformly distributed sequences are often considered as sequences showing random behavior; this is justified by the Glivenko–Cantelli theorem, which asserts that for a sequence (𝑈𝑘)𝑘≥1 of independent, uniformly [0, 1]-distributed ran- dom variables we have lim 𝑁→∞ 1 𝑁 𝑁 ∑ 𝑘=1 1[𝑎,𝑏)(𝑈𝑘) = 𝑏 − 𝑎 almost surely for all [𝑎, 𝑏) ⊂ [0, 1]. Thus a deterministic sequence (𝑥𝑘)𝑘≥1 which is u.d. mod 1 can be seen as a typical realization of a random (uniformly [0, 1]-distributed) sequence. The theory of uniform distribution was boosted by Weyl’s [43] seminal paper of 1916, which contains the celebrated Weyl criterion for uniform distribution of a se- quence: a sequence (𝑥𝑘)𝑘≥1 is u.d. mod 1 if and only if for all integers ℎ ̸ = 0 lim 𝑁→∞ 1 𝑁 𝑁 ∑ 𝑘=1 𝑒2𝜋𝑖ℎ𝑥𝑘 = 0 . (1.2) This criterion can be used to give an easy proof for the fact that the sequence ({𝑘𝑥})𝑘≥1 is u.d. mod 1 if and only if 𝑥 ̸ ∈ ℚ (here {⋅} stands for the fractional part function; usually this sequence is called 𝑛𝛼 rather than 𝑘𝑥, but for the sake of consistency of the notation with later parts of this article we will denote it by 𝑘𝑥). In fact, assume that 𝑥 ̸ ∈ ℚ; then, using the well-known formula for the geometric series, we have 1 𝑁 𝑁 ∑ 𝑘=1 𝑒2𝜋𝑖ℎ𝑘𝑥 = 1 𝑁 𝑒2𝜋𝑖ℎ(𝑁+1)𝑥 − 𝑒2𝜋𝑖ℎ𝑥 𝑒2𝜋𝑖ℎ𝑥 − 1 → 0 as 𝑁 → ∞ , where we used the fact that 𝑒2𝜋𝑖ℎ𝑥 − 1 ̸ = 0 for ℎ ̸ = 0 and 𝑥 ̸ ∈ ℚ. It is easy to see that for 𝑥 ∈ ℚ the sequence ({𝑘𝑥})𝑘≥1 is not u.d. mod 1; thus the problem of deciding for which 𝑥 ∈ [0, 1] the parametric sequence (𝑘𝑥)𝑘≥1 is u.d. mod 1 is completely solved. Weyl’s paper also contained a general result for parametric sequences of the form ({𝑛𝑘𝑥})𝑘≥1, where (𝑛𝑘)𝑘≥1 is a sequence of distinct positive integers and 𝑥 is a real number from [0, 1]: for almost all 𝑥 (in the sense of Lebesgue measure) the sequence ({𝑛𝑘𝑥})𝑘≥1 is u.d. mod 1. Accordingly, the general case ({𝑛𝑘𝑥})𝑘≥1 resembles the prop- erties of the case ({𝑘𝑥})𝑘≥1 insofar as in both cases the exceptional set is of measure
- 17. Metric number theory, lacunary series and systems of dilated functions | 3 zero; however, while in the latter case the exceptional set can be explicitly determined, it is generally very difficult to decide whether for a given sequence (𝑛𝑘)𝑘≥1 and a given parameter 𝑥 the sequence ({𝑛𝑘𝑥})𝑘≥1 is u.d. mod 1 or not (see also Section 2). In the following paragraph, we want to prove Weyl’s result that ({𝑛𝑘𝑥})𝑘≥1 is u.d. mod 1 for almost all 𝑥. Throughout this article, we will repeatedly use methods from probability theory; this makes perfect sense, since the unit interval, equipped with Borel sets and Lebesgue measure 𝜆, is a probability space (that is, a measure space (𝛺, F, ℙ) for which ℙ(𝛺) = 1). We will use the Rademacher–Menshov inequality, which states that for a real orthonormal system 𝜙1(𝑥), . . . , 𝜙𝑁(𝑥) and for real coeffi- cients 𝛼1, . . . , 𝛼𝑁 we have 1 ∫ 0 max 1≤𝑀≤𝑁 ( 𝑀 ∑ 𝑘=1 𝛼𝑘𝜙𝑘) 2 𝑑𝑥 ≤ (log2 𝑁 + 2)2 𝑁 ∑ 𝑘=1 𝛼2 𝑘 (1.3) (this inequality has been obtained independently by Rademacher [39] and Men- shov [36]; it can be proved quite easily using a dyadic splitting method, see e.g. [34]). Note that an equivalent formulation of the Weyl criterion (1.2) is lim 𝑁→∞ 1 𝑁 𝑁 ∑ 𝑘=1 cos 2𝜋ℎ𝑥𝑘 = 0 and lim 𝑁→∞ 1 𝑁 𝑁 ∑ 𝑘=1 sin 2𝜋ℎ𝑥𝑘 = 0 for all integers ℎ ̸ = 0. For integers 𝑚 ≥ 1 and ℎ ̸ = 0 we set 𝑆𝑚,ℎ = {𝑥 ∈ [0, 1]: max 1≤𝑀≤2𝑚 𝑀 ∑ 𝑘=1 cos 2𝜋ℎ𝑛𝑘𝑥 > 2𝑚/2 𝑚2 } . Then by (1.3), by the orthogonality of the trigonometric system and by the fact that by assumption the numbers (𝑛𝑘)𝑘≥1 are distinct we have 1 ∫ 0 max 1≤𝑀≤2𝑚 ( 𝑀 ∑ 𝑘=1 cos 2𝜋ℎ𝑛𝑘𝑥) 2 𝑑𝑥 ≪ 2𝑚 (log 2𝑚 )2 ≪ 2𝑚 𝑚2 (1.4) (where “≪” is the Vinogradov symbol). Chebyshev’s inequality states that for any square-integrable function 𝑓 on [0, 1] we have that for any 𝑡 > 0 𝜆(𝑥 ∈ [0, 1]: |𝑓(𝑥)| ≥ 𝑡) ≤ 1 𝑡2 1 ∫ 0 𝑓(𝑥)2 𝑑𝑥 . (1.5) Applying this inequality, by (1.4) we have 𝜆(𝑆𝑚,ℎ) ≪ 1 𝑚2 .
- 18. 4 | Christoph Aistleitner Thus ∞ ∑ 𝑚=1 𝜆(𝑆𝑚,ℎ) < ∞ , which by the first Borel–Cantelli lemma implies that with probability one (with re- spect to the Lebesgue measure on [0, 1]) only finitely many events 𝑆𝑚,ℎ occur. Thus for almost all 𝑥 ∈ [0, 1] there exists an 𝑚0 = 𝑚0(𝑥) such that max 1≤𝑀≤2𝑚 𝑀 ∑ 𝑘=1 cos 2𝜋ℎ𝑛𝑘𝑥 ≤ 2𝑚/2 𝑚2 for all 𝑚 ≥ 𝑚0; consequently, there also exists an 𝑁0 = 𝑁0(𝑥) such that 𝑁 ∑ 𝑘=1 cos 2𝜋ℎ𝑛𝑘𝑥 ≤ 2𝑁1/2 (log2 𝑁)2 for 𝑁 ≥ 𝑁0, which implies lim 𝑁→∞ 1 𝑁 𝑁 ∑ 𝑘=1 cos 2𝜋ℎ𝑛𝑘𝑥 = 0 . The same argument applies if we replace the function cos by sin. Consequently, ({𝑛𝑘𝑥})𝑘≥1 is u.d. mod 1 for almost all 𝑥 ∈ [0, 1]. 2 Metric number theory One of the aims of metric number theory is to describe properties which are typical for (real) numbers, where “typical” means that the exceptional set of numbers not possessing this property is small; in our case, we consider a property to be “typical” if it holds for almost all numbers in the sense of Lebesgue measure (but of course there are also other possibilities of deciding what a “typical” property is, for example by means of the Hausdorff dimension). An early result from metric number theory is due to Borel: he proved that almost all numbers are normal with respect to a given base 𝑏 ≥ 2 (𝑏 being an integer). Here, a real number 𝑥 ∈ [0, 1] is called “normal” if in its base-𝑏 expansion 𝑥 = ∞ ∑ 𝑖=1 𝑟𝑖𝑏−𝑖 each digit 0, 1, . . . , 𝑏 − 1 appears asymptotically with frequency 𝑏−1 , each block of 2 digits appears asymptotically withfrequency 𝑏−2 , and, generally, eachblock of 𝑑digits appears with asymptotic frequency 𝑏−𝑑 . Formally, this can be written as lim 𝑁→∞ 1 𝑁 𝑁 ∑ 𝑘=1 1[𝑎𝑏−𝑑,(𝑎+1)𝑏−𝑑)({𝑏𝑘−1 𝑥}) = 𝑏−𝑑 (1.6)
- 19. Metric number theory, lacunary series and systems of dilated functions | 5 for all integers 𝑑 ≥ 1 and all integers 𝑎 ∈ {0, . . . , 𝑏𝑑 − 1}. Historically, Borel’s result is the first appearance of what we call the strong law of large numbers, today. To see this, we choose 𝑏 = 2 for simplicity, and let for 𝑥 ∈ [0, 1] the function 𝑟𝑘(𝑥) be defined as the 𝑘-th digit (after the decimal point) of the binary expansion of 𝑥. Then it is an easy exercise to check that the functions (𝑟𝑘(𝑥))𝑘≥1, interpreted as random variables over the probability space ([0, 1], B([0, 1]), 𝜆), form a sequence of independent, identically distributed (i.i.d.) random variables (remember that by definition, a random variable is just a measurable function). Thus Borel’s theorem, which in the special case 𝑑 = 1 (that is, for single digits) states that lim 𝑁→∞ 1 𝑁 𝑁 ∑ 𝑘=1 𝑟𝑘(𝑥) = 1 2 = 1 ∫ 0 𝑟1(𝑥) 𝑑𝑥 = 𝔼𝜆(𝑟1) a.e. is the strong law of large numbers for i.i.d. fair Bernoulli-distributed random variables (the functions (𝑟𝑘(𝑥))𝑘≥1 are called Rademacher functions; the fact that they form a system of i.i.d. random variables was first observed by Steinhaus in the 1920s). We note, by the way, that Borel’s theorem can also be interpreted as an early appearance of the pointwise ergodic theorem (for the transformation 𝑇𝑥 = {𝑏𝑥}). Written in the form (1.6) (which is not Borel’s original notation) it is quite obvious that there is a connection between normal numbers and the criterion for uniform dis- tribution modulo 1 in (1.1). Surprisingly, this connection was not noted (or, at least, not rigorously proved) before 1949, when Wall [42] showed that a number 𝑥 is normal in a base 𝑏 if and only if the sequence ({𝑏𝑘 𝑥})𝑘≥1 is u.d. mod 1. Thus, Borel’s theorem can also be seen as a special case of Weyl’s metric theorem on the uniform distribution of ({𝑛𝑘𝑥})𝑘≥1 for a.e. 𝑥. Now we know that almost all numbers are normal (which was not so difficult to establish); on the other hand, constructing a normal number is rather difficult, and checking whether a given number is normal or not is (usually) absolutely infeasible. Most constructions of normal numbers are based on the principle of concatenating blocks of digits generated by “simple” functions; for example, Champernowne’s num- ber (in base 10) 0. 1 2 3 4 5 6 7 8 9 10 11 12 . . . is obtained by concatenating the decimal expansions of the positive integers in con- secutive order, the Copeland–Erdős number (again in base 10) is obtained by concate- nating the decimal expansions of the primes 0. 2 3 5 7 11 13 17 19 23 29 . . . , and there are several other constructions of this type (for example concatenating the values of polynomials [37] or other entire functions [35]). As mentioned before, check- ing whether a given number is normal or not is extremely difficult, and it is unknown whether constants such as √2, 𝑒, 𝜋 are normal. It is conjectured that all algebraic ir- rationals are normal, but no example or counterexample is known. For more details
- 20. 6 | Christoph Aistleitner on this problem, see [6]. A closely-related problem concerns sequences of the form ({𝑥𝑘 })𝑘≥1. By a result of Koksma [30] this sequence is u.d. mod 1 for almost all 𝑥 > 1; however, not a single explicit value of 𝑥 for which this is the case, is known. The sequence ({(3/2)𝑘 })𝑘≥1 has attracted particular attention, but it is not even known whether lim sup𝑘→∞{(3/2)𝑘 } − lim inf𝑘→∞{(3/2)𝑘 } ≥ 1/2 (Vijayaraghavan’s conjec- ture of 1940). For more information concerning this problem see [17, 18]. The most important open problem in metric number theory is probably the Duffin– Schaeffer conjecture in metric Diophantine approximation. For a nonnegative function 𝜓: ℕ → ℝ, let 𝑊(𝜓) denote the set of real numbers 𝑥 ∈ [0, 1] for which the in- equality |𝑛𝑥 − 𝑎| < 𝜓(𝑛) has infinitely many coprime solutions (𝑎, 𝑛). It is an easy application of the first Borel–Cantelli lemma to prove that 𝜆(𝑊(𝜓)) = 0 if ∞ ∑ 𝑛=1 𝜓(𝑛)𝜑(𝑛) 𝑛 < ∞ (1.7) (here 𝜑 denotes the Euler totient function); that means, divergence of the sum in (1.7) is a necessary condition for 𝜆(𝑊(𝜓)) = 1. The Duffin–Schaeffer conjecture, proposed by R. J. Duffin and A. C. Schaeffer [19] in 1941, asserts that divergence of the sum in (1.7) is also sufficient to have 𝜆(𝑊(𝜓)) = 1. Several special cases of the conjecture have been established (see for example [25]), but a complete solution of the problem seems to be far out of reach. More information on the problems discussed in this section can be found in the books of Bugeaud [13] and Harman [26]. 3 Discrepancy The notion of the discrepancy of a sequence has been introduced as a measure of the quality of the u.d. mod 1 of a sequence. For a finite sequence (𝑥1, . . . , 𝑥𝑁) of points in the unit interval, the discrepancy 𝐷𝑁 and the star-discrepancy 𝐷∗ 𝑁 are defined as 𝐷𝑁(𝑥1, . . . , 𝑥𝑁) = sup 0≤𝑎<𝑏≤1 ∑𝑁 𝑘=1 1[𝑎,𝑏)(𝑥𝑘) 𝑁 − (𝑏 − 𝑎) and 𝐷∗ 𝑁(𝑥1, . . . , 𝑥𝑁) = sup 0≤𝑎≤1 ∑𝑁 𝑘=1 1[0,𝑎)(𝑥𝑘) 𝑁 − 𝑎 . It is easy to see that these two discrepancies are equivalent in the sense that at all times 𝐷∗ 𝑁 ≤ 𝐷𝑁 ≤ 2𝐷∗ 𝑁, and that an infinite sequence (𝑥𝑘)𝑘≥1 is u.d. mod 1 if and only if 𝐷∗ 𝑁(𝑥1, . . . , 𝑥𝑁) → 0 as 𝑁 → ∞. An important inequality to estimate the discrepancy of a sequence is the Erdős–Turán inequality, which (in one out of many
- 21. Metric number theory, lacunary series and systems of dilated functions | 7 possible formulations) states that for any positive integer 𝐻 𝐷∗ 𝑁(𝑥1, . . . , 𝑥𝑁) ≤ 3 𝐻 + 3 𝐻 ∑ ℎ=1 1 ℎ 1 𝑁 𝑁 ∑ 𝑘=1 𝑒2𝜋𝑖ℎ𝑥𝑘 . (1.8) We will use this inequality in Section 5 to obtain an upper bound for the discrepancy of ({𝑛𝑘𝑥})𝑘≥1 for almost all 𝑥, by this means establishing a quantitative version of the theorem of Weyl mentioned in Section 1. Another important inequality concerning the discrepancy of sequences of points is Koksma’s inequality, which states that for any function 𝑓 which has bounded variation Var(𝑓) in the unit interval the estimate 1 𝑁 𝑁 ∑ 𝑘=1 𝑓(𝑥𝑘) − 1 ∫ 0 𝑓(𝑥) 𝑑𝑥 ≤ Var(𝑓) ⋅ 𝐷∗ 𝑁(𝑥1, . . . , 𝑥𝑁) (1.9) holds. The notions of u.d. mod 1 and discrepancy can be generalized in a natural way to the multidimensional setting, as can be the Erdős–Turán inequality and Koksma’s inequality (then called the Koksma–Hlawka inequality). The multidimensional ver- sion of (1.9) forms the foundation of the so-called Quasi-Monte Carlo method, which is based on the observation that sequences having small discrepancy can be used for numerical integration. By a result of Schmidt [41] there exists a positive constant 𝑐 such that for any infi- nite sequence (𝑥𝑘)𝑘≥1 of points in the unit interval the inequality 𝐷∗ 𝑁(𝑥1, . . . , 𝑥𝑁) > 𝑐 log 𝑁 𝑁 holds for infinitely many 𝑁. On the other hand, there exist several constructions of sequences satisfying 𝐷∗ 𝑁(𝑥1, . . . , 𝑥𝑁) = O((log 𝑁)𝑁−1 ) as 𝑁 → ∞, so in the one- dimensional case the problem of the optimal asymptotic order of the discrepancy is solved. On the contrary, determining the optimal asymptotic order of the discrepancy in the multidimensional case turned out to be a very difficult problem, which is still open (see [11] for a survey). More information on discrepancy theory and the Quasi-Monte Carlo method can be found in the books of Dick and Pillichshammer [15] and Drmota and Tichy [16]. 4 Lacunary series The word lacunary originates from the Latin lacuna (ditch, gap), which is the diminu- tive form of lacus (lake). Accordingly, a lacunary Fourier series is a series which has “gaps” in the sense that it is composed of trigonometric functions whose frequencies are far apart from each other. A classicalgap condition is the Hadamard gap condition, requiring that 𝑛𝑘+1 𝑛𝑘 ≥ 𝑞 > 1 , 𝑘 ≥ 1 ; (1.10)
- 22. 8 | Christoph Aistleitner thus a (Hadamard) lacunary Fourier series is of the form ∞ ∑ 𝑘=1 (𝑎𝑘 cos 2𝜋𝑛𝑘𝑥 + 𝑏𝑘 sin 2𝜋𝑛𝑘𝑥) (1.11) for (𝑛𝑘)𝑘≥1 satisfying (1.10). By a classical heuristics, lacunary sequences resemble many properties which are typical for sequences of independent random variables. For example, by Kolmogorov’s three series theorem, a sequence of centered and uni- formly bounded independent random variables (𝑋𝑘)𝑘≥1 is almost surely convergent if and only if the variances satisfy ∞ ∑ 𝑘=1 𝕍(𝑋𝑘) < ∞, (1.12) and by a counterpart for lacunary series, also due to Kolmogorov, the series (1.11) is almost convergent everywhere if and only if ∞ ∑ 𝑘=1 (𝑎2 𝑘 + 𝑏2 𝑘 ) < ∞ . (1.13) Note here that the variance of the function 𝑎𝑘 cos 2𝜋𝑛𝑘𝑥+𝑏𝑘 sin 2𝜋𝑛𝑘𝑥, considered as a random variable over the probability space ([0, 1], B([0, 1]), 𝜆), is simply given by 1 ∫ 0 (𝑎𝑘 cos 2𝜋𝑛𝑘𝑥 + 𝑏𝑘 sin 2𝜋𝑛𝑘𝑥) 2 𝑑𝑥 = 𝑎2 𝑘 + 𝑏2 𝑘 . Thus the almost sure (a.s.) convergence behavior of series of independent random variables and of lacunary trigonometric series are in perfect accordance. Many sim- ilar results of the same type exist: for example, by a classical result of Salem and Zyg- mund [40], under the gap condition (1.10) we have 𝜆 (𝑥 ∈ [0, 1]: 𝑁 ∑ 𝑘=1 cos 2𝜋𝑛𝑘𝑥 < 𝑡√𝑁/2) → 𝛷(𝑡) , where𝛷(𝑡)denotes thestandard normaldistributionfunction. In other words, the sys- tem (cos 2𝜋𝑛𝑘𝑥)𝑘≥1 satisfies the central limit theorem. By a result of Erdős and Gál [21], the same system also satisfies the law of the iterated logarithm (LIL), that is lim sup 𝑁→∞ ∑ 𝑁 𝑘=1 cos 2𝜋𝑛𝑘𝑥 √2𝑁 log log 𝑁 = 1 √2 a.e. The situation gets significantly more complicated if we consider the more general sequence (𝑓(𝑛𝑘𝑥))𝑘≥1 for a (in some sense) “nice” function 𝑓 satisfying 𝑓(𝑥 + 1) = 𝑓(𝑥) , 1 ∫ 0 𝑓(𝑥) 𝑑𝑥 = 0 , (1.14)
- 23. Metric number theory, lacunary series and systems of dilated functions | 9 instead of (cos 2𝜋𝑛𝑘𝑥)𝑘≥1. A striking result for this general setting is a theorem of Philipp [38], who confirmed the so-called Erdős–Gál conjecture by proving that un- der (1.10) we have 1 4√2 ≤ lim sup 𝑁→∞ 𝑁𝐷∗ 𝑁({𝑛1𝑥}, . . . , {𝑛𝑁𝑥}) √2𝑁 log log 𝑁 ≤ 𝐶𝑞 a.e.; (1.15) this is a counterpart of the Chung–Smirnov LIL for the Kolmogorov–Smirnov statistic in probability theory. As a consequence of (1.15) and Koksma’s inequality (1.9) we have lim sup 𝑁→∞ ∑ 𝑁 𝑘=1 𝑓(𝑛𝑘𝑥) √2𝑁 log log 𝑁 ≤ 𝐶𝑓,𝑞 a.e. (1.16) Calculating the precise value of the lim sup in (1.15) and (1.16) is a very difficult prob- lem, and depends on number-theoretic properties of (𝑛𝑘)𝑘≥1 and Fourier-analytic properties of 𝑓 (or of the indicator functions in the case of (1.15)) in a very delicate way. In the case of 𝑛𝑘 = 𝜃𝑘 for an integer 𝜃 the problem has been solved by Fukuyama [23]; he proved that almost everywhere lim sup 𝑁→∞ 𝑁𝐷∗ 𝑁({𝜃𝑥}, . . . , {𝜃𝑁 𝑥}) √2𝑁 log log 𝑁 = { { { { { { { √42 9 if 𝜃 = 2, √(𝜃+1)𝜃(𝜃−2) 2√(𝜃−1)3 if 𝜃 ≥ 4 is even, √𝜃+1 2√𝜃−1 if 𝜃 ≥ 3 is odd. In view of the results mentioned in Section 2, Fukuyama’s theorem establishes the typical asymptotic order of the discrepancy of normal numbers. In a sense, the se- quence (𝜃𝑘 )𝑘≥1 is a pathological example of a lacunary sequence, exhibiting an ex- tremely strong relation between its consecutive terms. For a lacunary sequence for which no such strong arithmetic relations exist, the LIL is satisfied in the form lim sup 𝑁→∞ 𝑁𝐷∗ 𝑁({𝑛1𝑥}, . . . , {𝑛𝑁𝑥}) √2𝑁 log log 𝑁 = 1 2 a.e. , which is in perfect accordance (including the value of the constant of the right-hand side) with the Chung–Smirnov LIL for i.i.d. random variables (see [1] for details). For more information on lacunary sequences in the context of metric discrepancy theory and probabilistic limit theorems, see the survey paper [3]. Lacunary functions are well known in analysis for several other interesting prop- erties, apart from their resemblance of the behavior of systems of independent random variables. For example, Weierstrass’ celebrated example of a nowhere differentiable function is defined by means of a lacunary trigonometric series. It should be noted that the notion of lacunary series does not only include lacunary trigonometric se- ries, but also other series such as for example lacunary Taylor series. For a survey, see [29].
- 24. 10 | Christoph Aistleitner 5 Almost everywhere convergence The Kolmogorov three series theorem gives a full characterization of the a.s. conver- gence behavior of sums of independent random variables. In general, the a.s. conver- gence condition comprises of three conditions about the convergence or divergence of certain series (which explains the name three series theorem), but in the case of cen- tered, uniformly-bounded random variables the criterion reduces to the simple condi- tion (1.12). As noted in the previous section, there exists an analogue of the three series theorem for the case of Hadamard lacunary trigonometric series; however, surpris- ingly, the requirement of considering a Fourier series which contains only frequencies along an exponentially growing subsequence canbe entirely dropped. This is the cele- brated Carleson’s theorem [14], which is considered as one of the major achievements of Fourier analysis in twentieth century mathematics: a Fourier series ∞ ∑ 𝑘=1 (𝑎𝑘 cos 2𝜋𝑘𝑥 + 𝑏𝑘 sin 2𝜋𝑘𝑥) is almost everywhere (a.e.) convergent, provided (1.13) holds. Moreover, for any func- tion 𝑓 ∈ 𝐿2 ([0, 1]) and 𝑓(𝑥) ∼ ∞ ∑ 𝑘=1 (𝑎𝑘 cos 2𝜋𝑘𝑥 + 𝑏𝑘 sin 2𝜋𝑘𝑥) , setting 𝑠𝑁(𝑓; 𝑥) = 𝑁 ∑ 𝑘=1 (𝑎𝑘 cos 2𝜋𝑘𝑥 + 𝑏𝑘 sin 2𝜋𝑘𝑥) we have 𝑠𝑁(𝑓; 𝑥) → 𝑓(𝑥) as 𝑁 → ∞ for a.e. 𝑥 . Carleson’s theorem has a breathtaking consequence: there exists an absolute con- stant 𝑐 such that for any function 𝑓 in 𝐿2 ([0, 1]) we have, writing ‖⋅‖ for the 𝐿2 ([0, 1]) norm, sup 𝑁≥1 𝑠𝑁(𝑓; 𝑥) ≤ 𝑐‖𝑓‖ . (1.17) Carleson’s theorem has been extended to the case 𝑓 ∈ 𝐿𝑝 ([0, 1]), 𝑝 > 1, by Hunt [28]. It is a very deep result, and although alternative proofs have been given by Feffer- man [22] and Lacey and Thiele [33], no “easy” proof exists. For a comprehensive treatment of the subject, see the monograph of Arias de Reyna [5] and the survey paper [32]. As an application of Carleson’s theorem, we will show how it can be used to obtain a quantitative version of the results on a.e. uniform distribution of Weyl mentioned in
- 25. Metric number theory, lacunary series and systems of dilated functions | 11 Section 1. This argument is due to Baker [7] and leads to the upper bound 𝐷∗ 𝑁({𝑛1𝑥}, . . . , {𝑛𝑁𝑥}) ≪ (log 𝑁)3/2+𝜀 √𝑁 a.e. (1.18) for any strictly-increasing sequence of positive integers (𝑛𝑘)𝑘≥1 and any 𝜀 > 0. To outline the similarity between this proof and the one given in Section 1, we will use a real version of the Erdős–Turán inequality (1.8). Let 𝜀 > 0. For integers 𝑚 ≥ 1 we set 𝑆𝑚 = {𝑥 ∈ [0, 1]: max 1≤𝑀≤2𝑚 ∑ 1≤ℎ≤2𝑚/2 1 ℎ 𝑀 ∑ 𝑘=1 cos 2𝜋ℎ𝑛𝑘𝑥 > 2𝑚/2 𝑚3/2+𝜀 } . Note that by (1.17) for any ℎ we have max 1≤𝑀≤2𝑚 𝑀 ∑ 𝑘=1 cos 2𝜋ℎ𝑛𝑘𝑥 ≤ 𝑐 2𝑚 ∑ 𝑘=1 cos 2𝜋ℎ𝑛𝑘𝑥 ≤ 𝑐2𝑚/2 (1.19) for an absolute constant 𝑐. Thus by Minkowski’s inequality we have max 1≤𝑀≤2𝑚 ∑ 1≤ℎ≤2𝑚/2 1 ℎ 𝑀 ∑ 𝑘=1 cos 2𝜋ℎ𝑛𝑘𝑥 ≤ ∑ 1≤ℎ≤2𝑚/2 1 ℎ max 1≤𝑀≤2𝑚 𝑀 ∑ 𝑘=1 cos 2𝜋ℎ𝑛𝑘𝑥 ≪ 𝑐𝑚2𝑚/2 . Consequently, by Chebyshev’s inequality (1.5) we have 𝜆(𝑆𝑚) ≪ 1 𝑚1+2𝜀 . Thus ∞ ∑ 𝑚=1 𝜆(𝑆𝑚) < ∞ , which by the first Borel–Cantelli lemma means that with probability one only finitely many events 𝑆𝑚 occur. Thus for almost all 𝑥 ∈ [0, 1] there exists an 𝑚0 = 𝑚0(𝑥) such that max 1≤𝑀≤2𝑚 ∑ 1≤ℎ≤2𝑚/2 1 ℎ 𝑀 ∑ 𝑘=1 cos 2𝜋ℎ𝑛𝑘𝑥 ≤ 2𝑚/2 𝑚3/2+𝜀 for all 𝑚 ≥ 𝑚0; consequently, there also exists an 𝑁0 = 𝑁0(𝑥) such that ∑ 1≤ℎ≤√𝑁 1 ℎ 𝑁 ∑ 𝑘=1 cos 2𝜋ℎ𝑛𝑘𝑥 ≤ 2𝑁1/2 (log2 𝑁)3/2+𝜀
- 26. 12 | Christoph Aistleitner for 𝑁 ≥ 𝑁0. The same result holds if we replace the function cos by sin, and using (1.8) (split into a real and imaginary part) we get (1.18). Carleson’s inequality (1.17) in the form (1.19) plays a key role in this proof, and if it is replaced by the Rademacher– Menshov inequality, which gives an additional logarithmic factor as in (1.4), one can only obtain (1.18) with the exponent 3/2 + 𝜀 replaced by 5/2 + 𝜀. The optimal exponent of the logarithmic term in (1.18) is an important open prob- lem in metric discrepancy theory. Note that by Koksma’s inequality (1.9) as a conse- quence of (1.18) we get 𝑁 ∑ 𝑘=1 𝑓(𝑛𝑘𝑥) ≪ √𝑁(log 𝑁)3/2+𝜀 a.e. (1.20) for any function 𝑓satisfying (1.14) which has bounded variation on [0, 1]. On the other hand, Berkes and Philipp [8] constructed a sequence (𝑛𝑘)𝑘≥1 for which lim sup 𝑁→∞ ∑ 𝑁 𝑘=1 cos 2𝜋𝑛𝑘𝑥 √𝑁 log 𝑁 = ∞ a.e. , which again by Koksma’s inequality implies that the exponent of the logarithmic term in (1.18) can in general not be reduced below 1/2. In the following section we will see that 1/2 is in fact the optimal exponent, at least if we consider only a single function 𝑓 (as in (1.20)) and not the discrepancy 𝐷∗ 𝑁. For more information on the a.e. convergence of sums of dilated functions, the in- terested reader is referred to the comprehensive survey article of Berkes and Weber [9]. 6 Sums involving greatest common divisors In the context of counting lattice points in right-angled triangles, around 1920 Hardy and Littlewood investigated the problem of finding good upper bounds for the asymp- totic order of 𝑁 ∑ 𝑘=1 ({𝑘𝑥} − 1/2) as 𝑁 → ∞ . (1.21) The 𝐿2 ([0, 1])-norm of (1.21) can be calculated using the formula 1 ∫ 0 ({𝑚𝑥} − 1/2) ({𝑛𝑥} − 1/2) 𝑑𝑥 = 1 12 (gcd(𝑚, 𝑛))2 𝑚𝑛 (1.22) for integers 𝑚, 𝑛(first stated by Franel and proved by Landau in 1924). The generalized problem of estimating 𝑁 ∑ 𝑘,𝑙=1 (gcd(𝑛𝑘, 𝑛𝑙))2 𝑛𝑘𝑛𝑙 (1.23)
- 27. Metric number theory, lacunary series and systems of dilated functions | 13 for an arbitrary sequence of distinct positive integers 𝑛1, . . . , 𝑛𝑁 was posed as a prize problem by the Scientific Society in Amsterdam in 1947 (following a suggestion of Erdős), and solved by Gál [24] in 1949. He proved that there exists an absolute con- stant 𝑐 such that (1.23) is bounded by 𝑐𝑁(log log 𝑁)2 , and that this upper bound is asymptotically optimal. Koksma [31] observed that as a consequence for any centered, periodically-extended indicator function 𝑓(𝑥) = 1(𝑎,𝑏)({𝑥}) − (𝑏 − 𝑎) (and in fact even for any 1-periodic function 𝑓 having mean zero and bounded variation on [0,1]) the estimate 1 ∫ 0 ( 𝑁 ∑ 𝑘=1 𝑓(𝑛𝑘𝑥)) 2 𝑑𝑥 ≪ 𝑁(log log 𝑁)2 (1.24) holds. This follows from a generalized version of (1.22), which we will deduce in the next few lines. Assume that 𝑓 satisfies (1.14) and is of bounded variation on [0, 1]. Let 𝑓(𝑥) ∼ ∞ ∑ 𝑗=1 𝑎𝑗 cos 2𝜋𝑗𝑥 denote the Fourier series of 𝑓 (for simplicity we assume that it is a pure cosine-series; the general case works in exactly the same way). Then |𝑎𝑗| ≪ 𝑗−1 (1.25) (see [44, p. 48]; this estimate can be easily proved using the fact that any function of bounded variation can be written as the sum of two bounded and monotone func- tions). Thus for integers 𝑚, 𝑛 we have, by (1.25) and the orthogonality of the trigono- metric system, and writing 𝛿(⋅, ⋅) for the Kronecker function, 1 ∫ 0 𝑓(𝑚𝑥)𝑓(𝑛𝑥) 𝑑𝑥 = ∞ ∑ 𝑗1,𝑗2=1 𝑎𝑗1 𝑎𝑗2 𝛿(𝑚𝑗1, 𝑛𝑗2) ≪ ∞ ∑ 𝑗1,𝑗2=1 1 𝑗1𝑗2 𝛿(𝑚𝑗1, 𝑛𝑗2) . (1.26) Now 𝑚𝑗1 = 𝑛𝑗2 is only possible if 𝑗1 = 𝑗𝑛/ gcd(𝑚, 𝑛) and 𝑗2 = 𝑗𝑚/ gcd(𝑚, 𝑛) for some integer 𝑗 ≥ 1. Consequently, (1.26) is at most ∞ ∑ 𝑗=1 gcd(𝑚, 𝑛) 𝑗𝑛 gcd(𝑚, 𝑛) 𝑗𝑚 ≪ gcd(𝑚, 𝑛)2 𝑚𝑛 , which implies, together with the aforementioned result of Gál, that (1.24) holds. Sums involving common divisors similar to (1.23) were studied by Dyer and Harman [20] in the context of metric Diophantine approximation. They investigated max 𝑛1<⋅⋅⋅<𝑛𝑁 𝑁 ∑ 𝑘,𝑙=1 (gcd(𝑛𝑘, 𝑛𝑙))2𝛼 (𝑛𝑘𝑛𝑙)𝛼 , 𝛼 ∈ [1/2, 1) , (1.27)
- 28. 14 | Christoph Aistleitner and, amongst other results, proved for the particularly interesting case 𝛼 = 1/2 the upper bound max 𝑛1<⋅⋅⋅<𝑛𝑁 𝑁 ∑ 𝑘,𝑙=1 gcd(𝑛𝑘, 𝑛𝑙) √𝑛𝑘𝑛𝑙 ≪ 𝑁 exp ( 5 log 𝑁 log log 𝑁 ) . Recently, Aistleitner, Berkes and Seip [4] obtained upper bounds for (1.27) which are essentially optimal. They proved that max 𝑛1<⋅⋅⋅<𝑛𝑁 𝑁 ∑ 𝑘,𝑙=1 (gcd(𝑛𝑘, 𝑛𝑙))2𝛼 (𝑛𝑘𝑛𝑙)𝛼 ≤ 𝐶𝜀𝑁 exp ((1 + 𝜀)𝑔(𝛼, 𝑁)) , where for 1/2 < 𝛼 < 1 we have 𝑔(𝛼, 𝑁) = ( 8 1−𝛼 + 16⋅2−𝛼 √2𝛼−1 ) (log 𝑁)1−𝛼 (log log 𝑁)𝛼 + (log 𝑁)(1−𝛼)/2 1 − 𝛼 , (1.28) for 𝛼 = 1/2 we have 𝑔(1/2, 𝑁) = 25√log 𝑁√log log 𝑁 , (1.29) and where 𝐶𝜀 is a constant only depending on 𝜀 > 0. Here the asymptotic order of 𝑔(𝛼, 𝑁) in (1.28) is optimal, and the asymptotic order of 𝑔(1/2, 𝑁) in (1.29) can per- haps be reduced from √log 𝑁√log log 𝑁 to √log 𝑁/√log log 𝑁 (but not below). As an application, Aistleitner, Berkes and Seip improved the exponent of the logarithmic term in (1.20) to 1/2 + 𝜀, which is optimal (up to the 𝜀). Another application of such GCD sums is concerning the a.e. convergence of series ∑∞ 𝑘=1 𝑐𝑘𝑓(𝑛𝑘𝑥) for functions 𝑓 of bounded variation or being Hölder-continuous (see [4]; cf. also [2]) or of so-called Davenport series (see [12]). There is also a close connection with certain properties of the Riemann zeta function, which requires further investigation (cf. [27]). As a consequence of (1.28) (and using a trick to modify the argument around (1.26) in such a way to get a generalized GCD sum of the form (1.27) instead of (1.23)) one can show that for any 𝑓 of bounded variation satisfying (1.14) and for any strictly increas- ing sequence (𝑛𝑘)𝑘≥1 of positive integers, the Carleson-type inequality max 1≤𝑀≤𝑁 𝑀 ∑ 𝑘=1 𝑐𝑘𝑓(𝑛𝑘𝑥) ≤ 𝑐(log log 𝑁)4 𝑁 ∑ 𝑘=1 𝑐2 𝑘 holds, which, using an argument similar to the one used to prove (1.18) in the previous section, leads to 𝑁 ∑ 𝑘=1 𝑓(𝑛𝑘𝑥) ≪ √𝑁(log 𝑁)1/2+𝜀 a.e. (1.30) As mentioned at the end of the previous section this means that the problem con- cerning the optimal (a.e.) asymptotic order of the sum ∑ 𝑓(𝑛𝑘𝑥) is solved; how- ever, the more difficult case of the precise a.e. asymptotic order of the discrepancy 𝐷∗ 𝑁({𝑛1𝑥}, . . . , {𝑛𝑁𝑥}) remains open.
- 29. Metric number theory, lacunary series and systems of dilated functions | 15 Concluding remark: There exists a close connection between discrepancy theory and harmonic analysis, which we have for example observed in Weyl’s criterion (1.2), in the Erdős–Turán inequality (1.8) and in the Carleson convergence theorem in Sec- tion 5. This connection goes far beyond the material contained in this article, and is comprehensively presented in a survey article of Dmitriy Bilyk in the present volume (see [10]). References [1] Christoph Aistleitner, On the law of the iterated logarithm for the discrepancy of lacunary se- quences, Trans. Amer. Math. Soc. 362 (2010), 5967–5982. [2] Christoph Aistleitner, Convergence of ∑ 𝑐𝑘𝑓(𝑘𝑥) and the Lip 𝛼 class, Proc. Amer. Math. Soc. 140 (2012), 3893–3903. [3] Christoph Aistleitner and Istvan Berkes, Probability and metric discrepancy theory, Stoch. Dyn. 11 (2011), 183–207. [4] Christoph Aistleitner, Istvan Berkes and Kristian Seip, GCD sums from Poisson integrals and systems of dilated functions, J. Eur. Math. Soc., to appear. Available at http://arxiv.org/abs/ 1210.0741. [5] Juan Arias de Reyna, Pointwise convergence of Fourier series, Lecture Notes in Mathematics 1785, Springer-Verlag, Berlin, 2002. [6] David H. Bailey and Richard E. Crandall, On the random character of fundamental constant expansions, Experiment. Math. 10 (2001), 175–190. [7] Roger C. Baker, Metric number theory and the large sieve, J. London Math. Soc. 24, No. 2 (1981), 34–40. [8] István Berkes and Walter Philipp, The size of trigonometric and Walsh series and uniform dis- tribution mod 1, J. London Math. Soc. 50, No. 2 (1994), 454–464. [9] István Berkes and Michel Weber, On the convergence of ∑ 𝑐𝑘𝑓(𝑛𝑘𝑥), Mem. Amer. Math. Soc. 201 (2009). [10] Dmitriy Bilyk, Discrepancy theory and harmonic analysis, contained in this volume. [11] Dmitriy Bilyk, On Roth’s orthogonal function method in discrepancy theory, Unif. Distrib. The- ory 6 (2011), 143–184. [12] Julien Brémont, Davenport series and almost-sure convergence, Q. J. Math. 62 (2011), 825– 843. [13] Yann Bugeaud, Distribution modulo one and Diophantine approximation, Cambridge Tracts in Mathematics 193, Cambridge University Press, Cambridge, 2012. [14] Lennart Carleson, On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1966), 135–157. [15] Josef Dick and Friedrich Pillichshammer, Digital nets and sequences, Cambridge University Press, Cambridge, 2010. [16] Michael Drmota and Robert F. Tichy, Sequences, discrepancies and applications, Lecture Notes in Mathematics 1651, Springer-Verlag, Berlin, 1997. [17] Artūras Dubickas, On the distance from a rational power to the nearest integer, J. Number The- ory 117 (2006), 222–239. [18] Artūras Dubickas, On the powers of 3/2 and other rational numbers, Math. Nachr. 281 (2008), 951–958. [19] Richard J. Duffin and Albert C. Schaeffer, Khintchine’s problem in metric Diophantine approxi- mation, Duke Math. J. 8 (1941), 243–255.
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- 31. József Beck Strong uniformity Abstract: We attempt to develop a new chapter of the theory of Uniform Distribution; we call it strong uniformity. Strong uniformity, in a nutshell, means that we combine Lebesguemeasurewiththeclassicaltheory of Uniform Distribution, basically founded by H. Weyl in his famous paper from 1916 [12], which is built around nice test sets, such as axis-parallel rectangles and boxes. We prove the continuous version of the well- known Khinchin’s conjecture (see [7]) in every dimension 𝑑 ≥ 2 (the discrete version turned out to be false – it was disproved by Marstrand [10]). We consider an arbitrarily complicated but fixed measurable test set 𝑆 in the 𝑑-dimensional unit cube, and study the uniformity of a typical member of some natural families of curves, such as all torus lines or billiard paths starting from the origin, with respect to 𝑆. In the 2-dimensional case we have the very surprising superuniformity of the typical torus lines and billiard paths. In dimensions ≥ 3 we still have strong uniformity, but not superuniformity. However, in dimension 3 we have the even more striking super-duper uniformity for 2- dimensional rays (replacing the torus lines). Finally, we briefly indicate how to exhibit superuniform motions on every “reasonable” plane region (e.g. the circular disk) and on every “reasonable” closed surface (sphere, torus and so on). Keywords: Discrepancy of curves, time discrepancy of motions, strong uniformity, su- peruniformity Mathematics Subject Classification 2010: 11K38 || József Beck: Mathematics Department, Busch Campus, Hill Center, Rutgers University, New Brunswick, NJ 08903, USA, e-mail: jbeck@math.rutgers.edu 1 Introduction In 1923, Khinchin [7] made the following conjecture: given a Lebesgue measurable set 𝑆 ⊂ [0, 1], the sequence 𝛼, 2𝛼, 3𝛼, . . . modulo 1 is uniformly distributed with respect to 𝑆 for almost every 𝛼. Formally, the conjecture states that lim 𝑛→∞ 1 𝑛 ∑ 1≤𝑘≤𝑛: {𝑘𝛼}∈𝑆 1 = meas(𝑆) for almost every 𝛼 . (2.1) Here 0 ≤ {𝑥} < 1 denotes the fractional part of a real number 𝑥, and meas stands for the 1-dimensional Lebesgue measure. Khinchin’s conjecture was motivated by the classical equidistribution theorem that (2.1) holds for every irrational 𝛼 if 𝑆 = [𝑎, 𝑏) is an arbitrary subinterval of [0, 1).
- 32. 18 | József Beck Formally, lim 𝑛→∞ 1 𝑛 ∑ 1≤𝑘≤𝑛: 𝑎≤{𝑘𝛼}<𝑏 1 = 𝑏 − 𝑎 (2.2) holds for every 0 < 𝑎 < 𝑏 < 1 and every irrational 𝛼. We obtain a second positive result if the sequence 𝛼, 2𝛼, 3𝛼, . . . is replaced by the translated copy 𝛽+𝛼, 𝛽+2𝛼, 𝛽+3𝛼, . . . , translated by a typical real number 𝛽, that is, if we start from a typical 𝛽 instead of 0. More precisely, for every Lebesgue measurable set 𝑆 ⊂ [0, 1] and for every irrational 𝛼, we have lim 𝑛→∞ 1 𝑛 ∑ 1≤𝑘≤𝑛: {𝛽+𝑘𝛼}∈𝑆 1 = meas(𝑆) for almost every 𝛽. (2.3) Note that (2.3) is a special case of Birkhoff’s well-known individual ergodic theo- rem. Also there is a simple way of deriving (2.3) directly from (2.2). A third positive result was proved by Raikov [11]: (2.1) holds if the linear sequence 𝑘𝛼 is replaced by the exponential sequence 2𝑘 𝛼, or in general, by any sequence 𝑞𝑘 𝛼, where 𝑞 is a fixed integer greater than one (𝑘 = 1, 2, 3, . . . ). Khinchin’s conjecture remained among the most famous open problems in the subject of Uniform Distribution for several decades. The likely reason why the conjec- ture resisted every attack is that people were convinced about its truth, and wanted to prove a positive result. However, despite the three positive results mentioned above (see (2.2), (2.3) and Raikov’s theorem), conjecture (2.1) turned out to be false. In 1970, Marstrand [10] proved that there exists an open set 𝑆 ⊂ [0, 1] with meas(𝑆) < 1 such that lim sup 𝑛→∞ 1 𝑛 ∑ 1≤𝑘≤𝑛: {𝑘𝛼}∈𝑆 1 = 1 for every 𝛼 The fact that open sets are the simplest in the Borel hierarchy makes Marstrand’s neg- ative result even more surprising. Marstrand’s result demonstrates that Khinchin was too optimistic. In this paper we show how to “save” Khinchin’s conjecture in the continuous case, i.e. by switching from a sequence (arithmetic progression) to the continuous torus line. Of course, we have to increase the dimension of the underlying set: we replace the unit torus [0, 1) with the 2-dimensional unit torus [0, 1)2 = 𝐼2 , and replace the arithmetic progres- sion 𝛼, 2𝛼, 3𝛼, . . . starting from 0 with the straight line (𝑡 cos 𝜃, 𝑡 sin 𝜃), 𝑡 ≥ 0 start- ing from the origin (0, 0) with a fixed angle 𝜃. The continuous version of the classical equidistribution theorem (Kronecker–Weyl) says that the torus line (𝑡 cos 𝜃, 𝑡 sin 𝜃), 𝑡 ≥ 0 modulo 1 is uniformly distributed with respect to every axis-parallel rectangle in the unit square [0, 1]2 = 𝐼2 if and only if the slope tan 𝜃 is irrational. The continuous analog of Khinchin’s conjecture goes as follows: What happens if one replaces the rectangle with an arbitrary Lebesgue measurable test set 𝑆 ⊂ [0, 1]2 ?
- 33. Strong uniformity | 19 Can one prove uniformity with respect to 𝑆 for almost every angle 𝜃? In other words, can one prove “strong uniformity” in the continuous case? Here, “strong uniformity” refers to the fact that Lebesgue measurable sets form the largest class of sets for which we can define measure (without measure we cannot define uniformity). The answer is yes; our Theorem 1.1 below proves strong uniformity in the contin- uous case. We actually prove much more: we prove superuniformity in the sense that the error term turns out to be shockingly small: we prove a polylogarithmic error term. More precisely, let 𝑆 ⊂ [0, 1]2 = 𝐼2 be an arbitrary Lebesgue measurable set in the unit square, and assume that 0 < area(𝑆) < 1, where area denotes the 2-dimensional Lebesgue measure. Let 𝜃 ∈ [0, 2𝜋)be an arbitraryangle, and consider the straight line (𝑡 cos 𝜃, 𝑡 sin 𝜃), 𝑡 ≥ 0 starting from the origin (0, 0) with angle 𝜃. Let 𝑇𝑆(𝜃) denote the time the line (𝑡 cos 𝜃, 𝑡 sin 𝜃) modulo 1 spends in the given set 𝑆 as 0 ≤ 𝑡 ≤ 𝑇 (line modulo 1 = torus line). Formally, 𝑇𝑆(𝜃) = meas {𝑡 ∈ [0, 𝑇]: ({𝑡 cos 𝜃}, {𝑡 sin 𝜃}) ∈ 𝑆} . Note that it may happen that the set {𝑡 ∈ [0, 𝑇]: ({𝑡 cos 𝜃}, {𝑡 sin 𝜃}) ∈ 𝑆} is not measurable for some particular angle 𝜃 ∈ [0, 2𝜋), and so 𝑇𝑆(𝜃) is not defined. But this technical nuisance happens only for a negligible set of angles: 𝑇𝑆(𝜃) is certainly well-defined for almost every 𝜃 ∈ [0, 2𝜋). This follows from some general results in Lebesgueintegration. Wearereferringto Fubini’s theorem and whatwecallthechange of variables in multiple integrals, applied in the special case of polar coordinates. (The relevance of polar coordinates is clear from the fact that the lines (𝑡 cos 𝜃, 𝑡 sin 𝜃) are passing through the origin.) Uniformity of the torus line (𝑡 cos 𝜃, 𝑡 sin 𝜃) modulo 1 in 𝑆 means that lim 𝑇→∞ 𝑇𝑆(𝜃) − area(𝑆)𝑇 𝑇 = 0 . (2.4) Our first result says that we can replace the factor of 𝑇 in the denominator of (2.4) with the much smaller (log 𝑇)3+𝜀 for almost every 𝜃. In this paper log 𝑥 and log2 𝑥 denote, respectively, the natural and the binary logarithm of 𝑥. Theorem 1.1. Let 𝑆 ⊂ [0, 1)2 be an arbitrary Lebesgue measurable set in the unit square with 0 < area(𝑆) < 1. Then for every 𝜀 > 0, lim 𝑇→∞ 𝑇𝑆(𝜃) − area(𝑆)𝑇 (log 𝑇)3+𝜀 = 0 for almost every angle 𝜃. We can of course rewrite Theorem 1.1 in the equivalent form 𝑇𝑆(𝜃) = area(𝑆)𝑇 + 𝑜 ((log 𝑇)3+𝜀 ) . Notice that the polylogarithmic error term is shockingly small compared to the linear main term area(𝑆)𝑇. This is why we call Theorem 1.1 a superuniformity result.
- 34. 20 | József Beck What makes the “continuous” Theorem 1.1 particularly interesting is the sharp contrast with the “discrete” version (Khinchin’s conjecture), where there is no unifor- mity at all. We actually prove a more general version of Theorem 1.1, extending the case of the 0–1 valued characteristic function 𝜒𝑆 of an arbitrary measurable subset 𝑆 of the unit square to an arbitrary real-valued square-integrable function 𝑓 ∈ 𝐿2 defined on the unit torus [0, 1)2 (one could also extend it to complex-valued functions 𝑓 ∈ 𝐿2). This generalization (from sets to functions) will be needed in some later applications, where we will study superuniform motions inside finite regions more general than the rectangle, and will also study superuniform motions on closed surfaces; see Section 3. Of course, a function defined on the unit torus (in any dimension) means that the function is periodic with period one in each variable. (Warning: for convenience I use 𝐼𝑑 for both the unit torus [0, 1)𝑑 and the unit cube [0, 1]𝑑 ; I hope this minor ambiguity does not confuse the reader.) Theorem 1.2. Let 𝑓 ∈ 𝐿2 be an arbitrary real-valued Lebesgue square-integrable func- tion on the unit torus [0, 1)2 = 𝐼2 . Then for every 𝜀 > 0, lim 𝑇→∞ ∫ 𝑇 0 𝑓(𝑡 cos 𝜃, 𝑡 sin 𝜃) 𝑑𝑡 − 𝑇 ∫𝐼2 𝑓(y) 𝑑y (log 𝑇)3+𝜀 = 0 for almost every angle 𝜃. Notice that Theorem 1.2 implies Theorem 1.1 in the special case 𝑓 = 𝜒𝑆. We derive Theorem 1.2 from the following quantitative result. Theorem 1.3. Let 𝑓 ∈ 𝐿2 be an arbitrary real-valued Lebesgue square-integrable func- tion on the unit torus [0, 1)2 = 𝐼2 . Then for every 𝜂 > 0 there is a subset A = A(𝑓; 𝜂) of the interval [0, 2𝜋) such that 1 2𝜋 meas(A) ≥ 1 − 𝜂 , and for every 𝜃 ∈ A and 𝑇 ≥ 8, 𝑇 ∫ 0 𝑓(𝑡 cos 𝜃, 𝑡 sin 𝜃) 𝑑𝑡 − 𝑇 ∫ 𝐼2 𝑓(y) 𝑑y ≤ 8 ∫ 0 |𝑓(𝑡 cos 𝜃, 𝑡 sin 𝜃)| 𝑑𝑡 + 12 ⋅ 104 𝜎0(𝑓) 𝜂 (log2 𝑇 + 1)3 (log(log2 𝑇 + 1))2 + 12𝜎0(𝑓) ,
- 35. Strong uniformity | 21 where 𝜎2 0(𝑓) = ∫ 𝐼2 (𝑓(y) − ∫ 𝐼2 𝑓(z) 𝑑z) 2 𝑑y is the “variance” of 𝑓. To derive Theorem 1.2, choose 𝜂 = 2−𝑛 , 𝑛 = 1, 2, 3, . . . , and consider the union set A = ∞ ⋃ 𝑛=1 A(𝑓; 2−𝑛 ) . Then meas(A) = 2𝜋; and by Theorem 1.3, for every 𝜀 > 0 and every 𝜃 ∈ A, lim 𝑇→∞ ∫ 𝑇 0 𝑓(𝑡 cos 𝜃, 𝑡 sin 𝜃) 𝑑𝑡 − 𝑇 ∫𝐼2 𝑓(y) 𝑑y (log 𝑇)3+𝜀 = 0 , proving Theorem 1.2. It is natural to ask what happens in higher dimensions, where we replace the unit square with the unit cube [0, 1]𝑑 , 𝑑 ≥ 3. Again we study the “strong” uniformity of typical torus lines starting from the origin. Here, “strong” means that the test set is an arbitrary measurable set. Let 𝑆 ⊂ [0, 1]𝑑 = 𝐼𝑑 be an arbitrary Lebesgue measurable set in the unit cube of dimension 𝑑 ≥ 3, and assume that 0 < volume(𝑆) < 1, where volume denotes the 𝑑-dimensional Lebesgue measure. Let e ∈ S𝑑−1 be an arbitrary unit vector in the 𝑑-dimensional Euclidean space ℝ𝑑 ; S𝑑−1 denotes the unit sphere in ℝ𝑑 . Consider the straight line 𝑡e, 𝑡 ≥ 0 starting from the origin 0 ∈ ℝ𝑑 . Let 𝑇𝑆(e) denote the time the line 𝑡e modulo 1 spends in the given set 𝑆 as 0 ≤ 𝑡 ≤ 𝑇 (line modulo 1=torus line). Uniformity of the torus line 𝑡e modulo 1 in 𝑆 means that lim 𝑇→∞ 𝑇𝑆(e) − volume(𝑆)𝑇 𝑇 = 0 . (2.5) In the 3-dimensional case we can replace the factor of 𝑇 in the denominator of (2.5) with the substantially smaller 𝑇1/4 (log 𝑇)3+𝜀 for almost every direction e ∈ S2 in the 3-space. In the 𝑑-dimensional case with 𝑑 ≥ 4 we can replace the factor of 𝑇 in the denominator of (2.5) with 𝑇 1 2 − 1 2(𝑑−1) (log 𝑇)3+𝜀 for almost every direction e ∈ S𝑑−1 in the 𝑑-space ℝ𝑑 . Theorem 1.4. (a) Let 𝑆 ⊂ [0, 1)3 be an arbitrary Lebesgue measurable set in the unit cube with 0 < volume(𝑆) < 1. Then for every 𝜀 > 0, lim 𝑇→∞ 𝑇𝑆(e) − volume(𝑆)𝑇 𝑇1/4(log 𝑇)3+𝜀 = 0 (2.6) for almost every direction e ∈ S2 in the 3-space.
- 36. 22 | József Beck (b) In the 𝑑-dimensional case 𝑆 ⊂ [0, 1)𝑑 with 𝑑 ≥ 4, we have the perfect analog of (2.6) where the factor of 𝑇1/4 in (2.6) is replaced by 𝑇 1 2 − 1 2(𝑑−1) for almost every direction e ∈ S𝑑−1 in the 𝑑-space ℝ𝑑 . Again the proof automatically works for arbitrary (real-valued) square-integrable functions 𝑓 ∈ 𝐿2 defined on the unit torus. Theorem 1.5. (a) Let 𝑓 ∈ 𝐿2 be an arbitrary real-valued Lebesgue square-integrable function on the unit torus [0, 1)3 = 𝐼3 . Then for every 𝜀 > 0, lim 𝑇→∞ ∫ 𝑇 0 𝑓(𝑡e) 𝑑𝑡 − 𝑇 ∫𝐼3 𝑓(y) 𝑑y 𝑇1/4(log 𝑇)3+𝜀 = 0 (2.7) for almost every direction e ∈ S2 in the 3-space. (b) In the 𝑑-dimensional case with 𝑑 ≥ 4, we have the perfect analog of (2.7) where the factor of 𝑇1/4 in (2.7) is replaced by 𝑇 1 2 − 1 2(𝑑−1) for almost every direction e ∈ S𝑑−1 in the 𝑑-space. Theorem 1.5 implies Theorem 1.4 in the special case of 0–1 valued characteristic func- tions 𝑓 = 𝜒𝑆. Theorem 1.5 can be derived from the following quantitative result exactly the same way as we derived Theorem 1.2 from Theorem 1.3. Theorem 1.6. (a) Let 𝑓 ∈ 𝐿2 be an arbitrary real-valued Lebesgue square-integrable function on the unit torus [0, 1)3 = 𝐼3 . Then for every 𝜂 > 0 there is a subset A = A(𝑓; 𝜂) of the unit sphere S2 such that 1 4𝜋 SurfaceArea(A) ≥ 1 − 𝜂 , and for every direction e ∈ A and 𝑇 ≥ 8, 𝑇 ∫ 0 𝑓(𝑡e) 𝑑𝑡 − 𝑇 ∫ 𝐼3 𝑓(y) 𝑑y ≤ 8 ∫ 0 |𝑓(𝑡e)| 𝑑𝑡 + 12 ⋅ 105 𝜎0(𝑓) 𝜂 𝑇1/4 (log2 𝑇 + 1)3 (log(log2 𝑇 + 1))2 + 12𝜎0(𝑓) , (2.8) where again 𝜎2 0(𝑓) = ∫ 𝐼3 (𝑓(y) − ∫ 𝐼3 𝑓(z) 𝑑z) 2 𝑑y is the “variance” of 𝑓.
- 37. Strong uniformity | 23 (b) In the 𝑑-dimensional case with 𝑑 ≥ 4, we have the perfect analog of (2.8) where the factor of 𝑇1/4 in (2.8) is replaced by 𝑐0(𝑑)𝑇 1 2 − 1 2(𝑑−1) , and 𝑐0(𝑑) is a positive absolute constant that depends only on the dimension. If we make the choice 𝑑 = 2 in Theorem 1.4 (or 1.5 or 1.6) then we obtain Theorem 1.1 (or 1.2 or 1.3). Perhaps the reader is wondering, why did we formulate two separate theorems if the first one is the special case of the second one in the 2-dimensional case. Well, the answer is that we wanted to emphasize superuniformity – meaning polylogarithmic error term – which is not the case in dimensions 𝑑 ≥ 3. This follows from the next result, which states that Theorem 1.4 (and 1.5 and 1.6) is best possible apart from polylogarithmic factor: the exponent 1 2 − 1 2(𝑑−1) of 𝑇 in the error term of Theorem 1.4 (and 1.5 and 1.6) is best possible. Note that in some problems it is natural to study very high dimensions; e.g. in statistical mechanics the phase space is 6𝑁-dimensional, where 𝑁 is the number of gas molecules in a container. Since 𝑁 is typically in the range of the Avogadro num- ber (close to 1024 ), it is natural to ask what happens if the dimension 𝑑 is much larger than the time parameter 𝑇. Theorem 1.6 (b) is about the general case of arbitrary di- mension 𝑑, but it does not help, because of the unspecified constant factor 𝑐0(𝑑) in the upper bound for the discrepancy. What we need is an upper bound on the discrep- ancy that does not depend on the dimension. I briefly mention such a dimension-free result. For simplicity I consider the special case 𝑓 = 𝜒𝑆, where 𝑆 is an arbitrary mea- surable test set in the 𝑑-dimensional unit cube 𝐼𝑑 = [0, 1)𝑑 , and write 0 < 𝑝 = volume(𝑆) < 1 and 𝑞 = 1 − 𝑝. Notice that the diameter of the 𝑑-dimensional unit cube [0, 1)𝑑 is √𝑑, and this explains why it is natural to modify the time-counting function (e ∈ S𝑑−1 ) 𝑇𝑆(e) = meas {𝑡 ∈ [0, 𝑇]: 𝑡e ∈ 𝑆 modulo 1} , and to replace it with 𝑇𝑆(𝑑; e) = meas {𝑡 ∈ [0, 𝑇]: 𝑡√𝑑e ∈ 𝑆 modulo 1} , which can be interpreted as speeding up the linear motion by a factor of √𝑑. (In this paper we focus on low dimensions such as 𝑑 = 2, 3, 4; then this switch is rather neg- ligible.) Here comes our “very high dimension vs. short time” result. Assume that 𝑇 > log2 𝑑 𝑝𝑞 , then ( ∫ e∈S𝑑−1 (𝑇𝑆(𝑑; e) − 𝑝𝑇) 2 𝑑𝑆𝐴(e)) 1/2 < 40√𝑝𝑞𝑇 (1 + 2 log2 𝑇 𝑑1/4 ) , (2.9) where 𝑑𝑆𝐴(e) denotes the integration with respect to the normalized surface area on the sphere e ∈ S𝑑−1 , i.e. 𝑆𝐴(S𝑑−1 ) = 1.
- 38. 24 | József Beck In view of the weak condition 𝑇 > log2 𝑑 𝑝𝑞 , the dimension 𝑑 can be much larger than the time parameter 𝑇, and still the quadratic average of the discrepancy (see the left-hand side of (2.9)) is in the square-root range √𝑇 (see the right-hand side of (2.9)); notice that the order of magnitude √𝑇 represents a high level of uniformity with re- spect to the given test set 𝑆. Combining (2.9) with Chebyshev’s inequality, we conclude that for the majority of the directions e ∈ S𝑑−1 , the torus line segment 𝑡e(mod 1), 0 ≤ 𝑡 ≤ 𝑇 is very uniform with respect to the test set 𝑆. (Note that Theorem 1.6 (b) does not say anything interesting in the case where the dimension 𝑑 is larger than the time parameter 𝑇). Similarly to Theorems 1.1 and 1.4, (2.9) is also “complexity-free”, i.e. the upper bound in (2.9) does not depend on the complexity (“ugliness”) of the test set 𝑆. We give a more detailed discussion of (2.9) in another paper, which is devoted to the applications of dimension-free and complexity-free strong uniformity in non- equilibrium statistical mechanics. Let us return to Theorems 1.1 and 1.4: they are about the torus line, which is the simplest curve on the torus. We can define a simple motion on the torus by assuming that a particle moves on the torus line with unit speed. Theorem 1.7 below is a general result about the limitations of the time discrepancy of a motion of a particle in the unit torus [0, 1)𝑑 , 𝑑 ≥ 3. Let 𝛤 = {x(𝑡) = ({𝑥1(𝑡)}, . . . , {𝑥𝑑(𝑡)}): 0 ≤ 𝑡 ≤ 𝑇} (2.10) be an arbitrary (continuous) parametrized curve on the 𝑑-dimensional unit torus [0, 1)𝑑 , 𝑑 ≥ 2 with arc-length 𝑇; here each coordinate 𝑥𝑗(𝑡), 1 ≤ 𝑗 ≤ 𝑑 is a contin- uous function of 𝑡, and {𝑥} denotes, as usual, the fractional part of a real number 𝑥. Note that the parametrized curve 𝛤 in (2.10) represents the motion of a particle on the torus, and we constantly use this interpretation below; we refer to 𝑇 as the “total traveling time”. Let 𝑆 ⊂ [0, 1)𝑑 be an arbitrary measurable subset, and let vol(𝑆) denote the volume, i.e. the 𝑑-dimensional Lebesgue measure (of course vol = area for 𝑑 = 2). Let 𝑇𝑆(𝛤) denote the time the particle spends in the given set 𝑆; formally, 𝑇𝑆(𝛤) = {0 ≤ 𝑡 ≤ 𝑇: x(𝑡) ∈ 𝑆} . We call 𝑇𝑆(𝛤) the actual time, and compare it to the expected time, which – assuming perfect uniformity – is proportional to the volume: expected time = 𝑇 ⋅ vol(𝑆). The differenceof theactualtimeand theexpected timeis called time discrepancy; formally, time discrepancy = D𝑆(𝛤) = 𝑇𝑆(𝛤) − 𝑇 ⋅ vol(𝑆) . Short Detour Note that the concept of time discrepancy is not new; in some sense it goes back to Weyl; see also Section 1.9 in the book of Kuipers–Niederreiter [9] and Section 2.3 in the book of Drmota and Tichy [3], which discuss the branch called “continuous uniform
- 39. Strong uniformity | 25 distribution mod 1”, studying arbitrary continuous parametrized curves. My main re- sults (Theorems 1.1 and 1.4)are about the simplest curves (e.g. lines) with the most nat- ural arc-length parametrization. If we allow arbitrary (continuous) parametrizations, then we can have, at first sight, very surprising results such as the time discrepancy can tend to zero arbitrarily fast; see e.g. Theorem 2.89 in [3]. But, unfortunately, this result is almost trivial: the proof is just a few lines, it takes advantage of the extreme generality of the definition, which allows using arbitrary parametrizations of a curve. Of course we cannot expect a similar result for the natural arc-length parametrization. The proofs of Theorems 1.1 and 1.4 (about the natural arc-length parametrization) are not simple; in fact, they are very long and complicated (see [2]). Just because Theorem 2.89 in [3] is almost trivial, it does not mean at all that the results in “continuous uniform distribution mod 1” are all simple. For example, The- orems 2.93 and 2.96 in [3] are very interesting deep results, but because they are not too close to our subject of strong uniformity, I stop this short detour about “continu- ous uniform distribution mod 1”. (Note that “continuous” discrepancy has been also studied in much more general situations, e.g. in compact Riemann manifolds, but that would lead us very far from our main topic.) To prove a nontrivial result, we need an extra assumption. We assume that “𝑇 = total traveling time = arc-length”, which is equivalent to the requirement that the av- erage speed is one. Theorem 1.7. For every integer 𝑑 ≥ 3 and real 𝑇 > 1, there exists an integer 𝑚 = 𝑚(𝑑, 𝑇) ≥ 2 such that we can construct 𝑚 measurable subsets 𝑆1, . . . , 𝑆𝑚 of the unit torus [0, 1)𝑑 with the following property: given any parametrized curve 𝛤 = {x(𝑡) = ({𝑥1(𝑡)}, . . . , {𝑥𝑑(𝑡)}): 0 ≤ 𝑡 ≤ 𝑇} of arc-length 𝑇 on the torus [0, 1)𝑑 (i.e. the average speed is one) D𝑆𝑗 (𝛤) > 𝑐1(𝑑)𝑇 1 2 − 1 2(𝑑−1) holds for at least two thirds of the 𝑚 subsets 𝑆1, . . . , 𝑆𝑚. Here 𝑐1(𝑑) > 0 is a constant depending only on the dimension 𝑑 ≥ 3. In particular, 𝑐1(3) = 1/500 is a good choice for 𝑑 = 3. Theorem 1.7 implies, via a standard average argument, that Theorem 1.6 is best pos- sible apart from a polylogarithmic factor of 𝑇. The explanation goes as follows. First note that every torus line in Theorem 1.6 is determined by its direction e ∈ S𝑑−1 , and using the ((𝑑 − 1)-dimensional) surface area on the unit sphere S𝑑−1 , it is meaningful to talk about the “majority of torus lines”, or more precisely, about “1 − 𝜀 part of all torus lines passing through the origin”. Now assume that, for some 𝑑 ≥ 3 and 𝑇 > 1, there exists a continuous family of parametrized curves {𝛤𝜔 : 𝜔 ∈ 𝛺} (2.11)
- 40. 26 | József Beck on the torus [0, 1)𝑑 such that there is a probability measure 𝜇 on the index-set 𝛺 (i.e. 𝜇(𝛺) = 1, so it is meaningful to talk about a (1 − 𝜀)-part), and the family of curves in (2.11) beats Theorem 1.6 in the following quantitative sense: Given any measurable subset 𝑆 ⊂ [0, 1)𝑑 , D𝑆(𝛤𝜔) < 𝑐1(𝑑)𝑇 1 2 − 1 2(𝑑−1) (2.12) holds for at least two thirds of the curves 𝛤𝜔, 𝜔 ∈ 𝛺 in the sense of the 𝜇-measure. We show that this contradicts Theorem 1.7. Indeed, we apply (2.12) for the 𝑚 = 𝑚(𝑑, 𝑇) ≥ 2 sets 𝑆𝑗, 1 ≤ 𝑗 ≤ 𝑚 whose existence is guaranteed by Theorem 1.7. Thus, for every 1 ≤ 𝑗 ≤ 𝑚 there exists a (measurable) subset 𝛺𝑗 of the index-set such that 𝜇(𝛺𝑗) ≥ 2/3, and D𝑆𝑗 (𝛤𝜔) < 𝑐1(𝑑)𝑇 1 2 − 1 2(𝑑−1) holds for all 𝜔 ∈ 𝛺𝑗. The fact 1 𝑚 𝑚 ∑ 𝑗=1 𝜇(𝛺𝑗) ≥ 2/3 immediately implies that there must exist an index 𝜔0 ∈ 𝛺 which is contained by at least 2𝑚/3 of the 𝑚 sets 𝛺𝑗, 1 ≤ 𝑗 ≤ 𝑚. In other words, there is a curve 𝛤𝜔0 such that (see (2.12)) D𝑆𝑗 (𝛤𝜔0 ) < 𝑐1(𝑑)𝑇 1 2 − 1 2(𝑑−1) (2.13) holds for at least two thirds of the 𝑚 sets 𝑆1, . . . , 𝑆𝑚. But (2.13) clearly contradicts The- orem 1.7, and this contradiction proves that Theorem 1.6 is nearly best possible in every dimension 𝑑 ≥ 3: we cannot replace the error term 𝑇 1 2 − 1 2(𝑑−1) (log 𝑇)3+𝜀 with 𝑜 (𝑇 1 2 − 1 2(𝑑−1) ) . In the next two sections we elaborate on superuniformity, and formulate more results (see Theorems 2.2–3.1). 2 Superuniformity and super-duper uniformity 2.1 Superuniformity of the typical billiard paths The message of Theorem 1.7 is that superuniformity (meaning polylogarithmic error) of the torus line is a 2-dimensional phenomenon: in dimensions 𝑑 ≥ 3 the error is greater than a power of 𝑇; more precisely, greater than a constant times 𝑇 1 2 − 1 2(𝑑−1) for “ugly” test sets.
- 41. Strong uniformity | 27 Note that the torus line does not represent a continuous motion inside a square (or a cube): when it hits the boundary, it comes back at the opposite side of the bound- ary – a big jump. There is a well-known way to switch from a torus line to a billiard path (see unfolding below). Using this, it is easy to extend the results of Section 1 about the torus line in a square (or a cube) to the billiard path in a square (or a cube). The ad- vantage is that the billiard path is a piecewise smooth, continuous curve; it represents a continuous motion. As a byproduct, we obtain that – roughly speaking – the set of billiard paths represents the “most strongly uniform family of curves” in [0, 1)𝑑 , 𝑑 ≥ 2 (we can of course extend it to rectangles and axis-parallel boxes). Roughly speaking, this means that we ignore polylogarithmic factors. For simplicity we focus on the 2-dimensional case; we assume that the under- lying rectangle is the unit square [0, 1]2 . Since every rectangle can be mapped to a square by a linear transformation of the plane, this is not a real restriction. Let x(𝑡) = (𝑥1(𝑡), 𝑥2(𝑡)), 0 < 𝑡 < ∞ represent an infinite billiard path with starting point x(0) = (𝑥1(0), 𝑥2(0)) = s ∈ [0, 1]2 and initial direction (=angle) 𝜃 ∈ [0, 2𝜋). The intuitively plausible concept of “billiard path” precisely means that a point- mass (representing a tiny billiard ball) moves freely along a straight line inside the unit square with unit speed until it hits the boundary (i.e. one of the four sides of the square). The reflection off the boundary is elastic, meaning the familiar law of reflection: the angle of incidence equals the angle of reflection. After the reflection, the point- billiard continues its linear motion with the new velocity (but of course the speed re- mains the same) until it hits the boundary again, and so forth (we ignore friction, air resistance, etc.). The initial condition, i.e. the starting point s ∈ [0, 1]2 of the billiard path and the initial direction 𝜃, uniquely determine an infinite piecewise linear bil- liard path x(𝑡) = (𝑥1(𝑡), 𝑥2(𝑡)), 0 < 𝑡 < ∞ in the unit square. The law of reflection implies that there are at most four different directions along the billiard path (the ini- tial direction is preserved modulo 𝜋/2, which is one-fourth of the whole angle 2𝜋; the same holds for any rectangle). Because of the unit speed, arc-length and time are the same. Formally, a billiard path in the unit square [0, 1]2 has the form x(𝑡) = (𝑥1(𝑡), 𝑥2(𝑡)) , 0 < 𝑡 < ∞ with 𝑥𝑗(𝑡) = 2 (𝑠𝑗 + 𝑡𝛽𝑗)/2 , 𝑗 = 1, 2 , where e = (𝛽1, 𝛽2) is a unit vector, and ‖𝑦‖ denotes the distance of a real number 𝑦 from the nearest integer. Here s = (𝑠1, 𝑠2) is the starting point, and e is the initial direction. So 𝜃 = arctan 𝛽2 𝛽1 (“inverse tangent”) is the initial angle. An alternative, more symmetric way is to replace [0, 1]2 with [−1 2 , 1 2 ]2 ; then x(𝑡) = (𝑥1(𝑡), 𝑥2(𝑡)) , 0 < 𝑡 < ∞ with 𝑥𝑗(𝑡) = ⟨𝑠𝑗 + 𝑡𝛽𝑗⟩ , 𝑗 = 1, 2 , where ⟨𝑦⟩ = ‖𝑦‖ if 𝑦 ≥ 0 , and − ‖ − 𝑦‖ if 𝑦 < 0 .
- 42. 28 | József Beck Next we recall the well-known geometric trick of unfolding a billiard path inside the unit square to a straight line in the entire plane. The idea is very simple and el- egant: we keep reflecting the square itself in the respective side, and this procedure unfolds a given piecewise linear billiard path to a straight line. The figure below illus- trates unfolding Two (straight) lines in the plane correspond to the same billiard path if and only if they differ by a translation through an integral vector where both coordinates are even, i.e. where the vector is from the “double” square lattice 2ℤ × 2ℤ. In other words, the problem of the distribution of a billiard path in the unit square is equivalent to the distribution of the corresponding torus-line in the 2 × 2 square. The problem of uniformity of a billiard path in the unit square with respect to a given test set 𝑆 is equivalent to the problem of uniformity of the corresponding torus- line in the 2 × 2 square, where each one of the four unit squares contains a reflected copy of the given test set 𝑆; see 𝑆1, 𝑆2, 𝑆3, 𝑆4 in the second figure below. At the end, we shrink the underlying 2 × 2 square to the unit square 𝐼2 = [0, 1)2 .
- 43. Strong uniformity | 29 In the general case, the test set is upgraded to a function 𝑓 ∈ 𝐿2(𝐼2 ), i.e. 𝑓(𝑥, 𝑦) is a periodic real-valued Lebesgue square-integrable function with period one in both variables. In the case of the picture below, 𝑓 means the 0–1 valued characteristic func- tion of the union 𝑆1 ∪ 𝑆2 ∪ 𝑆3 ∪ 𝑆4 of the four half-size reflected copies of the given subset 𝑆 ⊂ 𝐼2 = [0, 1)2 . S1 S S2 S4 = S3 A billiard pathintheunitcube[0, 1]𝑑 , 𝑑 ≥ 3canbedefined similarly. Theproblem of uniformity of a billiard path in the unit cube [0, 1)𝑑 with respect to a given test set 𝑆 is equivalent – via unfolding – to the problem of uniformity of the corresponding torus-line in the 2 × ⋅ ⋅ ⋅ × 2 cube, where each one of the 2𝑑 unit cubes contains a reflected copy of the given test set 𝑆. As far as I know, the first appearance of the geometric trick of unfolding is in a paper of D. König and A. Szücs from 1913 (see [8]), and it became widely known after Hardy and Wright included it in their well-known book on number theory [4]. The con- tinuous form of the equidistribution theorem (Kronecker–Weyl) implies the following elegant property of the torus line in the unit square: if the slope of the initial direction is rational, then the torus line is periodic, and if the slope of the initial direction is ir- rational, then the torus line is dense (so far this is Kronecker’s theorem). And what is more, for an irrational slope the torus line is uniformly distributed in the unit square, meaning that, for any axis-parallel subrectangle 𝑆 = [𝑎, 𝑏] × [𝑐, 𝑑] ⊂ [0, 1]2 , lim 𝑇→∞ 1 𝑇 measure {𝑡 ∈ [0, 𝑇]: x(𝑡) ∈ 𝑆} = area(𝑆) = (𝑏 − 𝑎)(𝑑 − 𝑐) , where x(𝑡), 0 < 𝑡 < ∞ denotes the torus line in the unit square, parametrized with the arc-length. (This upgrading of Kronecker’s theorem is due to H. Weyl.)
- 44. 30 | József Beck König and Szücs used the trick of unfolding, combined with the Kronecker–Weyl theorem, to prove the following analog property of the billiard path in the unit square: if the slope of the initial direction is rational, then the billiard path is periodic, and if the slope of the initial direction is irrational, then the billiard path is uniformly dis- tributed in the unit square. This settles the 2-dimensional case. There is an analog result in every dimension 𝑑 ≥ 3. To talk about a “typical” billiard path, we need a measure on the set of all initial conditions of the billiard paths. Since the initialcondition is the pair of a starting point s ∈ [0, 1]2 and an initial direction (angle) 𝜃 ∈ [0, 2𝜋), the corresponding measure is simply the (Cartesian) product of the 2-dimensional Lebesgue measure on the unit square and the normalized 1-dimensional Lebesgue measure. This way the term “1 − 𝜀 part of all billiard paths” in Theorem 2.1 below will become perfectly precise. The following result was proved in my first paper about superuniformity. Theorem 2.1 (Beck [1]). Let 𝑆 be an arbitrary Lebesgue measurable subset of the unit square [0, 1)2 with 2-dimensional Lebesgue measure area(𝑆) > 0, and let 𝑇 > 100 be an arbitrarily large but fixed real number. Let x(𝑡) = (𝑥1(𝑡), 𝑥2(𝑡)), 0 ≤ 𝑡 ≤ 𝑇 be a general billiard path of length 𝑇 (length=time) in the unit square (the starting point is not necessarily the origin). Let 𝑇𝑆 denote the time the billiard path spends in subset 𝑆: 𝑇𝑆 = measure {𝑡 ∈ [0, 𝑇]: x(𝑡) ∈ 𝑆} . Let 0 < 𝜂 < 1/2 be arbitrary. Then for the (1 − 𝜂)-part of all billiard paths of length 𝑇 in the square, the curve-discrepancy |𝑇𝑆 − 𝑇 ⋅ area(𝑆)| is estimated from above as follows: |𝑇𝑆 − 𝑇 ⋅ area(𝑆)| < 20 𝜂 √area(𝑆)(1 − area(𝑆)) ⋅ √log 𝑇 ⋅ log log 𝑇 . (2.14) Moreover, Theorem 2.1 remains true if “billiard path” is replaced with “torus line”. This is how the proof goes: we prove Theorem 2.1 for torus lines only – the case of bil- liard paths is a corollary via unfolding. This is true in general: if we have a theorem about torus lines (e.g. Theorem 1.1 or Theorem 1.4), the use of unfolding automatically extends it to billiard paths. To formulate the billiard versions of Theorem 1.3 and The- orem 1.6, we use the notation x𝜃(𝑡) and xe(𝑡) to denote, respectively, the billiard path starting from the origin with initial angle 𝜃 ∈ [0, 2𝜋) in the unit square [0, 1]2 and with initial direction e ∈ S𝑑−1 in the unit cube [0, 1]𝑑 , 𝑑 ≥ 3. Theorem 2.2. (a) Let 𝑓 ∈ 𝐿2 be an arbitrary real-valued Lebesgue square integrable function on the unit square [0, 1]2 = 𝐼2 . Then for every 𝜂 > 0 there is a subset A = A(𝑓; 𝜂) of the interval [0, 2𝜋) such that 1 2𝜋 meas(A) ≥ 1 − 𝜂 ,
- 45. Strong uniformity | 31 and for every 𝜃 ∈ A and 𝑇 ≥ 8, 𝑇 ∫ 0 𝑓(x𝜃(𝑡)) 𝑑𝑡 − 𝑇 ∫ 𝐼2 𝑓(y) 𝑑y ≤ 8 ∫ 0 |𝑓(x𝜃(𝑡))| 𝑑𝑡 + 12 ⋅ 104 𝜎0(𝑓) 𝜂 (log2 𝑇 + 1)3 (log(log2 𝑇 + 1))2 + 12𝜎0(𝑓) , where 𝜎2 0(𝑓) = ∫ 𝐼2 (𝑓(y) − ∫ 𝐼2 𝑓(z) 𝑑z) 2 𝑑y . (b) Let 𝑓 ∈ 𝐿2 be an arbitrary real-valued Lebesgue square integrable function on the unit cube [0, 1]3 = 𝐼3 . Then for every 𝜂 > 0 there is a subset A = A(𝑓; 𝜂) of the unit sphere S2 such that 1 4𝜋 SurfaceArea(A) ≥ 1 − 𝜂 , and for every direction e ∈ A and 𝑇 ≥ 8, 𝑇 ∫ 0 𝑓(xe(𝑡)) 𝑑𝑡 − 𝑇 ∫ 𝐼3 𝑓(y) 𝑑y ≤ 8 ∫ 0 |𝑓(xe(𝑡))| 𝑑𝑡 + 12 ⋅ 105 𝜎0(𝑓) 𝜂 𝑇1/4 (log2 𝑇 + 1)3 (log(log2 𝑇 + 1))2 + 12𝜎0(𝑓) , (2.15) where again 𝜎2 0(𝑓) = ∫ 𝐼3 (𝑓(y) − ∫ 𝐼3 𝑓(z) 𝑑z) 2 𝑑y . (c) In the case of the 𝑑-dimensional billiard with 𝑑 ≥ 4, we have the perfect analog of (2.15) where the factor of 𝑇1/4 in (2.15) is replaced by 𝑐0(𝑑)𝑇 1 2 − 1 2(𝑑−1) , and 𝑐0(𝑑) is a positive absolute constant that depends only on the dimension.
- 46. 32 | József Beck Let us return to Theorem 2.1: it was my first result proving superuniformity, i.e. su- peruniformity of curves. Comparing it to Theorem 1.3 we see two major weaknesses: (1) Theorem 2.1 works for a fixed 𝑇 instead of all 𝑇’s simultaneously, (2) the starting point is an unspecified “typical” point instead of a specified point such as the origin. The advantage of Theorem 2.1 is that the error term is substantially smaller: (log 𝑇) 1 2 +𝜀 instead of (log 𝑇) 3+𝜀 . Notice that the complexity of the test set 𝑆 does not appear in the explicit form of the error term – this is true for both Theorem 1.3 and Theorem 2.1. It is very surprising that, given an arbitrarily complicated measurable subset 𝑆 ⊂ [0, 1)2 (and measurable sets can be very, very complicated!), the curve-discrepancy |𝑇𝑆 − 𝑇 ⋅ area(𝑆)| is at most roughly square-root logarithmic for the majority of the billiard paths; see (2.14). Since square-root logarithmic is “almost” constant, the “ug- liness” (=complexity) of 𝑆 plays a negligible role in (2.14). Indeed, (2.14) is nearly sharp inthefollowingstrongsense. Evenif Theorem 2.1is restricted to thenicest possibletest sets – say, the family of axes-parallel subsquares – then (at least) constant size curve- discrepancy 𝑂(1) in (2.14) is still unavoidable, meaning that we cannot have 𝑜(1) in- stead; we explain it below. This shows that the complexity of the test set 𝐴 ⊂ [0, 1)2 in Theorem 2.1 is basically irrelevant: the curve-discrepancy is between roughly √log 𝑇 and 𝑂(1), i.e. it is almost the same independently of the complexity of the test set 𝑆 ⊂ [0, 1)2 . We now quickly prove the (almost trivial) fact that for the class of axes-parallel subsquares – as test sets – we must have (at least) constant curve-discrepancy 𝑂(1) in (2.14). Consider the two subsquares, 𝑆1 = [0, 1/3]2 and 𝑆2 = [2/3, 1]2 ; the distance between them is √2/3. Let x(𝑡)be an arbitrarycontinuous curve in the unit square; we always assume that the arc-length of every segment x(𝑡), 𝑇1 < 𝑡 < 𝑇2 is exactly 𝑇2 −𝑇1 (meaning: 𝑡 is the time and a point-mass moves along the curve with unit speed). For any real number 𝜏 > 0 write 𝑆𝑖(𝜏) = measure {𝑡 ∈ [0, 𝜏]: x(𝑡) ∈ 𝑆𝑖} , 𝑖 = 1, 2 , where𝑆𝑖, 𝑖 = 1, 2arethetwo subsquares mentioned above. Weshowthatthefollowing four curve-discrepancies: |𝑆𝑖(𝑇) − 𝑇 ⋅ area(𝑆𝑖)| , |𝑆𝑖(𝑇 + 𝑐) − (𝑇 + 𝑐) ⋅ area(𝑆𝑖)| , 𝑖 = 1, 2 , (2.16) where 𝑐 = √2/3 is the distance between the two given subsquares 𝑆1 and 𝑆2 (com- puted for the same curve!), cannot be all 𝑜(1). Indeed, the middle segment x(𝑡), 𝑇 < 𝑡 < 𝑇 + 𝑐 of the curve cannot visit both subsquares (because the arc-length is exactly the distance between 𝑆1 and 𝑆2); consequently, at least one of the four curve-discrep- ancies in (2.16) must be ≥ 1 2 𝑐 ⋅ area(𝑆𝑖) = 1 2 ⋅ √2 3 ⋅ 1 9 = √2 54 .
- 47. Strong uniformity | 33 This elementary argument shows that in Theorem 2.1 it is impossible to have curve- discrepancy 𝑜(1) in (2.14), not even for the simplest possible test sets. Next we give a slight improvement of Theorem 2.1: we recently succeeded to erase the iterated logarithmic factor log log 𝑇 in (2.14). We include it here, because we feel this latest result is the best possible. Again for later applications, we actually prove the general result extending the case of the characteristic function 𝜒𝑆 of a test set 𝑆 to an arbitrary square-integrable function 𝑓 ∈ 𝐿2 defined on the unit square. Theorem 2.3. Let 𝑓 ∈ 𝐿2 be an arbitrary real-valued Lebesgue square integrable func- tion on the unit square [0, 1)2 = 𝐼2 , and let 𝑇 > 100 be an arbitrarily large but fixed real number. Let x(𝑡) = (𝑥1(𝑡), 𝑥2(𝑡)), 0 ≤ 𝑡 ≤ 𝑇 be a general billiard path of length 𝑇 in the unit square (the starting point is not necessarily the origin). Let Bil𝑓(𝑇) = 𝑇 ∫ 0 𝑓(x(𝑡)) 𝑑 denote the integral of 𝑓 along the billiard path of length 𝑇 (Bil stands for billiard). We assume unit speed, so 𝑇 is also the time. Let 0 < 𝜂 < 1/2 be arbitrary. Then for the (1 − 𝜂)-part of all billiard paths of length 𝑇 in the square, the discrepancy discrepancy𝑓(𝑇) = Bil𝑓(𝑇) − 𝑇 ∫ 𝐼2 𝑓(z) 𝑑z is estimated from above by |discrepancy𝑓(𝑇)| < 30𝜎0(𝑓) 𝜂 √log 𝑇 , where, as usual, 𝜎2 0(𝑓) = ∫ 𝐼2 (𝑓(y) − ∫ 𝐼2 𝑓(z) 𝑑z) 2 𝑑y . The same result holds for the torus line instead of the billiard path. Remarks 2.4. In the special case where 𝑓 = 𝜒𝑆 is the characteristic function of an arbitrary measurable subset 𝑆 of the unit square, Bil𝑓(𝑇) denotes the time the billiard path spends in the given subset 𝑆 (denoted by 𝑇𝑆 in Theorem 2.1); also the “expecta- tion” 𝑇 ∫ 1 0 ∫ 1 0 𝑓(z) 𝑑z becomes 𝑇 ⋅ area(𝐴) (where area=2-dimensional Lebesgue mea- sure); and finally, the “variance” 𝜎2 0(𝑓) is simply area(𝑆)(1 − area(𝑆)). Here is a numerical illustration of Theorem 2.3. Physicists claim that the diameter of the observable universe is about 50 billion light years, which is about 1027 meters. Let
- 48. 34 | József Beck 𝑆 ⊂ [0, 1]2 be an arbitrary measurable test set in the unit square of side length one meter. We apply Theorem 2.3 with 𝑓 = 𝜒𝑆, 𝑇 = 1027 and 𝜂 = 1/5. Then for at least 80% of the initial conditions, if the point-billiard travels 𝑇 = 1027 meters (which is more than the diameter of the observable universe!), then the total distance traveled inside 𝑆 differs from the expected value 𝑇 ⋅ area(𝑆) by less than 30𝜎0(𝑓) 𝜂 √log 𝑇 ≤ 30 ⋅ 1/2 1/5 √log 1027 < 630 meters . Assume the point-billiard represents a gas molecule. A typical gas molecule moves with speed 103 m/s, and to cover the total distance 1027 meters, it takes 1024 seconds, which is a million times more than the estimated age of the universe. On the other hand, it takes less than one second for the gas molecule to cover 630 meters (=discrepancy). One second versus a million times the age of the universe is a strikingcontrast; itrepresents anastonishingprecision, especially thatthecomplexity of the given test set 𝑆 is irrelevant. Theorem 2.3 provides a continuous family of curves of length 𝑇 in a square – all billiard paths of length 𝑇 – that has strong discrepancy 𝑂(√log 𝑇). Here, strong means that we allow arbitrary measurable test sets, and want uniformity for the ma- jority of the curves. The analog problem in one less dimension – namely, the strong discrepancy of the most uniform continuous family of 𝑁-element point sets in the unit interval – exhibits a much worse quantitative behavior: we have √𝑁 instead of √log 𝑇, which is a super-exponential jump! The fact that discrepancy of size √𝑁 is inevitable follows from the following result, which is basically a discrete analog of Theorem 1.7. Theorem 2.5. For every integer 𝑁 ≥ 1 there exists another integer 𝑚 = 𝑚(𝑁) such that we can construct 𝑚 Lebesgue measurable subsets 𝑆1, . . . , 𝑆𝑚 of the unit interval [0, 1) with the following property: given any 𝑁-element subset X = {𝑥1, 𝑥2, . . . , 𝑥𝑁} in the unit interval [0, 1], for at least two thirds of the 𝑚 subsets 𝑆1, . . . , 𝑆𝑚, |X ∩ 𝑆𝑗| − 𝑁 ⋅ meas(𝑆𝑗) > √2 20 √𝑁 . Theorem 2.5 implies, via a standard average argument, that for continuous families of 𝑁-element point sets it is inevitable to have strong discrepancy of size √𝑁. Indeed, it follows from (basically) repeating the argument after Theorem 1.7. Assume that, for some 𝑁 > 1, there exists a continuous family of 𝑁-element point sets {X𝜔 : 𝜔 ∈ 𝛺} (2.17) on the unit interval [0, 1) such that there is a probability measure 𝜇 on the index-set 𝛺 (i.e. 𝜇(𝛺) = 1, so it is meaningful to talk about a (1−𝜀)-part), and the family of 𝑁-sets in (2.17) is strongly uniform in the following quantitative sense: Given any measurable
- 49. Strong uniformity | 35 subset 𝑆 ⊂ [0, 1), D𝑆(X𝜔) < √2 20 √𝑁 (2.18) holds for at least two-thirds of the 𝑁-sets X𝜔, 𝜔 ∈ 𝛺 in the sense of the 𝜇-measure. We show that this contradicts Theorem 2.5. Indeed, we apply (2.18) for the 𝑚 = 𝑚(𝑁) ≥ 2 sets 𝑆𝑗, 1 ≤ 𝑗 ≤ 𝑚 whose existence is guaranteed by Theorem 2.5. Thus for every 1 ≤ 𝑗 ≤ 𝑚 there exists a (measurable) subset 𝛺𝑗 of the index-set such that 𝜇(𝛺𝑗) ≥ 2/3, and D𝑆𝑗 (X𝜔) < √2 20 √𝑁 holds for all 𝜔 ∈ 𝛺𝑗. The fact 1 𝑚 𝑚 ∑ 𝑗=1 𝜇(𝛺𝑗) ≥ 2/3 implies that there must exist an index 𝜔0 ∈ 𝛺 which is contained by at least 2𝑚/3 sets 𝛺𝑗. In other words, there is an 𝑁-set X𝜔0 such that (2.18)) D𝑆𝑗 (X𝜔0 ) < √2 20 √𝑁 (2.19) holds for at least two-thirds of the 𝑚 sets 𝑆1, . . . , 𝑆𝑚. But (2.19) clearly contradicts The- orem 2.5, and this contradiction proves that an error of size 𝑜(√𝑁) is impossible. On the other hand, the family of arithmetic progressions 𝛽 + 𝑘𝛼, 0 ≤ 𝑘 < 𝑁 modulo 1 shows that the error √𝑁 is best possible, that is, Theorem 2.5 is sharp. This follows from the next two statements. Proposition 2.6. Let 𝑆 ⊂ [0, 1] be an arbitrary Lebesgue measurable subset of the unit interval, and let 𝑁 ≥ 1 be an arbitrarily large integer. Then for more than 75% of the pairs (𝛼, 𝛽) ∈ [0, 1]2 , ∑ 0≤𝑘≤𝑁−1: 𝑘𝛼+𝛽∈𝑆(mod 1) 1 − 𝑁 ⋅ meas(𝑆) ≤ √𝑁 , where meas stands for the 1-dimensional Lebesgue measure. Proposition 2.7. Let 𝑆𝑗 ⊂ [0, 1], 𝑗 = 1, 2, . . . , 𝑚 be an arbitrary finite sequence of Lebesgue measurable subsets of the unit interval, and let 𝑁 ≥ 1 be an arbitrarily large integer. Then there is a pair (𝛼0, 𝛽0) ∈ [0, 1]2 such that ∑ 0≤𝑘≤𝑁−1: 𝑘𝛼0+𝛽0∈𝑆𝑗(mod 1) 1 − 𝑁 ⋅ meas(𝑆𝑗) ≤ √𝑁 holds for more than 75% of the sets 𝑆𝑗, 𝑗 = 1, 2, . . . , 𝑚.
- 50. 36 | József Beck Of course, the term “75%” can be replaced with “90%” or “99.9%”, but then √𝑁 is multiplied by a constant factor larger than one. Proposition 2.6 and Proposition 2.7 are straightforward corollaries of the following simple result (we choose 𝑓 = 𝜒𝑆). Proposition 2.8. Suppose that 𝑓(𝑥) is a Lebesgue square-integrable function on the unit interval 0 ≤ 𝑥 < 1; for notational simplicity we extend 𝑓 over the entire real line ℝ periodically with period one. Then for any natural number 𝑛 ≥ 1, 1 ∫ 0 1 ∫ 0 ( 1 𝑁 𝑁−1 ∑ 𝑗=0 𝑓(𝑦 + 𝑗𝑥) − 1 ∫ 0 𝑓(𝑢) 𝑑𝑢) 2 𝑑𝑥 𝑑𝑦 = 𝜎2 0(𝑓) 𝑁 , (2.20) where 𝜎2 0(𝑓) = 1 ∫ 0 (𝑓(𝑧) − 1 ∫ 0 𝑓(𝑢) 𝑑𝑢) 2 𝑑𝑧 . The proof of Proposition 2.8 is almost trivial, and goes as follows. Notice that (2.20) is basically a “Pythagorean theorem”, so it suffices to prove orthogonality. What we have here is the stronger property of pairwise independence. Indeed, pairwise inde- pendence follows from the following: Simple Fact Let 0 ≤ 𝑘 < ℓ be arbitrary integers; then given any pair 0 ≤ 𝑧1 < 1, 0 ≤ 𝑧2 < 1 of real numbers, there exist(s) exactly ℓ − 𝑘 pair(s) (𝑥, 𝑦) of real numbers in 0 ≤ 𝑥 < 1, 0 ≤ 𝑦 < 1 such that 𝑘𝑥 + 𝑦 ≡ 𝑧1(mod 1) and ℓ𝑥 + 𝑦 ≡ 𝑧2(mod 1) . (2.21) Notice that ℓ − 𝑘 is exactly the determinant of the matrix of the linear equations in (2.21). To prove the Simple Fact, we just solve (2.21) for 𝑥 and 𝑦: taking the difference of the two equations, we have (ℓ − 𝑘)𝑥 ≡ 𝑧2 − 𝑧1(mod 1), which gives 𝑥 = 𝑧2 − 𝑧1 ℓ − 𝑘 + integer ℓ − 𝑘 , and there are exactly ℓ − 𝑘 numbers 𝑥 = 𝑥𝑗, 𝑗 = 1, 2, . . . , ℓ − 𝑘 of this type in the interval 0 ≤ 𝑥 < 1. For each one of these numbers 𝑥 = 𝑥𝑗, 𝑗 = 1, 2, . . . , ℓ − 𝑘 there is a uniquely determined 0 ≤ 𝑦 = 𝑦𝑗 < 1 satisfying 𝑘𝑥𝑗 + 𝑦𝑗 ≡ 𝑧1 (mod 1), completing the proof of Proposition 2.8.
- 51. Strong uniformity | 37 2.2 Super-duper uniformity of the 2-dimensional ray Next we make a “dimension one increase” in Theorem 2.3: we approximate the vol- ume of an arbitrarily complicated measurable test set 𝑆 inthe 3-dimensional unit cube with a 2-dimensional ray. In this case we can prove an even stronger result – see The- orem 2.9 below – as follows: (1) we can prove the absolute minimum error, constant error, instead of the square- root logarithmic error in Theorem 2.3, (2) it holds simultaneously for all time 1 < 𝑇 < ∞; and, finally, (3) the error remains less than the same constant for all initial positions, i.e. the initial position is irrelevant. We can say, therefore, that Theorem 2.9 below represents “super-duper uniformity”. The concept of a 2-dimensional ray is motivated by the spatial coherence of the laser. We define the torus version; the billiard version is reduced to the torus version via unfolding. A 2-dimensional ray, or simply 2−𝑑-ray, is a flat surface, a parallelogram in the 3-space (considered modulo 1) such that one of its vertices is fixed, which is called the starting point. More precisely, given two real numbers 𝑇1 > 0 and 𝑇2 > 0, two unit vectors b1 = (𝛽1,1, 𝛽2,1, 𝛽3,1) and b2 = (𝛽1,2, 𝛽2,2, 𝛽3,2), and a third vector s in the 3-space, let 𝛤(𝑇1, 𝑇2; b1, b2; s) denote the following torus-parallelogram: 𝛤(𝑇1, 𝑇2 > 0; b1, b2; s) = {s + x(𝑡1, 𝑡2)(mod 1): 0 ≤ 𝑡1 ≤ 𝑇1, 0 ≤ 𝑡2 ≤ 𝑇2} , (2.22) where x(𝑡1, 𝑡2) = (𝑥1(𝑡1, 𝑡2), 𝑥2(𝑡1, 𝑡2), 𝑥3(𝑡1, 𝑡2)) 𝑥1(𝑡1, 𝑡2) = 𝛽1,1𝑡1 + 𝛽1,2𝑡2 , 𝑥2(𝑡1, 𝑡2) = 𝛽2,1𝑡1 + 𝛽2,2𝑡2 and 𝑥3(𝑡1, 𝑡2) = 𝛽3,1𝑡1 + 𝛽3,2𝑡2 , |b1|2 = 𝛽2 1,1+𝛽2 2,1+𝛽2 3,1 = 1, |b2|2 = 𝛽2 1,2+𝛽2 2,2+𝛽2 3,2 = 1. Theparameters 𝑇1, 𝑇2, b1, b2, s uniquely determine the torus-parallelogram (2.22); we call 𝑇1, 𝑇2 the size parameters; we call the pair of unit vectors (b1, b2) the directions; we call s the starting point; and, finally, we call the torus-parallelogram (2.22) a 2 − 𝑑-ray. (We like to visualize (2.22) in the special case where 𝑇2 is a fixed small constant and 𝑇1 → ∞: it looks like a long-and-narrow flat laser beam.)
- 52. 38 | József Beck We choose a test set 𝑆: it is an arbitrary Lebesgue measurable subset of the unit torus [0, 1)3 with 3-dimensional Lebesgue measure 0 < vol(𝑆) < 1. Let 𝑆(𝑇1, 𝑇2; b1, b2; s) = 𝛤(𝑇1, 𝑇2; b1, b2; s) ∩ 𝑆 denote the intersection of the torus-parallelogram (2.22) with the given subset 𝑆; the area of the intersection equals area(𝑆(𝑇1, 𝑇2; b1, b2; s)) = |b1 × b2| ⋅ area {(𝑡1, 𝑡2) ∈ [0, 𝑇1] × [0, 𝑇2]: x(𝑡1, 𝑡2) ∈ 𝑆(mod 1)} , (2.23) where b1 × b2 denotes the usual cross product of 3-dimensional vectors. In view of (2.23) it is natural to call |b1 × b2|𝑇1𝑇2 ⋅ vol(𝑆) the expectation: in the case of perfect uniformity the actual intersection area of the 2 − 𝑑-ray area(𝑆(𝑇1, 𝑇2; b1, b2; s)) equals the expectation. Theorem 2.9. Let 𝑆 be an arbitrary measurable subset of the unit cube [0, 1)3 with 3- dimensional Lebesgue measure 0 < vol(𝑆) < 1. Let 0 < 𝜂 < 1/2 be arbitrary. Then for a (1 − 𝜂)-part of all directions (b1, b2) ∈ S2 × S2 , discrepancy = | area(𝑆(𝑇1, 𝑇2; b1, b2; s)) − |b1 × b2|𝑇1𝑇2 ⋅ vol(𝑆)| < 1000 log2 (1/𝜂) + 200 𝜂2 ⋅ √vol(𝑆)(1 − vol(𝑆)) , (2.24) for all values of the size parameters 𝑇1, 𝑇2 and for all starting points s ∈ ℝ3 . Remarks 2.10. (1) An interesting special case is where one of the size parameters is fixed, and the other one tends to infinity. For example, let 𝑇2 = 10−3 and 𝑇1 = 𝑇 → ∞. Then the 2 − 𝑑-ray in (2.22) represents a “long-and-narrow flat laser beam” traveling inside a cube, reflecting on the walls (this “billiard beam” is reduced to the “torus beam” via unfolding). Equation (2.24) describes the super- duper uniformity of the narrow flat laser beam: the error term remains uniformly bounded for all 0 < 𝑇 < ∞. (2) As usual, the “ugliness” (= complexity) of 𝑆 plays absolutely no role here; see (2.24). Given an arbitrarily complicated subset 𝑆 ⊂ [0, 1)3 , the typical surface-dis- crepancy (see (2.24)) is less than an absolute constant. (Note that I did not make a serious effort to find the best values of the constants in the last line of (2.24).) The constant upper bound is sharp: even if Theorem 2.9 is restricted to the simplest subsets, say, to the family of axes-parallel subcubes, then (at least) constant sur- face-discrepancy 𝑂(1) in (2.24) is still unavoidable – I explain it in (3) below. This means that the complexity of the test set 𝑆 ⊂ [0, 1)3 in Theorem 2.9 is completely irrelevant.
- 53. Strong uniformity | 39 (3) I outline the almost trivial fact why even for the narrow class of axes-parallel sub- cubes we must have (at least) constant surface-discrepancy 𝑂(1) in (2.24). The following argument is a straightforward adaptation of the argument in part (3) of the Remarks after Theorem 2.1. Consider the two subcubes 𝑆1 = [0, 1/3]3 and 𝑆2 = [2/3, 1]3 ; the distance between them is √3/3. We show that the following four surface-discrepancies cannot be all very small like 𝑜(1): let s = 0 and con- sider | area(𝑆𝑖(𝑇1, 𝑇2; b1, b2; 0)) − 𝑇1𝑇2|b1 × b2| ⋅ vol(𝑆𝑖) where 𝑖 = 1, 2 and (𝑇1, 𝑇2) runs through the pair (𝑇, 10−3 ) and (𝑇 + 𝑐, 10−3 ); here 𝑐 > 0 is a sufficiently small positive constant. Indeed, the difference set [𝑇, 𝑇 + 𝑐] × [0, 10−3 ] (2.25) is a “small” rectangle, and in terms of the torus parallelogram (2.22), the equation (𝑡1, 𝑡2) ∈ rectangle (2.25) gives a “small” torus parallelogram of diameter less than √3/3, i.e. the diameter is less than the distance between the two subcubes 𝑆1 and 𝑆2, if 𝑐 > 0 is a small enough absolute constant. Therefore, because of the small diameter, this “small” constant size torus parallelogram (see (2.25)) cannot intersect both 𝑆1 and 𝑆2, which yields a constant size surface-discrepancy. This proves that in Theorem 2.9 we cannot expect that the surface-discrepancy tends to zero as the product 𝑇1𝑇2 tends to infinity, not even for the simplest families of subsets, such as the family of all axes-parallel subcubes. (4) In Theorem 2.9 we test the uniformity of a 𝑘-dimensional ray with respect to a (measurable) subset 𝑆 ⊂ [0, 1]𝑑 , where 𝑘 = 2 and 𝑑 = 3 (Theorem 1.1 corresponds to the case 𝑘 = 1and 𝑑 = 2). Consider now the general case with arbitrary 1 ≤ 𝑘 < 𝑑: 𝑘-dimensional ray in the 𝑑-dimensional unit torus [0, 1)𝑑 . By a straightforward adaptation of our proof technique, one can show that the corresponding upper bound for the discrepancy is 𝑂(1) if 𝑘 > 𝑑/2, and it is 𝑇 1 2 − 𝑘 2(𝑑−𝑘) +𝜀 (2.26) if 𝑘 ≤ 𝑑/2, where 𝑇 denotes the total 𝑘-dimensional surface area (arc-length for 𝑘 = 1). Moreover, the exponent 1 2 − 𝑘 2(𝑑−𝑘) in (2.26) is sharp: it cannot be replaced by any smaller constant 1 2 − 𝑘 2(𝑑−𝑘) − 𝜀. What is going on here? To give a guiding intuition, let 𝑆 ⊂ [0, 1)𝑑 be an arbitrary Lebesgue measurable subset, and consider the Fourier series of the characteristic function 𝜒𝑆: 𝜒𝑆(u) = ∑ r∈ℤ𝑑 𝑎r𝑒2𝜋ir⋅u with 𝑎r = ∫ 𝑆 𝑒−2𝜋ir⋅y 𝑑y , where 𝑎0 = vol(𝑆)(𝑑-dimensional Lebesgue measure), and by Parseval’s formula, ∑ r∈ℤ𝑑0 |𝑎r|2 = vol(𝑆) − vol 2 (𝑆) . (2.27)
- 54. 40 | József Beck Assume we want to test the uniformity of a 𝑘-dimensional ray with respect to the given 𝑆 ⊂ [0, 1]𝑑 . The following sum plays a critical role: ∑ r∈ℤ𝑑0 |𝑎r| |r|𝑘 . (2.28) Using the Cauchy–Schwarz inequality in (2.28), we obtain ∑ r∈ℤ𝑑0 |𝑎r| |r|𝑘 ≤ ( ∑ r∈ℤ𝑑0 |𝑎r|2 ) 1/2 ( ∑ r∈ℤ𝑑0 |r|−2𝑘 ) 1/2 = √vol(𝑆)(1 − vol(𝑆)) ( ∑ r∈ℤ𝑑0 |r|−2𝑘 ) 1/2 , where we used (2.27). Simple lattice point counting shows that the infinite sum ∑ r∈ℤ𝑑0 |r|−2𝑘 (2.29) is convergent if 2𝑘 > 𝑑. This is when we have “super-duper uniformity” (like Theorem 2.9). If 2𝑘 = 𝑑, then (2.29) has a very slow, logarithmic divergence. This is when we still have superuniformity (like Theorems 1.1 and 2.3). (5) As we explained in (4) above, “super-duper uniformity” is very common in higher dimensions. On the other hand, there is no similar “super-duper uniformity” in the discrete case. We reduce the dimension by 2: the 2 − 𝑑-ray becomes a point sequence, and the unit cube is reduced to the unit interval. The discrete analog of a ray is the 𝑛𝛼 sequence 𝛼, 2𝛼, 3𝛼, . . . , 𝑛𝛼, . . . (mod 1). Consider the simplest possible test set in the unit interval: let 𝐼 = [𝑎, 𝑏) ⊂ [0, 1) be a subinterval. It is easy to see that for all 𝑁 ≥ 1, ∑ 1≤𝑛≤𝑁: 𝑛𝛼∈𝐼(mod 1) 1 − 𝑁(𝑏 − 𝑎) ≤ 1 holds for every subinterval 𝐼 of the unit torus that has length 𝛼 > 0. In general, sup 𝑁≥1 ∑ 1≤𝑛≤𝑁: 𝑛𝛼∈𝐼(mod 1) 1 − 𝑁(𝑏 − 𝑎) < ∞ (2.30) holds for every subinterval 𝐼 of the unit torus that has length 𝑘𝛼 (mod 1) for some integer 𝑘. A well-known theorem of Kesten [6] proves the converse of (2.30): if
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- 56. The Plunge Across the Ai same time they were attacked by General Inouye's field batteries in front. Once again the fierce and destructive character of the cannonade is revealed by the dispatches of the Russian commanders. Just as General Kashtalinsky, referring to the bombardment of April 30th, described it as "extraordinarily violent and prolonged," so General Sassulitch used similar terms in regard to this new bombardment. Before the day was over the Russian Commander had more opportunities of appreciating the "extraordinary" quality of the troops whose powers he, in common with more highly placed officers in the service of the Czar, had so fatally despised; but it seems clear from the use of the same phrase independently by the two generals that the artillery tactics of General Kuroki caused them more surprise than almost anything else in the whole of these surprising operations. It goes to prove that the Intelligence Department on the Russian side was not well equipped, for the possession by their enterprising foe of heavy guns so far north in Korea seems never to have been suspected by them. Supported by this tremendous cannonade, the infantry of the 12th Division pressed steadily forward. The survivors of the first line melted into the second line, which was advancing quickly behind, and careless of death, the gallant little Japs plunged into the waters of the Ai up to their breasts, and waded across the ford. Notwithstanding the raking fire, however, from General Kuroki's batteries, the Russians stuck to their posts like heroes, and the field guns of the 3rd Battery, assisted by a number of machine guns, ploughed up the ranks of the Mikado's troops, doing terrible execution. But the Japanese were in overwhelming force, and though men were falling on every hand, the main body pressed resistlessly forward, crossed the river, and took up a position on the right bank, at the base of the hills. Not a moment was wasted. As the column reached the shore, it diverged regiment by regiment to right and left, spreading out in wider formation for the task of scaling the heights. The evolution was executed with great speed, but with the precision and steadiness of parade; and if anything could be more impressive than the gallantry
- 57. Overwhelming Legions The Circling Ring of Fate Devastating Artillery Bombardment of the Japanese rank and file, it was the skill and coolness of their officers from General down to company commander. Though it was exposed to a withering fire at comparatively close quarters, the movements of the whole force were executed like those of a machine. It will be remembered that there are two fords over the Ai river, the one leading from a position near Yulchasan, on the left bank, to a position slightly north of Yukushan, on the right bank; the other opposite to Tiger Hill, and a little to the north of Makau. It was opposite to this latter ford that the bulk of General Kashtalinsky's force was stationed, and here in consequence, the greatest losses befell the Japanese. But while a fierce engagement was raging at Makau, the decisive movement was taking place on the extreme left of the Russians at Yushukau. The defence at that spot was entrusted to only one battalion of the 22nd Regiment of Sharpshooters, and it was impossible for such a small contingent, gallantly as it held its ground for a time, finally to withstand the overwhelming legions which were hurled against it. For slowly but steadily the Japanese lines encircled the hills with a ring of fate, creeping up the sides with infinite nimbleness and dexterity, pausing now to take cover and return the Russian fire, then up again and climbing from rock to rock with indomitable courage and resolution. On the other hand, General Kashtalinsky bravely fought on against his advancing foe. With the force at his command, it was obviously a desperate undertaking, and he had sent for reinforcements. But they came not, and for hours he had to do the best he could without them. The fact was, of course, that General Sassulitch himself was so busily engaged both on the right wing and at the centre that he could spare little assistance to his subordinate. For almost simultaneously with the advance of the 12th Division across the Ai the Imperial Guards under General Hasegawa had forced the passage of
- 58. Black Mass of Human Figures the stream on the left, at the foot of the slope which led up to the village of Chiu-lien-cheng, while the 2nd Division, led by General Nishi, crossed lower down and menaced the Russian right. Four batteries of howitzers had been ferried across the stream from the left bank of the Yalu to the Island of Cheun-song-do, and as the skirmishing line of both divisions moved forward in a fan-like formation these powerful pieces of ordnance opened a destructive fire upon the enemy. A sharp rattle of musketry was the first sign that the Russians were prepared to contest the passage of the river in this quarter, but their field artillery remained silent, and it turned out afterwards that all the guns which had survived the bombardment of the previous day had been removed to the rear, or to strengthen General Kashtalinsky's position. As it was, the rifle fire from the trenches was very galling, and the Japanese lost a great many men, but the devastating effects of General Kuroki's artillery bombardment were beyond anything that the Russians could produce in return. It was in one of these trenches on the ridge of the hills to the northeast of Chiu-lien-cheng that the greatest damage was wrought. As the Japanese infantry steadily advanced, General Sassulitch ordered forward a body of his supports from the immediate rear to occupy this trench. In order to obey this command they had to round a small spur of the hill and pass across the open. Their appearance against the sky-line provided a target which the Japanese gunners were not likely to neglect. Instantly a rain of shell and shrapnel was directed upon the black mass of human figures. Men were seen falling thick and fast under this withering fire; but still the Russians pressed on indomitably, and at the expense of great loss of life occupied the trench, whence they in turn poured a fierce rifle-fire upon the enemy below them. By this time, however, the Guards were swarming over the lower slopes of the hills around Chiu-lien-cheng, and General Hasegawa sent a strong force to the left of the Russian position to turn General Sassulitch's flank. At the same time General Nishi's men
- 59. The Blood-Red Banner were climbing steadily up the ridge further south, and were threatening the Russian right. It is interesting to note that the somewhat drab aspect of warfare which many of the operations in the South African war assumed, accustoming us to the idea that all picturesqueness had departed from modern combat, and that the ancient gauds and trappings so dear to the soldier's heart had been abandoned for ever, was entirely absent from this great battle in the Far East. The opposing forces were not separated from one another by illimitable distances of rolling veldt and brown hills. They were, on the contrary, so near as to recall the fighting in the Franco-German War, or the bloody combats around Plevna in the great struggle between Turkey and Russia nearly thirty years ago. And more remarkable still, the regimental colors which in our army are kept for ceremonial purposes in times of peace, and do not accompany the troops into the field, were carried by the Japanese in the front of the fighting line. Their presence must have assisted the fire of the enemy considerably; but there can be no doubt, on the other hand, of the inspiriting effect on the Mikado's men of seeing the blood-red banner of their race floating in the van and beckoning them forward to victory.
- 60. A LAST GALLANT STAND OF RUSSIAN GUNNERS. Steadily indeed, and without pause, those flaming banners advanced upon the doomed Russian position. The swing round of General Hasewaga's troops to the left of Chiu-lien-cheng decided the fate of General Sassulitch's centre, and after four hours' fighting the Japanese, climbing up the ridges like cats, charged into the Russian trenches. All the defenders who remained to contest the charge
- 61. Fight Desperately Against Fate General Sassulitch's Retreat were bayonetted or taken prisoners, but the main body of the 9th and 10th Siberian Regiments retreated stubbornly towards Hoh-mu- tang, contesting every inch of the ground. The heights, however, in this part of the field were won, and at 9 o'clock a great shout of "Banzai"—the Japanese form of "hurrah"—went up all along the line, as the banners of the Rising Sun were planted upon the ridge and waved proudly in the breeze. On their left the Russians under General Kashtalinsky were, as we have shown, making a more desperate resistance; but unable to obtain reinforcements in time, that gallant officer was compelled to retire before the advance of General Inouye's Division, which, by driving the battalion of the 22nd Regiment in rout before it at Yushukau, had completely crumpled up his flank. He therefore fell back slowly towards Hoh-mu-tang, fighting desperately against overwhelming odds opposed to him. It was not till noon, seven hours after the battle began, that reinforcements were at last sent to him. Then General Sassulitch ordered to his assistance the 11th Regiment, which all this time had been held in reserve well in the rear together with the 2nd Battery of the 6th Brigade of Field Artillery, under Lieutenant-Colonel Mouravsky. With this new force General Kashtalinsky set about the heavy task of covering the retreat of the 12th and the 22nd Regiments, or as much of them as was left, and also of checking the Japanese advance if possible until the 9th and 10th Regiments had made sure of their communications along the road to Feng-hwang-cheng. It was now that the fiercest and bloodiest fighting of the day took place, and that the Russians in particular suffered their heaviest losses. For no sooner had General Kuroki captured the whole ridge from Antung and Antushan in the south to Yukushan in the north than he ordered his force, strengthened by the reserves, to hasten at full speed along three lines in the direction of the Feng-whang-cheng road to cut off General Sassulitch's retreat.
- 62. The Japanese Chase The Last Gallant Stand A strong detachment from General Inouye's Division, therefore, crossed westwards to Tan-lang- fang; the Imperial Guards marched rapidly along the main road from Chiu-lien-cheng; and the 2nd Division spread out towards Antung and pursued the retiring 9th and 10th Regiments. It was the Guards Division and the 12th Division with whom General Kashtalinsky had to deal in this last brave stand. He ordered the 11th Regiment under his chief of staff to assume a commanding position in the rear, from which they could fire upon the enemy from two sides. Lieutenant-Colonel Mouravsky's battery he held in reserve; and then he ordered the wearied troops of the 12th Regiment, the 22nd Regiment, and the 3rd Battery of the 6th Brigade to retire under cover of the fire of the 11th. But before this manœuvre could be effected the fierce pursuit of the Japanese had gained its object. Both the Guards and the 12th Division reached the spot by 1 o'clock, and approaching from opposite sides, surrounded the hapless Russians. An enfilading fire made it impossible for the 3rd Battery to retire. Its horses were killed, and, therefore, Colonel Mouravsky, who assumed the command, ordered the gunners to take up a position where they stood and return the Japanese fire at close quarters. This they did with the greatest gallantry. They fought on steadily till not a man was left standing, their brave commander, Colonel Mouravsky, himself being among the last to fall. In the meanwhile, a company with machine guns had been ordered up to the assistance of the 3rd Battery. The officer in command, seeing the difficult situation of Colonel Mouravsky, took up a position, in the words of General Kashtalinsky's dispatch, "on his own initiative." He was no more fortunate than his superior officer. He, too, had entered the fatal ring of fire, and half his men and horses were shot down before he could render any effective service. An attempt to bring away his guns by hand and to take them under shelter of the hills under the terrible cross fire to which he was exposed, was no more successful, and the guns ultimately fell into the hands of the enemy. The case being evidently hopeless, the 2nd Battery, which had been
- 63. Rifle Fire and Cold Steel brought up as a reinforcement to the 11th Regiment, was ordered back to rejoin the reserve by another road, but half its horses, too, were killed, and, finding it impossible to ascend the slopes without them, the officer in command brought his guns back to their original position, and there bravely, but unavailingly, received the Japanese attack. Now ensued a fierce and bloody hand-to-hand combat, in which the utmost heroism was displayed on both sides. Closer and closer pressed the Japanese till the opposing forces were almost looking into one another's eyes, and rifle-fire was abandoned for cold steel. Again and again the Japanese desperately dashed themselves upon the serried ranks opposed to them, and again and again, in spite of the fearful execution wrought by each charge, they were hurled back. But bayonet charge followed bayonet charge, and at last the devoted band of Russians could hold out no more. In some quarters of the field the white flag was hoisted and numbers of men surrendered. But the main body, shattered as it was and a mere shadow of its former strength, fought its way through. A broken remnant of the 12th Regiment cut its way through and carried off the colors in safety, torn and riddled indeed, but not disgraced. The same fate befell the 11th Regiment, a small body of which, after several hours' fighting, forced a passage out of the melee and retreated to Hoh-mu-tang with its colors preserved. But the losses of this regiment were enormous. Colonel Laming, the Colonel Commandant, Lieutenant-Colonels Dometti and Raievsky, and forty subordinate officers were left dead upon the field, and 5,000 non- commissioned officers and men were killed or wounded. More than 30 officers and 400 men surrendered. The casualties sustained by the Japanese were nearly 1,000 killed and wounded.
- 64. Russian Demoralization T CHAPTER VII. Russian Demoralization—On the Heels of the Enemy—Remarkable Japanese Strategy—The Paper Army—The Thin Black Line of Reinforcements—Position of the Russian Army—Kuropatkin Tied to his Railway—The Second Scheme of Attack—A Model of Organization— Perfect Secrecy of Plans—Cutting off Port Arthur—Alexeieff's Command of Language—And the Sober Truth—Third Blocking Attempt—Lurid Flashing of Searchlights—On the Bones of their Predecessors—Half the Passage Blocked—Honored but Unarmed—Russian Acknowledgements—Terrific Casualties—Togo for Liao-tung—The Japanese Landings—Escape of Alexeieff—Port Arthur Isolated. he signal victory of the despised Japanese at the Yalu River filled official circles in St. Petersburg with the liveliest dismay and shook that determined optimism which had survived even the unexampled series of naval disasters sustained by the power of the Czar in the Far East. There seems never to have been the least doubt among the Grand Dukes and the Bureaucrats by whom the Emperor was surrounded that whatever fate might befall the fleet, the "yellow monkeys," as they elegantly called their foes, would fly headlong before the onslaught of the Russian soldiery, accustomed as it was to victory on many a bloody field in Europe. The fatuity of this overweening confidence now stood revealed, and it was at last tardily recognized that as stern a task awaited the Russian forces on land as at sea. But St. Petersburg officialdom, wounded in pride and shaken in nerve as it was, still preserved a bold front to the world, and excuses for the disaster that had befallen the Russian arms were as prolific as ever. The army at the disposal of General Sassulitch, it was explained, was but a small one; that commander had blundered, and by giving battle to an
- 65. overwhelmingly superior force, had disobeyed or misunderstood the orders of General Kuropatkin; and in any case, although severe losses were admitted, the main body had retreated in good order to Feng-hwang-cheng, and the morale of the troops was unshaken. The plea that General Sassulitch was solely responsible for the defeat which had befallen the Muscovite arms, and that he had failed to follow the instructions of his superior, has already been dealt with, and its extreme improbability has been demonstrated, though, even if it were accurate, it would throw a very unflattering light upon the powers of Russian leadership in the higher commands. It was soon, however, to be shown that the suggestion that the army of the Yalu had retired in good order and with unshaken morale was equally devoid of truth. As a matter of fact, the fierce pursuit of the Japanese and the heavy losses which they inflicted upon the retreating Russians at Hoh-mu-tang and elsewhere on the road to Feng-hwang-cheng reduced the defeat to an utter rout, and it became impossible for Sassulitch to make a stand at the latter point, naturally strong as it was and admirably calculated to resist an attack.
- 66. AFTER THREE MONTHS. The war began with the night attack on Port Arthur on February 8, but it was not until two months later that the Japanese appeared on the south-eastern border of Manchuria. On April 4 they occupied Wiju, on the 21st troops began to land at Tatungkau, and on May 1 took place the first great battle of the campaign, when the Japanese forced the passage of the Yalu, and drove the Russians back upon Feng-wang-cheng. On May 6 the latter place was occupied without resistance.
- 67. On the Heels of the Enemy The shaded portion shows the Japanese advance. After a day or two spent in recuperating his tired troops, whose tremendous exertions during the previous week must have tested their powers of endurance to the utmost, and also in bringing his heavy guns and supply train across the river from Wiju, in preparation for the march General Kuroki began a forward movement into Manchuria with his whole army. The cavalry led the advance, operating over a wide area of country and sweeping the scattered units of the Russians before it. Some sharp skirmishes took place at Erh-tai-tsu and San-tai-tsu, but no real difficulty was interposed in the way of the victorious Japanese, who drove the enemy in flight before them. On May 6th the foremost cavalry vedettes reached Feng-hwang-cheng, and instead of finding the strongly held entrenchments which the Russian press was even then busily assuring a sceptical Europe would prevent any further advance on the part of the presumptuous foe, they discovered that the troops of General Sassulitch had been withdrawn, and they entered the deserted town without having to fire a shot. The leading columns of the infantry, following quickly behind, marched in and took possession on the same day. Before his hurried departure General Sassulitch had ordered the magazine to be blown up, but large quantities of hospital and other stores fell into the hands of the Japanese. General Kuroki's main body was not far in the rear, and the position of the whole army was soon securely established at this important point. Feng-hwang-cheng is situated at a mountain pass on the Liao-yang road, at a distance of about 25 miles from the Yalu. As already stated, it possesses great strategical importance. It is the centre at which the roads meet, coming from Liao-yang, Haicheng, and Kaiping, places which are situated at about equal distances from one another along the Manchurian railway from north to south, and it therefore constitutes a point d'appui from which a force could be thrown against any of them, while it is itself a position of great strength. General Kuroki immediately began to entrench himself strongly at this spot and to consolidate his forces, while he waited
- 68. Remarkable Japanese Strategy The Paper Army for the highly important developments which were now to take place in other quarters of the theatre of war. IN THE RUSSIAN TRENCHES. A wide view of the position of affairs as they now stood over the entire field of operations is necessary at this point in order to make clear the remarkable events that followed, and to throw into full relief the extraordinary qualities of the Japanese strategy—a strategy conceived after the most patient study of all the conditions of the problems and worked out in practice with almost machine-like regularity and precision. When General Kuropatkin arrived at Mukden at the end of March and took over the command from General Linevitch, he had on paper an army of over 250,000 men. It was made up as follows: 223,000 infantry; 21,764
- 69. cavalry; 4,000 engineers; and artillery consisting of 496 field guns, 30 horse artillery guns, and 24 machine guns. This large force was organized in four Army Corps, each with divisions of infantry and its quota of artillery and cavalry; while there were also two independent divisions of Cossacks, four brigades of Frontier Guards, railway troops, fortress artillery and a number of small units not allotted. The First Army Corps was under the command of General Baron Stackelberg, the Second under General Sassulitch, the Third under General Stoessel, and the Fourth under General Zarubaieff. It was an imposing force, this army of Manchuria, calculated to strike terror into the hearts of an Oriental enemy, but unfortunately for the Russians it lacked one thing, and that was reality. The actual position of affairs was indeed very different. To begin with, the greater part of the troops were not near the front at all when the Commander-in- Chief appeared upon the scene to direct operations, but were being pushed along the Siberian Railway with a feverish haste which at the same time did not denote proportionate speed. When they did arrive they arrived in detached fragments, and the desperate necessities of the case did not admit of adherence to the paper arrangements. For instance, the 7th and 8th Divisions, which should have formed part of the Second Army Corps under General Sassulitch, were, as a matter of fact, sent to assist in garrisoning Port Arthur and Vladivostock. Port Arthur, it will be remembered, was by this time under the command of General Stoessel, who was therefore unable to direct the operations of the Third Army Corps, which properly should have been entrusted to him. On the other hand, the 3rd East Siberian Rifle Division, which belonged to that Corps, and the 6th East Siberian Rifle Division, which should have been attached to the First Army Corps, were sent to the Yalu, where, as we have already seen, they took part in the ill-fated conflict of the 1st of May. It will be observed from these shifts—only a few of the most noticeable out of many—that the Army Corps system of the Manchurian Army had completely broken down, and that the ideal of a coherent fighting force, with officers and men trained together in peace under the conditions to which they would be subjected in war, had not been attained in the slightest degree. The lack of organization which
- 70. The Thin Black Line of Reinforcements prevailed in the distribution of the larger commands was equally manifest in the mobilization of the units of which they were composed. Regiments were not complete; hastily-formed levies had to be added to bring them up to their nominal strength; and the ranks of the officers had to be filled up in many cases with volunteers from regiments in other parts of the Empire. The result was a composite force very different indeed in fighting power from the splendid machine which the Mikado's strategists had been carefully perfecting in time of peace in readiness for the struggle which they had so long foreseen. In bringing even this haphazard collection of unco- ordinated units to the front in Manchuria, the greatest difficulties had been experienced. All that European observers had predicted about the working capacity of a railway like the Trans-Siberian for the conveyance of a huge army for thousands of miles came true to the letter. Prince Khilkoff, the Director-General of Russian Railways, undoubtedly did wonders, and the tremendous efforts which he and his staff put forth, especially in surmounting the great natural obstacle presented by Lake Baikal, were worthy of all praise. But to carry an army of 250,000 men, with all its necessary supplies and munitions of war, into Manchuria in the time required for the purpose of striking an effective blow at an enemy like the Japanese was a task beyond the powers of any railway staff in the world. The rickety single line, with infrequent sidings, which stretches across the steppes of Siberia from Harbin to the Urals was quite inadequate for such a feat of transport. By the middle of May, therefore, the position in which General Kuropatkin found himself—a position partly created by himself, as Minister of War, and partly created for him by the ineptitude of others—was widely different from that which the easy and thoughtless optimists in St. Petersburg had anticipated when the war broke out. The Fourth Army Corps was not across Lake Baikal; 30,000 or 40,000 men were shut up in the fortresses of Port Arthur and Vladivostock, and were not only useless for field operations, but were themselves liable to siege and capture; and,
- 71. Position of the Russian Army Kuropatkin Tied to His Railway allowing the highest possible estimate, the Russian Commander-in- Chief had at his disposal for assuming the offensive in Manchuria no more than 100,000 men with 260 guns. With this army he was holding the railway line from Mukden to Port Arthur, a distance of about 230 miles. His headquarters were at Liao-yang, and he held Haicheng and Kaiping in force, while a detachment was thrown out to the south-west and occupied Niuchwang. In the extreme south Port Arthur, though closely blockaded from the sea by the watchful Togo, was as yet open to communication by land, and no attempt had hitherto been made by the Japanese to secure a footing on the Liao-tung Peninsula. On the east of the Liao-yang—Kaiping line the Russian troops occupied three important passes, namely, Ta- ling, about 50 miles distant, in a northeasterly direction, from Liao- yang; the Motien-ling, about 25 miles away on the main road to Feng-hwang-cheng; and Fen-chu-ling, half way on the road from Tashihchao to Siuyen. Tashihchao is on the railway midway between Haicheng and Kaiping. The Motien-ling Pass was the scene of a sanguinary combat between the Chinese and the Japanese in the war of 1894, and on that occasion the Mikado's forces had the greatest trouble in capturing it. Besides holding these passes General Kuropatkin had pushed forward his Cossack patrols to scour the country as far as Feng-hwang-cheng, and constant small encounters took place between them and General Kuroki's outposts during the ensuing six weeks. It is clear from this brief statement of the Russian position that the Japanese, always provided that they could retain the command of the sea, were placed at a great strategical advantage compared with their enemy. Holding their First Army poised at Feng-hwang-cheng, they could throw their Second and Third Armies upon the coast at any point that suited them best for the purpose of making a great combined movement. On the other hand, Kuropatkin was practically tied to the railway, and, with the inadequate force at his disposal, could not advance against Kuroki to destroy him in detail before the arrival of
- 72. The Second Scheme of Attack A Model of Organization fresh armies from Japan. He was liable to attack at any point, and it was the peculiar difficulty of his situation that he could not tell which point would be selected. As a matter of fact, when the blow fell, as it soon did with crushing effect, he was powerless to prevent it. The chapter of strategy which now opens is a fascinating one to any student of war, and fortunately its main features can be readily appreciated also by any layman who makes an intelligent study of a map of Manchuria and the Liao-tung Peninsula. The prime object of the Japanese plainly was to cut General Kuropatkin's extended line of communications, isolate Port Arthur, and then attempt to envelope his main force by advancing simultaneously from the south, the east, and the northeast. It was consequently necessary, as a preliminary, to establish the First Army securely in Manchuria, it being clear that with this menace on his left flank, General Kuropatkin would not be able to detach many troops to the south to prevent the investment of Port Arthur. Everything, therefore, depended on the fortune that would attend the advance of General Kuroki across the Yalu, and the Moltkes at Tokio, after a patient study of all the conditions of an intricate problem, had thought out two great alternative schemes to meet the eventuality either of victory or defeat. In case of General Kuroki's finding the task of crossing the Yalu unaided to be an insuperable one, the Second Army, under General Oku, was to be landed at Takushan, a port on the coast some miles to the west of the mouth of the river, and thence to strike a blow at General Sassulitch's right flank. On the other hand, if Kuroki met with success, Oku's army was to be landed at a point on the Liao-tung Peninsula to cut Kuropatkin's communications and invest Port Arthur. As we have seen, General Kuroki's signal triumph at the Yalu River rendered the first alternative unnecessary, and opened the way for the more decisive and dramatic stroke involved in the second scheme. But before anything could be done to land the Second Army, either at Takushan or on the Liao- tung Peninsula, it was imperatively necessary to
- 73. Perfect Secrecy of Plans disarm the Russian Fleet at Port Arthur, and prevent even the remotest possibility of its interfering with the operations. Here, as always, the two services, the army and the navy, had to work in close correspondence and interdependence. From the beginning of the war these separate branches of the Japanese forces had fitted into one another like parts of the same piece of machinery, the whole directed by one uniform purpose and striving towards one great common end. The joint schemes of the naval and military strategists at Tokio will ever provide an invaluable object-lesson to all students of the art of war; and it may be predicted that they will prove of valuable assistance to the strategists of our own army and navy. One of the most remarkable features of the war has been the certainty and precision with which the Japanese have worked out their complex plans; it is no less remarkable, and affords a further striking evidence of their efficiency, that they felt able, absolutely, to count upon that certainty and precision, and to make arrangements long beforehand, which with a less carefully organized scheme and less trustworthy commanders to carry it out would have been foolhardy, or at least wasteful. Failure in any real sense does not seem to have entered into their calculations. One portion of the plan, indeed, might miscarry, but, as we have seen, partial failure had been provided against, and a rapid modification of strategy to meet the case would have been possible. It was, in fact, one of the most interesting examples of the application of brains to war that have ever been seen in the history of the world. In the action and inter-action, then, of this great double machine, the army had done all that it was possible for it to do for the moment; and once again it came round to the turn of the navy to make the next decisive move. Upon the success of this move may be said to have depended the whole success of the after operations, but, calculating with absolute confidence upon the skill of Admiral Togo, the Mikado's strategists had already put the Second Army into a state of complete preparation, and had even ordered it to be conveyed to a place from which it could be transferred to the front at any quarter at a
- 74. Cutting off Port Arthur moment's notice. Arrangements for its embarkation were begun as soon as General Kuroki reached Wiju with the First Army in the early days of April. When that commander was able to report that his dispositions for the attack upon the Russian entrenchments on the right bank of the Yalu were well advanced, the process of embarking General Oku's troops was started at once. Not a hint was allowed to escape as to their destination; even if the press correspondents, chafing under their enforced inaction at Tokio, had learnt the name, the censor would not have let it pass to the outer world; but, as a matter of fact, it is safe to say that the secret was safely locked in the breasts of half a dozen men. By April 22nd the whole army with its transports, commissariat, ammunition train, and hospital corps, had been put on board ship, and said farewell to the shores of Japan, vanishing, for all the world could tell, into the inane. For more than a fortnight nothing further was heard of it No one could report its landing anywhere, no one could say what it was doing, and day by day the mystery grew more mysterious. Only on May 7th was the veil lifted, when this great army fell upon the coast of Liao-tung as if from the heavens, and proceeded to the investment of Port Arthur. The truth was that during this fortnight it had been lying perdu on some small islands close to the west coast of Korea, called the Sir James Hall group, and distant 160 miles in a southeastern direction from the shores of Liao-tung. Here, briefly stated, is the manner in which the scheme worked out. On May 1st General Kuroki triumphantly crossed the Yalu and stormed the heights above Chiu-lien-cheng. On May 2nd Admiral Togo descended once more upon Port Arthur, and blocked the harbor completely by sinking eight steamers at the entrance to the channel. On the afternoon of May 3rd, having made sure of the thoroughness of the work, he set off at full speed for the Sir James Hall Islands, reaching his destination by early morning on the 4th. Everything there was in readiness for the expedition, and within a few hours the whole of the transports, escorted by the fleet, set sail for the east coast of Liao-tung. At dawn the next day they reached the point on the
- 75. Alexeieff's Command of Language peninsula which had been selected for the landing—Yentoa Bay—and in a few short hours a considerable portion of the force had been disembarked, the resistance offered by a small detachment of Cossacks, the only force possessed by the Russians in the neighborhood, being entirely negligible. On the 6th the railway line was severed, and in a few days more the Japanese were sitting securely astride of the peninsula, and Port Arthur was cut off from the world. The scheme had been carried out like the combinations of a skilful chess player, or like the successive steps of a mathematical problem. A DESPERATE ENCOUNTER AT PORT ARTHUR. It is necessary now to follow the development of these operations more in detail. The first that falls to be described is the successful attempt, the third of the series, to block the entrance to the harbor of
- 76. Port Arthur. But before giving the real version of this thrilling enterprise it may be interesting to quote the report sent to the Grand Admiral unconquerable Alexeieff, whose optimism rose superior to every disaster and the alchemy of whose dispatches could still transmute defeat into signal victory. Here is the message, so soothing to the nerves of his fellow-countrymen, in which he announced the event that enabled the Japanese to land troops at any point they desired up their enemy's coasts:— "I respectfully report to your Highness that a fresh attack made by the enemy last night with the object of obstructing the entrance to the port was successfully repelled. "At 1 o'clock in the morning five torpedo-boats were perceived near the coast from the eastern batteries. Under the fire of our batteries and warships they retreated southward. "At 1.45 the first fireship, escorted by several torpedo-boats, came in sight. We opened fire upon it from our batteries and warships. Three-quarters of an hour afterwards our searchlights revealed a number of fireships making for the entrance to the harbor from the east and southeast. The Otvajni, the Giliak, the Gremiashtchi, and the batteries on the shore repulsed each Japanese ship by a well- directed fire. "Altogether eight ships were sunk by our vigorous cannonade, by Whitehead torpedoes launched from our torpedo-boats, and by the explosion of several submarine mines. "Further, according to the reports of the officers commanding the batteries and the warship Giliak, two Japanese torpedo-boats were destroyed. "After 4 a. m., the batteries and gunboats ceased fire, subsequently firing only at intervals on the enemy's torpedo-boats, which were visible on the horizon. "All the fireships carried quick-firing guns, with which the enemy maintained a constant fire.
- 77. And the Sober Truth Third Blocking Attempt "Up to the present thirty men, including two mortally wounded officers who sought refuge in the launches, or were rescued from the fireships by us, have been picked up. The inspection of the roadstead and the work of saving drowning men are hindered by the heavy sea which is running. "We suffered no casualties with the exception of a seaman belonging to the torpedo-boat destroyer Boevoi." No one reading this remarkable account could imagine that it described an operation which ultimately sealed the doom of Port Arthur. For a more sober but a more accurate narrative we must turn to the dispatches of Admiral Togo. On May 2nd, as already recounted, the Japanese Naval Commander-in-Chief received the news of the successful crossing of the Yalu. His plans were already laid and his preparations were complete. Eight merchant steamers this time had been secured for the service, and upwards of 20,000 men volunteered for the glorious duty of manning them and dying for their country. Of these, 159 were ultimately selected. The names of the steamers were the Mikawa, Sakura, Totomi, Yedo, Otaru, Sagami, Aikoku, and Asagawo. The vessels ordered to escort the doomed hulks were the gunboats Akagi and Chokai, the 2nd, 3rd, 4th, and 5th destroyer flotillas, and the 9th, 10th, and 14th torpedo- boat flotillas. The whole force, which was under the command of Commander Hayashi, started for its destination on the night of May 2nd. It is a melancholy circumstance, typical of the sombre, but ofttimes splendid, tragedy of war, that of this third and most successful attempt to block the harbor the narrative is necessarily the most fragmentary and obscure, owing to the loss of life which it entailed. On the two previous occasions, reckless as was the gallantry of the Japanese and enormous as were the risks they ran, the casualties were surprisingly small, and the majority of the men engaged were able to return to their ships and tell the story of their enterprise. On this
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