Marton's Conjecture in abelian groups with bounded torsion
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arXiv
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| Hauptverfasser: | , , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866929350473940992 |
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| author | Gowers, W. T. Green, Ben Manners, Freddie Tao, Terence |
| author_facet | Gowers, W. T. Green, Ben Manners, Freddie Tao, Terence |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_02244 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Marton's Conjecture in abelian groups with bounded torsion Gowers, W. T. Green, Ben Manners, Freddie Tao, Terence Number Theory Combinatorics 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. |
| title | Marton's Conjecture in abelian groups with bounded torsion |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2404.02244 |