Differences of subgroups in subgroups

Abstract

We prove, in particular, that if A,G are two arbitrary multiplicative subgroups of the prime field f_p, |G| < p^{3/4} such that the difference A-A is contained in G then |A| \ll |\G|^{1/3+o(1)}. Also, we obtain that for any eps>0 and a sufficiently large subgroup G with |G| \ll p^{1/2-eps} there is no representation G as G = A+B, where A is another subgroup, and B is an arbitrary set, |A|,|B|>1. Finally, we study the number of collinear triples containing in a set of f_p and prove a "dual" sum-products estimate.