Sumsets and entropy revisited

Abstract

The entropic doubling σ ent [ X ] $${\sigma}_{\mathrm{ent}}\left[X\right]$$ of a random variable X $$X$$ taking values in an abelian group G $$G$$ is a variant of the notion of the doubling constant σ [ A ] $$\sigma \left[A\right]$$ of a finite subset A $$A$$ of G $$G$$ , 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 Pálvölgyi and Zhelezov on the “skew dimension” of subsets of Z D $${\mathbf{Z}}^D$$ with small doubling; (2) A new proof, and an improvement, of a result of the second author on the dimension of subsets of Z D $${\mathbf{Z}}^D$$ with small doubling; (3) A proof that the Polynomial Freiman–Ruzsa conjecture over F 2 $${\mathbf{F}}_2$$ implies the (weak) Polynomial Freiman–Ruzsa conjecture over Z $$\mathbf{Z}$$