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 $\alpha_r\in \mathbb K_\infty$ for $r\in \mathcal K\cup \{0\}$. We show that the polynomial $\sum_{r\in \mathcal K\cup\{0\}}\alpha_ru^r$ is equidistributed in $\mathbb T$ whenever $\alpha_k$ is irrational for some $k\in \mathcal K$ satisfying $p\nmid k$, and also $p^vk\not\in \mathcal K$ for any positive integer $v$. This conclusion resolves in full a conjecture made jointly by the third, fourth and fifth authors.