A polynomial Freiman-Ruzsa inverse theorem for function fields

TL;DR

This paper extends the polynomial Freiman-Ruzsa theorem to function fields, showing that sets with small sumsets are structured and providing bounds on their size and structure, with applications to sumsets involving transcendental elements.

Contribution

It proves a polynomial Freiman-Ruzsa inverse theorem for function fields, adapting recent finite field results to the setting of $\, ext{function fields}$ and establishing bounds on sumset sizes.

Findings

Sets with small sumsets are covered by a bounded number of generalized arithmetic progressions.

Provides an optimal lower bound for the size of sumsets involving transcendental elements.

Extends polynomial Freiman-Ruzsa results from finite fields to 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 $A+\xi A$ where $A\subset\mathbb{F}_p((1/t))$ is a finite set and $\xi\in \mathbb{F}_p((1/t))$ is transcendental over $\mathbb{F}_p[t]$.