A polynomial Freiman-Ruzsa inverse theorem for 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 $|A+tA|\le K|A|$ 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 $K^{O(1)}|A|$. As an application we give an optimal lower bound for the size of $A+\hat{I}¾A$ where $A\subset {\mathbb{F}}_p((1/t))$ is a finite set and $\hat{I}¾\in {\mathbb{F}}_p((1/t))$ is transcendental over ${\mathbb{F}}_p[t]$.

11 pages