A polynomial Freiman-Ruzsa inverse theorem for function fields

2025/01/20 by Bloom, Thomas F.
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2501.11580

Abstract

Using the recent proof of the polynomial Freiman-Ruzsa conjecture over \mathbbF_pⁿ 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⊂\mathbbF_p[t] satisfies | A+tA|≤ 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+ξA where A⊂\mathbbF_p((1/t)) is a finite set and ξ∈ \mathbbF_p((1/t)) is transcendental over \mathbbF_p[t].

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