Improved Exponent for Marton's Conjecture in F₂ⁿ

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=F₂ⁿ. Specifically, for G=F₂ⁿ, they proved that any set A G with |A+A| K|A| can be covered by at most 2KC 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.