Improved Bounds for the Freiman-Ruzsa Theorem

Abstract

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

29 pages, Comments welcome! Update improves the bound in the main theorem, removing the logloglog(K) factor, and fixes some typos