A universal lower bound for the discrepancies of actions of a locally compact group

Antoine Pinochet Lobos, Christophe Pittet

Published 2022-08-12Version 1

We prove a universal lower bound for the discrepancies of measure-preserving actions of a locally compact group on an atomless probability space. It generalizes the universal lower bound for the discrepancies of measure-preserving actions of a discrete group. Many examples show that the generalization from discrete groups to locally compact groups requires some additional hypothesis on the action (we detail some examples due to Margulis). Well-known examples and results of Kazdhan and Zimmer show that the discrepancies of some actions of Lie groups on homogeneous spaces match exactly the universal lower bounds we prove.