Marton's Conjecture in Finite Fields of Odd Characteristic via a Polynomial Stability Lemma (2512.04433v1)

Abstract: We study small-doubling subsets of finite-dimensional vector spaces over finite fields of odd characteristic. Let $A \subset \mathbb{F}_p^n$ be non-empty with $|A+A| \le K|A|$. We prove that $A$ can be covered by at most $K^{O(1)}$ cosets of a subspace $H \le \mathbb{F}_p^n$ with $|H| \le K^{O(1)}|A|$, giving a polynomial Freiman--Ruzsa (PFR/Marton) theorem in $\mathbb{F}_p^n$ for odd primes $p$. The key input is a polynomial stability lemma which yields a dichotomy: either the $L^4$ Fourier mass of $1_A$ concentrates on a span of dimension $\operatorname{poly}(K)$, or in a quotient of codimension $\operatorname{poly}(K)$ the doubling constant decreases by at least $K^{-C}$. Iterating the latter alternative and combining it with standard covering arguments gives the polynomial-structured conclusion. In conjunction with the characteristic-$2$ results of Green, Gowers, Manners, and Tao, our methods provide an alternative finite-field route to Marton-type Freiman--Ruzsa theorems across all characteristics.