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

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 $C=9$.