An Approximate Structure Theorem for Small Sumsets

Abstract

Let A and B be randomly chosen s-subsets of the first n integers such that their sumset \(A+B\) has size at most Ks. We show that asymptotically almost surely A and B are almost fully contained in arithmetic progressions \(P_A\) and \(P_B\) with the same common difference and cardinalities approximately Ks/2. The result holds for \(s = \omega (\log ^3 n)\) and \(2 \le K = o(s/\log ^3 n)\). Our main tool is an asymmetric version of the method of hypergraph containers which was recently used by Campos to prove the result in the special case \(A=B\).