Marton's Conjecture in abelian groups with bounded torsion
| Autor: | Gowers, W. T., Green, Ben, Manners, Freddie, Tao, Terence |
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| Rok vydání: | 2024 |
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| Druh dokumentu: | Working Paper |
| Popis: | We prove a Freiman--Ruzsa-type theorem with polynomial bounds in arbitrary abelian groups with bounded torsion, thereby proving (in full generality) a conjecture of Marton. Specifically, let $G$ be an abelian group of torsion $m$ (meaning $mg=0$ for all $g \in G$) and suppose that $A$ is a non-empty subset of $G$ with $|A+A| \leq K|A|$. Then $A$ can be covered by at most $(2K)^{O(m^3)}$ translates of a subgroup of $H \leq G$ of cardinality at most $|A|$. The argument is a variant of that used in the case $G = \mathbf{F}_2^n$ in a recent paper of the authors. Comment: 33 pages, to be submitted; v2 corrects some minor issues pointed out by readers of v1 |
| Databáze: | arXiv |
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