On entropy Marton-type inequalities and small symmetric differences with cosets of abelian groups
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866911919279964160 |
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| author | Karam, Thomas |
| author_facet | Karam, Thomas |
| contents | We recognise that an entropy inequality akin to the main intermediate goal of recent works (Gowers, Green, Manners, Tao [3],[2]) regarding a conjecture of Marton provides a black box from which we can also through a short deduction recover another description: if a finite subset $A$ of an abelian group $G$ is such that the distribution of the sums $a+b$ with $(a,b) \in A \times A$ is only slightly more spread out than the uniform distribution on $A$, then $A$ has small symmetric difference with some finite coset of $G$. The resulting bounds are necessarily sharp up to a logarithmic factor. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_10872 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On entropy Marton-type inequalities and small symmetric differences with cosets of abelian groups Karam, Thomas Information Theory Combinatorics Group Theory Number Theory Probability 05B10, 94A17 (Primary) 11B30, 11B13 (Secondary) We recognise that an entropy inequality akin to the main intermediate goal of recent works (Gowers, Green, Manners, Tao [3],[2]) regarding a conjecture of Marton provides a black box from which we can also through a short deduction recover another description: if a finite subset $A$ of an abelian group $G$ is such that the distribution of the sums $a+b$ with $(a,b) \in A \times A$ is only slightly more spread out than the uniform distribution on $A$, then $A$ has small symmetric difference with some finite coset of $G$. The resulting bounds are necessarily sharp up to a logarithmic factor. |
| title | On entropy Marton-type inequalities and small symmetric differences with cosets of abelian groups |
| topic | Information Theory Combinatorics Group Theory Number Theory Probability 05B10, 94A17 (Primary) 11B30, 11B13 (Secondary) |
| url | https://arxiv.org/abs/2406.10872 |