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

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 $α_r\in \mathbb K_\infty$ for $r\in \mathcal K\cup {0}$. We show that the polynomial $\sum_{r\in \mathcal K\cup{0}}α_ru^r$ is equidistributed in $\mathbb T$ whenever $α_k$ 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.

💡 Research Summary

The paper studies equidistribution of polynomial sequences over function fields. Let 𝔽_q be a finite field of characteristic p, and let 𝕂_∞ = 𝔽_q((1/t)) be the completion of the rational function field with respect to the degree norm. The compact additive group 𝕋 = 𝕂_∞ / 𝔽_q