A bilinear Bogolyubov theorem, with applications

Transcription

1 A bilinear Bogolyubov theorem, with applications Thái Hoàng Lê 1 & Pierre-Yves Bienvenu 2 1 University of Mississippi 2 Institut Camille Jordan November 11, 2018 Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

2 Part I: A bilinear Bogolyubov theorem Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

3 Subspaces in difference sets If A X, then the density of A in X is A X. Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

4 Subspaces in difference sets If A X, then the density of A in X is A X. If A F n p has density α > 0, then A A should contain a large subspace. Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

5 Green 2005, Sanders 2010: If A F n 2 has density α > 0, then A A contains a subspace of dimension Ω(αn). Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

6 Green 2005, Sanders 2010: If A F n 2 has density α > 0, then A A contains a subspace of dimension Ω(αn). However, finite codimension (i.e. dimension n c(α)) is impossible. Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

7 Green 2005, Sanders 2010: If A F n 2 has density α > 0, then A A contains a subspace of dimension Ω(αn). However, finite codimension (i.e. dimension n c(α)) is impossible. Ruzsa 1991, Green 2005: The largest subspace guaranteed to be in A A cannot have codimension c(α) n (i.e. dimension n c(α) n). Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

8 Bogolyubov s theorem The more sumsets we have, the larger subspace we can find. Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

9 Bogolyubov s theorem The more sumsets we have, the larger subspace we can find. Theorem (Bogolyubov 1939) If A F n p has density α > 0, then A + A A A contains a subspace of codimension c(α). Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

10 Bogolyubov s theorem The more sumsets we have, the larger subspace we can find. Theorem (Bogolyubov 1939) If A F n p has density α > 0, then A + A A A contains a subspace of codimension c(α). Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

11 Bogolyubov s theorem The more sumsets we have, the larger subspace we can find. Theorem (Bogolyubov 1939) If A F n p has density α > 0, then A + A A A contains a subspace of codimension c(α). Bogolyubov proved for A Z, with subspaces replaced by Bohr sets. Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

12 Bogolyubov s theorem The more sumsets we have, the larger subspace we can find. Theorem (Bogolyubov 1939) If A F n p has density α > 0, then A + A A A contains a subspace of codimension c(α). Bogolyubov proved for A Z, with subspaces replaced by Bohr sets. Bogolyubov s proof gives c(α) = O ( 1 α 2 ). Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

13 Bogolyubov s theorem The more sumsets we have, the larger subspace we can find. Theorem (Bogolyubov 1939) If A F n p has density α > 0, then A + A A A contains a subspace of codimension c(α). Bogolyubov proved for A Z, with subspaces replaced by Bohr sets. Bogolyubov s proof gives c(α) = O ( ) 1 α. ( ) 2 Sanders 2010: c(α) = O log 4 1 α. Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

14 Bogolyubov s theorem The more sumsets we have, the larger subspace we can find. Theorem (Bogolyubov 1939) If A F n p has density α > 0, then A + A A A contains a subspace of codimension c(α). Bogolyubov proved for A Z, with subspaces replaced by Bohr sets. Bogolyubov s proof gives c(α) = O ( ) 1 α. ( ) 2 Sanders 2010: c(α) = O log 4 1 α. If A is a subspace of density α, then codim(a) = log p 1 α and A + A A A = A. Thus we cannot do better than O ( log 1 α). Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

15 We are interested in structures arising from subsets A F n p F n p of density α > 0. Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

16 We are interested in structures arising from subsets A F n p F n p of density α > 0. Define φ h A = {(x 1 x 2, y) : (x 1, y), (x 2, y) A}, φ v A = {(x, y 1 y 2 ) : (x, y 1 ), (x, y 2 ) A}. Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

17 We are interested in structures arising from subsets A F n p F n p of density α > 0. Define φ h A = {(x 1 x 2, y) : (x 1, y), (x 2, y) A}, φ v A = {(x, y 1 y 2 ) : (x, y 1 ), (x, y 2 ) A}. For a sequence w 1, w 2,..., w k of h s and v s, φ w1 w 2 w k φ w1 φ w2 φ wk. denotes Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

18 A bilinear Bogolyubov theorem Theorem (Bienvenu-L. 2017, Gowers-Milićević 2017) If A F n p F n p has density α > 0, then φ hhvvhh (A) contains a set {(x, y) W 1 W 2 : Q 1 (x, y) =... = Q r (x, y) = 0} where Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

19 A bilinear Bogolyubov theorem Theorem (Bienvenu-L. 2017, Gowers-Milićević 2017) If A F n p F n p has density α > 0, then φ hhvvhh (A) contains a set {(x, y) W 1 W 2 : Q 1 (x, y) =... = Q r (x, y) = 0} where W 1 F n p is a subspace of codimension r 1, Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

20 A bilinear Bogolyubov theorem Theorem (Bienvenu-L. 2017, Gowers-Milićević 2017) If A F n p F n p has density α > 0, then φ hhvvhh (A) contains a set {(x, y) W 1 W 2 : Q 1 (x, y) =... = Q r (x, y) = 0} where W 1 F n p is a subspace of codimension r 1, W 2 F n p is a subspace of codimension r 2, Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

21 A bilinear Bogolyubov theorem Theorem (Bienvenu-L. 2017, Gowers-Milićević 2017) If A F n p F n p has density α > 0, then φ hhvvhh (A) contains a set {(x, y) W 1 W 2 : Q 1 (x, y) =... = Q r (x, y) = 0} where W 1 F n p is a subspace of codimension r 1, W 2 F n p is a subspace of codimension r 2, Q 1,..., Q r are bilinear forms: W 1 W 2 F p, Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

22 A bilinear Bogolyubov theorem Theorem (Bienvenu-L. 2017, Gowers-Milićević 2017) If A F n p F n p has density α > 0, then φ hhvvhh (A) contains a set {(x, y) W 1 W 2 : Q 1 (x, y) =... = Q r (x, y) = 0} where W 1 F n p is a subspace of codimension r 1, W 2 F n p is a subspace of codimension r 2, Q 1,..., Q r are bilinear forms: W 1 W 2 F p, Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

23 A bilinear Bogolyubov theorem Theorem (Bienvenu-L. 2017, Gowers-Milićević 2017) If A F n p F n p has density α > 0, then φ hhvvhh (A) contains a set {(x, y) W 1 W 2 : Q 1 (x, y) =... = Q r (x, y) = 0} where W 1 F n p is a subspace of codimension r 1, W 2 F n p is a subspace of codimension r 2, Q 1,..., Q r are bilinear forms: W 1 W 2 F p, and max(r 1, r 2, r) c(α). Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

24 A bilinear Bogolyubov theorem Theorem (Bienvenu-L. 2017, Gowers-Milićević 2017) If A F n p F n p has density α > 0, then φ hhvvhh (A) contains a set where {(x, y) W 1 W 2 : Q 1 (x, y) =... = Q r (x, y) = 0} W 1 F n p is a subspace of codimension r 1, W 2 F n p is a subspace of codimension r 2, Q 1,..., Q r are bilinear forms: W 1 W 2 F p, and max(r 1, r 2, r) c(α). Recall Bogolyubov s theorem: if A F n p has density α, then A + A A A contains a subspace of codimension r c (α) (= zero set of r linear forms). Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

25 Quantitative bounds φ hhvvhh (A) {(x, y) W 1 W 2 : Q 1 (x, y) =... = Q r (x, y) = 0} W 1 F n p is a subspace of codimension r 1, Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

26 Quantitative bounds φ hhvvhh (A) {(x, y) W 1 W 2 : Q 1 (x, y) =... = Q r (x, y) = 0} W 1 F n p is a subspace of codimension r 1, W 2 F n p is a subspace of codimension r 2, Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

27 Quantitative bounds φ hhvvhh (A) {(x, y) W 1 W 2 : Q 1 (x, y) =... = Q r (x, y) = 0} W 1 F n p is a subspace of codimension r 1, W 2 F n p is a subspace of codimension r 2, Q 1,..., Q r are bilinear forms: W 1 W 2 F p. Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

28 Quantitative bounds φ hhvvhh (A) {(x, y) W 1 W 2 : Q 1 (x, y) =... = Q r (x, y) = 0} W 1 F n p is a subspace of codimension r 1, W 2 F n p is a subspace of codimension r 2, Q 1,..., Q r are bilinear forms: W 1 W 2 F p. Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

29 Quantitative bounds φ hhvvhh (A) {(x, y) W 1 W 2 : Q 1 (x, y) =... = Q r (x, y) = 0} W 1 F n p is a subspace of codimension r 1, W 2 F n p is a subspace of codimension r 2, Q 1,..., Q r are bilinear forms: W 1 W 2 F p. ( ( ( ))) Gowers-Milićević: max(r 1, r 2, r) = O exp exp log O(1) 1 α. Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

30 Quantitative bounds φ hhvvhh (A) {(x, y) W 1 W 2 : Q 1 (x, y) =... = Q r (x, y) = 0} W 1 F n p is a subspace of codimension r 1, W 2 F n p is a subspace of codimension r 2, Q 1,..., Q r are bilinear forms: W 1 W 2 F p. ( ( ( ))) Gowers-Milićević: max(r 1, r 2, r) = O exp exp log O(1) 1 α. Bienvenu-L.: max(r 1, r) = O(log O(1) 1 α ), Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

31 Quantitative bounds φ hhvvhh (A) {(x, y) W 1 W 2 : Q 1 (x, y) =... = Q r (x, y) = 0} W 1 F n p is a subspace of codimension r 1, W 2 F n p is a subspace of codimension r 2, Q 1,..., Q r are bilinear forms: W 1 W 2 F p. ( ( ( ))) Gowers-Milićević: max(r 1, r 2, r) = O exp exp log O(1) 1 α. Bienvenu-L.: ( ( max(r ( 1, r) ( = O(log O(1) )))) 1 α ), and r 2 = O exp exp exp log O(1) 1 α. Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

32 Motivated by Sanders bound r = O(log 4 1 α ), and since the roles of r 1 and r 2 are symmetric, it is natural to conjecture. Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

33 Motivated by Sanders bound r = O(log 4 1 α ), and since the roles of r 1 and r 2 are symmetric, it is natural to conjecture. Conjecture (Bienvenu-L. 2017) There exists a sequence w 1 w 2... w k of length O(1) such that max(r 1, r 2, r) = O(log O(1) 1 α ). Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

34 Motivated by Sanders bound r = O(log 4 1 α ), and since the roles of r 1 and r 2 are symmetric, it is natural to conjecture. Conjecture (Bienvenu-L. 2017) There exists a sequence w 1 w 2... w k of length O(1) such that max(r 1, r 2, r) = O(log O(1) 1 α ). Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

35 Motivated by Sanders bound r = O(log 4 1 α ), and since the roles of r 1 and r 2 are symmetric, it is natural to conjecture. Conjecture (Bienvenu-L. 2017) There exists a sequence w 1 w 2... w k of length O(1) such that max(r 1, r 2, r) = O(log O(1) 1 α ). Simple examples show that we cannot do better than this. Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

36 Motivated by Sanders bound r = O(log 4 1 α ), and since the roles of r 1 and r 2 are symmetric, it is natural to conjecture. Conjecture (Bienvenu-L. 2017) There exists a sequence w 1 w 2... w k of length O(1) such that max(r 1, r 2, r) = O(log O(1) 1 α ). Simple examples show that we cannot do better than this. Theorem (Hosseini-Lovett 2018) The conjecture is true for the sequence hvvhvvvhh, and max(r 1, r 2, r) = O(log 80 1 α ). Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

37 Part II: Applications Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

38 Gowers and Milićević used a variant of the bilinear Bogolyubov theorem to prove a quantitative bound for the inverse theorem for the U 4 -norm (first proved by Bergelson, Tao and Ziegler using qualitative methods). Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

39 Gowers and Milićević used a variant of the bilinear Bogolyubov theorem to prove a quantitative bound for the inverse theorem for the U 4 -norm (first proved by Bergelson, Tao and Ziegler using qualitative methods). We used the bilinear Bogolyubov theorem to prove an instance of the Möbius randomness principle in function fields. Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

40 The Möbius randomness principle Recall { ( 1) k if n = p µ(n) = 1 p 2 p k is a product of k distinct primes, 0 if n is not squarefree. Thus the sequence {µ(n)} is 1, 1, 1, 0, 1, 1, 1, 0, 0, 1,.... Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

41 The Möbius randomness principle Recall { ( 1) k if n = p µ(n) = 1 p 2 p k is a product of k distinct primes, 0 if n is not squarefree. Thus the sequence {µ(n)} is 1, 1, 1, 0, 1, 1, 1, 0, 0, 1,.... The Möbius randomness principle states that µ is random-like, Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

42 The Möbius randomness principle Recall { ( 1) k if n = p µ(n) = 1 p 2 p k is a product of k distinct primes, 0 if n is not squarefree. Thus the sequence {µ(n)} is 1, 1, 1, 0, 1, 1, 1, 0, 0, 1,.... The Möbius randomness principle states that µ is random-like, i.e. for any bounded, simple or structured function F, we have N µ(n)f(n) = o(n). n=1 Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

43 Examples: 1 If F(n) = 1, then PNT is equivalent to N is equivalent to for any ɛ > 0. N n=1 n=1 µ(n) = O ɛ ( N 1/2+ɛ) µ(n) = o(n) and RH Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

44 Examples: 1 If F(n) = 1, then PNT is equivalent to N is equivalent to for any ɛ > 0. N n=1 n=1 µ(n) = O ɛ ( N 1/2+ɛ) µ(n) = o(n) and RH 2 If F(n) is periodic with period q, then N n=1 µ(n)f(n) = o(n) is equivalent to PNT in arithmetic progressions. Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

45 Examples: 1 If F(n) = 1, then PNT is equivalent to N is equivalent to for any ɛ > 0. N n=1 n=1 µ(n) = O ɛ ( N 1/2+ɛ) µ(n) = o(n) and RH 2 If F(n) is periodic with period q, then N n=1 µ(n)f(n) = o(n) is equivalent to PNT in arithmetic progressions. 3 We can formulate the Möbius randomness principle in terms of dynamical systems (Sarnak) or computational complexity (Kalai). Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

46 Exponential sums Davenport/Vinogradov (1937): for any A > 0, N µ(n)e(nα) A n=1 N log A N uniformly in α R/Z. Here e(x) = e 2πix. The implied constant is ineffective. Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

47 Exponential sums Davenport/Vinogradov (1937): for any A > 0, N µ(n)e(nα) A n=1 N log A N uniformly in α R/Z. Here e(x) = e 2πix. The implied constant is ineffective. Baker-Harman (1991), Montgomery-Vaughan (unpublished): Assuming GRH, we have N µ(n)e(nα) ɛ N 3/4+ɛ n=1 uniformly in α R/Z, for any ɛ > 0. Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

48 Since 1 0 N µ(n)e(nα) n=1 we cannot do better than N 1/2. 2 dα = N µ(n) 2 N, n=1 Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

49 Since 1 0 N µ(n)e(nα) n=1 we cannot do better than N 1/2. 2 dα = N µ(n) 2 N, Green-Tao (2008, 2012): If F(n) is a nilsequence, then for any A > 0, N µ(n)f (n) A n=1 n=1 N log A N. Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

50 Since 1 0 N µ(n)e(nα) n=1 we cannot do better than N 1/2. 2 dα = N µ(n) 2 N, Green-Tao (2008, 2012): If F(n) is a nilsequence, then for any A > 0, N µ(n)f (n) A n=1 n=1 N log A N. Nilsequences include F(n) = e ( αn 2 + βn ) and F(n) = e ( nα βn). Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

51 Function field analogy Let F q be the finite field on q elements. It has been known since Dedekind-Weber (1882) that F q [t] is similar to Z in many aspects. For example, both are unique factorization domains. Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

52 Function field analogy Let F q be the finite field on q elements. It has been known since Dedekind-Weber (1882) that F q [t] is similar to Z in many aspects. For example, both are unique factorization domains. For f F q [t], define ( 1) k if f = cp 1 P 2 P k, P i distinct monic irreducibles, µ(f ) = c F q, 0 if f is not squarefree. Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

53 Function field analogy Let F q be the finite field on q elements. It has been known since Dedekind-Weber (1882) that F q [t] is similar to Z in many aspects. For example, both are unique factorization domains. For f F q [t], define ( 1) k if f = cp 1 P 2 P k, P i distinct monic irreducibles, µ(f ) = c F q, 0 if f is not squarefree. RH is true in F q [t]: for n 2, deg f =n, f monic µ(f ) = 0. Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

54 Function field analogy Let F q be the finite field on q elements. It has been known since Dedekind-Weber (1882) that F q [t] is similar to Z in many aspects. For example, both are unique factorization domains. For f F q [t], define ( 1) k if f = cp 1 P 2 P k, P i distinct monic irreducibles, µ(f ) = c F q, 0 if f is not squarefree. RH is true in F q [t]: for n 2, deg f =n, f monic µ(f ) = 0. Furthermore, GRH is true in F q [t] (Weil 1948). Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

55 Fix e q : F q {z C : z = 1} to be a nontrivial additive character of F q, i.e. e q (x + y) = e q (x)e q (y) for any x, y F q. Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

56 Fix e q : F q {z C : z = 1} to be a nontrivial additive character of F q, i.e. e q (x + y) = e q (x)e q (y) for any x, y F q. Problem. Let k 1 and Q F q [x 0, x 1,..., x n 1 ] be a polynomial of degree k. Show that µ(f )e q (Q(f )) = o q,k (q n ) deg f <n uniformly in Q of degree k. Here Q(f ) is Q evaluated at the coefficients of f. Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

57 Fix e q : F q {z C : z = 1} to be a nontrivial additive character of F q, i.e. e q (x + y) = e q (x)e q (y) for any x, y F q. Problem. Let k 1 and Q F q [x 0, x 1,..., x n 1 ] be a polynomial of degree k. Show that µ(f )e q (Q(f )) = o q,k (q n ) deg f <n uniformly in Q of degree k. Here Q(f ) is Q evaluated at the coefficients of f. Since we have GRH, we may expect O q (q c k n ) for some constant c k < 1, or even O q,k,ɛ (q (1/2+ɛ)n ). Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

58 Our results Theorem (k = 1) For any ɛ > 0, we have deg f <n µ(f )e q (L(f )) ɛ,q q (3/4+ɛ)n uniformly in linear forms L F q [x 0,..., x n 1 ]. Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

59 Our results Theorem (k = 1) For any ɛ > 0, we have deg f <n µ(f )e q (L(f )) ɛ,q q (3/4+ɛ)n uniformly in linear forms L F q [x 0,..., x n 1 ]. Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

60 Our results Theorem (k = 1) For any ɛ > 0, we have deg f <n µ(f )e q (L(f )) ɛ,q q (3/4+ɛ)n uniformly in linear forms L F q [x 0,..., x n 1 ]. Recall Baker-Harman and Montgomery-Vaughan s bound in Z (under GRH) N µ(n)e(αn) N 3/4+ɛ. n=1 Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

61 Our results Theorem (k = 1) For any ɛ > 0, we have deg f <n µ(f )e q (L(f )) ɛ,q q (3/4+ɛ)n uniformly in linear forms L F q [x 0,..., x n 1 ]. Recall Baker-Harman and Montgomery-Vaughan s bound in Z (under GRH) N µ(n)e(αn) N 3/4+ɛ. n=1 Our argument is different from the proof in Z in some respects. Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

62 Theorem (k = 2) Suppose q is odd. There exists an absolute constant c (c = 1/161 will do) such that µ(f )e q (Q(f )) q q n nc. deg f <n uniformly in quadratic polynomials Q F q [x 0,..., x n 1 ]. Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

63 Theorem (k = 2) Suppose q is odd. There exists an absolute constant c (c = 1/161 will do) such that µ(f )e q (Q(f )) q q n nc. deg f <n uniformly in quadratic polynomials Q F q [x 0,..., x n 1 ]. Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

64 Theorem (k = 2) Suppose q is odd. There exists an absolute constant c (c = 1/161 will do) such that µ(f )e q (Q(f )) q q n nc. deg f <n uniformly in quadratic polynomials Q F q [x 0,..., x n 1 ]. This is better than what is known in Z for nilsequences of step 2: N µ(n)f (n) F,A n=1 N log A N for any A > 0. Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

65 The circle method The classical circle method deals with exponential sums like N µ(n)e(αn). n=1 To estimate such sums we need to distinguish two cases. α is close to a rational with small denominator (α is in the major arcs): use our knowledge about the distribution of primes in arithmetic progressions. Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

66 The circle method The classical circle method deals with exponential sums like N µ(n)e(αn). n=1 To estimate such sums we need to distinguish two cases. α is close to a rational with small denominator (α is in the major arcs): use our knowledge about the distribution of primes in arithmetic progressions. α is not in the major arcs (α is in the minor arcs): use combinatorial machinery (Vaughan s identity, Vinogradov s Type I/Type II sums) and Cauchy-Schwarz. Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

67 In the case of µ(f )e q (Φ(f )), deg f <n where Φ(x) = x T Mx and M is a symmetric matrix, the major arcs and minor arcs correspond to low rank and high rank matrices M. This is because e q (x T Mx) qn rank(m)/2. x F n q Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

68 We want to show that where δ = q nc. deg f <n µ(f )e q (Φ(f )) δqn Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

69 We want to show that where δ = q nc. deg f <n µ(f )e q (Φ(f )) δqn If rank(m) is small, we can reduce our problem to the linear case. Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

70 We want to show that where δ = q nc. deg f <n µ(f )e q (Φ(f )) δqn If rank(m) is small, we can reduce our problem to the linear case. Suppose deg f <n µ(f )e q(φ(f )) δq n. We will show that rank(m) is small, which is a contradiction. Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

71 After using Vaughan s identity, Vinogradov s Type I/Type II sums, Cauchy-Schwarz and some combinatorial reasoning, we find that for some n k n, the set of pairs P s := {(a, b) : a, b G k+1 G k+1 : rank M a,b s} is large (has size q O(nc) q 2k+2 ) for some s = O(n c ). Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

72 After using Vaughan s identity, Vinogradov s Type I/Type II sums, Cauchy-Schwarz and some combinatorial reasoning, we find that for some n k n, the set of pairs P s := {(a, b) : a, b G k+1 G k+1 : rank M a,b s} is large (has size q O(nc) q 2k+2 ) for some s = O(n c ). Here G m = {f : deg f < m}, M a,b = L T a ML b + L T b ML a and L a is the matrix of the map G n k G n, f af. Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

73 We know P s := {(a, b) : a, b G k+1 G k+1 : rank M a,b s} is large, where M a,b = L T a ML b + L T b ML a. Want to show that rank M is small. Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

74 We know P s := {(a, b) : a, b G k+1 G k+1 : rank M a,b s} is large, where M a,b = L T a ML b + L T b ML a. Want to show that rank M is small. If rank M a,b, rank M a,b s, then rank M a a,b = rank (M a,b M a,b) 2s. Similarly for the second coordinate. Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24

75 We know P s := {(a, b) : a, b G k+1 G k+1 : rank M a,b s} is large, where M a,b = L T a ML b + L T b ML a. Want to show that rank M is small. If rank M a,b, rank M a,b s, then rank M a a,b = rank (M a,b M a,b) 2s. Similarly for the second coordinate. By repeatedly applying the operations φ h and φ v on P s, the bilinear Bogolyubov theorem implies that P 2 9 s contains a bilinear structure. By exploiting this, we can show that M has low rank, as desired. Thái Hoàng Lê & Pierre-Yves Bienvenu A bilinear Bogolyubov theorem MS Discrete Workshop / 24