The Typical Structure of Sets with Small Sumset

TL;DR

This paper characterizes the typical structure of integer sets with small sumsets, showing most such sets are contained in specific arithmetic progressions, extending previous results in additive combinatorics.

Contribution

It improves existing bounds and describes the typical structure of sets with bounded doubling for larger set sizes and sumset constraints.

Findings

Most sets with small sumsets are contained in arithmetic progressions.

The results extend previous bounds to larger set sizes.

Almost all such sets have a predictable, structured form.

Abstract

In this paper we determine the number and typical structure of sets of integers with bounded doubling. In particular, improving recent results of Green and Morris, and of Mazur, we show that the following holds for every fixed $\lambda > 2$ and every $k \geqslant (\log n)^4$: if $\omega \to \infty$ as $n \to \infty$ (arbitrarily slowly), then almost all sets $A \subset [n]$ with $|A| = k$ and $|A + A| \leqslant \lambda k$ are contained in an arithmetic progression of length $\lambda k/2 + \omega$.