The typical approximate structure of sets with bounded sumset

TL;DR

This paper demonstrates that sets with bounded sumsets are typically structured as subsets of arithmetic progressions, extending understanding of their typical form in abelian groups.

Contribution

It introduces a new asymptotic structural characterization of sets with small sumsets and develops a hypergraph container method for this purpose.

Findings

Sets with bounded sumsets are almost fully contained in arithmetic progressions.

The results apply to subsets of integers and arbitrary abelian groups.

A counting theorem for such sets is established.

Abstract

Let $A_1$ and $A_2$ be randomly chosen subsets of the first $n$ integers of cardinalities $s_2\geq s_1 = \Omega(s_2)$, such that their sumset $A_1+A_2$ has size $m$. We show that asymptotically almost surely $A_1$ and $A_2$ are almost fully contained in arithmetic progressions $P_1$ and $P_2$ with the same common difference and cardinalities approximately $s_i m/(s_1+s_2)$. We also prove a counting theorem for such pairs of sets in arbitrary abelian groups. The results hold for $s_i = \omega(\log^3 n)$ and $s_1+s_2 \leq m = o(s_2/\log^3 n)$. 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 $A=B$.