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| Main Author: | |
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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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| _version_ | 1866916995384999936 |
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| author | Bloom, Thomas F. |
| author_facet | Bloom, Thomas F. |
| 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 $\mathbb{F}_p[t]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_11580 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A polynomial Freiman-Ruzsa inverse theorem for function fields Bloom, Thomas F. Number Theory 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 $\mathbb{F}_p[t]$. |
| title | A polynomial Freiman-Ruzsa inverse theorem for function fields |
| topic | Number Theory |
| url | https://arxiv.org/abs/2501.11580 |