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arXiv · 2306.13403

Sumsets and entropy revisited

Ben Green·Freddie Manners·Terence Tao
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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 $\mathbf{Z}$.

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Ben Green, Freddie Manners, Terence Tao. 2023-06-23. Sumsets and entropy revisited. https://doi.org/10.1002/rsa.21252

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