Improved Bounds for the Freiman-Ruzsa Theorem

Let $A$ be a finite subset of an abelian group $G$, and suppose that $|A+A|\leq K|A|$. We show that for any $\epsilon>0$, there exists a constant $C_\epsilon$ such that $A$ can be covered by at most $\exp(C_\epsilon \log(2K)^{1+\epsilon})$ translates of a convex coset progression with dimension at most $C_\epsilon \log(2K)^{1+\epsilon}$ and size at most $\exp(C_\epsilon \log(2K)^{1+\epsilon})|A|$. This falls just short of the Polynomial Freiman-Ruzsa conjecture, which asserts that this statement is true for $\epsilon=0$, and improves on results of Sanders and Konyagin, who showed that this statement is true for all $\epsilon>2$. To prove this result, we use a mixture of entropy methods and Fourier analysis.