Croatian-German meeting on analysis and mathematical physics, March 25, 2021

1. Density theorems for Euclidean point
configurations
Vjekoslav Kovač (University of Zagreb)
Joint work with P. Durcik, K. Falconer, L. Rimanić, and A. Yavicoli
Supported by HRZZ UIP-2017-05-4129 (MUNHANAP)
Croatian–German meeting on analysis
and mathematical physics
March 25, 2021
1/63

2. Euclidean density theorems
There are patterns in large but otherwise arbitrary structures!
The main idea behind Ramsey theory (⊆ combinatorics), but also
widespread in other areas of mathematics.
2/63

3. Euclidean density theorems
Euclidean density theorems could belong to:
• geometric measure theory (28A12, etc.);
• Ramsey theory (05D10);
• arithmetic combinatorics (11B25, 11B30, etc.).
Harmonic analysis seems to be the most powerful tool for attacking
this type of problems.
3/63

4. Density theorems study “large” measurable sets
When is a measurable set A considered large?
• For A ⊆ [0, 1]d this means
|A| > 0
(the Lebesgue measure).
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5. Density theorems study “large” measurable sets
When is a measurable set A considered large?
• For A ⊆ Rd this means
δ(A) := lim sp
R→∞
sp
x∈Rd
|A ∩ (x + [0, R]d)|
Rd
> 0
(the upper Banach density).
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6. Classical results
A question by Székely (1982)
For every measurable set A ⊆ R2 satisfying δ(A) > 0 is there a
number λ0 = λ0(A) such that for each λ ∈ [λ0, ∞) there exist
points x, x0 ∈ A satisfying |x − x0| = λ?
6/63

7. Classical results
A question by Székely (1982)
For every measurable set A ⊆ R2 satisfying δ(A) > 0 is there a
number λ0 = λ0(A) such that for each λ ∈ [λ0, ∞) there exist
points x, x0 ∈ A satisfying |x − x0| = λ?
Yes:
Furstenberg, Katznelson, and Weiss (1980s);
Falconer and Marstrand (1986);
Bourgain (1986).
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8. Classical results — simplices
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.
8/63

9. Classical results — simplices
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).
Open question #1 (Bourgain?)
When n ≥ 2, does the same hold for subsets of Rn?
9/63

10. Classical results
Bourgain (1986) also gave a counterexample for 3-term arithmetic
progressions, i.e., isometric copies of dilates of {0, 1, 2}.
A :=
{︀
x ∈ Rd
: (∃m ∈ Z)
(︀
m − ϵ < |x|2
< m + ϵ
)︀}︀
10/63

11. Classical results
Bourgain (1986) also gave a counterexample for 3-term arithmetic
progressions, i.e., isometric copies of dilates of {0, 1, 2}.
A :=
{︀
x ∈ Rd
: (∃m ∈ Z)
(︀
m − ϵ < |x|2
< m + ϵ
)︀}︀
• Take some 0 < ϵ < 1/8. We have δ(A) = 2ϵ > 0.
• The parallelogram law:
|x|2
− 2|x + y|2
+ |x + 2y|2
= |y|2
.
• x, x + y, x + 2y ∈ A
dist(|x|2, Z), dist(|x + y|2, Z), dist(|x + 2y|2, Z) < ϵ
=⇒ dist(2|y|2, Z) < 4ϵ < 1/2
=⇒ not all large numbers are attained by |y|
11/63

12. Generalizations to other point configurations
Open question #2 (Graham? Furstenberg?)
Which point configurations P have the following property: for some
(sufficiently large) dimension d and every measurable A ⊆ Rd with
δ(A) > 0 there exists λ0 = λ0(P, A) ∈ (0, ∞) such that for every
λ ≥ λ0 the set A contains an isometric copy of λP?
The most general known positive result is due to Lyall and Magyar
(2019): this holds for products of vertex-sets of nondegenerate
simplices Δ1 × · · · × Δm.
The most general known negative result is due to Graham (1993): this
fails for configurations that cannot be inscribed in a sphere.
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13. Compact formulations
Δ = the set of vertices of a non-degenerate n-dimensional simplex
Theorem (simplices — compact formulation, Bourgain (1986))
Take δ ∈ (0, 1/2], A ⊆ [0, 1]n+1 measurable, |A| ≥ δ.
Then the set of “scales”
{λ ∈ (0, 1] : A contains an isometric copy of λΔ}
contains an interval of length at least
(︀
exp(δ−C(Δ,n))
)︀−1
.
Such a formulation is qualitatively weaker, but it is quantitative.
Such formulations initiate the race to find better dependencies on δ.
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14. General scheme of the approach
Abstracted from: Cook, Magyar, and Pramanik (2017)
N 0
λ
= configuration “counting” form, identifies the configuration
associated with the parameter λ > 0 (i.e., of “size” λ)
N 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
14/63

15. General scheme of the approach (continued)
Decompose:
N 0
λ
= N 1
λ
+
(︀
N ϵ
λ
− N 1
λ
)︀
+
(︀
N 0
λ
− N ϵ
λ
)︀
.
N 1
λ
= structured part,
N ϵ
λ
− N 1
λ
= error part,
N 0
λ
− N ϵ
λ
= uniform part.
15/63

16. General scheme of the approach (continued)
For the structured part N 1
λ
we need a lower bound
N 1
λ
≥ c(δ)
that is uniform in λ, but this should be a simpler/smoother problem.
For the uniform part N 0
λ
− N ϵ
λ
we want
lim
ϵ→0
⃒
⃒N 0
λ
− N ϵ
λ
⃒
⃒ = 0
uniformly in λ; this usually leads to some oscillatory integrals.
For the error part N ϵ
λ
− N 1
λ
one tries to prove
J
∑︁
j=1
⃒
⃒N ϵ
λj
− N 1
λj
⃒
⃒ ≤ C(ϵ)o(J)
for lacunary scales λ1 < · · · < λJ; this usually leads to some
multilinear singular integrals. 16/63

17. General scheme of the approach (continued)
We argue by contradiction. Take sufficiently many lacunary scales
λ1 < · · · < λJ such that N 0
λj
= 0 for each j.
The structured part
N 1
λj
≥ c(δ)
dominates the uniform part
⃒
⃒N 0
λj
− N ϵ
λj
⃒
⃒ 1 (for sufficiently small ϵ)
and the error part
⃒
⃒N ϵ
λj
− N 1
λj
⃒
⃒ C(ϵ) (for some j by pigeonholing)
for at least one index j. This contradicts N 0
λj
= 0.
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18. 1. Rectangular boxes
Newer results... a warm-up 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.
Previous particular cases by:
Lyall and Magyar (2016), for n = 2;
Durcik and K. (2018), general n, but in (R5)n.
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19. 1. Rectangular boxes — quantitative strengthening
Fix b1, . . . , bn 0 (box sidelengths).
Theorem (Durcik and K. (2020))
For 0 δ ≤ 1/2 and measurable A ⊆ ([0, 1]2)n with |A| ≥ δ there
exists an interval I = I(A, b1, . . . , bn) ⊆ (0, 1] of length at least
(︀
exp(δ−C(n)
)
)︀−1
s. t. for every λ ∈ I one can find x1, . . . , xn, y1, . . . , yn ∈ R2 satisfying
(x1 + r1y1, x2 + r2y2, . . . , xn + rnyn) ∈ A for (r1, . . . , rn) ∈ {0, 1}n
;
|yi| = λbi for i = 1, . . . , n.
This improves the bound of Lyall and Magyar (2019) of the form
(︀
exp(exp(· · · exp(C(n)δ−3·2n
) · · · ))
)︀−1
(a tower of height n). 19/63

20. 1. Rectangular boxes
σ = circle measure in R2, f = 1A
Configuration counting form:
N 0
λ
(f) :=
ˆ
(R2)2n
(︁ ∏︁
(r1,...,rn)∈{0,1}n
f(x1+r1y1, . . . , xn+rnyn)
)︁(︁ n
∏︁
k=1
dxk dσλbk
(yk)
)︁
N ϵ
λ
can be obtained by “heating up” N 0
λ
.
g = standard Gaussian, k = Δg.
The approach benefits from the heat equation
∂
∂t
(︀
gt(x)
)︀
=
1
2πt
kt(x).
20/63

21. 1. Rectangular boxes
Smoothened counting form:
N ϵ
λ
(f) :=
ˆ
(R2)2n
(︁
· · ·
)︁(︁ n
∏︁
k=1
(σ ∗ gϵ)λbk
(yk) dxk dyk
)︁
=
ˆ
(R2)2n
F (x)
(︁ n
∏︁
k=1
(σ ∗ gϵ)λbk
(x0
k
− x1
k
)
)︁
dx
F (x) :=
∏︁
(r1,...,rn)∈{0,1}n
f(xr1
1
, . . . , xrn
n
), dx := dx0
1
dx1
1
dx0
2
dx1
2
· · · dx0
n
dx1
n
21/63

22. 1. Rectangular boxes
It is sufficient to show
N 1
λ
(1A) δ2n
,
J
∑︁
j=1
⃒
⃒N ϵ
λj
(1A) − N 1
λj
(1A)
⃒
⃒ . ϵ−C
,
⃒
⃒N 0
λ
(1A) − N ϵ
λ
(1A)
⃒
⃒ . ϵc
for each scale λ 0 and lacunary scales λ1 · · · λJ.
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23. 1. Rectangular boxes — structured part
Partition “most” of the cube ([0, 1]2)n into rectangular boxes
Q1 × · · · × Qn, where
Qk = [lλbk, (l + 1)λbk) × [l0
λbk, (l0
+ 1)λbk)
We only need the box–Gowers–Cauchy–Schwarz inequality:
Q1×Q1×···×Qn×Qn
F (x) dx ≥
(︁
Q1×...×Qn
f
)︁2n
23/63

24. 1. Rectangular boxes — error part
N α
λ
(f) − N β
λ
(f) =
n
∑︁
m=1
L α,β,m
λ
(f)
L α,β,m
λ
(f) := −
1
2π
ˆ β
α
ˆ
(R2)2n
F (x) (σ ∗ kt)λbm (x0
m
− x1
m
)
×
(︁ ∏︁
1≤k≤n
k6=m
(σ ∗ gt)λbk (x0
k
− x1
k
)
)︁
dx
dt
t
24/63

25. 1. Rectangular boxes — error part (continued)
From
∑︀J
j=1
⃒
⃒N ϵ
λj
(1B) − N 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).
25/63

26. 1. Rectangular boxes — uniform part
Split F = F 0(x0
1
)F 0(x1
1
).
The uniform part boils down to the decay of ̂︀
σ:
⃒
⃒̂︀
σ(ξ)
⃒
⃒ . (1 + |ξ|)−1/2
.
ˆ
(R2)2
F (σλ ∗ ktλ)(x0
1
− x1
1
) dx0
1
dx1
1
=
ˆ
R2
(F 0
∗ σλ ∗ ktλ)(x0
1
) F 0
(x0
1
) dx0
1
=
ˆ
R2
(
F 0 ∗ σλ ∗ ktλ)(ξ) ̂︁
F 0(ξ) dξ
=
ˆ
R2
⃒
⃒ ̂︁
F 0(ξ)
⃒
⃒2
̂︀
σ(λξ) ̂︀
k(tλξ) dξ
26/63

27. 2. Anisotropic configurations
Polynomial generalizations?
• There are no triangles with sides λ, λ2, and λ3 for large λ
• One can look for triangles with sides λ, λ2 and a fixed angle θ
between them.
27/63

28. 2. Anisotropic configurations
We will be working with anisotropic power-type scalings
(x1, . . . , xn) 7→ (λa1
b1x1, . . . , λan
bnxn).
Here a1, a2, . . . , an, b1, b2, . . . , bn 0 are fixed parameters.
Crucial observation: the heat equation is unaffected by a power-type
change of the time variable
∂
∂t
(︀
gtab(x)
)︀
=
a
2πt
ktab(x).
28/63

29. 2.1 Anisotropic 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.
29/63

30. 2.1 Anisotropic simplices
Pattern counting form:
N 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:
N 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
30/63

31. 2.1 Anisotropic simplices (continued)
It is sufficient to show:
N 1
λ
(1B) δn+1
Rn+1
,
J
∑︁
j=1
⃒
⃒N ϵ
λj
(1B) − N 1
λj
(1B)
⃒
⃒ . ϵ−C
J1/2
Rn+1
,
⃒
⃒N 0
λ
(1B) − N 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.)
31/63

32. 2.1 Anisotropic simplices — structured part
σH = the spherical measure inside a subspace H.
N 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
.
32/63

33. 2.1 Anisotropic simplices — error part
N α
λ
(f) − N β
λ
(f) =
n
∑︁
m=1
L α,β,m
λ
(f)
L 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
k6=m
(f ∗ g(tλ)ak bk
)(x + λak
bkUek)
)︁
dμ(U) dx
dt
t
L 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
33/63

34. 2.1 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.
34/63

35. 2.1 Anisotropic simplices — error part (continued)
From
∑︀J
j=1
⃒
⃒N ϵ
λj
(1B) − N 1
λj
(1B)
⃒
⃒ 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.
35/63

36. 2.1 Anisotropic simplices — uniform part
⃒
⃒L 0,ϵ,n
λ
(f)
⃒
⃒ .
ˆ ϵ
0
ˆ
(Rn+1)n−1
kfkL2(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
⃒
⃒L 0,ϵ,n
λ
(f)
⃒
⃒ . kfkL2(Rn+1)
ˆ ϵ
0
(︁ˆ
Rn+1
⃒
⃒̂︀
f(ξ)
⃒
⃒2⃒
⃒̂︀
k(tan
λan
bnξ)
⃒
⃒2
I (λan
bnξ)dξ
)︁1/2 dt
t
I (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
⃒
⃒L 0,ϵ,n
λ
(f)
⃒
⃒ . kfk2
L2(Rn+1)
ˆ ϵ
0
tc
dt
t
36/63

37. 2.2 Anisotropic 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.
37/63

38. 2.3 Anisotropic trees
T = (V, E) be a finite tree with vertices V and edges E
Theorem (K. (2020))
For every measurable set A ⊆ R2 satisfying δ(A) 0 there is a
number λ0 = λ0(A, T , a1, . . . , an, b1, . . . , bn) such that for each
λ ∈ [λ0, ∞) one can find a set of points
{xv : v ∈ V} ⊆ A
satisfying
|xu − xv| = λak
bk for each edge k ∈ E joining vertices u, v ∈ V.
This is an anisotropic variant of the result by Lyall and Magyar (2018)
on certain distance graphs.
38/63

39. 2.3 Anisotropic configurations — error part
One would like to handle more general distance graphs (generalizing
simplices, boxes, and trees).
For the error part there is a lot of potential in applying entangled
multilinear singular integrals associated with bipartite graphs or
r-partite r-regular hypergraphs.
The only cases studied so far are:
the so-called “twisted paraproduct operator,”
K. (2010.); Durcik and Roos (2018);
the operators with cubical structure,
Durcik (2014, 2015); Durcik and Thiele (2018).
Dyadic models of these operators are significantly easier:
K. (2011); Stipčić (2019).
39/63

40. 3. Arithmetic progressions
Reformulation of Szemerédi’s theorem
For n ≥ 3 and d ≥ 1 there exists a constant C(n, d) such that for
0 δ ≤ 1/2 and a measurable set A ⊆ [0, 1]d with |A| ≥ δ one has
ˆ
[0,1]d
ˆ
[0,1]d
n−1
∏︁
k=0
1A(x + ky) dy dx
≥
⎧
⎨
⎩
(︀
exp(δ−C(n,d))
)︀−1
when 3 ≤ n ≤ 4,
(︀
exp(exp(δ−C(n,d)))
)︀−1
when n ≥ 5.
Follows from the best known type of bounds in Szemerédi’s theorem:
n = 3, log by Heath-Brown (1987)
n = 4, log by Green and Tao (2017)
n ≥ 5, log log by Gowers (2001)
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41. 3. Arithmetic progressions in other ℓp
-norms
Bourgain’s counterexample applies.
Cook, Magyar, and Pramanik (2015) decided to measure gap lengths
in the ℓp-norm for p 6= 1, 2, ∞.
Theorem (Cook, Magyar, and Pramanik (2015))
If p 6= 1, 2, ∞, d sufficiently large, A ⊆ Rd measurable, δ(A) 0,
then ∃λ0 = λ0(p, d, A) ∈ (0, ∞) such that for every λ ≥ λ0 one
can find x, y ∈ Rd satisfying x, x + y, x + 2y ∈ A and kykℓp = λ.
Open question #3 (Cook, Magyar, and Pramanik (2015))
Is it possible to lower the dimensional threshold all the way to d = 2
or d = 3?
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42. 3. Arithmetic progressions
Open question #4 (Durcik, K., and Rimanić)
Prove or disprove: if n ≥ 4, p 6= 1, 2, . . . , n − 1, ∞, d sufficiently
large, A ⊆ Rd measurable, δ(A) 0, then
∃λ0 = λ0(n, p, d, A) ∈ (0, ∞) such that for every λ ≥ λ0 one can
find x, y ∈ Rd satisfying x, x + y, . . . , x + (n − 1)y ∈ A and kykℓp = λ.
It is necessary to assume p 6= 1, 2, . . . , n − 1, ∞.
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43. 3. Arithmetic progressions — compact formulation
Theorem (Durcik and K. (2020))
Take n ≥ 3, p 6= 1, 2, . . . , n − 1, ∞, d ≥ D(n, p), δ ∈ (0, 1/2],
A ⊆ [0, 1]d measurable, |A| ≥ δ. Then the set of ℓp-norms of the
gaps of n-term APs in the set A contains an interval of length at least
⎧
⎨
⎩
(︀
exp(exp(δ−C(n,p,d)))
)︀−1
when 3 ≤ n ≤ 4,
(︀
exp(exp(exp(δ−C(n,p,d))))
)︀−1
when n ≥ 5.
Modifying Bourgain’s example → sharp regarding the values of p
One can take D(n, p) = 2n+3(n + p) → certainly not sharp
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44. 3. Arithmetic progressions
σ(x) = δ(kxkp
ℓp − 1) = a measure supported on the unit sphere in
the ℓp-norm
N 0
λ
(A) :=
ˆ
Rd
ˆ
Rd
n−1
∏︁
k=0
1A(x + ky) dσλ(y) dx
N 0
λ
(A) 0 =⇒ (∃x, y)
(︀
x, x+y, . . . , x+(n−1)y ∈ A, kykℓp = λ
)︀
N ϵ
λ
(A) :=
ˆ
Rd
ˆ
Rd
n−1
∏︁
k=0
1A(x + ky)(σλ ∗ φϵλ)(y) dy dx
for a smooth φ ≥ 0 with
´
Rd φ = 1
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45. 3. Arithmetic progressions — structured part
Proposition handling the structured part
If λ ∈ (0, 1], δ ∈ (0, 1/2], A ⊆ [0, 1]d, |A| ≥ δ, then
N 1
λ
(A) ≥
⎧
⎨
⎩
(︀
exp(δ−C)
)︀−1
when 3 ≤ n ≤ 4,
(︀
exp(exp(δ−C))
)︀−1
when n ≥ 5.
This is essentially the analytical reformulation of Szemerédi’s
theorem (just at scale λ).
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46. 3. Arithmetic progressions — error part
Proposition handling the error part
If J ∈ N, λj ∈ (2−j, 2−j+1] for j = 1, 2, . . . , J, ϵ ∈ (0, 1/2],
A ⊆ [0, 1]d measurable, then
J
∑︁
j=1
⃒
⃒N ϵ
λj
(A) − N 1
λj
(A)
⃒
⃒ ≤ ϵ−C
J1−2−n+2
.
Note the gain of J−2−n+2
over the trivial estimate.
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47. 3. Arithmetic progressions — uniform part
Proposition handling the uniform part
If d ≥ D(n, p), λ, ϵ ∈ (0, 1], A ⊆ [0, 1]d measurable, then
⃒
⃒N 0
λ
(A) − N ϵ
λ
(A)
⃒
⃒ ≤ Cϵ1/3
.
Consequently, the uniform part can be made arbitrarily small by
choosing a sufficiently small ϵ.
Here is where the assumption p 6= 1, 2, . . . , n − 1, ∞ is needed.
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48. 3. Arithmetic progressions — back to the error part
The most interesting part for us is the error part.
We need to estimate
J
∑︁
j=1
κj
(︀
N ϵ
λj
(A) − N 1
λj
(A)
)︀
for arbitrary scales λj ∈ (2−j, 2−j+1] and arbitrary complex signs κj,
with a bound that is sub-linear in J.
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49. 3. Arithmetic progressions — error part
It can be expanded as
ˆ
Rd
ˆ
Rd
n−1
∏︁
k=0
1A(x + ky)K(y) dy dx,
where
K(y) :=
J
∑︁
j=1
κj
(︀
(σλj ∗ φϵλj )(y) − (σλj ∗ φλj )(y)
)︀
is a translation-invariant Calderón–Zygmund kernel.
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50. 3. Arithmetic progressions — error part
If d = 1 and K(y) is a truncation of 1/y, then this becomes the
(dualized and truncated) multilinear Hilbert transform,
ˆ
R
ˆ
[−R,−r]∪[r,R]
n−1
∏︁
k=0
fk(x + ky)
dy
y
dx.
• When n ≥ 4, no Lp-bounds uniform in r, R are known.
• Tao (2016) showed a bound of the form o(J), where
J ∼ log(R/r) is the “number of scales” involved.
• Reproved and generalized by Zorin-Kranich (2016).
• Durcik, K., and Thiele (2016) showed a bound O(J1−ϵ
).
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51. 4. Some other arithmetic configurations
Allowed symmetries play a major role.
Note a difference between:
the so-called corners: (x, y), (x + s, y), (x, y + s) (harder),
isosceles right triangles: (x, y), (x + s, y), (x, y + t)
with kskℓ2 = ktkℓ2 (easier).
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52. 4.1 Corners
Theorem (Durcik, K., and Rimanić (2016))
If p 6= 1, 2, ∞, d sufficiently large, A ⊆ Rd × Rd measurable,
δ(A) 0, then ∃λ0 = λ0(p, d, A) ∈ (0, ∞) such that for every
λ ≥ λ0 one can find x, y, s ∈ Rd satisfying
(x, y), (x + s, y), (x, y + s) ∈ A and kskℓp = λ.
Generalizes the result of Cook, Magyar, and Pramanik (2015) via the
skew projection (x, y) 7→ y − x.
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53. 4.2 AP-extended boxes
Consider the configuration in Rd1 × · · · × Rdn consisting of:
(x1 + k1s1, x2 + k2s2, . . . , xn + knsn), k1, k2, . . . , kn ∈ {0, 1},
(x1+2s1, x2, . . . , xn), (x1, x2+2s2, . . . , xn), . . . , (x1, x2, . . . , xn+2sn).
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54. 4.2 AP-extended boxes
Consider the configuration in Rd1 × · · · × Rdn consisting of:
(x1 + k1s1, x2 + k2s2, . . . , xn + knsn), k1, k2, . . . , kn ∈ {0, 1},
(x1+2s1, x2, . . . , xn), (x1, x2+2s2, . . . , xn), . . . , (x1, x2, . . . , xn+2sn).
Fix b1, . . . , bn 0 and p 6= 1, 2, ∞.
Theorem (Durcik and K. (2018))
There exists a dimensional threshold dmin such that for any d1, d2, . . . ,
dn ≥ dmin and any measurable set A with δ(A) 0 one can find
λ0 0 with the property that for any λ ≥ λ0 the set A contains the
above 3AP-extended box with ksikℓp = λbi, i = 1, 2, . . . , n.
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55. 4.3 Corner-extended boxes
Consider the config. in Rd1 × Rd1 × · · · × Rdn × Rdn consisting of:
(x1 + k1s1, . . . , xn + knsn, y1, y2, . . . , yn), k1, k2, . . . , kn ∈ {0, 1},
(x1, x2, . . . , xn, y1+s1, y2, . . . , yn), . . . , (x1, x2, . . . , xn, y1, y2, . . . , yn+sn).
Fix b1, . . . , bn 0 and p 6= 1, 2, ∞.
Theorem (Durcik and K. (2018))
There exists a dimensional threshold dmin such that for any d1, . . . ,
dn ≥ dmin and any measurable set A with δ(A) 0 one can find
λ0 0 with the property that for any λ ≥ λ0 the set A contains the
above corner-extended box with ksikℓp = λbi, i = 1, 2, . . . , n.
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56. 5. Very dense sets
Theorem (Falconer, K., and Yavicoli (2020))
If d ≥ 2 and A ⊆ Rd is measurable with δ(A) 1 − 1
n−1
, then for
every n-point configuration P there exists λ0 0 s. t. for every
λ ≥ λ0 the set A contains an isometric copy of λP.
The result would be trivial for δ(A) 1 − 1
n
and rotations would not
even be needed there.
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57. 5. Very dense sets — lower bound
What can one say about the lower bound for such density threshold
(depending on n)?
Let us return to arithmetic progressions!
Theorem (Falconer, K., and Yavicoli (2020))
For all n, d ≥ 2 there exists a measurable set A ⊆ Rd of density at
least
1 −
10 log n
n1/5
s.t. there are arbitrarily large values of λ for which A contains no
congruent copy of λ{0, 1, . . . , n − 1}.
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58. 5. Very dense sets — lower bound
It is hard to come up with anything better than Bourgain’s
(counter)example.
Take ϵ := 10n−1/5 log n ∈ (0, 1) and define:
A :=
{︀
x ∈ Rd
: dist(|x|2
, Z) (1 − ϵ)/2
}︀
.
It has density 1 − ϵ.
If x0, x1, . . . , xn−1 ∈ A stand in an arithmetic progression, then the
parallelogram law gives
ak+2 − 2ak+1 + ak = 2λ2
for ak = |xk|2.
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59. 5. Very dense sets — lower bound
The main idea is the following:
• On the one hand, the first n terms of (ak) modulo 1 completely
avoid the interval [(1 − ϵ)/2, (1 + ϵ)/2] ⊆ T of length ϵ.
• On the other hand, for sufficiently large n, the first n terms of
the quadratic sequence (ak) should be “sufficiently uniformly
distributed” over T.
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60. 5. Very dense sets — lower bound
The main idea is the following:
• On the one hand, the first n terms of (ak) modulo 1 completely
avoid the interval [(1 − ϵ)/2, (1 + ϵ)/2] ⊆ T of length ϵ.
• On the other hand, for sufficiently large n, the first n terms of
the quadratic sequence (ak) should be “sufficiently uniformly
distributed” over T.
Take λ such that
λ2
∈ −1+
p
5
2
+ Z
which is “badly approximable” by rationals.
Use some inequality for the discrepancy, or the Erdős–Turán
inequality and work out van der Corput’s trick.
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61. 5. Very dense sets
Open question #5 (Falconer, K., and Yavicoli)
What is the smallest 0 ≤ δmin(d, n) 1 such that every measurable
set A ∈ Rd of upper density δmin(d, n) contains all sufficiently
large scale similar copies of all n-point configurations?
Previous results give
1 −
10 log n
n1/5
≤ δmin(d, n) ≤ 1 −
1
n − 1
.
Is it possible to improve either one of the two asymptotic bounds
1 − O(n−1/5 log n) and 1 − O(n−1) as n → ∞?
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62. 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 and oscillatory integrals.
• It gives superior quantitative bounds.
• It can be an overkill in relation to problems without any
arithmetic structure — it was primarily devised to handle
arithmetic progressions and similar patterns.
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63. Thank you for your attention!
63/63