An algorithmic Polynomial Freiman-Ruzsa theorem

2026-04-06

AI Summary

This work addresses the problem of efficiently constructing low-dimensional subspaces that cover a set \( A \subseteq \mathbb{F}_2^n \) with small doubling constant. We present the first polynomial-time algorithm that outputs a subspace \( V \) of controlled dimension such that \( A \) is covered by at most \( 2K^{O(1)} \) translates of \( V \), thereby achieving the first effective algorithmic realization of the Polynomial Freiman–Ruzsa conjecture. Our approach combines an improved optimal quadratic Goldreich–Levin algorithm, quadratic Fourier analysis, and tools from symplectic geometry, establishing an explicit connection between these frameworks. This result not only yields practical algorithms for approximate Freiman homomorphism classification and structure–randomness decomposition, but also provides a computational foundation for Gowers inverse theorems and equivalent formulations of quadratic structure decomposition.

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.

Problem

• Polynomial Freiman-Ruzsa theorem
• algorithmic
• doubling constant
• subspace covering
• quadratic Fourier analysis

Innovation

• Polynomial Freiman-Ruzsa theorem
• quadratic Goldreich-Levin algorithm
• symplectic geometry
• quadratic Fourier analysis
• algorithmic additive combinatorics