![University Printing House, Cambridge CB2 8BS, United Kingdom
Cambridge University Press is part of the University of Cambridge.
It furthers the University’s mission by disseminating knowledge in the pursuit of
education, learning, and research at the highest international levels of excellence.
www.cambridge.org
Information on this title: www.cambridge.org/9781107552371
c
Cambridge University Press 2016
This publication is in copyright. Subject to statutory exception
and to the provisions of relevant collective licensing agreements,
no reproduction of any part may take place without the written
permission of Cambridge University Press.
First published 2016
Printed in the United Kingdom by Clays, St Ives plc
A catalogue record for this publication is available from the British Library.
Library of Congress Cataloging-in-Publication Data
Names: Durham Easter School (2014 : University of Durham) | Badziahin,
Dmitry (Dmitry A.), editor. | Gorodnik, Alexander,
1975– editor. | Peyerimhoff, Norbert, 1964– editor.
Title: Dynamics and analytic number theory : proceedings of the Durham Easter
School 2014 / edited by Dzmitry Badziahin, University of Durham, Alexander
Gorodnik, University of Bristol, Norbert Peyerimhoff, University of Durham.
Description: Cambridge : Cambridge University Press, [2016] |
Series: London Mathematical Society lecture note series ; 437 |
Includes bibliographical references and index.
Identifiers: LCCN 2016044609 | ISBN 9781107552371 (alk. paper)
Subjects: LCSH: Number theory – Congresses. | Dynamics – Congresses.
Classification: LCC QA241 .D87 2014 | DDC 512.7/3–dc23
LC record available at https://lccn.loc.gov/2016044609
ISBN 978-1-107-55237-1 Paperback
Cambridge University Press has no responsibility for the persistence or accuracy of
URLs for external or third-party Internet Web sites referred to in this publication
and does not guarantee that any content on such Web sites is, or will remain,
accurate or appropriate.](https://image.slidesharecdn.com/90206-250313212418-feee36d4/85/Dynamics-and-Analytic-Number-Theory-1st-Edition-Dzmitry-Badziahin-11-320.jpg)
![xii Preface
Chapter 3 is devoted to the study of exponential sums. Given a real
polynomial P(x) = ak xk + · · · + a1x + a0, the Weyl sums are defined as
WN :=
N−1
n=0
e2πi P(n)
.
The study of Weyl sums has a long history that goes back to foundational
works of Hardy, Littlewood, and Weyl. When the coefficients of the polyno-
mial P(X) satisfy a suitable irrationality condition, then it is known that for
some w ∈ (0, 1),
WN = O(N1−w
) as N → ∞,
and improving the value of the exponent in this estimate is a topic of current
research. This problem has been approached recently by several very different
methods. The method of Wooley is based on refinements of the Vinogradov
mean value theorem and a new idea of efficient congruencing, and the method
of Flaminio and Forni involves the investigation of asymptotic properties of
flows on nilmanifolds using renormalisation techniques. It is quite remarkable
that the exponents w obtained by the Flaminio–Forni approach, which is deter-
mined by optimal scaling of invariant distributions, essentially coincide with
the exponents derived by Wooley using his method of efficient congruencing.
As discussed in Chapter 3, flows on nilmanifolds provide a very convenient
tool for investigating the distribution of polynomial sequences modulo one and
modelling Weyl sums. We illustrate this by a simple example. Let
N :=
⎧
⎨
⎩
[p, q, r] :=
⎛
⎝
1 p r
0 1 q
0 0 1
⎞
⎠ : p, q,r ∈ R
⎫
⎬
⎭
denote the three-dimensional Heisenberg group, and be the subgroup con-
sisting of matrices with integral entries. Then the factor space M := N
provides the simplest example of a nilmanifold. Given an upper triangular
nilpotent matrix X = (xi j ), the flow generated by X is defined by
φX
t (m) = m exp(t X) with m ∈ M.
More explicitly, exp(t X) = [x12t, x23t, x13t + x12x23t2/2]. The space M con-
tains a two-dimensional subtorus T defined by the condition q = 0. If we
take x23 = 1, then the intersection of the orbit φX
t (e) with this torus gives
the sequence of points [x12n, 0, x13n + x12n2/2] with n ∈ N. Hence, choos-
ing suitable matrices X, the flows φX
t can be used to model values of general
quadratic polynomials P modulo one. Moreover, this relation can be made](https://image.slidesharecdn.com/90206-250313212418-feee36d4/85/Dynamics-and-Analytic-Number-Theory-1st-Edition-Dzmitry-Badziahin-19-320.jpg)
![Preface xiii
much more precise. In particular, with a suitable choice of a test function F on
M and m ∈ M,
N−1
n=0
e2πi P(n)
=
N
0
F(φX
t (m)) dt + O(1).
This demonstrates that quadratic Weyl sums are intimately related to averages
of one-parameter flows on the Heisenberg manifold. A more elaborate con-
struction discussed in detail in Chapter 3 shows that general Weyl sums can
be approximated by integrals along orbits on higher-dimensional nilmanifolds.
Chapter 3 discusses asymptotic behaviour of orbits averages on nilmanifolds
and related estimates for Weyl sums.
Dynamical systems techniques also provide powerful tools to analyse com-
binatorial structures of large subsets of integers and of more general groups.
This active research field fusing ideas from Ramsey Theory, Additive Combi-
natorics, and Ergodic Theory is surveyed in Chapter 4. We say that a subset
E ⊂ Z has positive upper density if
d̄(E) := lim sup
N−M→∞
|E ∩ [M, N]|
N − M
0.
Surprisingly, this soft analytic condition on the set E has profound combina-
torial consequences, one of the most remarkable of which is the Szemerédi
theorem. It states that every subset of positive density contains arbitrarily long
arithmetic progressions: namely, configurations of the form a, a + n, . . . , a +
(k − 1)n with arbitrary large k. It should be noted that the existence of three-
term arithmetic progressions had previously been established by Roth using
a variant of the circle method, but the case of general progressions required
substantial new ideas. Shortly after Szemerédi’s work appeared, Furstenberg
discovered a very different ingenious approach to this problem that used
ergodic-theoretic techniques. He realised that the Szemerédi theorem is equiv-
alent to a new ergodic-theoretic phenomenon called multiple recurrence. This
unexpected connection is summarised by the Furstenberg correspondence prin-
ciple which shows that, given a subset E ⊂ Z, one can construct a probability
space (X, μ), a measure-preserving transformation T : X → X, and a
measurable subset A ⊂ X such that μ(A) = d̄(E) and
d̄(E ∩(E −n)∩· · ·∩(E −(k −1)n)) ≥ μ(A ∩ T −n
(A)∩· · ·∩ T−(k−1)n
(A)).
This allows the proof of Szemerédi’s theorem to be reduced to establishing the
multiple recurrence property, which shows that if μ(A) 0 and k ≥ 1, then
there exists n ≥ 1 such that
μ(A ∩ T −n
A ∩ · · · ∩ T −(k−1)n
A) 0.](https://image.slidesharecdn.com/90206-250313212418-feee36d4/85/Dynamics-and-Analytic-Number-Theory-1st-Edition-Dzmitry-Badziahin-20-320.jpg)
![Preface xv
and a subset of the space Xd+1 defined as
Xd+1(ε) := { : ∩ [−ε, ε]d+1
= {0}}.
Let gQ := diag(Q−d, Q, . . . , Q). If we establish that the orbit gQ visits the
subset Xd+1(ε), then this will imply that the systems of inequalities
|q| ≤ εQd
and qv − p∞ ≤
ε
Q
have a non-trivial integral solution (q, p) ∈ Z×Zd. When ε ≥ 1, the existence
of solutions is a consequence of the classical Dirichlet theorem, but for ε 1,
this is a delicate property which was studied by Davenport and Schmidt. Vec-
tors for which the above system of inequalities has a non-trivial solution for
some ε ∈ (0, 1) and all sufficiently large Q are called Dirichlet-improvable.
Chapter 5 explains how to study this property using dynamical systems tools
such as the theory of unipotent flow. This approach proved to be very success-
ful. In particular, it was used by Shah to solve the problem posed by Davenport
and Schmidt in the 60s. He proved that if φ : (0, 1) → Rd is an analytic curve
whose image is not contained in a proper affine subspace, then the vector φ(t)
is not Dirichlet-improvable for almost all t. Chapter 5 explains Shah’s proof of
this result.
Chapter 5 also discusses how dynamical systems techniques can be used
to derive asymptotic counting results. Although this approach is applicable in
great generality, its essence can be illustrated by a simple example: counting
points in lattice orbits on the hyperbolic upper half-plane H. We recall that the
group G = PSL2(R) acts on H by isometries. Given = PSL2(Z) (or, more
generally, a discrete subgroup of G with finite covolume), we consider the
orbit · i in H. We will be interested in asymptotics of the counting function
N(R) := |{γ · i : dH(γ · i, i) R, γ ∈ }|,
where dH denotes the hyperbolic distance in H. Since H G/K with K =
PSO(2), the following diagram:
suggests that the counting function N(R) can be expressed in terms of the
space
X := G.](https://image.slidesharecdn.com/90206-250313212418-feee36d4/85/Dynamics-and-Analytic-Number-Theory-1st-Edition-Dzmitry-Badziahin-22-320.jpg)
![xvi Preface
This idea, which goes back to the work of Duke, Rudnick, and Sarnak, is
explained in detail in Chapter 5. Ultimately, one shows that N(R) can be
approximated by combinations of averages along orbits Kg as g varies over
some subset of G. This argument reduces the original problem to analysing the
distribution of the sets Kg inside the space X which can be carried out using
dynamical systems techniques.
The space X introduced above is of fundamental importance in the The-
ory of Dynamical Systems and Geometry because it can be identified with
the unit tangent bundle of the modular surface H. Of particular inter-
est is the geodesic flow defined on this space, which plays a central role
in Chapter 6. This chapter discusses recent striking applications of the
sieving theory for thin groups, developed by Bourgain and Kontorovich,
to the arithmetics of continued fractions and the distribution of periodic
geodesic orbits. It is well known in the theory of hyperbolic dynamical
systems that one can construct periodic geodesic orbits with prescribed
properties. In particular, a single periodic geodesic orbit may exhibit a
very peculiar behaviour. Surprisingly, it turns out that the finite packets
of periodic geodesic orbits corresponding to a given fundamental discrim-
inant D become equidistributed as D → ∞. This remarkable result was
proved in full generality by Duke, generalising previous works of Linnik
and Skubenko. While Duke’s proof uses elaborate tools from analytic num-
ber theory (in particular, the theory of half-integral modular forms), now
there is also a dynamical approach developed by Einsiedler, Lindenstrauss,
Michel, and Venkatesh. They raised a question whether there exist infinitely
many periodic geodesic orbits corresponding to fundamental discriminants
which are contained in a fixed bounded subset of X. Chapter 6 outlines an
approach to this problem, which uses that the geodesic flow dynamics is
closely related to the symbolic dynamics of the continued fractions expan-
sions. In particular, a quadratic irrational with a periodic continued fraction
expansion
α = [a0, a1, . . . , a ]
corresponds to a periodic geodesic orbit. Moreover, the property of having
a fundamental discriminant can be characterised in terms of the trace of the
matrix
Mα :=
a0 1
1 0
a1 1
1 0
· · ·
a 1
1 0
,](https://image.slidesharecdn.com/90206-250313212418-feee36d4/85/Dynamics-and-Analytic-Number-Theory-1st-Edition-Dzmitry-Badziahin-23-320.jpg)
![Preface xvii
and the corresponding geodesic orbit lies in a fixed bounded set of X if ai ≤
A for all i for a fixed A 0. Hence, the original question reduces to the
investigation of the semigroup
A :=
a 1
1 0
: a ≤ A
+
∩ SL2(R),
and the trace map tr : A → N. The semigroup A arises naturally in con-
nection with several other deep problems involving periodic geodesic orbits
and continued fractions. Chapter 6 outlines a promising approach to the Arith-
metic Chaos Conjecture formulated by McMullen, which predicts that there
exists a fixed bounded subset of the space X such that, for all real quadratic
fields K, the closure of the set of periodic geodesic orbits defined over K
and contained in this set has positive entropy. Equivalently, in the language of
continued fractions, McMullen’s conjecture predicts that for some A ∞,
the set
{α = [a0, a1, . . . , a ] ∈ K : all aj ≤ A}
has exponential growth as → ∞. Since
α ∈ Q(
tr(Mα)2 − 4),
this problem also reduces to the analysis of the map tr : A → N. Chap-
ter 6 also discusses progress on the Zaremba conjecture regarding continued
fraction expansions of rational fractions. As is explained in Chapter 6, all these
problems can be unified by the far-reaching Local-Global Conjectures describ-
ing the distribution of solutions of F(γ ) = n, γ ∈ A, where F is a suitable
polynomial map.
We hope that this book will help to communicate the exciting material
written by experts in the field and covering a wide range of different top-
ics which are, nevertheless, in many ways connected to a broad circle of
young researchers as well as to other experts working in Number Theory or
Dynamical Systems.
Dmitry Badziahin
Department of Mathematical Sciences, Durham University
Durham, DH1 3LE, UK.
email: dzmitry.badziahin@durham.ac.uk
www.maths.dur.ac.uk/users/dzmitry.badziahin/
DB_webpage.htm](https://image.slidesharecdn.com/90206-250313212418-feee36d4/85/Dynamics-and-Analytic-Number-Theory-1st-Edition-Dzmitry-Badziahin-24-320.jpg)
Dynamics and Analytic Number Theory 1st Edition Dzmitry Badziahin
- 1. Visit https://ebookultra.com to download the full version and explore more ebooks or textbooks Dynamics and Analytic Number Theory 1st Edition Dzmitry Badziahin _____ Click the link below to download _____ https://ebookultra.com/download/dynamics-and-analytic- number-theory-1st-edition-dzmitry-badziahin/ Explore and download more ebooks or textbooks at ebookultra.com
- 2. Here are some recommended products that we believe you will be interested in. You can click the link to download. Duality in Analytic Number Theory 1st Edition Peter D. T. A. Elliott https://ebookultra.com/download/duality-in-analytic-number-theory-1st- edition-peter-d-t-a-elliott/ Algorithmic number theory lattices number fields curves and cryptography 1st Edition J.P. Buhler https://ebookultra.com/download/algorithmic-number-theory-lattices- number-fields-curves-and-cryptography-1st-edition-j-p-buhler/ Fluid Transport Theory Dynamics and Applications Theory Dynamics and Applications 1st Edition Emma T. Berg https://ebookultra.com/download/fluid-transport-theory-dynamics-and- applications-theory-dynamics-and-applications-1st-edition-emma-t-berg/ Recurrence in Ergodic Theory and Combinatorial Number Theory Harry Furstenberg https://ebookultra.com/download/recurrence-in-ergodic-theory-and- combinatorial-number-theory-harry-furstenberg/
- 3. Elliptic Tales Curves Counting and Number Theory Avner Ash https://ebookultra.com/download/elliptic-tales-curves-counting-and- number-theory-avner-ash/ Number Theory Fourier Analysis and Geometric Discrepancy London Mathematical Society Student Texts Series Number 81 1st Edition Giancarlo Travaglini https://ebookultra.com/download/number-theory-fourier-analysis-and- geometric-discrepancy-london-mathematical-society-student-texts- series-number-81-1st-edition-giancarlo-travaglini/ Elementary Number Theory with Applications 2nd Edition Koshy https://ebookultra.com/download/elementary-number-theory-with- applications-2nd-edition-koshy/ Combinatorics and Number Theory of Counting Sequences 1st Edition Istvan Mezo (Author) https://ebookultra.com/download/combinatorics-and-number-theory-of- counting-sequences-1st-edition-istvan-mezo-author/ Current Trends in Number Theory 1st Edition Sukumar Das Adhikari https://ebookultra.com/download/current-trends-in-number-theory-1st- edition-sukumar-das-adhikari/
- 5. Dynamics and Analytic Number Theory 1st Edition Dzmitry Badziahin Digital Instant Download Author(s): Dzmitry Badziahin, Alexander Gorodnik, Norbert Peyerimhoff (eds.) ISBN(s): 9781107552371, 1107552370 Edition: 1 File Details: PDF, 1.88 MB Year: 2017 Language: english
- 8. LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES Managing Editor: Professor M. Reid, Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom The titles below are available from booksellers, or from Cambridge University Press at http://www.cambridge.org/mathematics 314 Spectral generalizations of line graphs, D. CVETKOVIĆ, P. ROWLINSON & S. SIMIĆ 315 Structured ring spectra, A. BAKER & B. RICHTER (eds) 316 Linear logic in computer science, T. EHRHARD, P. RUET, J.-Y. GIRARD & P. SCOTT (eds) 317 Advances in elliptic curve cryptography, I.F. BLAKE, G. SEROUSSI & N.P. SMART (eds) 318 Perturbation of the boundary in boundary-value problems of partial differential equations, D. HENRY 319 Double affine Hecke algebras, I. CHEREDNIK 320 L-functions and Galois representations, D. BURNS, K. BUZZARD & J. NEKOVÁŘ (eds) 321 Surveys in modern mathematics, V. PRASOLOV & Y. ILYASHENKO (eds) 322 Recent perspectives in random matrix theory and number theory, F. MEZZADRI & N.C. SNAITH (eds) 323 Poisson geometry, deformation quantisation and group representations, S. GUTT et al (eds) 324 Singularities and computer algebra, C. LOSSEN & G. PFISTER (eds) 325 Lectures on the Ricci flow, P. TOPPING 326 Modular representations of finite groups of Lie type, J.E. HUMPHREYS 327 Surveys in combinatorics 2005, B.S. WEBB (ed) 328 Fundamentals of hyperbolic manifolds, R. CANARY, D. EPSTEIN & A. MARDEN (eds) 329 Spaces of Kleinian groups, Y. MINSKY, M. SAKUMA & C. SERIES (eds) 330 Noncommutative localization in algebra and topology, A. RANICKI (ed) 331 Foundations of computational mathematics, Santander 2005, L.M PARDO, A. PINKUS, E. SÜLI & M.J. TODD (eds) 332 Handbook of tilting theory, L. ANGELERI HÜGEL, D. HAPPEL & H. KRAUSE (eds) 333 Synthetic differential geometry (2nd Edition), A. KOCK 334 The Navier–Stokes equations, N. RILEY & P. DRAZIN 335 Lectures on the combinatorics of free probability, A. NICA & R. SPEICHER 336 Integral closure of ideals, rings, and modules, I. SWANSON & C. HUNEKE 337 Methods in Banach space theory, J.M.F. CASTILLO & W.B. JOHNSON (eds) 338 Surveys in geometry and number theory, N. YOUNG (ed) 339 Groups St Andrews 2005 I, C.M. CAMPBELL, M.R. QUICK, E.F. ROBERTSON & G.C. SMITH (eds) 340 Groups St Andrews 2005 II, C.M. CAMPBELL, M.R. QUICK, E.F. ROBERTSON & G.C. SMITH (eds) 341 Ranks of elliptic curves and random matrix theory, J.B. CONREY, D.W. FARMER, F. MEZZADRI & N.C. SNAITH (eds) 342 Elliptic cohomology, H.R. MILLER & D.C. RAVENEL (eds) 343 Algebraic cycles and motives I, J. NAGEL & C. PETERS (eds) 344 Algebraic cycles and motives II, J. NAGEL & C. PETERS (eds) 345 Algebraic and analytic geometry, A. NEEMAN 346 Surveys in combinatorics 2007, A. HILTON & J. TALBOT (eds) 347 Surveys in contemporary mathematics, N. YOUNG & Y. CHOI (eds) 348 Transcendental dynamics and complex analysis, P.J. RIPPON & G.M. STALLARD (eds) 349 Model theory with applications to algebra and analysis I, Z. CHATZIDAKIS, D. MACPHERSON, A. PILLAY & A. WILKIE (eds) 350 Model theory with applications to algebra and analysis II, Z. CHATZIDAKIS, D. MACPHERSON, A. PILLAY & A. WILKIE (eds) 351 Finite von Neumann algebras and masas, A.M. SINCLAIR & R.R. SMITH 352 Number theory and polynomials, J. MCKEE & C. SMYTH (eds) 353 Trends in stochastic analysis, J. BLATH, P. MÖRTERS & M. SCHEUTZOW (eds) 354 Groups and analysis, K. TENT (ed) 355 Non-equilibrium statistical mechanics and turbulence, J. CARDY, G. FALKOVICH & K. GAWEDZKI 356 Elliptic curves and big Galois representations, D. DELBOURGO 357 Algebraic theory of differential equations, M.A.H. MACCALLUM & A.V. MIKHAILOV (eds) 358 Geometric and cohomological methods in group theory, M.R. BRIDSON, P.H. KROPHOLLER & I.J. LEARY (eds) 359 Moduli spaces and vector bundles, L. BRAMBILA-PAZ, S.B. BRADLOW, O. GARCÍA-PRADA & S. RAMANAN (eds) 360 Zariski geometries, B. ZILBER 361 Words: Notes on verbal width in groups, D. SEGAL 362 Differential tensor algebras and their module categories, R. BAUTISTA, L. SALMERÓN & R. ZUAZUA 363 Foundations of computational mathematics, Hong Kong 2008, F. CUCKER, A. PINKUS & M.J. TODD (eds) 364 Partial differential equations and fluid mechanics, J.C. ROBINSON & J.L. RODRIGO (eds) 365 Surveys in combinatorics 2009, S. HUCZYNSKA, J.D. MITCHELL & C.M. RONEY-DOUGAL (eds) 366 Highly oscillatory problems, B. ENGQUIST, A. FOKAS, E. HAIRER & A. ISERLES (eds) 367 Random matrices: High dimensional phenomena, G. BLOWER 368 Geometry of Riemann surfaces, F.P. GARDINER, G. GONZÁLEZ-DIEZ & C. KOUROUNIOTIS (eds) 369 Epidemics and rumours in complex networks, M. DRAIEF & L. MASSOULIÉ 370 Theory of p-adic distributions, S. ALBEVERIO, A.YU. KHRENNIKOV & V.M. SHELKOVICH 371 Conformal fractals, F. PRZYTYCKI & M. URBAŃSKI 372 Moonshine: The first quarter century and beyond, J. LEPOWSKY, J. MCKAY & M.P. TUITE (eds) 373 Smoothness, regularity and complete intersection, J. MAJADAS & A. G. RODICIO 374 Geometric analysis of hyperbolic differential equations: An introduction, S. ALINHAC 375 Triangulated categories, T. HOLM, P. JØRGENSEN & R. ROUQUIER (eds)
- 9. 376 Permutation patterns, S. LINTON, N. RUŠKUC & V. VATTER (eds) 377 An introduction to Galois cohomology and its applications, G. BERHUY 378 Probability and mathematical genetics, N. H. BINGHAM & C. M. GOLDIE (eds) 379 Finite and algorithmic model theory, J. ESPARZA, C. MICHAUX & C. STEINHORN (eds) 380 Real and complex singularities, M. MANOEL, M.C. ROMERO FUSTER & C.T.C WALL (eds) 381 Symmetries and integrability of difference equations, D. LEVI, P. OLVER, Z. THOMOVA & P. WINTERNITZ (eds) 382 Forcing with random variables and proof complexity, J. KRAJÍČEK 383 Motivic integration and its interactions with model theory and non-Archimedean geometry I, R. CLUCKERS, J. NICAISE & J. SEBAG (eds) 384 Motivic integration and its interactions with model theory and non-Archimedean geometry II, R. CLUCKERS, J. NICAISE & J. SEBAG (eds) 385 Entropy of hidden Markov processes and connections to dynamical systems, B. MARCUS, K. PETERSEN & T. WEISSMAN (eds) 386 Independence-friendly logic, A.L. MANN, G. SANDU & M. SEVENSTER 387 Groups St Andrews 2009 in Bath I, C.M. CAMPBELL et al (eds) 388 Groups St Andrews 2009 in Bath II, C.M. CAMPBELL et al (eds) 389 Random fields on the sphere, D. MARINUCCI & G. PECCATI 390 Localization in periodic potentials, D.E. PELINOVSKY 391 Fusion systems in algebra and topology, M. ASCHBACHER, R. KESSAR & B. OLIVER 392 Surveys in combinatorics 2011, R. CHAPMAN (ed) 393 Non-abelian fundamental groups and Iwasawa theory, J. COATES et al (eds) 394 Variational problems in differential geometry, R. BIELAWSKI, K. HOUSTON & M. SPEIGHT (eds) 395 How groups grow, A. MANN 396 Arithmetic differential operators over the p-adic integers, C.C. RALPH & S.R. SIMANCA 397 Hyperbolic geometry and applications in quantum chaos and cosmology, J. BOLTE & F. STEINER (eds) 398 Mathematical models in contact mechanics, M. SOFONEA & A. MATEI 399 Circuit double cover of graphs, C.-Q. ZHANG 400 Dense sphere packings: a blueprint for formal proofs, T. HALES 401 A double Hall algebra approach to affine quantum Schur–Weyl theory, B. DENG, J. DU & Q. FU 402 Mathematical aspects of fluid mechanics, J.C. ROBINSON, J.L. RODRIGO & W. SADOWSKI (eds) 403 Foundations of computational mathematics, Budapest 2011, F. CUCKER, T. KRICK, A. PINKUS & A. SZANTO (eds) 404 Operator methods for boundary value problems, S. HASSI, H.S.V. DE SNOO & F.H. SZAFRANIEC (eds) 405 Torsors, étale homotopy and applications to rational points, A.N. SKOROBOGATOV (ed) 406 Appalachian set theory, J. CUMMINGS & E. SCHIMMERLING (eds) 407 The maximal subgroups of the low-dimensional finite classical groups, J.N. BRAY, D.F. HOLT & C.M. RONEY-DOUGAL 408 Complexity science: the Warwick master’s course, R. BALL, V. KOLOKOLTSOV & R.S. MACKAY (eds) 409 Surveys in combinatorics 2013, S.R. BLACKBURN, S. GERKE & M. WILDON (eds) 410 Representation theory and harmonic analysis of wreath products of finite groups, T. CECCHERINI-SILBERSTEIN, F. SCARABOTTI & F. TOLLI 411 Moduli spaces, L. BRAMBILA-PAZ, O. GARCÍA-PRADA, P. NEWSTEAD & R.P. THOMAS (eds) 412 Automorphisms and equivalence relations in topological dynamics, D.B. ELLIS & R. ELLIS 413 Optimal transportation, Y. OLLIVIER, H. PAJOT & C. VILLANI (eds) 414 Automorphic forms and Galois representations I, F. DIAMOND, P.L. KASSAEI & M. KIM (eds) 415 Automorphic forms and Galois representations II, F. DIAMOND, P.L. KASSAEI & M. KIM (eds) 416 Reversibility in dynamics and group theory, A.G. O’FARRELL & I. SHORT 417 Recent advances in algebraic geometry, C.D. HACON, M. MUSTAŢĂ & M. POPA (eds) 418 The Bloch–Kato conjecture for the Riemann zeta function, J. COATES, A. RAGHURAM, A. SAIKIA & R. SUJATHA (eds) 419 The Cauchy problem for non-Lipschitz semi-linear parabolic partial differential equations, J.C. MEYER & D.J. NEEDHAM 420 Arithmetic and geometry, L. DIEULEFAIT et al (eds) 421 O-minimality and Diophantine geometry, G.O. JONES & A.J. WILKIE (eds) 422 Groups St Andrews 2013, C.M. CAMPBELL et al (eds) 423 Inequalities for graph eigenvalues, Z. STANIĆ 424 Surveys in combinatorics 2015, A. CZUMAJ et al (eds) 425 Geometry, topology and dynamics in negative curvature, C.S. ARAVINDA, F.T. FARRELL & J.-F. LAFONT (eds) 426 Lectures on the theory of water waves, T. BRIDGES, M. GROVES & D. NICHOLLS (eds) 427 Recent advances in Hodge theory, M. KERR & G. PEARLSTEIN (eds) 428 Geometry in a Fréchet context, C. T. J. DODSON, G. GALANIS & E. VASSILIOU 429 Sheaves and functions modulo p, L. TAELMAN 430 Recent progress in the theory of the Euler and Navier-Stokes equations, J.C. ROBINSON, J.L. RODRIGO, W. SADOWSKI & A. VIDAL-LÓPEZ (eds) 431 Harmonic and subharmonic function theory on the real hyperbolic ball, M. STOLL 432 Topics in graph automorphisms and reconstruction (2nd Edition), J. LAURI & R. SCAPELLATO 433 Regular and irregular holonomic D-modules, M. KASHIWARA & P. SCHAPIRA 434 Analytic semigroups and semilinear initial boundary value problems (2nd Edition), K. TAIRA 435 Graded rings and graded Grothendieck groups, R. HAZRAT 436 Groups, graphs and random walks, T. CECCHERINI-SILBERSTEIN, M. SALVATORI & E. SAVA-HUSS (eds) 437 Dynamics and analytic number theory, D. BADZIAHIN, A. GORODNIK & N. PEYERIMHOFF (eds)
- 10. London Mathematical Society Lecture Note Series: 437 Dynamics and Analytic Number Theory Proceedings of the Durham Easter School 2014 Edited by DZMITRY BADZIAHIN University of Durham ALEXANDER GORODNIK University of Bristol NORBERT PEYERIMHOFF University of Durham
- 11. University Printing House, Cambridge CB2 8BS, United Kingdom Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107552371 c Cambridge University Press 2016 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2016 Printed in the United Kingdom by Clays, St Ives plc A catalogue record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication Data Names: Durham Easter School (2014 : University of Durham) | Badziahin, Dmitry (Dmitry A.), editor. | Gorodnik, Alexander, 1975– editor. | Peyerimhoff, Norbert, 1964– editor. Title: Dynamics and analytic number theory : proceedings of the Durham Easter School 2014 / edited by Dzmitry Badziahin, University of Durham, Alexander Gorodnik, University of Bristol, Norbert Peyerimhoff, University of Durham. Description: Cambridge : Cambridge University Press, [2016] | Series: London Mathematical Society lecture note series ; 437 | Includes bibliographical references and index. Identifiers: LCCN 2016044609 | ISBN 9781107552371 (alk. paper) Subjects: LCSH: Number theory – Congresses. | Dynamics – Congresses. Classification: LCC QA241 .D87 2014 | DDC 512.7/3–dc23 LC record available at https://lccn.loc.gov/2016044609 ISBN 978-1-107-55237-1 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party Internet Web sites referred to in this publication and does not guarantee that any content on such Web sites is, or will remain, accurate or appropriate.
- 12. Contents List of contributors page vii Preface ix 1 Metric Diophantine Approximation: Aspects of Recent Work 1 Victor Beresnevich, Felipe Ramírez and Sanju Velani 1.1 Background: Dirichlet and Bad 1 1.2 Metric Diophantine Approximation: The Classical Lebesgue Theory 9 1.3 Metric Diophantine Approximation: The Classical Hausdorff Theory 18 1.4 The Higher-Dimensional Theory 27 1.5 Ubiquitous Systems of Points 46 1.6 Diophantine Approximation on Manifolds 53 1.7 The Badly Approximable Theory 74 2 Exponents of Diophantine Approximation 96 Yann Bugeaud 2.1 Introduction and Generalities 96 2.2 Further Definitions and First Results 99 2.3 Overview of Known Relations Between Exponents 106 2.4 Bounds for the Exponents of Approximation 109 2.5 Spectra 114 2.6 Intermediate Exponents 121 2.7 Parametric Geometry of Numbers 124 2.8 Real Numbers Which Are Badly Approximable by Algebraic Numbers 126 2.9 Open Problems 127 v
- 13. vi Contents 3 Effective Equidistribution of Nilflows and Bounds on Weyl Sums 136 Giovanni Forni 3.1 Introduction 137 3.2 Nilflows and Weyl Sums 142 3.3 The Cohomological Equation 152 3.4 The Heisenberg Case 159 3.5 Higher-Step Filiform Nilflows 174 4 Multiple Recurrence and Finding Patterns in Dense Sets 189 Tim Austin 4.1 Szemerédi’s Theorem and Its Relatives 189 4.2 Multiple Recurrence 192 4.3 Background from Ergodic Theory 197 4.4 Multiple Recurrence in Terms of Self-Joinings 212 4.5 Weak Mixing 222 4.6 Roth’s Theorem 230 4.7 Towards Convergence in General 238 4.8 Sated Systems and Pleasant Extensions 242 4.9 Further Reading 248 5 Diophantine Problems and Homogeneous Dynamics 258 Manfred Einsiedler and Tom Ward 5.1 Equidistribution and the Gauss Circle Problem 258 5.2 Counting Points in SL2(Z) · i ⊆ H 267 5.3 Dirichlet’s Theorem and Dani’s Correspondence 278 6 Applications of Thin Orbits 289 Alex Kontorovich 6.1 Lecture 1: Closed Geodesics, Binary Quadratic Forms, and Duke’s Theorem 289 6.2 Lecture 2: Three Problems in Continued Fractions: ELMV, McMullen, and Zaremba 306 6.3 Lecture 3: The Thin Orbits Perspective 310 Index 318
- 14. Contributors Tim Austin Courant Institute, NYU New York, NY 10012, USA Email: tim@cims.nyu.edu Victor Beresnevich Department of Mathematics, University of York Heslington, York, Y010 5DD, United Kingdom Email: victor.beresnevich@york.ac.uk Yann Bugeaud Département de mathématiques, Université de Strasbourg F-67084 Strasbourg, France Email: bugeaud@math.unistra.fr Manfred Einsiedler Departement Mathematik, ETH Zürich 8092 Zürich, Switzerland Email: manfred.einsiedler@math.ethz.ch Giovanni Forni Department of Mathematics, University of Maryland College Park, MD 20742-4015, USA Email: gforni@math.umd.edu Alex Kontorovich Department of Mathematics, Rutgers University Piscataway, NJ 08854, USA Email: alex.kontorovich@rutgers.edu vii
- 15. viii List of contributors Felipe Ramírez Department of Mathematics and Computer Science Wesleyan University, Middletown, CT 06459, USA Email: framirez@wesleyan.edu Sanju Velani Department of Mathematics, University of York Heslington, York, Y010 5DD, United Kingdom Email: slv3@york.ac.uk Tom Ward Executive Office, Palatine Centre, Durham University Durham DH1 3LE, United Kingdom Email: t.b.ward@durham.ac.uk
- 16. Preface This book is devoted to some of the interesting recently discovered interac- tions between Analytic Number Theory and the Theory of Dynamical Systems. Analytical Number Theory has a very long history. Many people associate its starting point with the work of Dirichet on L-functions in 1837, where he proved his famous result about infinitely many primes in arithmetic progres- sions. Since then, analytical methods have played a crucial role in proving many important results in Number Theory. For example, the study of the Riemann zeta function allowed to uncover deep information about the distri- bution of prime numbers. Hardy and Littlewood developed their circle method to establish first explicit general estimates for the Waring problem. Later, Vinogradov used the idea of the circle method to create his own method of exponential sums which allowed him to solve, unconditionally of the Rie- mann hypothesis, the ternary Goldbach conjecture for all but finitely many natural numbers. Roth also used exponential sums to prove the existence of three-term arithmetic progressions in subsets of positive density. One of the fundamental questions which arise in the investigation of exponential sums, as well as many other problems in Number Theory, is how rational numbers/vec- tors are distributed and how well real numbers/vectors can be approximated by rationals. Understanding various properties of sets of numbers/vectors that have prescribed approximational properties, such as their size, is the subject of the metric theory of Diophantine approximation, which involves an inter- esting interplay between Arithmetic and Measure Theory. While these topics are now considered as classical, the behaviour of exponential sums is still not well understood today, and there are still many challenging open problems in Diophantine approximation. On the other hand, in the last decades there have been several important breakthroughs in these areas of Number Theory where progress on long-standing open problems has been achieved by utilising techniques which originated from the Theory of Dynamical Systems. These ix
- 17. x Preface developments have uncovered many profound and very promising connections between number-theoretic and dynamical objects that are at the forefront of current research. For instance, it turned out that properties of exponential sums are intimately related to the behaviour of orbits of flows on nilmanifolds; the existence of given combinatorial configurations (e.g. arithmetic progressions) in subsets of integers can be established through the study of multiple recur- rence properties for dynamical systems; and Diophantine properties of vectors in the Euclidean spaces can be characterised in terms of excursions of orbits of suitable flows in the space of lattices. The material of this book is based on the Durham Easter School, ‘Dynam- ics and Analytic Number Theory’, that was held at the University of Durham in Spring 2014. The intention of this school was to communicate some of these remarkable developments at the interface between Number Theory and Dynamical Systems to young researchers. The Easter School consisted of a series of mini-courses (with two to three lectures each) given by Tim Austin, Manfred Einsiedler, Giovanni Forni, Alex Kontorovich, Sanju Velani and Trevor Wooley, and a talk by Yann Bugeaud presenting a collection of recent results and open problems in Diophantine approximation. The event was very well received by more than 60 participants, many of them PhD students from all around the world. Because of the great interest of young researchers in this topic, we decided to encourage the speakers to write contributions to this Proceedings volume. One of the typical examples where both classical and dynamical approaches are now actively developing and producing deep results is the theory of Dio- phantine approximation. One of the classical problems in this area asks how well a given n-dimensional vector x ∈ Rn can be approximated by vectors with rational coefficients. More specifically, one can ask: what is the supremum λ(x) of the values λ such that the inequality ||qx − p||∞ Q−λ (1) has infinitely many integer solutions Q ∈ N, q ∈ N, p ∈ Zn satisfying q ≤ Q? This type of problem is referred to as a simultaneous Diophantine approxima- tion. There is also a dual Diophantine approximation problem which asks for the supremum ω(x) of the values ω such that the inequality |(x, q) − p| Q−ω (2) has infinitely many solutions Q ∈ N, q ∈ Zn, p ∈ Z with q = 0 and ||q||∞ ≤ Q. It turns out that there are various relations between the exponents λ(x) and ω(x). Chapter 2 provides an overview of known relations between these and some other similar exponents. It mostly concentrates on the case where x lies
- 18. Preface xi on the so-called Veronese curve which is defined by x(t) := (t, t2, . . . , tn) with real t. This case is of particular importance for number theorists since it has implications for the question about the distribution of algebraic numbers of bounded degree. For example, condition (2) in this case transforms to |P(t)| Q−ω where P(t) is a polynomial with integer coefficients. For large Q this implies that x is very close to the root of P, which is an algebraic number. Metric theory of Diophantine approximation does not work with particular vectors x. Instead it deals with the sets of all vectors x satisfying inequali- ties like (1) or (2) for infinitely many Q ∈ N, q ∈ N, p ∈ Zn (respectively, Q ∈ N, q ∈ Zn, p ∈ Z, q = 0). The central problem is to estimate the mea- sure and the Hausdorff dimension of such sets. This area of Number Theory was founded at the beginning of the twentieth century with Khintchine’s work which was later generalised by Groshev. In the most general way they showed that, given a function ψ : R≥0 → R≥0, the set of m × n matrices A which satisfy the inequality ||Aq − p||∞ ψ(||q||∞) with p ∈ Zn and q ∈ Zm, has either zero or full Lebesgue measure. The matrices A satisfying this property are usually called ψ-well approximable. Furthermore, with some mild conditions on ψ, the Lebesgue measure of the set of ψ-well approximable matrices is determined by the convergence of a certain series which involves ψ. Later, many other results of this type were established, some of them with help of the classical methods and others by using the ideas from homogeneous dynamics. Chapter 1 describes several powerful ‘classical’ techniques used in metric theory of Diophantine approximation, such as the Mass Transference Prin- ciple, ubiquitous systems, Cantor sets constructions and winning sets. The Mass Transference Principle allows us to get results about the more sensi- tive Hausdorff measure and Hausdorff dimension of sets of well approximable matrices or similar objects as soon as results about their Lebesgue measure are known. Ubiquitous systems provide another powerful method originat- ing from works of A. Baker and W. Schmidt. It enables us to obtain the ‘full Lebesgue measure’-type results in various analogues of the Khintchine– Groshev theorem. Finally, Chapter 1 introduces the generalised Cantor set construction technique, which helps in investigating badly approximable num- bers or vectors. It also relates such sets with so-called winning sets developed by W. Schmidt. The winning sets have several surprising properties. For example, they have the maximal possible Hausdorff dimension and, even though such sets may be null in terms of Lebesgue measure, their countable intersection must also be winning.
- 19. xii Preface Chapter 3 is devoted to the study of exponential sums. Given a real polynomial P(x) = ak xk + · · · + a1x + a0, the Weyl sums are defined as WN := N−1 n=0 e2πi P(n) . The study of Weyl sums has a long history that goes back to foundational works of Hardy, Littlewood, and Weyl. When the coefficients of the polyno- mial P(X) satisfy a suitable irrationality condition, then it is known that for some w ∈ (0, 1), WN = O(N1−w ) as N → ∞, and improving the value of the exponent in this estimate is a topic of current research. This problem has been approached recently by several very different methods. The method of Wooley is based on refinements of the Vinogradov mean value theorem and a new idea of efficient congruencing, and the method of Flaminio and Forni involves the investigation of asymptotic properties of flows on nilmanifolds using renormalisation techniques. It is quite remarkable that the exponents w obtained by the Flaminio–Forni approach, which is deter- mined by optimal scaling of invariant distributions, essentially coincide with the exponents derived by Wooley using his method of efficient congruencing. As discussed in Chapter 3, flows on nilmanifolds provide a very convenient tool for investigating the distribution of polynomial sequences modulo one and modelling Weyl sums. We illustrate this by a simple example. Let N := ⎧ ⎨ ⎩ [p, q, r] := ⎛ ⎝ 1 p r 0 1 q 0 0 1 ⎞ ⎠ : p, q,r ∈ R ⎫ ⎬ ⎭ denote the three-dimensional Heisenberg group, and be the subgroup con- sisting of matrices with integral entries. Then the factor space M := N provides the simplest example of a nilmanifold. Given an upper triangular nilpotent matrix X = (xi j ), the flow generated by X is defined by φX t (m) = m exp(t X) with m ∈ M. More explicitly, exp(t X) = [x12t, x23t, x13t + x12x23t2/2]. The space M con- tains a two-dimensional subtorus T defined by the condition q = 0. If we take x23 = 1, then the intersection of the orbit φX t (e) with this torus gives the sequence of points [x12n, 0, x13n + x12n2/2] with n ∈ N. Hence, choos- ing suitable matrices X, the flows φX t can be used to model values of general quadratic polynomials P modulo one. Moreover, this relation can be made
- 20. Preface xiii much more precise. In particular, with a suitable choice of a test function F on M and m ∈ M, N−1 n=0 e2πi P(n) = N 0 F(φX t (m)) dt + O(1). This demonstrates that quadratic Weyl sums are intimately related to averages of one-parameter flows on the Heisenberg manifold. A more elaborate con- struction discussed in detail in Chapter 3 shows that general Weyl sums can be approximated by integrals along orbits on higher-dimensional nilmanifolds. Chapter 3 discusses asymptotic behaviour of orbits averages on nilmanifolds and related estimates for Weyl sums. Dynamical systems techniques also provide powerful tools to analyse com- binatorial structures of large subsets of integers and of more general groups. This active research field fusing ideas from Ramsey Theory, Additive Combi- natorics, and Ergodic Theory is surveyed in Chapter 4. We say that a subset E ⊂ Z has positive upper density if d̄(E) := lim sup N−M→∞ |E ∩ [M, N]| N − M 0. Surprisingly, this soft analytic condition on the set E has profound combina- torial consequences, one of the most remarkable of which is the Szemerédi theorem. It states that every subset of positive density contains arbitrarily long arithmetic progressions: namely, configurations of the form a, a + n, . . . , a + (k − 1)n with arbitrary large k. It should be noted that the existence of three- term arithmetic progressions had previously been established by Roth using a variant of the circle method, but the case of general progressions required substantial new ideas. Shortly after Szemerédi’s work appeared, Furstenberg discovered a very different ingenious approach to this problem that used ergodic-theoretic techniques. He realised that the Szemerédi theorem is equiv- alent to a new ergodic-theoretic phenomenon called multiple recurrence. This unexpected connection is summarised by the Furstenberg correspondence prin- ciple which shows that, given a subset E ⊂ Z, one can construct a probability space (X, μ), a measure-preserving transformation T : X → X, and a measurable subset A ⊂ X such that μ(A) = d̄(E) and d̄(E ∩(E −n)∩· · ·∩(E −(k −1)n)) ≥ μ(A ∩ T −n (A)∩· · ·∩ T−(k−1)n (A)). This allows the proof of Szemerédi’s theorem to be reduced to establishing the multiple recurrence property, which shows that if μ(A) 0 and k ≥ 1, then there exists n ≥ 1 such that μ(A ∩ T −n A ∩ · · · ∩ T −(k−1)n A) 0.
- 21. xiv Preface This result is the crux of Furstenberg’s approach, and in order to prove it, a deep structure theorem for general dynamical systems is needed. Furstenberg’s work has opened a number of promising vistas for future research and started a new field of Ergodic Theory – ergodic Ramsey theory, which explores the existence of combinatorial structures in large subsets of groups. This is the subject of Chapter 4. In view of the above connection it is of fundamental importance to explore asymptotics of the averages 1 N N−1 n=0 μ(A ∩ T−n (A) ∩ · · · ∩ T−(k−1)n (A)), and more generally, the averages 1 N N−1 n=0 ( f1 ◦ T n ) · · · ( fk−1 ◦ T (k−1)n ) (3) for test functions f1, . . . , fk−1 ∈ L∞(μ). The existence of limits for these averages was established in the groundbreaking works of Host, Kra, and Ziegler. Chapter 4 explains an elegant argument of Austin which permits the proof of the existence of limits for these multiple averages as well as multiple averages for actions of the group Zd. A number of important applications of the Theory of Dynamical Systems to Number Theory involve analysing the distribution of orbits on the space of unimodular lattices in Rd+1. This space, which will be denoted by Xd+1, consists of discrete cocompact subgroups of Rd+1 with covolume one. It can be realised as a homogeneous space Xd+1 SLd+1(R)/SLd+1(Z). which allows us to equip Xd+1 with coordinate charts and an invariant finite measure. Some of the striking applications of dynamics on the space Xd+1 to problems of Diophantine approximation are explored in Chapter 5. It was realised by Dani that information about the distribution of suitable orbits on Xd+1 can be used to investigate the existence of solutions of Diophantine inequalities. In particular, this allows a convenient dynamical characterisation of many Diophantine classes of vectors in Rd discussed in Chapters 1 and 2 to be obtained, such as, for instance, badly approximable vectors, very well approximable vectors, singular vectors. This connection is explained by the following construction. Given a vector v ∈ Rd, we consider the lattice v := {(q, qv + p) : (q, p) ∈ Z × Zd },
- 22. Preface xv and a subset of the space Xd+1 defined as Xd+1(ε) := { : ∩ [−ε, ε]d+1 = {0}}. Let gQ := diag(Q−d, Q, . . . , Q). If we establish that the orbit gQ visits the subset Xd+1(ε), then this will imply that the systems of inequalities |q| ≤ εQd and qv − p∞ ≤ ε Q have a non-trivial integral solution (q, p) ∈ Z×Zd. When ε ≥ 1, the existence of solutions is a consequence of the classical Dirichlet theorem, but for ε 1, this is a delicate property which was studied by Davenport and Schmidt. Vec- tors for which the above system of inequalities has a non-trivial solution for some ε ∈ (0, 1) and all sufficiently large Q are called Dirichlet-improvable. Chapter 5 explains how to study this property using dynamical systems tools such as the theory of unipotent flow. This approach proved to be very success- ful. In particular, it was used by Shah to solve the problem posed by Davenport and Schmidt in the 60s. He proved that if φ : (0, 1) → Rd is an analytic curve whose image is not contained in a proper affine subspace, then the vector φ(t) is not Dirichlet-improvable for almost all t. Chapter 5 explains Shah’s proof of this result. Chapter 5 also discusses how dynamical systems techniques can be used to derive asymptotic counting results. Although this approach is applicable in great generality, its essence can be illustrated by a simple example: counting points in lattice orbits on the hyperbolic upper half-plane H. We recall that the group G = PSL2(R) acts on H by isometries. Given = PSL2(Z) (or, more generally, a discrete subgroup of G with finite covolume), we consider the orbit · i in H. We will be interested in asymptotics of the counting function N(R) := |{γ · i : dH(γ · i, i) R, γ ∈ }|, where dH denotes the hyperbolic distance in H. Since H G/K with K = PSO(2), the following diagram: suggests that the counting function N(R) can be expressed in terms of the space X := G.
- 23. xvi Preface This idea, which goes back to the work of Duke, Rudnick, and Sarnak, is explained in detail in Chapter 5. Ultimately, one shows that N(R) can be approximated by combinations of averages along orbits Kg as g varies over some subset of G. This argument reduces the original problem to analysing the distribution of the sets Kg inside the space X which can be carried out using dynamical systems techniques. The space X introduced above is of fundamental importance in the The- ory of Dynamical Systems and Geometry because it can be identified with the unit tangent bundle of the modular surface H. Of particular inter- est is the geodesic flow defined on this space, which plays a central role in Chapter 6. This chapter discusses recent striking applications of the sieving theory for thin groups, developed by Bourgain and Kontorovich, to the arithmetics of continued fractions and the distribution of periodic geodesic orbits. It is well known in the theory of hyperbolic dynamical systems that one can construct periodic geodesic orbits with prescribed properties. In particular, a single periodic geodesic orbit may exhibit a very peculiar behaviour. Surprisingly, it turns out that the finite packets of periodic geodesic orbits corresponding to a given fundamental discrim- inant D become equidistributed as D → ∞. This remarkable result was proved in full generality by Duke, generalising previous works of Linnik and Skubenko. While Duke’s proof uses elaborate tools from analytic num- ber theory (in particular, the theory of half-integral modular forms), now there is also a dynamical approach developed by Einsiedler, Lindenstrauss, Michel, and Venkatesh. They raised a question whether there exist infinitely many periodic geodesic orbits corresponding to fundamental discriminants which are contained in a fixed bounded subset of X. Chapter 6 outlines an approach to this problem, which uses that the geodesic flow dynamics is closely related to the symbolic dynamics of the continued fractions expan- sions. In particular, a quadratic irrational with a periodic continued fraction expansion α = [a0, a1, . . . , a ] corresponds to a periodic geodesic orbit. Moreover, the property of having a fundamental discriminant can be characterised in terms of the trace of the matrix Mα := a0 1 1 0 a1 1 1 0 · · · a 1 1 0 ,
- 24. Preface xvii and the corresponding geodesic orbit lies in a fixed bounded set of X if ai ≤ A for all i for a fixed A 0. Hence, the original question reduces to the investigation of the semigroup A := a 1 1 0 : a ≤ A + ∩ SL2(R), and the trace map tr : A → N. The semigroup A arises naturally in con- nection with several other deep problems involving periodic geodesic orbits and continued fractions. Chapter 6 outlines a promising approach to the Arith- metic Chaos Conjecture formulated by McMullen, which predicts that there exists a fixed bounded subset of the space X such that, for all real quadratic fields K, the closure of the set of periodic geodesic orbits defined over K and contained in this set has positive entropy. Equivalently, in the language of continued fractions, McMullen’s conjecture predicts that for some A ∞, the set {α = [a0, a1, . . . , a ] ∈ K : all aj ≤ A} has exponential growth as → ∞. Since α ∈ Q( tr(Mα)2 − 4), this problem also reduces to the analysis of the map tr : A → N. Chap- ter 6 also discusses progress on the Zaremba conjecture regarding continued fraction expansions of rational fractions. As is explained in Chapter 6, all these problems can be unified by the far-reaching Local-Global Conjectures describ- ing the distribution of solutions of F(γ ) = n, γ ∈ A, where F is a suitable polynomial map. We hope that this book will help to communicate the exciting material written by experts in the field and covering a wide range of different top- ics which are, nevertheless, in many ways connected to a broad circle of young researchers as well as to other experts working in Number Theory or Dynamical Systems. Dmitry Badziahin Department of Mathematical Sciences, Durham University Durham, DH1 3LE, UK. email: dzmitry.badziahin@durham.ac.uk www.maths.dur.ac.uk/users/dzmitry.badziahin/ DB_webpage.htm
- 25. xviii Preface Alex Gorodnik School of Mathematics, University of Bristol Bristol BS8 1SD, UK. email: a.gorodnik@bristol.ac.uk www.maths.bris.ac.uk/~mazag/ Norbert Peyerimhoff Department of Mathematical Sciences, Durham University Durham, DH1 3LE, UK. email: norbert.peyerimhoff@durham.ac.uk www.maths.dur.ac.uk/~dma0np/ Acknowledgements We, as part of the organising committee of the Durham Easter School 2014, which followed a one-day LMS Northern Regional Meeting dedicated to the same topic, would like to thank all the people who helped to make this event highly enjoyable and successful. We are also grateful to everybody who made this Proceedings volume possible. Specifically, we would like to acknowledge the help of: ● our Easter School co-organisers, T. Ward, A. Ghosh, and B. Weiss, for their support in this event and the selection of a very impressive list of speakers; ● Sam Harrison and his colleagues from Cambridge University Press for all their encouragement and support in producing this book; ● our distinguished speakers for their interesting talks; ● all participants of the Easter School for their interest and help to create a positive and inspiring atmosphere; ● all contributors to this volume, for their work in producing their chapters within the given timeframe. Particular thanks are due to Durham University for its great hospitality in hosting the Easter School. Last, but not least, we would like to stress the fact that the Easter School would not have been possible without the generous funding received, via the University of Bristol, from the ERC grant 239606 and the financial support of the London Mathematical Society.
- 26. Other documents randomly have different content
- 27. THE VULTURES AND THE PIGEONS.
- 28. I should want breath for the detail, If I told how with tooth and nail They battled. Many chiefs fell dead, Many a dauntless hero bled; Prometheus on his mountain sighed, And hoped Jove nearly satisfied. 'Twas pleasure to observe their pains— 'Twas sad to see the corpse-strewn plains. Valour, address, and stratagem, By turns were tried by all of them; By folks so brave no means were lost To fill each spare place on the coast Of Styx. Each varied element Ghosts to the distant realm had sent. This fury roused, at last, deep pity, Within the pigeons' quiet city; They—of the neck of changing hue, The heart so tender and so true— Resolved, as well became their nation, To end the war by mediation. Ambassadors they chose and sent, Who worked with such a good intent, The Vultures cried, A truce, at last, And wars red horrors from them cast. Alas! the Pigeons paid for it; Their heart was better than their wit; The cursed race upon them fell, And made a carnage terrible; Dispeopled every farm and town, And struck the unwise people down. In this, then, always be decided: Keep wicked people still divided; The safety of the world depends On that—sow war among their friends;
- 29. Contract no peace with such, I say, But this is merely by the way. FABLE CXXXII THE COURT OF THE LION. His Majesty Leo, in order to find The extent of his varied and ample dominions, Had summoned his vassals of every kind, Of all colours and shapes, and of divers opinions. A circular, signed by His Majesty's hand. Was the means of conveying the King's
- 30. invitation— He promised festivities regally grand (With an evident eye to self-glorification). His palace was open, of course, to the throng; What a place!—a mere slaughter-house, putting it plainly, Where visitors met with an odour so strong, That they strove to protect their olfactories vainly. The Bear in disgust put a paw to his nose; He had scarcely the time to repent his grimaces; For Leo at once in a fury arose, And consigned the poor brute to the Styx, to make faces. The Monkey, true courtier, approved of the deed— Said the palace was fit for a king's habitation, And thought neither amber nor musk could exceed The rich odour that gave him such gratification. His fulsome behaviour had little success; He was treated the same as the previous aspirant (His Leonine Majesty, let us confess, Was Caligula-like, and a bit of a tyrant). The Fox trotted up, very servile and sly; Said the monarch, No shuffling, but answer me frankly; Beware how you venture to give your reply: Do you notice that anything smells rather rankly? But Reynard was more than a match for his king,
- 31. And replied that his cold being rather a bad one, He could not at present distinguish a thing By its odour, or even assert that it had one. There's a hint for plain-speakers and flatterers here— You should ne'er be too servile nor over- sincere; And to answer sometimes in a round-about way, Is a dozen times better than plain yea or nay. FABLE CXXXIII. THE MILK-MAID AND THE MILK-PAIL.
- 32. Perette, her Milk-pail balanced on her head, Tripped gaily and without hindrance down the road, So slim and trim, and gay she nimbly sped. For more agility, with such a load, She'd donned her shortest kirtle and light shoes. And as she went she counted up her gains— Her future gains—with her twice one, twice twos. How long division racked her little brains! First buy a hundred eggs, then triple broods; With care like mine the money soon will grow;
- 33. THE MILK-MAID AND THE MILK-PAIL.
- 34. No fox so clever in our neighbour's woods But must leave me enough, as well I know, To buy a pig, 'twill fatten very soon; I buy him large, and for a good round sum I sell him, mark you that some afternoon; A cow and calf into our stable come; Who'll prevent that? that's what I mean to say. I see the calf skipping among the herd. Then Perette skipped for joy. Alack-a-day! Down came the milk, I give you my sworn word: Adieu cow, calf, pig, chicken, all the rest. She left with tearful eye her fortune lost, And ran to tell her husband, dreading lest He'd beat her, when in anger tempest tossed. The neighbours, doubling up with laughter, Called her the Milk-pail ever after. Who has not raised his tower in Spain, And in a cloud-land longed to reign? Picrocolles, Pyrrhus have so done, Sages or fools, just like this one. All dream by turns; the dream is sweet; The world lies prostrate at our feet: Our souls yield blindly to the vision, Ours beauty, honour, fields Elysian. 'Tis I alone the bravest smite, The dethroned Sophy owns my might; They choose me king, in crowds I'm led; Gold crowns come raining on my head. A fly soon wakes me up once more, And I am Big John, as before.
- 35. FABLE CXXXIV. THE CURATE AND THE CORPSE. A Dead man, on his mournful way. To his last lodging went one day. A Curé, bustling gaily, came In due form, to inter the same. Deceased was in a coach, with care Packed snugly from the sun and air; Clad in a robe, alas! ye proud, Summer or winter, called a shroud; To change it no one is allowed. The pastor sat the dead beside, Reciting, without grief or pride, Lessons, responses, and those done,
- 36. The funeral psalms; yes, every one. Good Mr. Dead-man, let them chant, The salary is all they want. The Curé Chouart shut the eyes Of his dead man, lest he surprise The priest who snatched from him a prize. His looks they seemed to say, My friend, From you I'll have, before I end, This much in silver, that in wax, And many another little tax; That soon would bring our good divine A small cask of the choicest wine; His pretty niece a new silk gown, And Paquette something from the town. Just as his pleasant thoughts took flight, There came a crash... Curé, good night! The leaden coffin strikes his head. Parishioner, lapped up in lead, Politely you went first, you see, Now comes the priest for company. Such is our life, as in this tale: See Curé Chouart counting on his fee, Like the poor girl with the milk-pail.
- 37. FABLE CXXXV. THE MAN WHO RUNS AFTER FORTUNE, AND THE MAN WHO WAITS FOR HER. Is there a man beneath the sun, Who does not after Fortune run? I would I were in some snug place, And high enough to watch the race Of the long, scuffling, struggling train That hunt Dame Fortune all in vain. The phantom flies from land to land, They follow with an outstretched hand. Now they have almost caught her. No; She's vanished like the April bow. Poor creatures! Pity them, I do: Fools deserve pity—the whole crew, By no means rage—You see, we hope; That cabbage-planter made a Pope. Are we not quite as good? they cry. Twenty times better, my reply. But what avails your mighty mind, When Fortune is so densely blind? Besides, what use the Papacy? It is not worth the price, may be. Rest, rest; a treasure that's so great
- 38. 'Twas once for gods reserved by Fate; How rarely fickle Fortune sends Such gifts unto her trusting friends. Seek not the goddess, stay at home; Then like her sex she's sure to come. Two friends there lived in the same place, Who were by no means in bad case. One sighed for Fortune night and day: Let's quit our sojourn here, I pray, He to the other said, You know, Prophets in their own country go Unhonoured; let us seek elsewhere. Seek! said the other; I'll stay here. I wish no better land or sky: Content yourself, and I will try To sleep the time out patiently. The friend—ambitious, greedy soul!— Set out to reach the wished-for goal; And on the morrow sought a place Where Fortune ought to show her face, And frequently—the Court, I mean; So there he halts, to view the scene; Still seeking early, seeking late, The hours propitious to Fate; But yet, though seeking everywhere, He only found regret and care. It's of no use, at last he cried; Queen Fortune elsewhere must abide; And yet I see her, o'er and o'er, Enter by this and that man's door: And how, then, is it I can never Meet her, though I seek her ever? These sort of people, I'm afraid, Ambition find a losing trade. Adieu, my lords; my lords, adieu; Follow the shadow ruling you.
- 39. Fortune at Surat temples boasts; Let's seek those distant Indian coasts, Ye souls of bronze who e'er essayed This voyage; nay, diamond arms arrayed The man who first crossed the abyss. Many a time our friend, I wis, Thought of his village and his farm, Fearing incessantly some harm From pirates, tempests, rocks and sands, All friends of death. In many lands Man seeks his foeman, round and round, Who soon enough at home is found. In Tartary they tell the man That Fortune's busy at Japan: Then off he hurries, ne'er downcast. Seas weary of the man at last, And all the profit that he gains Is this one lesson for his pains: Japan, no more than Tartary, Brought good to him or wealthy fee. At last he settles it was shame To leave his home, and takes the blame. Then he returns: the well-loved place Makes tears of joy run down his face. Happy, he cries, the man at ease, Who lives at home himself to please; Ruling his passions, by report Knowing alone of sea or Court, Or Fortune, of thy empire, Jade, Which has by turns to all displayed Titles and wealth, that lead us on From rising to the setting sun; And yet thy promises astray Still lead us to our dying day. Henceforth I will not budge again, And shall do better, I see plain.
- 40. While he thus schemed, resolved, and planned, And against Fortune clenched his hand, He found her in the open air At his friend's door, and sleeping there. FABLE CXXXVI. THE TWO FOWLS. Two Barn-door Fowls in peace spent all their life, Until, at last, love, love lit up the strife: War's flames burst out. O Love! that ruined Troy, 'Twas thou who, by fierce quarrel, banished joy, And stained with blood and crime the Xanthus' tide! Long, long the combat raged 'tween wrath and pride, Until the rumour spread the whole town through, And all the crested people ran to view.
- 41. Many a well-plumed Helen was the prize Of him who conquered; but the vanquished flies— Skulks to the darkest and most hidden place,
- 42. THE TWO FOWLS.
- 43. And mourns his love with a dejected face. His rival, proud of recent victory, Exulting crows, and claims the sovereignty. The conquered rival, big with rage, dilates, Sharpens his beak, and Fortune invocates, Clapping his wings, while, maddened by defeat, The other skulks and plans a safe retreat. The victor on the roof is perched, to crow; A vulture sees the bragger far below. Adieu! love, pride, and glory, all are vain Beneath the vulture's beak;—so ends that reign. The rival soon returns to make his court To the fair dame, and victory to report, As he had half-a-dozen other wives, to say the least, You'll guess the chattering at his wedding feast. Fortune always rejoices in such blows: Insolent conquerors, beware of those. Still mistrust Fate, and dread security, Even the evening after victory.
- 44. FABLE CXXXVII. THE COACH AND THE FLY. Up a long dusty hill, deep sunk in sand, Six sturdy horses drew a Coach. The band Of passengers were pushing hard behind: Women, old men, and monks, all of one mind. Weary and spent they were, and faint with heat; Straight on their heads the sunbeams fiercely beat. In the hot air, just then, came buzzing by, Thinking to rouse the team, a paltry Fly. Stings one, and then another; views the scene: Believing that this ponderous machine Is by his efforts moved, the pole bestrides; And now upon the coachman's nose he rides. Soon as the wheels begin again to grind The upward road, and folks to push behind, He claims the glory; bustles here and there, Fussy and fast, with all the toil and care With which a general hurries up his men, To charge the broken enemy again,
- 45. And victory secure. The Fly, perplexed With all the work, confessed that she was vexed No one was helping, in that time of need. The monk his foolish breviary would read: He chose a pretty time! a woman sang: Let her and all her foolish songs go hang! Dame Fly went buzzing restless in their ears, And with such mockery their journey cheers. After much toil, the Coach moves on at last: Now let us breathe; the worst of it is past, The Fly exclaimed; it is quite smooth, you know; Come, my good nags, now pay me what you owe. So, certain people give themselves great airs, And meddlers mix themselves with one's affairs; Try to be useful, worry more and more, Until, at last, you show the fools the door.
- 46. FABLE CXXXVIII. THE INGRATITUDE AND INJUSTICE OF MEN TOWARDS FORTUNE. A merchant, trading o'er the seas, Became enriched by every trip. No gulf nor rock destroyed his ease; He lost no goods, from any ship. To others came misfortunes sad, For Fate and Neptune had their will. Fortune for him safe harbours had; His servants served with zeal and skill. He sold tobacco, sugar, spices, Silks, porcelains, or what you please; Made boundless wealth (this phrase suffices), And lived to clutch the golden keys. 'Twas luxury that gave him millions: In gold men almost talked to him. Dogs, horses, carriages, postillions, To give this man seemed Fortune's whim. A Friend asked how came all this splendour: I know the 'nick of time,' he said, When to be borrower and lender: My care and talent all this made. His profit seemed so very sweet, He risked once more his handsome gains; But, this time, baffled was his fleet: Imprudent, he paid all the pains.
- 47. One rotten ship sank 'neath a storm, And one to watchful pirates fell; A third, indeed, made port in form, But nothing wanted had to sell. Fortune gives but one chance, we know: All was reversed,—his servants thieves. Fate came upon him with one blow, And made the mark that seldom leaves. The Friend perceived his painful case. Fortune, alas! the merchant cries. Be happy, says his Friend, and face The world, and be a little wise. To counsel you is to give health: I know that all mankind impute To Industry their peace and wealth, To Fortune all that does not suit. Thus, if each time we errors make, That bring us up with sudden halt, Nothing's more common than to take Our own for Fate or Fortune's fault. Our good we always make by force, The evil fetters us so strong; For we are always right, of course, And Destiny is always wrong.
- 48. AN ANIMAL IN THE MOON.
- 49. FABLE CXXXIX. AN ANIMAL IN THE MOON. Some sages argue that all men are dupes, And that their senses lead the fools in troops; Other philosophers reverse this quite, And prove that man is nearly always right. Philosophy says true, senses mislead, If we judge only by them without heed; But if we mark the distance and reflect On atmosphere and what it will effect, The senses cheat none of us; Nature's wise: I'll give an instance. With my naked eyes I see the sun; how large is it, think you? Three feet at farthest? It appears so, true! But could I see it from a nearer sky, 'Twould seem of our vast universe the eye: The distance shows its magnitude, you see; My hand discovers angles easily. Fools think the earth is flat; it's round, I know; Some think it motionless, it moves so slow. Thus, in a word, my eyes have wisdom got,
- 50. The illusions of the senses cheat me not. My soul, beneath appearances, sees deep; My eye's too quick, a watch on it I keep; My ear, not slow to carry sounds, betrays; When water seems to bend a stick ten ways, My reason helps me out, and if my sight Lies always, yet it never cheats me quite: If I would trust my senses, very soon They'd tell me of the woman in the moon. What is there really?—No, mistrust your eyes, For what you see are inequalities. The surface of the moon has many regions, Here spread the plains, there mountains rise in legions.
- 51. AN ANIMAL IN THE MOON (2).
- 52. In light and shade strange figures you can trace—- An elephant, an ox, a human face. Not long ago, in England men perplexed, Saw, in a telescope, what savants vexed, A monster in this planet's mirror fair; Wild cries of horror filled the midnight air. Some change was pending—some mysterious change, Predicting wars, or a misfortune strange. The monarch came, he favoured learned men; The wondrous monster showed itself again: It was a mouse between the glasses shut— The source of war—the nibbler of a nut. The people laughed—oh, nation blessed with ease, When will the French have time for toils like these? Mars brings us glory's harvests; still the foe Shrinks down before us, dreading every blow; 'Tis we who seek them, sure that victory, Slave to our Louis, follows ceaselessly His flag; his laurels render us renowned: Yet memory has not left this mortal round. We wish for peace—for peace alone we sigh; Charles tastes the joys of rest: he would in war Display his valour, and his flag bear far, To reach the tranquil joy that now he shares. Would he could end our quarrels and our cares! What incense would be his, what endless fame! Did not Augustus win a glorious name,
- 53. Equal to Cæsar's in its majesty, And worthy of like reverence, may be? Oh, happy people, when will Peace come down, To dower our nation with her olive-crown? FABLE CXL. THE FORTUNE-TELLER. Opinion is the child of Chance, And this Opinion forms our taste. Against all people I advance These words. I find the world all haste— Infatuation; justice gone;
- 54. A torrent towards a goal unseen. We only know things will be done In their own way, as they have been. In Paris lived a Sorceress, Who told the people of their fate. All sought her:—men; girls loverless; A husband whom his wife thought late In dying; many a jealous woman. Ill-natured mothers, by the score, Came—for they all were simply human— To hear what Fortune had in store. Her tricks of trade were hardihood, Some terms of art, a neat address. Sometimes a prophecy proved good, And then they thought her nothing less Than Delphi's Pythoness of yore: Though ignorance itself was she; And made her wretched garret floor Highway for gullibility. Grown rich, she took a house, and bought A place of profit for her lord. The witch's garret soon was sought By a young girl, who never soared To witchery, save by eyes and voice. But yet they all came, as of old— The lucky, who in wealth rejoice, And poor—to have their fortunes told. The regulation had been made For this poor place, by her who late Had been its tenant; and the shade Sybillic hovered o'er its state.
- 55. In vain the maiden said, You mock. Read Fate!—I scarcely know my letters! But though such words, of course, might shock, They never could convince her betters. Predict—divine;—here's gold in pay, More than the learned get together. What wonder if the maid gave way, Despite herself, such gold to gather? For fortune-telling seemed the place All tumble-down, and weird, and broken: A broomstick, for the witches' chase, And many another mystic token; The witches' sabbath; all suggested The change of body, and of face; And so in Fate fools still invested. But what of her who made the place? She seeks the golden prize to gain, In gorgeous state, like any parrot; But people jeer and pass. In vain; They all go rushing to the garret. 'Tis custom governs everything. I've often seen, in courts of law, Some stupid barrister, who'll bring Briefs such as clever men ne'er saw. All a mistake: his eyes may glisten; They'll take him for some other man: One unto whom the world will listen. Explain me this, now, if you can.
- 56. FABLE CXLI. THE COBBLER AND THE BANKER. A Cobbler, who would sing from dawn to dark (A very merry soul to hear and see, As satisfied as all the Seven Wise Men could be), Had for a neighbour, not a paltry clerk, But a great Banker, who could roll in gold: A Crœsus, singing little, sleeping less; Who, if by chance he had the happiness, Just towards morning, to drop off, I'm told, Was by the Cobbler's merry singing woke. Loud he complain'd that Heaven did not keep For sale, in market-places, soothing sleep.
- 57. He sent, then, for the Cobbler ('twas no joke):— What, Gregory, do you earn in the half- year? Half-year, sir! said the Cobbler, very gaily; I do not reckon so. I struggle daily For the day's bread, and only hunger fear. Well, what a day?—what is your profit, man? Now more, now less;—the worst thing is those fêtes. Why, without them—and hang their constant dates!— The living would be tidy—drat the plan! Monsieur the Curé always a fresh saint Stuffs in his sermon every other week. The Banker laughed to hear the fellow speak, And utter with such naïveté his complaint. I wish, he said, to mount you on a throne; Here are a hundred crowns, knave—keep them all, They'll serve you well, whatever ill befall. The Cobbler thought he saw before him thrown All money in the earth that had been found. Home went he to conceal it in a vault, Safe from discovery and thieves' assault. There, too, he buried joy,—deep under ground; No singing now: he'd lost his voice from fear.
- 58. THE COBBLER AND THE BANKER.
- 59. Welcome to our website – the ideal destination for book lovers and knowledge seekers. With a mission to inspire endlessly, we offer a vast collection of books, ranging from classic literary works to specialized publications, self-development books, and children's literature. Each book is a new journey of discovery, expanding knowledge and enriching the soul of the reade Our website is not just a platform for buying books, but a bridge connecting readers to the timeless values of culture and wisdom. With an elegant, user-friendly interface and an intelligent search system, we are committed to providing a quick and convenient shopping experience. Additionally, our special promotions and home delivery services ensure that you save time and fully enjoy the joy of reading. Let us accompany you on the journey of exploring knowledge and personal growth! ebookultra.com