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An algorithmic Polynomial Freiman-Ruzsa theorem

Davi Castro-Silva, Jop Briët, Srinivasan Arunachalam, Arkopal Dutt, Tom Gur

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.

An algorithmic Polynomial Freiman-Ruzsa theorem

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 with doubling constant , returns a subspace of size such that can be covered by translates of , for a universal constant . 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.

Paper Structure

This paper contains 43 sections, 52 theorems, 138 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Algorithmic PFR
  3. An algorithmic polynomial Gowers inverse theorem
  4. Symplectic geometry and quadratic Fourier analysis
  5. Algorithms for approximate algebraic structure
  6. Quantum algorithms.
  7. Structure of the paper
  8. Notation
  9. Symplectic geometry and quadratic Fourier analysis
  10. Basic notions in finite symplectic geometry
  11. The Heisenberg group
  12. The Weyl operators
  13. In odd characteristics
  14. The $U^3$ norm via Weyl operators
  15. In odd characteristics
  16. ...and 28 more sections

Key Result

Theorem 1.1

Let $n\geq 1$ be an integer and let $A \subseteq \mathbb{F}_2^{n}$ be a set satisfying $|A + A| \leq K |A|$. Then, there exists a subspace $V \leq \mathbb{F}_2^n$ of size $|V|\leq |A|$ such that $A$ can be covered by $\hbox{\rm poly}(K)$ translates of $V$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (68)

  • Theorem 1.1: $\textsf{PFR}$
  • Theorem 1.2: Algorithmic PFR
  • Theorem 1.3: $\textsf{PGI}$
  • Theorem 1.4: Algorithmic $\textsf{PGI}$
  • Theorem 1.5: Quadratic Goldreich--Levin algorithm
  • Theorem 1.6
  • Definition 2.1: The Heisenberg group over $\mathbb{F}_2$
  • Definition 2.2: Weyl operators
  • Remark 2.3
  • Remark 2.4
  • ...and 58 more