Freiman-Ruzsa-type theory for 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 $|V|\le |A|$. Finally, we classify all binary sets with small doubling constant.

9 pages

References in corpus (2)

• Freiman's theorem in finite fields via extremal set theory
• A note on the Freiman and Balog-Szemeredi-Gowers theorems in finite fields