Sumsets and entropy revisited

Résumé : The entropic doubling ${\sigma }_{\mathrm{ent}}[X]$ 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 $\mathbf{Z}$.