Improved Bounds for the Freiman-Ruzsa Theorem

2025/12/12 by Raghavan, Rushil · 1 citation
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2512.11217

openalex publication_date 2025/12/12 · openalex created_date 2025/12/16 · openalex updated_date 2026/07/28

Abstract

Let A be a finite subset of an abelian group G, and suppose that |A+A|≤ 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_εlog(2K)^1+ε) translates of a convex coset progression with dimension at most C_εlog(2K)^1+ε and size at most exp(C_εlog(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.

Citations

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