Equidistribution of polynomial sequences in function fields, with applications

TL;DR

This paper establishes a function field analog of Weyl's theorem, demonstrating equidistribution of polynomial sequences even when the degree exceeds or equals the field's characteristic, with applications to various sets in function fields.

Contribution

It extends Weyl's equidistribution theorem to function fields, including cases where polynomial degree surpasses the characteristic, addressing a key barrier in the field.

Findings

Proves equidistribution of polynomial sequences in function fields.

Addresses the case where polynomial degree ≥ characteristic.

Provides applications to van der Corput, intersective, and Glasner sets.

Abstract

We prove a function field analog of Weyl's classical theorem on equidistribution of polynomial sequences. Our result covers the case in which the degree of the polynomial is greater than or equal to the characteristic of the field, which is a natural barrier when applying the Weyl differencing process to function fields. We also discuss applications to van der Corput, intersective and Glasner sets in function fields.