Irregularity of Distribution in Wasserstein Distance

We study the non-uniformity of probability measures on the interval and 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 circle. Furthermore, we prove an L-p-adapted Erdos-Turan inequality, and use it to extend a well-known bound of Polya and Vinogradov on the equidistribution of quadratic residues in finite fields.