Sumsets and entropy revisited

TL;DR

This paper explores the properties of entropic doubling in abelian groups, providing new proofs and improvements for results related to small doubling sets and connecting the Polynomial Freiman--Ruzsa conjecture over different fields.

Contribution

It advances the theory of entropic doubling, offers new proofs and improvements for existing results, and links conjectures across different algebraic settings.

Findings

New proof of skew dimension result for small doubling sets in Z^D

Improved proof of dimension bounds for small doubling sets in Z^D

Demonstration that F_2 Polynomial Freiman--Ruzsa conjecture implies the Z version

Abstract

The entropic doubling $\sigma_{\operatorname{ent}}[X]$ of a random variable $X$ taking values in an abelian group $G$ is a variant of the notion of the doubling constant $\sigma[A]$ of a finite subset $A$ of $G$, but it enjoys somewhat better properties; for instance, it contracts upon applying a homomorphism. In this paper we develop further the theory of entropic doubling and give various applications, including: (1) A new proof of a result of P\'alv\"olgyi and Zhelezov on the ``skew dimension'' of subsets of $\mathbf{Z}^D$ with small doubling; (2) A new proof, and an improvement, of a result of the second author on the dimension of subsets of $\mathbf{Z}^D$ with small doubling; (3) A proof that the Polynomial Freiman--Ruzsa conjecture over $\mathbf{F}_2$ implies the (weak) Polynomial Freiman--Ruzsa conjecture over $\mathbf{Z}$.