An algorithmic Polynomial Freiman-Ruzsa theorem

TL;DR

This paper develops polynomial-time algorithms for the Polynomial Freiman-Ruzsa theorem in finite fields, connecting quadratic Fourier analysis with symplectic geometry, and extends to various related structural theorems.

Contribution

It introduces an efficient algorithmic framework based on quadratic Fourier analysis and symplectic geometry for the Polynomial Freiman-Ruzsa theorem and related results.

Findings

Polynomial-time algorithm for covering sets with subspaces

Efficient algorithms for inverse theorems and structure decompositions

New connection between quadratic Fourier analysis and symplectic geometry

Abstract

We provide algorithmic versions of the Polynomial Freiman-Ruzsa theorem of Gowers, Green, Manners, and Tao (Ann. of Math., 2025). In particular, we give a polynomial-time algorithm that, given a set $A \subseteq \mathbb{F}_2^n$ with doubling constant $K$, returns a subspace $V \subseteq \mathbb{F}_2^n$ of size $|V| \leq |A|$ such that $A$ can be covered by $2K^C$ translates of $V$, for a universal constant $C>1$. We also provide efficient algorithms for several "equivalent" formulations of the Polynomial Freiman-Ruzsa theorem, such as the polynomial Gowers inverse theorem, the classification of approximate Freiman homomorphisms, and quadratic structure-vs-randomness decompositions. Our algorithmic framework is based on a new and optimal version of the Quadratic Goldreich-Levin algorithm, which we obtain using ideas from quantum learning theory. This framework fundamentally relies on a connection between quadratic Fourier analysis and symplectic geometry, first speculated by Green and Tao (Proc. of Edinb. Math. Soc., 2008) and which we make explicit in this paper.