A Note on Freĭman's Theorem in Vector Spaces

Abstract

A famous result of Freĭman describes the sets A, of integers, for which |A+A| ≤ K|A|. In this short note we address the analogous question for subsets of vector spaces over $\mathbb{F}_2$. Specifically we show that if A is a subset of a vector space over $\mathbb{F}_2$ with |A+A| ≤ K|A| then A is contained in a coset of size at most 2O(K3/2 log K)|A|, which improves upon the previous best, due to Green and Ruzsa, of 2O(K2)|A|. A simple example shows that the size may need to be at least 2Ω(K)|A|.