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

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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  • 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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