A polynomial Freiman-Ruzsa inverse theorem for function fields

TL;DR

This work develops a function-field analogue of the polynomial Freiman–Ruzsa inverse theorem by transferring the Gowers–Green–Manners–Tao framework to $\mathbb{F}_q[t]$. It proves that a finite set $A\subset \mathbb{F}_q[t]$ with $|A+tA|\le K|A|$ is efficiently covered by a bounded number of translates of a generalized $\mathbb{F}_q[t]$-progression of rank $O(\log K)$ and size $|P|\le K^{O(1)}|A|$, with a controlled ambient structure. A central technical contribution is a self-contained proof that for a finite $\mathbb{F}_q[t]$-vector space $V$ with $|V+tV|\le q^k|V|$, the weak, structural, and strong arithmetic dimensions all equal $k$, implying $V$ lies in a generalized progression of rank $O(k)$. As an application, the paper derives an optimal lower bound for $|A+\xi A|$ when $A\subset \mathbb{F}_p((t^{-1}))$ and $\xi$ is transcendental over $\mathbb{F}_p[t]$, and discusses higher-parameter bounds with multiple transcendental dilates, highlighting the strength of inverse-sumset phenomena in function fields.

Abstract

Using the recent proof of the polynomial Freiman-Ruzsa conjecture over $\mathbb{F}_p^n$ by Gowers, Green, Manners, and Tao, we prove a version of the polynomial Freiman-Ruzsa conjecture over function fields. In particular, we prove that if $A\subset\mathbb{F}_p[t]$ satisfies $\lvert A+tA\rvert\leq K\lvert A\rvert$ then $A$ is efficiently covered by at most $K^{O(1)}$ translates of a generalised arithmetic progression of rank $O(\log K)$ and size at most $K^{O(1)}\lvert A\rvert$. As an application we give an optimal lower bound for the size of $A+ξA$ where $A\subset\mathbb{F}_p((1/t))$ is a finite set and $ξ\in \mathbb{F}_p((1/t))$ is transcendental over $\mathbb{F}_p[t]$.

Theorems & Definitions (13)

• Definition 1: Generalised arithmetic progressions
• Theorem 1
• Corollary 1
• Theorem 2
• Theorem 3
• Remark 1
• proof : Proof of Theorem \ref{['th-gen']}
• Lemma 1
• proof
• proof : Proof of Theorem \ref{['th:vecspace']}