Sumsets and entropy revisited

Abstract

The entropic doubling of a random variable taking values in an abelian group is a variant of the notion of the doubling constant of a finite subset of , 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 Pálvölgyi and Zhelezov on the “skew dimension” of subsets of with small doubling; (2) A new proof, and an improvement, of a result of the second author on the dimension of subsets of with small doubling; (3) A proof that the Polynomial Freiman–Ruzsa conjecture over implies the (weak) Polynomial Freiman–Ruzsa conjecture over .