Differences of subgroups in subgroups

Abstract

We prove, in particular, that if [Formula: see text], [Formula: see text] are two arbitrary multiplicative subgroups satisfying [Formula: see text], then [Formula: see text]. Also, we obtain that for any [Formula: see text] and a sufficiently large subgroup [Formula: see text] with [Formula: see text] there is no representation [Formula: see text] as [Formula: see text], where [Formula: see text] is another subgroup, and [Formula: see text] is an arbitrary set, [Formula: see text]. Finally, we study the number of collinear triples containing in a set of [Formula: see text] and prove a variant of sum–product estimate.