Sumsets of sets of positive density in the integers

A central object of study in additive combinatorics is the sumset A+B of two sets A and B. Two of the basic questions one may ask are direct questions (“how large must A+B be in terms of the sizes of A and B?”) and inverse questions (“if A+B is small, what can be deduced about the structure of A and B?”). When A and B are infinite subsets of the integers with size quantified by natural density d(·), Kneser (1953) proved the direct theorem that d(A+B) ≥ d(A) + d(B) unless A and B have certain modular obstructions. Erdős and Graham (1980) asked for a corresponding inverse theorem classifying sets with d(A+B) = d(A) + d(B). In this talk, we will present a new result characterizing the pairs of sets satisfying d(A+B) = d(A) + d(B) in the absence of modular obstructions. This talk is based on joint work with Florian K. Richter.