Structure of sets with few sums

Abstract

We want to describe sets that have few sums. If |A|=m, then clearly |A+A|≥m in every group (with equality for cosets), which can be improved to 2m-1 for sets of integers (or torsion-free groups in general). What can we say if we know that |A+A|≤αm, where α is constant or grows slowly as n→∞? That is, we are looking for statements of the form |A|=m, |A+A|≤αm ⇒ (...).