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

TL;DR

The paper resolves a key conjecture on equidistribution of polynomial sequences in function fields by showing that, under a single irrational coefficient and a precise non-divisibility condition, the values f(u) become equidistributed in the function-field analogue of the unit interval. It develops a robust framework using additive polynomials, the tau-map, and a substitution that eliminates p-multiples, enabling the use of Vinogradov-type mean-value and large-sieve methods to obtain quantitative bounds. By situating the result within the Carlitz criterion and prior LLW2025 work, the authors achieve a best-possible function-field Weyl-type theorem in the stated regime and clarify its limitations with a constructed counterexample to BL/Ack2025 arguments. The work advances understanding of equidistribution in positive characteristic and provides tools potentially applicable to related additive-structure problems in function fields.

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 $α_r\in \mathbb K_\infty$ 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.

Theorems & Definitions (25)

• Definition 1.1
• Theorem 1.2
• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Lemma 2.3
• proof
• Lemma 3.1
• proof