University of Cambridge（剑桥大学）；Centrum Wiskunde & Informatica (CWI)（数学与计算机科学中心）；Leiden University（莱顿大学）；IBM Research, Silicon Valley Lab（IBM硅谷实验室）；IBM Research, Cambridge（IBM剑桥研究中心）
Davi Castro-Silva, Jop Briët, Srinivasan Arunachalam, Arkopal Dutt, Tom Gur
备注：
This submission incorporates and extends the earlier versions arXiv:2509.02338 and arXiv:2505.13134
我们为Gowers、Green、Manners和Tao（Ann. of Math., 2025）的多项式自由-鲁茨萨定理提供了算法版本。特别是，我们提供了一个多项式时间算法，给定一个具有倍增常数K的集合A⊆F_2^n，返回一个子空间V⊆F_2^n，其大小|V|≤|A|，使得A可以被2K^C个V的翻译覆盖，其中C是一个通用常数>1。我们还为多项式自由-鲁茨萨定理的几种等效形式提供了高效的算法，例如多项式Gowers逆定理、近似自由同态分类和二次结构-随机性分解。我们的算法框架基于一个新的最优版本的二次Goldreich-Levin算法，该算法利用了量子学习理论中的想法。此框架本质上依赖于二次傅里叶分析与辛几何之间的联系，这一联系最初由Green和Tao（Proc. of Edinb. Math. Soc., 2008）推测，并在本文中明确阐述。
英文摘要：
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.
