Density theorems for anisotropic point configurations

Density theorems for anisotropic point
conﬁgurations
Vjekoslav Kovač (University of Zagreb)
Supported by HRZZ UIP-2017-05-4129 (MUNHANAP)
Analysis and PDE Webinar
IMPA, Rio de Janeiro, July 21, 2020
1/24

A type of problems in the Euclidean Ramsey theory
Euclidean density theorems of the “strongest” type
• Bourgain (1986)
•
...
• Cook, Magyar, and Pramanik (2017)
•
...
We study positive density measurable sets A ⊆ Rd:
δ(A) := lim sp
R→∞
sp
x∈Rd
|A ∩ (x + [0, R]d)|
Rd
> 0
(upper Banach density)
2/24

Classical results
Theorem (Furstenberg, Katznelson, and Weiss (1980s);
Falconer and Marstrand (1986))
For every measurable set A ⊆ R2 satisfying δ(A) > 0 there is a
number λ0 = λ0(A) such that for each λ ∈ [λ0, ∞) there exist
points x, x ∈ A satisfying |x − x | = λ.
Δ = the set of vertices of a non-degenerate n-dimensional simplex
Theorem (Bourgain (1986))
For every measurable set A ⊆ Rn+1 satisfying δ(A) > 0 there is a
number λ0 = λ0(A, Δ) such that for each λ ∈ [λ0, ∞) the set A
contains an isometric copy of λΔ.
Alternative proofs by Lyall and Magyar (2016, 2018, 2019) 3/24

Another example
= the set of vertices of an n-dimensional rectangular box
Theorem (Lyall and Magyar (2019))
For every measurable set A ⊆ R2 × · · · × R2 = (R2)n satisfying
δ(A) > 0 there is a number λ0 = λ0(A, ) such that for each
λ ∈ [λ0, ∞) the set A contains an isometric copy of λ with sides
parallel to the distinguished 2-dimensional coordinate planes.
Alternative proofs by Durcik and K. (2018: in (R5)n, 2020)
Lyall and Magyar (2019) also handle products of nondegenerate
simplices Δ1 × · · · × Δn
4/24

Polynomial generalizations?
• There are no triangles with sides λ, λ2, and λ3 for large λ
• One can look for right-angled triangles with legs of length λ, λ2
We will be working with anisotropic power-type scalings
(x1, . . . , xn) → (λa1
b1x1, . . . , λan
bnxn)
Here a1, a2, . . . , an, b1, b2, . . . , bn > 0 are ﬁxed parameters
a1 = · · · = an = 1 is the (classical) “linear” case
5/24

Anisotropic result for (right-angled) simplices
Theorem (K. (2020))
For every measurable set A ⊆ Rn+1 satisfying δ(A) > 0 there is a
number λ0 = λ0(A, a1, . . . , an, b1, . . . , bn) such that for each
λ ∈ [λ0, ∞) one can ﬁnd a point x ∈ Rn+1 and mutually orthogonal
vectors y1, y2, . . . , yn ∈ Rn+1 satisfying
{x, x + y1, x + y2, . . . , x + yn} ⊆ A
and
|yi| = λai
bi; i = 1, 2, . . . , n.
6/24

Anisotropic result for boxes
Theorem (K. (2020))
For every measurable set A ⊆ (R2)n satisfying δ(A) > 0 there is a
number λ0 = λ0(A, a1, . . . , an, b1, . . . , bn) such that for each
λ ∈ [λ0, ∞) one can ﬁnd points x1, . . . , xn, y1, . . . , yn ∈ R2 satisfying
{︀
(x1 + r1y1, x2 + r2y2, . . . , xn + rnyn) : (r1, . . . , rn) ∈ {0, 1}n
}︀
⊆ A
and
|yi| = λai
bi; i = 1, 2, . . . , n.
7/24

General scheme of the approach
Abstracted from: Cook, Magyar, and Pramanik (2017)
0
λ
= pattern counting form, identiﬁes the conﬁguration associated
with the parameter λ > 0
ϵ
λ
= smoothened counting form; the picture is blurred up to scale
0 < ϵ ≤ 1
The largeness–smoothness multiscale approach:
• λ = scale of largeness
• ϵ = scale of smoothness
8/24

General scheme of the approach (continued)
Decompose:
0
λ
= 1
λ
+
(︀ ϵ
λ
− 1
λ
)︀
+
(︀ 0
λ
− ϵ
λ
)︀
1
λ
= structured part; its lower bound is a simpler problem
0
λ
− ϵ
λ
= uniform part; oscillatory integrals guarantee that it
→ 0 as ϵ → 0 uniformly in λ
ϵ
λ
− 1
λ
= error part; one tries to prove
J∑︁
j=1
⃒
⃒ ϵ
λj
− 1
λj
⃒
⃒ ≤ C(ϵ)o(J)
for lacunary scales λ1 < · · · < λJ using multilinear singular integrals
9/24

General scheme of the approach (continued)
Durcik and K. (2020): ϵ
λ
could be obtained by “heating up” 0
λ
g = standard Gaussian, k = Δg
The present talk mainly beneﬁts from the fact that the heat equation
∂
∂t
(︀
gt(x)
)︀
=
1
2πt
kt(x)
is unaﬀected by a power-type change of the time variable
∂
∂t
(︀
gtab(x)
)︀
=
a
2πt
ktab(x)
10/24

Anisotropic simplices
Pattern counting form:
0
λ
(f) :=
ˆ
Rn+1
ˆ
SO(n+1,R)
f(x)
(︁ n∏︁
k=1
f(x + λak
bkUek)
)︁
dμ(U) dx
Smoothened counting form:
ϵ
λ
(f) :=
ˆ
Rn+1
ˆ
SO(n+1,R)
f(x)
(︁ n∏︁
k=1
(f∗g(ϵλ)ak bk
)(x+λak
bkUek)
)︁
dμ(U) dx
11/24

Anisotropic simplices (continued)
It is suﬃcient to show:
1
λ
(1B) δn+1
Rn+1
J∑︁
j=1
⃒
⃒ ϵ
λj
(1B) − 1
λj
(1B)
⃒
⃒ ϵ−C
J1/2
Rn+1
⃒
⃒ 0
λ
(1B) − ϵ
λ
(1B)
⃒
⃒ ϵc
Rn+1
λ > 0, J ∈ N, 0 < λ1 < · · · < λJ satisfy λj+1 ≥ 2λj,
R > 0 is suﬃciently large, 0 < δ ≤ 1,
B ⊆ [0, R]n+1 has measure |B| ≥ δRn+1
(We take B := (A − x) ∩ [0, R]n+1 for appropriate x, R)
12/24

Anisotropic simplices — structured part
σH = the spherical measure inside a subspace H
ϵ
λ
(f) =
ˆ
(Rn+1)n+1
f(x)
(︁ n∏︁
k=1
(f ∗ g(ϵλ)ak bk
)(x + yk)
)︁
dσ{y1,...,yn−1}⊥
λan bn
(yn)
dσ{y1,...,yn−2}⊥
λan−1 bn−1
(yn−1) · · · dσ{y1}⊥
λa2 b2
(y2) dσRn+1
λa1 b1
(y1) dx
σH
∗ g ≥
(︁
min
B(0,2)
g
)︁
1B(0,1) φ := |B(0, 1)|−1
1B(0,1)
Bourgain’s lemma (1988):
[0,R]d
f(x)
(︁ n∏︁
k=1
(f ∗ φtk )(x)
)︁
dx
(︁
[0,R]d
f(x) dx
)︁n+1
13/24

Anisotropic simplices — error part
α
λ
(f) − β
λ
(f) =
n∑︁
m=1
α,β,m
λ
(f)
α,β,m
λ
(f) := −
am
2π
ˆ β
α
ˆ
Rn+1
ˆ
SO(n+1,R)
f(x) (f ∗ k(tλ)am bm
)(x + λam
bmUem)
×
(︁ ∏︁
1≤k≤n
k=m
(f ∗ g(tλ)ak bk
)(x + λak
bkUek)
)︁
dμ(U) dx
dt
t
α,β,n
λ
(f) = −
an
2π
ˆ β
α
ˆ
(Rn+1)n+1
f(x)
(︁ n−1∏︁
k=1
(f ∗ g(tλ)ak bk
)(x + yk)
)︁
× (f ∗ k(tλ)an bn
)(x + yn) dσ{y1,...,yn−1}⊥
λan bn
(yn) · · · dσ{y1}⊥
λa2 b2
(y2) dσRn+1
λa1 b1
(y1) dx
dt
t
14/24

Anisotropic simplices — error part (continued)
Elimination of measures σH:
convolution with a Gaussian Schwartz tail
a superposition of dilated Gaussians
The last inequality is the Gaussian domination trick of Durcik (2014)
The same trick also nicely converts the discrete scales λj into
continuous scales
— not strictly needed here, but convenient for tracking down
quantitative dependence on ϵ
15/24

Anisotropic simplices — error part (continued)
From
∑︀J
j=1
⃒
⃒ ϵ
λj
(1B) − 1
λj
(1B)
⃒
⃒p
we are lead to study
ΛK(f0, . . . , fn) :=
ˆ
(Rd)n+1
K(x1 − x0, . . . , xn − x0)
(︁ n∏︁
k=0
fk(xk) dxk
)︁
Multilinear C–Z operators: Coifman and Meyer (1970s); Grafakos and
Torres (2002)
Here K is a C–Z kernel, but with respect to the quasinorm associated
with our anisotropic dilation structure
16/24

Anisotropic simplices — uniform part
⃒
⃒ 0,ϵ,n
λ
(f)
⃒
⃒
ˆ ϵ
0
ˆ
(Rn+1)n−1
f L2(Rn+1)
⃦
⃦f ∗
(︀
σ{y1,...,yn−1}⊥
∗ ktan
)︀
λan bn
⃦
⃦
L2(Rn+1)
dσ{y1,...,yn−2}⊥
λan−1 bn−1
(yn−1) · · · dσ{y1}⊥
λa2 b2
(y2) dσRn+1
λa1 b1
(y1)
dt
t
⃒
⃒ 0,ϵ,n
λ
(f)
⃒
⃒ f L2(Rn+1)
ˆ ϵ
0
(︁ˆ
Rn+1
⃒
⃒̂︀f(ξ)
⃒
⃒2⃒
⃒̂︀k(tan
λan
bnξ)
⃒
⃒2
(λan
bnξ)dξ
)︁1/2 dt
t
(ζ) :=
ˆ
(Rn+1)n−1
⃒
⃒̂︀σ{y1,...,yn−1}⊥
(ζ)
⃒
⃒2
dσ{y1,...,yn−2}⊥
(yn−1) · · · dσRn+1
(y1)
⃒
⃒̂︀σ{y1,...,yn−1}⊥
(ζ)
⃒
⃒ dist
(︀
ζ, span({y1, . . . , yn−1})
)︀−1/2
⃒
⃒ 0,ϵ,n
λ
(f)
⃒
⃒ f 2
L2(Rn+1)
ˆ ϵ
0
tc
dt
t
17/24

Anisotropic boxes
Pattern counting form (σ = circle measure in R2):
0
λ
(f) :=
ˆ
(R2)2n
(︁ ∏︁
(r1,...,rn)∈{0,1}n
f(x1+r1y1, . . . , xn+rnyn)
)︁(︁ n∏︁
k=1
dxk dσλak bk
(yk)
)︁
Smoothened counting form:
ϵ
λ
(f) :=
ˆ
(R2)2n
(︁
· · ·
)︁(︁ n∏︁
k=1
(σ ∗ gϵak )λak bk
(yk) dxk dyk
)︁
=
ˆ
(R2)2n
(x)
(︁ n∏︁
k=1
(σ ∗ gϵak )λak bk
(x0
k
− x1
k
)
)︁
dx
(x) :=
∏︁
(r1,...,rn)∈{0,1}n
f(xr1
1
, . . . , xrn
n
), dx := dx0
1
dx1
1
dx0
2
dx1
2
· · · dx0
n
dx1
n
18/24

Anisotropic boxes (continued)
It is suﬃcient to show
1
λ
(1B) δ2n
R2n
J∑︁
j=1
⃒
⃒ ϵ
λj
(1B) − 1
λj
(1B)
⃒
⃒ ϵ−C
R2n
⃒
⃒ 0
λ
(1B) − ϵ
λ
(1B)
⃒
⃒ ϵc
R2n
B ⊆ ([0, R]2)n has measure |B| ≥ δR2n
19/24

Anisotropic boxes — structured part
Partition “most” of the cube ([0, R]2)n into rectangular boxes
Q1 × · · · × Qn, where
Qk = [lλak
bk, (l + 1)λak
bk) × [l λak
bk, (l + 1)λak
bk)
We only need the box–Gowers–Cauchy–Schwarz inequality:
Q1×Q1×···×Qn×Qn
(x) dx ≥
(︁
Q1×...×Qn
f
)︁2n
20/24

Anisotropic boxes — error part
α
λ
(f) − β
λ
(f) =
n∑︁
m=1
α,β,m
λ
(f)
α,β,m
λ
(f) := −
am
2π
ˆ β
α
ˆ
(R2)2n
(x) (σ ∗ ktam )λam bm (x0
m
− x1
m
)
×
(︁ ∏︁
1≤k≤n
k=m
(σ ∗ gtak )λak bk (x0
k
− x1
k
)
)︁
dx
dt
t
21/24

Anisotropic boxes — error part (continued)
From
∑︀J
j=1
⃒
⃒ ϵ
λj
(1B) − 1
λj
(1B)
⃒
⃒ we are lead to study
ΘK((fr1,...,rn )(r1,...,rn)∈{0,1}n )
:=
ˆ
(Rd)2n
∏︁
(r1,...,rn)∈{0,1}n
fr1,...,rn (x1 + r1y1, . . . , xn + rnyn)
)︁
K(y1, . . . , yn)
(︁ n∏︁
k=1
dxk dyk
)︁
Entangled multilinear singular integral forms with cubical structure:
Durcik (2014), K. (2010), Durcik and Thiele (2018: entangled
Brascamp–Lieb)
22/24

Anisotropic boxes — uniform part
Exactly the same as for the simplices
One only needs some decay of ̂︀σ
23/24

Conclusion
• The largeness–smoothness multiscale approach is quite ﬂexible
• Its applicability largely depends on the current state of the art
on estimates for multilinear singular integrals
• It gives superior bounds (when one cares about quantitative
aspects)
• It can be an overkill in relation to problems without any
arithmetic structure — it was devised to handle arithmetic
progressions and similar patterns
Thank you for your attention!
24/24

Density theorems for anisotropic point configurations

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