Equidistribution in multiplicative subgroups of

Transcription

1 Equidistribution in multiplicative subgroups of F p S. Bringer Université Jean Monnet Tuesday 9th July 2013 S.Bringer (Univ Jean Monnet) Densities and their applications 1 / 14

2 Contents 1 Denitions S.Bringer (Univ Jean Monnet) Densities and their applications 2 / 14

3 Contents 1 Denitions 2 Equidistribution of multi-sets S.Bringer (Univ Jean Monnet) Densities and their applications 2 / 14

4 Contents 1 Denitions 2 Equidistribution of multi-sets 3 Further applications S.Bringer (Univ Jean Monnet) Densities and their applications 2 / 14

5 Notations Notations 1 An interval I of F p is dened by I = { ax +b mod p 0 x I 1 } for some a F p and b F p. 2 A multi-set of F p is a nite sequence of elements of F p. Focus on multi-sets of the form I.H = (xh x I,h H), where I and H are two subsets of F p. 3 The cardinality of a multi-set A is written A. S.Bringer (Univ Jean Monnet) Densities and their applications 3 / 14

6 Denitions ε-equidistribution Denition 2.1. Let ε > 0. Let p be a prime number. Let A be a multi-set of F p. A is ε-equidistributed modulo p if, for all interval I included in F p : A I A I p < ε. S.Bringer (Univ Jean Monnet) Densities and their applications 4 / 14

7 Denitions Counterexample Let p = 2 n 1 be a Mersenne prime and A = { 2 k, with k = 0,...,n 1 }. If I = { x + 2 n 1,x = 1,...,2 n 1 1 }, then A I =. Hence A is not ε-equidistributed modulo p for ε < 1 2. S.Bringer (Univ Jean Monnet) Densities and their applications 5 / 14

8 Denitions Another denition Denition 2.2. Let e p (x) = e 2iπx p. Let ε > 0. Let p be a prime number. Let A be a multi-set of F p. A is Sε 1 -equidistributed modulo p if : max x F p a A e p (ax) ε A. S.Bringer (Univ Jean Monnet) Densities and their applications 6 / 14

9 Denitions a Weyl-like criterion Proposition 2.1. Let A be a multi-set of F p. Let ε > 0, there exists f (ε) > 0 such that if A is S 1 -equidistributed modulo p, then A is ε-equidistributed f (ε) modulo p. Let η > 0, there exists g(η) > 0 such that if A is η-equidistributed modulo p then A is S 1 g(η) -equidistributed modulo p. S.Bringer (Univ Jean Monnet) Densities and their applications 7 / 14

10 Denitions Result by Bourgain-Glibichuk-Konyagin (2006) Theorem 2.1. Let p be a prime number, there exist positive constants C 1 and C 2 such that for any δ > 0, any subgroup G of F p with G pδ and γ = exp( C1 δ C 2 ), Corollary 2.1. max x F p ep(gx) G p γ. g G Let ε > 0, p be a prime number, there exist positive constants C 1 and C 2 such that, for any subgroup G of F p with 1 C ln G > C 2 lnp, 1 ( ) G is ε-equidistributed. 1 lnlnp lnln 1 C2 ε S.Bringer (Univ Jean Monnet) Densities and their applications 8 / 14

11 Equidistribution of multi-sets Result by Bourgain (2009) Let I.H = (xh x I,h H) be a multi-set. Theorem 3.1. For all ε > 0 and γ > 0, there exist p 0 (ε,γ) and (ε,γ) such that for any prime p p 0 : any multiplicative subgroup H of F p with H > (lnp), any interval I of F p with I > p γ, I.H is ε-equidistributed. S.Bringer (Univ Jean Monnet) Densities and their applications 9 / 14

12 Equidistribution of multi-sets Sketch of the proof Let I = { αx + β mod p 0 x I 1 } be an interval of F p. Goal : To bound max e a F p p (ahx). h H x I S.Bringer (Univ Jean Monnet) Densities and their applications 10 / 14

13 Equidistribution of multi-sets Sketch of the proof Let I = { αx + β mod p 0 x I 1 } be an interval of F p. Goal : To bound max e a F p p (ahx). h H x I ( e p (ahx) min I, aαh ) 1 p x I S.Bringer (Univ Jean Monnet) Densities and their applications 10 / 14

14 Equidistribution of multi-sets Sketch of the proof Let I = { αx + β mod p 0 x I 1 } be an interval of F p. Goal : To bound max e a F p p (ahx). h H x I ( e p (ahx) min I, aαh ) 1 p x I Result by Bourgain(2009) : { Let Ω aα,ξ = h H : aαh p < p 1 ξ}. If Ωaα,ξ > ρ H then H < (lnp). S.Bringer (Univ Jean Monnet) Densities and their applications 10 / 14

15 Further applications Result by Hegyvári and Hennecart (2010) Theorem 4.1. For all ε > 0 and γ > 0, for all positive integer d,there exist p 0 (ε,γ,d) and (ε,γ,d) such that for any prime p p 0, any subset H of F p with H.H < 2 H and H > (lnp), any interval I of F p with I > p γ, for all f F p [X ], with degree d, the multi-set f (I ).H is ε-equidistributed. S.Bringer (Univ Jean Monnet) Densities and their applications 11 / 14

16 Further applications H is an approximate multiplicative subgroup Theorem 4.2. For all ε > 0,γ > 0 and K > 0, for all positive integer d, there exist p 0 (ε,γ,k,d) and (ε,γ,k,d) such that for any prime p p 0, any subset H of F p with H.H < K H and H > (lnp), any interval I of F p with I > p γ, for all f F p [X ], with degree d, the multi-set f (I ).H is ε-equidistributed. S.Bringer (Univ Jean Monnet) Densities and their applications 12 / 14

17 Further applications I is almost an additive subgroup Theorem 4.3. For all ε > 0,γ > 0, K > 0, for all positive integers n and d, there exist p 0 (ε,γ,n,k,d) and (ε,γ,n,k,d) such that for any prime p p 0, any subset H of F p with H.H < K H and H > (lnp), any proper generalized arithmetic progression I = {r 0 +r 1 u r d u d : 0 u i U i 1} of F p, with r i F p and U i a positive integer for all i {1,...,d}, and I > p γ, for all f F p [X ], with degree n, the multi-set f (I ).H is ε-equidistributed. S.Bringer (Univ Jean Monnet) Densities and their applications 13 / 14

18 Further applications Prospect Conjecture. For all ε > 0,γ > 0, K > 0,K > 0, for all positive integer d, there exist p 0 (ε,γ,k,k,d) and (ε,γ,k,k,d) such that for any prime p p 0, any subset H of F p with H.H < K H and H > (lnp), any subset I of F p with I +I < K I, and I > p γ, for all f F p [X ], with degree d, the multi-set f (I ).H is ε-equidistributed. S.Bringer (Univ Jean Monnet) Densities and their applications 14 / 14