THE TYPICAL APPROXIMATE STRUCTURE OF SETS WITH BOUNDED SUMSET*

Let A1 and A2 be randomly chosen subsets of the first n positive integers of cardinalities s2 \geq s1 = \Omega (s2), such that their sumset A1 + A2 has size m. We show that asymptotically almost surely A1 and A2 are almost fully contained in arithmetic progressions P1 and P2 with the same common difference and cardinalities approximately sim/(s1 + s2). We also prove a counting theorem for such pairs of sets in arbitrary abelian groups. The results hold for si = \omega (log3 n) and s1 + s2 \leq m = o(s2/ log3 n). Our main tool is an asymmetric version of the method of hypergraph containers which was recently used by Campos to prove similar results in the special case A = B.