Wasserstein metrics and quantitative equidistribution of exponential sums over finite fields

TL;DR

This paper explores the use of Wasserstein distances as a natural measure for quantifying how well exponential sums over finite fields are distributed, providing new quantitative equidistribution results.

Contribution

It introduces Wasserstein metrics as an intrinsic tool for measuring equidistribution and applies them to exponential sums, extending previous work and classical theorems.

Findings

Wasserstein distance effectively quantifies equidistribution of exponential sums.

New quantitative bounds for ultra-short exponential sums.

Application of Wasserstein metrics to Deligne and Katz's equidistribution theorems.

Abstract

The Wasserstein distance between probability measures on compact spaces provides a natural invariant quantitative measure of equidistribution, which is partly similar to the classical discrepancy appearing in Erd\"os-Tur\'an type inequalities in the case of tori, but is a more intrinsic quantity. We recall the basic properties of Wasserstein distances and present applications to quantitative forms of equidistribution of exponential sums in two examples, one related to our previous work on the equidistribution of ultra-short exponential sums, and the second a quantitative form of the equidistribution theorems of Deligne and Katz.