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2024

Improved Exponent for Marton's Conjecture in $\mathbb{F}_2^n$

Liao, Jyun-Jie

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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 $C=9$.

Improved Exponent for Marton's Conjecture in $\mathbb{F}_2^n$ · Around

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An analog of freiman’s theorem in groups
Imre Ruzsa · 1999
Earlier work this paper cites.
Additive combinatorics
Terence Tao and Van H Vu · 2006
Earlier work this paper cites.
Sumset and inverse sumset theory for shannon entropy
Terence Tao · 2010
Earlier work this paper cites.

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