New bounds in the Bogolyubov-Ruzsa lemma

2023/11/07 by Tomasz Kosciuszko, Kosciuszko, Tomasz, Tomasz Schoen +1
Computer Science · Engineering · Mathematics · #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2311.04125

openalex publication_date 2023/11/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We establish new bounds in the Bogolyubov-Ruzsa lemma, demonstrating that if A is a subset of a finite abelian group with density alpha, then 3A-3A contains a Bohr set of rank O(log² (2/alpha)) and radius Omega(log^-2 (2/alpha)). The Bogolyubov-Ruzsa lemma is one of the deepest results in additive combinatorics, with a plethora of important consequences. In particular, we obtain new results toward the Polynomial Freiman-Ruzsa conjecture and improved bounds in Freiman's theorem.

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