A note on polynomial equidistribution and recurrence in finite characteristic

TL;DR

This paper corrects errors in prior work on polynomial equidistribution in finite characteristic and introduces new characterizations of intersective polynomials using algebraic, combinatorial, and dynamical methods.

Contribution

It identifies and corrects two errors in a key theorem and provides novel characterizations of intersective polynomials in finite characteristic.

Findings

Corrected previous equidistribution theorem in finite characteristic

Established new algebraic and dynamical characterizations of intersective polynomials

Enhanced understanding of polynomial recurrence in function fields

Abstract

This paper addresses the topic of equidistribution and recurrence for polynomial sequences over function fields. The main focus is to note and correct two small errors in [V. Bergelson and A. Leibman, A Weyl-type equidistribution theorem in finite characteristic, Adv. Math. 289 (2016) 928-950], contextualized within the broader developing literature on number theory and additive combinatorics in function fields. Connected with the resolution of these issues, we also prove new results characterizing intersective polynomials in finite characteristic in terms of various algebraic, combinatorial, and dynamical properties.