A Freiman-type Theorem for restricted sumsets

Abstract

Let [Formula: see text] and [Formula: see text] be nonempty finite subsets of [Formula: see text]. Freiman’s [Formula: see text] Theorem states that if [Formula: see text], then [Formula: see text] is contained in a short arithmetic progression. Freiman generalized his theorem establishing that if [Formula: see text], then [Formula: see text] and [Formula: see text] are contained in short arithmetic progressions with common difference. Take [Formula: see text] and write [Formula: see text]. There have been several attempts to generalize Freiman’s statements for restricted sumsets [Formula: see text]. In the last few years, there have been some results establishing that (under reasonable technical conditions) if [Formula: see text] for an absolute constant [Formula: see text] and [Formula: see text] such that [Formula: see text] whenever [Formula: see text], then there are arithmetic progressions [Formula: see text] and [Formula: see text] with common difference such that [Formula: see text] and [Formula: see text] are small (in terms of [Formula: see text]), [Formula: see text] and [Formula: see text] where [Formula: see text] whenever [Formula: see text]. Furthermore, in some of the papers where these results appear, it was conjectured that the best possible value of [Formula: see text] such that the same conclusion is reached is [Formula: see text]. In this paper we confirm this conjecture.