A polynomial Freiman-Ruzsa inverse theorem for function fields

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 $\mid A+tA\mid \le K\mid A\mid$ 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 $K^{O(1)}\mid A\mid$. As an application we give an optimal lower bound for the size of $A+\xi A$ where $A\subset {\mathbb{F}}_p((1\mathrm{/}t))$ is a finite set and $\xi \in {\mathbb{F}}_p((1\mathrm{/}t))$ is transcendental over ${\mathbb{F}}_p[t]$.