Improved Bounds for the Freiman-Ruzsa Theorem

Paper

Abstract

Let

be a finite subset of an abelian group

, and suppose that

. We show that for any

, there exists a constant

such that

can be covered by at most

translates of a convex coset progression with dimension at most

and size at most

. This falls just short of the Polynomial Freiman-Ruzsa conjecture, which asserts that this statement is true for

, and improves on results of Sanders and Konyagin, who showed that this statement is true for all

. To prove this result, we use a mixture of entropy methods and Fourier analysis.