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

💡 Research Summary

The paper addresses the quantitative structure of finite subsets A of an abelian group G that satisfy a small-doubling condition |A+A| ≤ K|A|. The classical Freiman–Ruzsa theorem guarantees that such a set can be covered by a bounded number of translates of a convex coset progression, but the dependence on K has been far from optimal. The long‑standing Polynomial Freiman–Ruzsa conjecture predicts that the covering number, the dimension of the progression, and its size can all be bounded by exp(O(log K)) and O(log K) respectively (i.e., essentially polynomial in K). Prior work by Sanders and Konyagin achieved bounds of the form exp(O(log³ K·log log K)), which left a substantial gap.

The author improves this dramatically: for any ε>0 there exists a constant Cε such that A can be covered by at most exp(Cε·(log (2K))^{1+ε}) translates of a convex coset progression of dimension at most Cε·(log (2K))^{1+ε} and size at most exp(Cε·(log (2K))^{1+ε})·|A|. In other words, the function f(K) in the quantitative formulation satisfies 2f(K)=O(log K·log log K), which is essentially optimal up to the ε‑loss and brings the result within a logarithmic factor of the conjectured bound.

The proof is divided into two complementary parts.

Entropy side.
The author introduces the Shannon entropy H(X) of a G‑valued random variable X and the entropic Ruzsa distance d