Equidistribution of polynomial sequences in function fields: resolution of a conjecture

Abstract

Let ${\mathbb{F}}_q$ be the finite field of $q$ elements having characteristic $p$, and denote by ${\mathbb{K}}_{\infty }={\mathbb{F}}_q((1/t))$ the field of formal Laurent series in $1/t$. We consider the equidistribution in $\mathbb{T}={\mathbb{K}}_{\infty }/{\mathbb{F}}_q[t]$ of the values of polynomials $f(u)\in {\mathbb{K}}_{\infty }[u]$ as $u$ varies over ${\mathbb{F}}_q[t]$. Let $\mathcal{K}$ be a finite set of positive integers, and suppose that $\hat{I}{\pm }_r\in {\mathbb{K}}_{\infty }$ for $r\in \mathcal{K}\cup \{0\}$. We show that the polynomial ${\sum }_{r\in \mathcal{K}\cup \{0\}}\hat{I}{\pm }_r{u}^r$ is equidistributed in $\mathbb{T}$ whenever $\hat{I}{\pm }_k$ is irrational for some $k\in \mathcal{K}$ satisfying $p\nmid k$, and also $p^{v}k\notin \mathcal{K}$ for any positive integer $v$. This conclusion resolves in full a conjecture made jointly by the third, fourth and fifth authors.

17 pages