Abstract

Let A and B be subsets of an abelian group G with |B| ≤ |A|. Their sumset A + B is the subset consisting of all a + b with a ∈ A and b ∈ B. There are many results along the lines of small sumset implies structure. For instance, when |A + B| < |A| + 2|B| − 3 with G equal to the integers, then A and B must each be large subsets of arithmetic progressions (having common difference) with very precise bounds on just how large. This result derives from classical work started by Freiman, but decades later it is still an open question to fully extend it to an analogous result for G cyclic of prime order p. In this talk, I will overview the basic situation and methods used in this regime, and then focus on recent work attaining such precise inverse structure for sumsets with very high density |A + B| = (1 − ϵ)p.