Irregularity of distribution in Wasserstein distance

TL;DR

This paper investigates the irregularity of probability measures on intervals and circles using Wasserstein distances, providing sharp estimates and an adapted Erdős–Turán inequality to quantify distribution non-uniformity.

Contribution

It establishes a connection between Wasserstein-$p$ distances and classical discrepancy measures, offering new sharp bounds and inequalities for distribution irregularity.

Findings

Wasserstein-$p$ distance equals classical $L^p$-discrepancy on the interval.

Derived sharp estimates for distribution irregularity in Wasserstein distances.

Proved an $L^p$-adapted Erd ext{"o}s–Turán inequality.

Abstract

We study the non-uniformity of probability measures on the interval and the circle. On the interval, we identify the Wasserstein-$p$ distance with the classical $L^p$-discrepancy. We thereby derive sharp estimates in Wasserstein distances for the irregularity of distribution of sequences on the interval and the circle. Furthermore, we prove an $L^p$-adapted Erd\H{o}s$-$Tur\'{a}n inequality.