Wasserstein metrics and quantitative equidistribution of exponential sums over finite fields

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.