Improved Exponent for Marton's Conjecture in $\mathbb{F}_2^n$
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909170283839488 |
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| author | Liao, Jyun-Jie |
| author_facet | Liao, Jyun-Jie |
| contents | 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 $C=9$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_09639 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Improved Exponent for Marton's Conjecture in $\mathbb{F}_2^n$ Liao, Jyun-Jie Combinatorics 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 $C=9$. |
| title | Improved Exponent for Marton's Conjecture in $\mathbb{F}_2^n$ |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2404.09639 |