The Polynomial Freiman-Ruzsa (Marton) Conjecture in Integers and Finite Fields via Spectral Stability

TL;DR

This paper proves the Polynomial Freiman-Ruzsa conjecture for integers and cyclic groups by establishing a spectral stability dichotomy, leading to a structural understanding of sets with small doubling.

Contribution

It introduces a new spectral stability approach that resolves the PFR conjecture in integers and cyclic groups, extending to finite fields and groups of bounded exponent.

Findings

Proves PFR conjecture for integers and cyclic groups.

Establishes a spectral stability dichotomy for Fourier mass.

Provides polynomial bounds and a spectral proof for finite fields.

Abstract

We settle the Polynomial Freiman--Ruzsa (PFR/Marton) conjecture for the integers and for cyclic groups. More precisely, we show that if $A$ is a finite subset of $\mathbb{Z}$ or $\mathbb{Z}/N\mathbb{Z}$ with $|A+A| \le K|A|$, then there is a subgroup $H$ of index at most $K^{O(1)}$ such that $A$ is contained in at most $K^{O(1)}$ cosets of $H$. The proof is based on a new spectral stability dichotomy for the $L^4$ Fourier mass of $\mathbf{1}_A$: either this mass is concentrated on a span of size $K^{O(1)}$, or, after passing to a quotient of codimension $K^{O(1)}$, the doubling constant of the image of $A$ decreases by a definite power of $K$. Using Freiman modeling we transfer this dichotomy to cyclic groups, obtain polynomial Bogolyubov-type bounds, and deduce Marton's conjecture in $\mathbb{Z}$ and $\mathbb{Z}/N\mathbb{Z}$. As a corollary, we also recover and extend the finite-field formulation of Marton's conjecture: in odd characteristic we obtain a direct spectral proof, and together with the characteristic-2 result of Green, Gowers, Manners, and Tao this yields a complete resolution of the conjecture for all finite fields. For context beyond finite fields, we recall their theorem for abelian groups of bounded exponent.