Improved Bounds for the Freiman-Ruzsa Theorem

Let A be a finite subset of an abelian group which is "approximately closed under addition" in the sense that |A+A| < K|A|. To what extent is A approximately algebraically structured? We show that A can be covered efficiently by translates of subgroups and sets arising from convex bodies, with bounds slightly worse than those proposed in the Polynomial Freiman-Ruzsa Conjecture. To prove this result, we use a mixture of entropy methods and Fourier analysis.