Sumsets and entropy revisited

Journal article

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 $$ \mathbf{Z} $$ .

Authors: Green, B; Manners, F; Tao, T

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Wiley
• Journal: Random Structures and Algorithms
• Publication date: 2024-07-31
• Acceptance date: 2024-07-20
• DOI: 10.1002/rsa.21252
• ISSN: 1042-9832, 1098-2418
• Language: English
• Keywords: entropy; Freiman–Ruzsa; Sumsets