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

TL;DR

This paper proves that sets in binary vector spaces with small sumsets must significantly intersect certain low-dimensional subspaces, revealing structural properties of such sets.

Contribution

It establishes a new lower bound on the intersection size of sets with small doubling in $\,\mathbb{F}_2^n$ with subspaces of logarithmic dimension.

Findings

Sets with small sumsets intersect some low-dimensional subspace in many elements.

The intersection size is at least $|A|^{\eta - o(1)}$ for such sets.

The result links small doubling to strong structural intersection properties.

Abstract

We show that any set $A$ in $\mathbb F_2^n$ with $|A+A| \le |A|^{2-\eta}$ must intersect a subspace of dimension $O_{\eta}(\log |A|)$ in at least $|A|^{\eta - o(1)}$ elements.