{"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/improved-exponent-for-marton-s-conjecture-in","title":"Improved Exponent for Marton's Conjecture in $\\mathbb{F}_2^n$","arxiv_id":"2404.09639","date":"2024-04-15","proceeding":null,"authors":["Jyun-Jie Liao"],"abstract":"A conjecture of Marton, widely known as the polynomial Freiman-Ruzsa conjecture, was recently proved by Gowers, Green, Manners and Tao for any bounded-torsion Abelian group $G$. In this paper we show a few simple modifications that improve their bound in $G=\\mathbb{F}_2^n$. Specifically, for $G=\\mathbb{F}_2^n$, they proved that any set $A\\subseteq G$ with $|A+A|\\le K|A|$ can be covered by at most $2K^C$ cosets of a subgroup $H$ of $G$ of cardinality at most $|A|$, with $C=12$. In this paper we prove the same statement for $C=9$.","url_abs":"https://arxiv.org/abs/2404.09639v1","url_pdf":"https://arxiv.org/pdf/2404.09639v1.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":"improved-exponent-for-marton-s-conjecture-in","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}