{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/marton-s-conjecture-in-abelian-groups-with","title":"Marton's Conjecture in abelian groups with bounded torsion","arxiv_id":"2404.02244","date":"2024-04-02","proceeding":null,"authors":["W. T. Gowers","Ben Green","Freddie Manners","Terence Tao"],"abstract":"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.","url_abs":"https://arxiv.org/abs/2404.02244v2","url_pdf":"https://arxiv.org/pdf/2404.02244v2.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"marton-s-conjecture-in-abelian-groups-with","repo_url":"https://github.com/teorth/pfr","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}