Sumsets and entropy revisited
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 s...
| Main Authors: | , , |
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| Format: | Journal article |
| Language: | English |
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Wiley
2024
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| _version_ | 1811140387710959616 |
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| author | Green, B Manners, F Tao, T |
| author_facet | Green, B Manners, F Tao, T |
| author_sort | Green, B |
| collection | OXFORD |
| description | 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 $$ \mathbf{Z} $$ . |
| first_indexed | 2024-09-25T04:21:11Z |
| format | Journal article |
| id | oxford-uuid:d242ef65-58ac-4463-bee9-c9745fb0d9a3 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-09-25T04:21:11Z |
| publishDate | 2024 |
| publisher | Wiley |
| record_format | dspace |
| spelling | oxford-uuid:d242ef65-58ac-4463-bee9-c9745fb0d9a32024-08-01T19:35:00ZSumsets and entropy revisitedJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:d242ef65-58ac-4463-bee9-c9745fb0d9a3EnglishJisc Publications RouterWiley2024Green, BManners, FTao, TThe 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 $$ \mathbf{Z} $$ . |
| spellingShingle | Green, B Manners, F Tao, T Sumsets and entropy revisited |
| title | Sumsets and entropy revisited |
| title_full | Sumsets and entropy revisited |
| title_fullStr | Sumsets and entropy revisited |
| title_full_unstemmed | Sumsets and entropy revisited |
| title_short | Sumsets and entropy revisited |
| title_sort | sumsets and entropy revisited |
| work_keys_str_mv | AT greenb sumsetsandentropyrevisited AT mannersf sumsetsandentropyrevisited AT taot sumsetsandentropyrevisited |