An algorithmic Polynomial Freiman-Ruzsa theorem
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| author | Castro-Silva, Davi Briët, Jop Arunachalam, Srinivasan Dutt, Arkopal Gur, Tom |
| author_facet | Castro-Silva, Davi Briët, Jop Arunachalam, Srinivasan Dutt, Arkopal Gur, Tom |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_04547 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An algorithmic Polynomial Freiman-Ruzsa theorem Castro-Silva, Davi Briët, Jop Arunachalam, Srinivasan Dutt, Arkopal Gur, Tom Combinatorics Data Structures and Algorithms 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. |
| title | An algorithmic Polynomial Freiman-Ruzsa theorem |
| topic | Combinatorics Data Structures and Algorithms |
| url | https://arxiv.org/abs/2604.04547 |