Moderate-doubling sets in ${\mathbb{F}}_2^n$ intersect subspaces

Abstract

We show that any set $A$ in ${\mathbb{F}}_2^n$ with $|A+A|\le |A{|}^{2-\hat{I}\cdot }$ must intersect a subspace of dimension $O_{\hat{I}}\cdot (\log |A|)$ in at least $|A{|}^{\hat{I}\cdot -o(1)}$ elements.

20 pages