Introduction
Our results
Applications
Concluding remarks
Subsets of Zp with small Wiener norm and
arithmetic progressions
...
Introduction
Our results
Applications
Concluding remarks
Littlewood conjecture
Let A = {a1 < · · · < an} ⊂ Z.
Fourier tran...
Introduction
Our results
Applications
Concluding remarks
If A = {1, 2, . . . , n} then
1
0
n
j=1
e2πiaj u
du ∼ log n .
A i...
Introduction
Our results
Applications
Concluding remarks
f : Z → C .
Fourier transform
ˆf (u) =
x∈Z
f (x)e2πixu
.
Wiener n...
Introduction
Our results
Applications
Concluding remarks
General setting
G = Z, ˆG = R/(2πZ).
Green–Sanders, general abeli...
Introduction
Our results
Applications
Concluding remarks
Finite fields setting
Let p be a prime number.
Fp = Z/pZ has just ...
Introduction
Our results
Applications
Concluding remarks
Wiener norm for subsets of Fp
Littlewood conjecture in Fp.
Theore...
Introduction
Our results
Applications
Concluding remarks
Smaller sets
Theorem 1
Let p be a prime number, A ⊂ Zp, 0 < η = |...
Introduction
Our results
Applications
Concluding remarks
So, a nontrivial bound in Littlewood conjecture in Zp for
|A| ≫
p...
Introduction
Our results
Applications
Concluding remarks
Indeed, for any A = {a1, . . . , an} ⊆ Fp, s.t. n ≪ log p.
By Dir...
Introduction
Our results
Applications
Concluding remarks
Very small sets
Theorem 2
Let p be a prime number, A ⊂ Fp, and
|A...
Introduction
Our results
Applications
Concluding remarks
Additive dimension
Dissociated sets
Let G be an abelian group. A ...
Introduction
Our results
Applications
Concluding remarks
Main Lemma
Let A ⊆ G be a set, and 1A W (G) ≤ K. Then
dim(A) ≪ K2...
Introduction
Our results
Applications
Concluding remarks
Using Dirichlet Theorem for elements of Λ we find q s.t.
qλj
p
≤
1...
Introduction
Our results
Applications
Concluding remarks
Main Lemma follows from Lemma on additive relations
between eleme...
Introduction
Our results
Applications
Concluding remarks
Put B = Λ, dim(A) = |Λ|.
|Λ|2k
K2k−2|A|
≤ [Lemma on additive rela...
Introduction
Our results
Applications
Concluding remarks
Medium size
Theorem 3
Let p be a prime number, A ⊂ Fp,
exp (log p...
Introduction
Our results
Applications
Concluding remarks
Proof of Theorem 3
Methods of additive combinatorics.
Lemma on ad...
Introduction
Our results
Applications
Concluding remarks
Balog–Szemer´edi–Gowers
Theorem (Balog–Szemer´edi–Gowers)
Let G b...
Introduction
Our results
Applications
Concluding remarks
Theorem (Freiman, 1973)
Let A ⊆ Z, and |A + A| ≤ K|A|. Then there...
Introduction
Our results
Applications
Concluding remarks
Modern form of Freiman’s theorem
Theorem (Konyagin)
Let A ⊆ Z, an...
Introduction
Our results
Applications
Concluding remarks
Good subset of a set with small Wiener norm
Good subset
Let A ⊂ F...
Introduction
Our results
Applications
Concluding remarks
Recalling
f : T → C , T = R/(2πZ) .
Fourier transform
ˆf (k) = (2...
Introduction
Our results
Applications
Concluding remarks
Theorem (Beurling–Helson, 1953)
Let ϕ : T → T be a continuous map...
Introduction
Our results
Applications
Concluding remarks
Conjecture (Kahane, 1962)
Let ϕ : T → T be a continuous map. Supp...
Introduction
Our results
Applications
Concluding remarks
Theorem (Lebedev, 2012)
Let ϕ : T → T be a continuous map. Suppos...
Introduction
Our results
Applications
Concluding remarks
Discretization
Let ϕ : T → T be a continuous map, N be an integer...
Introduction
Our results
Applications
Concluding remarks
Θ(m) := max
|n|≤m
einϕ
A(T) .
Lemma (Lebedev)
Let ϕ : T → T be a ...
Introduction
Our results
Applications
Concluding remarks
By Dirichlet Theorem, we have
ϕ∗
(k) ≈
pk
Q
:= ϕ′
(k) .
For any n...
Introduction
Our results
Applications
Concluding remarks
Put As = A ∩ (A + s).
A U3 :=
s
E(As ) .
Criterium
E3(A) and A U3...
Introduction
Our results
Applications
Concluding remarks
Corollary
Put As = A ∩ (A + s).
Rough structure
For any set A wit...
Introduction
Our results
Applications
Concluding remarks
Weak counterexample to Gowers construction
Recall As = A ∩ (A + s...
Introduction
Our results
Applications
Concluding remarks
Thank you for your attention!
S. V. Konyagin, I. D. Shkredov Subs...
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Ilya Shkredov – Subsets of Z/pZ with small Wiener norm and arithmetic progressions

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It is proved that any subset of Z/pZ, p is a prime number, having small Wiener norm (l_1-norm of its Fourier transform) contains a subset which is close to be an arithmetic progression. We apply the obtained results to get some progress in so-called Littlewood conjecture in Z/pZ as well as in a quantitative version of Beurling-Helson theorem.

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Ilya Shkredov – Subsets of Z/pZ with small Wiener norm and arithmetic progressions

  1. 1. Introduction Our results Applications Concluding remarks Subsets of Zp with small Wiener norm and arithmetic progressions S. V. Konyagin, I. D. Shkredov Steklov Mathematical Institute S. V. Konyagin, I. D. Shkredov Subsets of Zp with small Wiener norm and arithmetic progressi
  2. 2. Introduction Our results Applications Concluding remarks Littlewood conjecture Let A = {a1 < · · · < an} ⊂ Z. Fourier transform ˆ1A(u) := n j=1 e2πiaj u , u ∈ [0, 1] . Littlewood conjecture 1 0 n j=1 e2πiaj u du ≫ log n . Littlewood conjecture was proved independently by S.V. Konyagin (1981) and O.C. McGehee, L. Pigno, B. Smith (1981). S. V. Konyagin, I. D. Shkredov Subsets of Zp with small Wiener norm and arithmetic progressi
  3. 3. Introduction Our results Applications Concluding remarks If A = {1, 2, . . . , n} then 1 0 n j=1 e2πiaj u du ∼ log n . A is an arithmetic progression. Littlewood conjecture is a direct question. Inverse question : small sum imply the structure of A. S. V. Konyagin, I. D. Shkredov Subsets of Zp with small Wiener norm and arithmetic progressi
  4. 4. Introduction Our results Applications Concluding remarks f : Z → C . Fourier transform ˆf (u) = x∈Z f (x)e2πixu . Wiener norm f W (Z) := 1 0 |ˆf (u)|du . Banach algebra fg W (Z) ≤ f W (Z) g W (Z) , AW (Z) = {f : f W (Z) < ∞} . S. V. Konyagin, I. D. Shkredov Subsets of Zp with small Wiener norm and arithmetic progressi
  5. 5. Introduction Our results Applications Concluding remarks General setting G = Z, ˆG = R/(2πZ). Green–Sanders, general abelian groups Let G be any abelian group and 1A W (G) ≤ K. Then 1A(x) = L j=1 ±1Hj +xj (x) where xj ∈ G, Hj ⊆ G are subgroups and L ≤ eeCK4 . For any subgroup H ⊆ G, we have 1H W (G) = 1 . S. V. Konyagin, I. D. Shkredov Subsets of Zp with small Wiener norm and arithmetic progressi
  6. 6. Introduction Our results Applications Concluding remarks Finite fields setting Let p be a prime number. Fp = Z/pZ has just two subgroups {0} and Fp. Let f : Fp → C. Fourier transform ˆf (x) := 1 p k∈Fp f (k)e−2πikx/p . Wiener norm f W (Fp) := x∈Fp |ˆf (x)| . S. V. Konyagin, I. D. Shkredov Subsets of Zp with small Wiener norm and arithmetic progressi
  7. 7. Introduction Our results Applications Concluding remarks Wiener norm for subsets of Fp Littlewood conjecture in Fp. Theorem (Green–Konyagin, Sanders) Let p be a prime number, A ⊂ Fp, 0 < η = |A|/p < 1/2. Suppose that η ≫ 1 (log p)0.24 . Then 1A W (Fp) ≫ η3/2 log1/2 p (log log p)3/2 . So, even in the case η ≫ 1, we have 1A W (Zp) ≫ log1/2 p (log log p)3/2 . S. V. Konyagin, I. D. Shkredov Subsets of Zp with small Wiener norm and arithmetic progressi
  8. 8. Introduction Our results Applications Concluding remarks Smaller sets Theorem 1 Let p be a prime number, A ⊂ Zp, 0 < η = |A|/p < 1/2. If η ≥ (log p)−1/4 (log log p)1/2 then 1A W (Fp) ≫ (log p)1/2 (log log p)−1 η3/2 × × 1 + log η2 (log p)1/2 (log log p)−1 −1/2 , and if η < (log p)−1/4 (log log p)1/2 then 1A W (Fp) ≫ η1/2 (log p)1/4 (log log p)−1/2 . S. V. Konyagin, I. D. Shkredov Subsets of Zp with small Wiener norm and arithmetic progressi
  9. 9. Introduction Our results Applications Concluding remarks So, a nontrivial bound in Littlewood conjecture in Zp for |A| ≫ p log log p log1/2 p . Proof : combining Sanders’ method with random shifts. On the other hand for very small sets |A| ≪ log p it is easy to see that A W ≫ log |A| by Konyagin, McGehee–Pigno–Smith result. S. V. Konyagin, I. D. Shkredov Subsets of Zp with small Wiener norm and arithmetic progressi
  10. 10. Introduction Our results Applications Concluding remarks Indeed, for any A = {a1, . . . , an} ⊆ Fp, s.t. n ≪ log p. By Dirichlet theorem there is q = 0 aj q p ≤ 1 3 Thus, qA ⊆ [−p/3, p/3] and we apply Konyagin, McGehee–Pigno–Smith result in Z, we obtain A W ≫ log |A| S. V. Konyagin, I. D. Shkredov Subsets of Zp with small Wiener norm and arithmetic progressi
  11. 11. Introduction Our results Applications Concluding remarks Very small sets Theorem 2 Let p be a prime number, A ⊂ Fp, and |A| ≤ exp (log p/ log log p)1/3 . Then 1A W (Fp) ≫ log |A| . Proof : computing additive dimension of A. S. V. Konyagin, I. D. Shkredov Subsets of Zp with small Wiener norm and arithmetic progressi
  12. 12. Introduction Our results Applications Concluding remarks Additive dimension Dissociated sets Let G be an abelian group. A set Λ = {λ1, . . . , λd } ⊆ G is called dissociated if any equation of the form d j=1 εj λj = 0 , where εj ∈ {0, ±1} implies εj = 0 for all j. Exm. G = Fn 2. S ⊆ G be a finite set. The additive dimension of S is the size of a maximal dissociated subset Λ ⊆ S. S. V. Konyagin, I. D. Shkredov Subsets of Zp with small Wiener norm and arithmetic progressi
  13. 13. Introduction Our results Applications Concluding remarks Main Lemma Let A ⊆ G be a set, and 1A W (G) ≤ K. Then dim(A) ≪ K2 log |A| K2 . Thus, any a ∈ A can be represented as a = dim(A) j=1 εj λj , εj ∈ {0, −1, 1} , λj ∈ Λ , and dim(A) = |Λ| ≪ K2 log |A| K2 . S. V. Konyagin, I. D. Shkredov Subsets of Zp with small Wiener norm and arithmetic progressi
  14. 14. Introduction Our results Applications Concluding remarks Using Dirichlet Theorem for elements of Λ we find q s.t. qλj p ≤ 1 p|Λ| , ∀λj ∈ Λ , we get qa p ≤ dim(A) j=1 qλj p ≤ 1 3p , ∀a ∈ A . Thus, qA ⊆ [−p/3, p/3] and we apply Littlewood in Z. S. V. Konyagin, I. D. Shkredov Subsets of Zp with small Wiener norm and arithmetic progressi
  15. 15. Introduction Our results Applications Concluding remarks Main Lemma follows from Lemma on additive relations between elements of any subset of S. Lemma on additive relations Let A ⊆ G be a set, and 1A W (G) ≤ K. Then for any B ⊆ A, we have |{b1 + · · · + bk = b′ 1 + · · · + b′ k : bj , b′ j ∈ B}| ≥ |B|2k K2k−2|A| . The quantity above is called Tk(B). Proof : H¨older inequality. S. V. Konyagin, I. D. Shkredov Subsets of Zp with small Wiener norm and arithmetic progressi
  16. 16. Introduction Our results Applications Concluding remarks Put B = Λ, dim(A) = |Λ|. |Λ|2k K2k−2|A| ≤ [Lemma on additive relations] ≤ |{λ1 + · · · + λk = λ′ 1 + · · · + λ′ k : λj , λ′ j ∈ Λ}| ≤ (Ck)k |Λ|k Hence |Λ| ≪ K2 k |A| K2 1/k . Putting k ∼ log |A| K2 , we get |Λ| ≪ K2 log |A| K2 . S. V. Konyagin, I. D. Shkredov Subsets of Zp with small Wiener norm and arithmetic progressi
  17. 17. Introduction Our results Applications Concluding remarks Medium size Theorem 3 Let p be a prime number, A ⊂ Fp, exp (log p/ log log p)1/3 ≤ |A| ≤ p/3. Then 1A W ≫ (log(p/|A|))1/3 (log log(p/|A|))−1+o(1) . Corollary 1) A nontrivial lower bound of Wiener norm for any A ⊂ Fp. 2) Weak universal bound, |A| ≤ p/2. 1A W ≫ (log log |A|)1/3−o(1) . S. V. Konyagin, I. D. Shkredov Subsets of Zp with small Wiener norm and arithmetic progressi
  18. 18. Introduction Our results Applications Concluding remarks Proof of Theorem 3 Methods of additive combinatorics. Lemma on additive relations, again Let A ⊆ G be a set, and 1A W (G) ≤ K. Then for any B ⊆ A, we have |{b1 + · · · + bk = b′ 1 + · · · + b′ k : bj , b′ j ∈ B}| ≥ |B|2k K2k−2|A| . In particular, T2(A) ≥ |A|3 /K2 . S. V. Konyagin, I. D. Shkredov Subsets of Zp with small Wiener norm and arithmetic progressi
  19. 19. Introduction Our results Applications Concluding remarks Balog–Szemer´edi–Gowers Theorem (Balog–Szemer´edi–Gowers) Let G be an abelian group, and A ⊆ G be a finite set. Suppose that T2(A) ≥ |A|3 /L. Then there is A∗ ⊆ A such that |A∗| ≥ |A|/C1(L) , and |A∗ + A∗| ≤ C2(L)|A∗| , where C1, C2 depend on L polynomially. Examples: arithmetic progressions A = P = {a, a + s, . . . , a + d(k − 1)} , generalized arithmetic progressions (GAP) A = P1 + · · · + Pd , large subsets of GAP. S. V. Konyagin, I. D. Shkredov Subsets of Zp with small Wiener norm and arithmetic progressi
  20. 20. Introduction Our results Applications Concluding remarks Theorem (Freiman, 1973) Let A ⊆ Z, and |A + A| ≤ K|A|. Then there is a GAP Q = P1 + · · · + Pd such that A ⊆ Q and |Q| ≤ C|A| , where d, C depend on K only. Thus, A is a large subset of a generalized arithmetic progression. S. V. Konyagin, I. D. Shkredov Subsets of Zp with small Wiener norm and arithmetic progressi
  21. 21. Introduction Our results Applications Concluding remarks Modern form of Freiman’s theorem Theorem (Konyagin) Let A ⊆ Z, and |A + A| ≤ K|A|. Then there is Q = P1 + · · · + Pd , |Q| ≤ |A| such that |A Q| ≥ |A|exp(−(log K)3+o(1) ) , and where dim(Q) ≤ (log K)3+o(1) . S. V. Konyagin, I. D. Shkredov Subsets of Zp with small Wiener norm and arithmetic progressi
  22. 22. Introduction Our results Applications Concluding remarks Good subset of a set with small Wiener norm Good subset Let A ⊂ Fp be set with 1A W ≤ K. Put d = log3+o(1) K and m = dp |A| p 1/d . Then there exist x0 ∈ Fp and q ∈ F∗ p such that for the set B = q(A − x0) = {q(x − x0) : x ∈ A} we have |B ∩ [−m, m]| ≥ |A|e−d . S. V. Konyagin, I. D. Shkredov Subsets of Zp with small Wiener norm and arithmetic progressi
  23. 23. Introduction Our results Applications Concluding remarks Recalling f : T → C , T = R/(2πZ) . Fourier transform ˆf (k) = (2π)−1 T f (t)e−ikt dt , k ∈ Z . Norm f W (T) := k∈Z |ˆf (k)| Banach algebra AW (T) = {f : f A(T) < ∞} . S. V. Konyagin, I. D. Shkredov Subsets of Zp with small Wiener norm and arithmetic progressi
  24. 24. Introduction Our results Applications Concluding remarks Theorem (Beurling–Helson, 1953) Let ϕ : T → T be a continuous map. Suppose that einϕ W (T) = O(1) , n ∈ Z, |n| → ∞ . Then ϕ(t) = νt + ϕ(0), ν ∈ Z. Corollary (Beurling–Helson, 1953) Any endomorphism of AW (T) is trivial f (t) → f (νt + t0) , ν ∈ Z . S. V. Konyagin, I. D. Shkredov Subsets of Zp with small Wiener norm and arithmetic progressi
  25. 25. Introduction Our results Applications Concluding remarks Conjecture (Kahane, 1962) Let ϕ : T → T be a continuous map. Suppose that einϕ W (T) = o(log |n|) , |n| → ∞ . Then ϕ is a linear function. Theorem (Kahane, 1976) Let ϕ : T → T be continuous piecewise linear but not linear. Then einϕ W (T) ≍ log |n| , |n| → ∞ . S. V. Konyagin, I. D. Shkredov Subsets of Zp with small Wiener norm and arithmetic progressi
  26. 26. Introduction Our results Applications Concluding remarks Theorem (Lebedev, 2012) Let ϕ : T → T be a continuous map. Suppose that einϕ W (T) = o log log |n| (log log log |n|) 1/12 , |n| → ∞ . Then ϕ is a linear function. Theorem (Konyagin–Shkredov, 2014) Let ϕ : T → T be a continuous map. Suppose that einϕ W (T) = o log1/22 |n| (log log |n|)3/11 , |n| → ∞ . Then ϕ is a linear function. S. V. Konyagin, I. D. Shkredov Subsets of Zp with small Wiener norm and arithmetic progressi
  27. 27. Introduction Our results Applications Concluding remarks Discretization Let ϕ : T → T be a continuous map, N be an integer. ϕ(x) → ϕ 2πik N = ϕ∗ (k) , k = 0, 1, . . . , N . Then einϕ∗ A(ZN ) ≤ einϕ A(T) , where for f : ZN → C, we put ˆf (x) := 1 N k∈ZN f (k)e−2πikx/N . By Dirichlet Theorem |ϕ∗ (k) − pk Q | ≤ 1 QN , k ∈ ZN , Q ≤ NN . S. V. Konyagin, I. D. Shkredov Subsets of Zp with small Wiener norm and arithmetic progressi
  28. 28. Introduction Our results Applications Concluding remarks Θ(m) := max |n|≤m einϕ A(T) . Lemma (Lebedev) Let ϕ : T → T be a continuous map. Let Q ∈ N, N be a prime number Qϕ∗ (x) ≤ 1/N , ∀x ∈ ZN . Then under some technical conditions Θ5 (Q) ≫ (log N)1/2 log log N . Putting Q = NN , we get Lebedev’s result with double log. S. V. Konyagin, I. D. Shkredov Subsets of Zp with small Wiener norm and arithmetic progressi
  29. 29. Introduction Our results Applications Concluding remarks By Dirichlet Theorem, we have ϕ∗ (k) ≈ pk Q := ϕ′ (k) . For any n = 1, . . . , Q 1 = einϕ′ 2 = einϕ′ 2 ≤ einϕ′ 1/3 1 einϕ′ 2/3 4 . Summing over n = 1, . . . , Q, we get N3 Θ2(Q) ≪ # x + y = z + w ϕ′ (x) + ϕ′ (y) = ϕ′ (z) + ϕ′ (w) Φ(x, y, z) := ϕ′ (x) + ϕ′ (y) − ϕ′ (z) − ϕ′ (x + y − z) . Project Φ, considering y, z fixed and use Sanders’ Theorem in ZN . S. V. Konyagin, I. D. Shkredov Subsets of Zp with small Wiener norm and arithmetic progressi
  30. 30. Introduction Our results Applications Concluding remarks Put As = A ∩ (A + s). A U3 := s E(As ) . Criterium E3(A) and A U3 are M–critical E3(A) ∼M A U3 + some technical conditions iff there is A′ ⊆ A, |A′ | ≫M |A| and A′ = H1 H2 · · · Hk , all Hj ⊆ Asj with small doubling, |Hj | ≫M |Asj |. S. V. Konyagin, I. D. Shkredov Subsets of Zp with small Wiener norm and arithmetic progressi
  31. 31. Introduction Our results Applications Concluding remarks Corollary Put As = A ∩ (A + s). Rough structure For any set A with Wiener norm 1A W = K there is A′ ⊆ A, |A′ | ≫K |A| s.t. A′ = H1 H2 · · · Hk , all Hj ⊆ Asj with small doubling, |Hj | ≫K |Asj |. S. V. Konyagin, I. D. Shkredov Subsets of Zp with small Wiener norm and arithmetic progressi
  32. 32. Introduction Our results Applications Concluding remarks Weak counterexample to Gowers construction Recall As = A ∩ (A + s). Existence of As with small energy Let A ⊆ G be a set, T2(A) = |A|3 /K, |As | ≤ M|A| K , where M ≥ 1 is a real number. Then ∃s = 0, |As | ≥ |A| 2K s.t. T2(As) ≪ M93/79 K1/198 · |As|3 . S. V. Konyagin, I. D. Shkredov Subsets of Zp with small Wiener norm and arithmetic progressi
  33. 33. Introduction Our results Applications Concluding remarks Thank you for your attention! S. V. Konyagin, I. D. Shkredov Subsets of Zp with small Wiener norm and arithmetic progressi

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