On additive shifts of multiplicative subgroups

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|.