Algorithmizing the entropy-minimization proof of PFR

Determine whether the entropy-minimization techniques underlying the Gowers–Green–Manners–Tao (Annals of Mathematics, 2025) proof of the Polynomial Freiman–Ruzsa theorem can be converted into an efficient algorithm that, given a set A ⊆ F2^n with doubling constant K, outputs in time polynomial in n (and with polynomial dependence on K) a basis of a subspace V ≤ F2^n with |V| ≤ |A| such that A is covered by poly(K) translates of V.

Background

The authors’ Algorithmic PFR result is obtained via a new framework leveraging quadratic Fourier analysis and a connection to symplectic geometry, rather than by directly algorithmizing the recent information-theoretic proof of PFR by Gowers, Green, Manners, and Tao.

They note that the PFR proof relies heavily on entropy-minimization methods, raising the question of whether those methods themselves admit an efficient, constructive translation. Resolving this would clarify whether the original (non-algorithmic) proof can yield an algorithm with comparable guarantees without changing the proof framework.