A Weyl equidistribution theorem over function fields

TL;DR

This work proves a function-field analogue of Weyl's equidistribution theorem and resolves a conjecture of Lê, Liu, and Wooley by showing that a polynomial sequence $P(u)=\sum_{r\in\mathcal{K}\cup\{0\}} \alpha_r u^r$ is well-distributed in the torus $\mathcal{T}$ provided there exists a $k$ with $p\nmid k$, no $p^v k$ in $\mathcal{K}$ for any $v\ge0$, and $\alpha_k\notin\mathcal{Q}$. The proof uses the Bergelson–Leibman equidistribution theorem by decomposing $P$ into additive components and exploiting an additive term with an irrational coefficient to force $\mathcal{F}(\eta)=\mathcal{T}$, yielding full equidistribution. The paper also extends the result to multivariable polynomials and discusses extensions to global function fields via integral-basis identifications, clarifying the role of Frobenius-related obstructions in positive characteristic. Overall, it provides a robust framework for equidistribution in dual groups of function-field rings and advances the understanding of when such sequences fail to be equidistributed due to characteristic-$p$ phenomena.

Abstract

A classical theorem of Weyl states that any polynomial with an irrational coefficient other than the constant term is uniformly distributed mod 1. We prove a new function field analogue of this statement, confirming a conjecture of Lê, Liu, and Wooley.

Theorems & Definitions (12)

• Theorem 1.1: weyl
• Theorem 1.2
• Remark 1.3
• Theorem 2.1: bl, Theorem 0.3
• Remark 2.2
• Proposition 2.3: bl
• proof : Proof of Theorem \ref{['thm: main']}
• Theorem 3.1
• Theorem 3.2: bl
• proof : Proof of Theorem \ref{['thm: multivariable']}