Freiman-Ruzsa-type theory for small doubling constant

TL;DR

This paper investigates the structure of subsets in binary vector spaces with small doubling constants, demonstrating they are contained in small affine subspaces, coverable by few shifts, and classifiable.

Contribution

It extends Freiman-Ruzsa theory to binary spaces, providing structural results and classifications for sets with doubling constant less than 2.

Findings

Sets with doubling constant < 2 are contained in small affine subspaces.

Such sets can be covered by at most four shifts of a subspace.

Complete classification of binary sets with small doubling constant.

Abstract

In this paper, we study the linear structure of sets $A \subset \mathbb{F}_2^n$ with doubling constant $\sigma(A)<2$, where $\sigma(A):=\frac{|A+A|}{|A|}$. In particular, we show that $A$ is contained in a small affine subspace. We also show that $A$ can be covered by at most four shifts of some subspace $V$ with $|V|\leq |A|$. Finally, we classify all binary sets with small doubling constant.