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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2501.11580 |
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Table of Contents:
- 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 $\mathbb{F}_p[t]$.