SETS WITH SMALL SUMSET AND RECTIFICATION

Abstract

We study the extent to which sets $A \subseteq \mathbb{Z}/N\mathbb{Z}, N$ prime, resemble sets of integers from the additive point of view (‘up to Freiman isomorphism’). We give a direct proof of a result of Freiman, namely that if $|A + A| \leq K|A|$ and $|A| < c(K)N$, then $A$ is Freiman isomorphic to a set of integers. Because we avoid appealing to Freiman's structure theorem, we obtain a reasonable bound: we can take $\;c(K) \geq (32K)^{-12K^2}$. As a byproduct of our argument we obtain a sharpening of the second author's result on sets with small sumset in torsion groups. For example, if $A \subseteq \mathbb{F}_2^n$, and if $|A + A| \leq K|A|$, then $A$ is contained in a coset of a subspace of size no more than $K^22^{2K^2 - 2}|A|$.