On additive shifts of multiplicative subgroups

2011/02/06 by Shkredov I. D., D., Shkredov I., Vyugin I. V +1
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.1102.1172

18 pages

arxiv created 2011/02/06 · arxiv updated 2011/02/08

Abstract

Generalizing a result of S.V. Konyagin and D.R. Heath--Brown, we prove, in particular, that for any multiplicative subgroup R of Z/pZ and any nonzero elements mu₁,...,mu_k the following holds |R ∩ (R+mu₁) ∩ ... ∩ (R+mu_k)| ≪_k |R|^1/2+alpha_k, provided by 1 ≪_k |R| ≪_k p^1-β_k, where alpha_k, beta_k are some sequences of positive reals and alpha_k, beta_k tend to zero. Besides we show that for an arbitrary subgroup R, |R| < p^1/2 one have |R± R| > |R|^5/3 log^-1/2 |R|.

Related

• Multiplicative irreducibility of shifted multiplicative subgroups in the extremal case
• On a question of Cochrane and Pinner concerning multiplicative subgroups
• Some new inequalities in additive combinatorics
• When is A + x A =ℝ
• On tripling constant of multiplicative subgroups