The distribution of $\alpha \pi$ modulo one in $\mathbb{F}q(T)$: beyond Matomäki's exponent

Abstract

Let $\alpha$ be an element of $\Fq((1/T))$ which is not in $\Fq(T)$. We ask how small $\nrm{απ}$ can be made when $\pi$ runs over irreducible polynomials, and write $\theta (\alpha )$ for the largest exponent $\theta$ such that $\nrm{απ}<|π|^{-θ}$ has infinitely many solutions. The analogue of Matomäki's exponent, $\theta (\alpha )\ge \frac{1}{3}$, was proved by Baier and the author, and it is the natural limit of \chg{methods based on} the Riemann Hypothesis. In this paper we go beyond it. For odd characteristic $p$ and $q$ large in terms of $p$ and $\theta$, we prove that $\theta (\alpha )\ge \theta$ for every $\alpha$ and every $\theta <\sqrt{2}-1=0.414\dots$, and for every $\theta <\frac{1}{2}$ when $\alpha$ has bounded partial quotients. The main tools are an exact count of points in affine subspaces of $\Fq[T]$, a splitting of these subspaces into long arithmetic progressions, and the results of Sawin and Shusterman on the Möbius function along such progressions.

arXiv comment25 pages