On additive shifts of multiplicative subgroups

TL;DR

This paper extends results on the structure of multiplicative subgroups in finite fields, providing bounds on intersections of shifted subgroups and sumset sizes, revealing new additive combinatorial properties.

Contribution

It generalizes previous results by establishing bounds on multiple shifted intersections and sumsets of multiplicative subgroups in finite fields.

Findings

Bound on intersection size of shifted subgroups: |R ∩ (R+μ_1) ∩ ...| ≪ |R|^{1/2+α_k}

Sumset size lower bound: |R ± R| > |R|^{5/3} / log^{1/2} |R| for |R| < p^{1/2}

Conditions on subgroup size: |R| ≪ p^{1-β_k} with α_k, β_k → 0

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 mu_1,...,mu_k the following holds |R \cap (R+mu_1) \cap ... \cap (R+mu_k)| \ll_k |R|^{1/2+alpha_k}, provided by 1 \ll_k |R| \ll_k p^{1-\beta_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\pm R| > |R|^{5/3} \log^{-1/2} |R|.