The exact convergence rate in the ergodic theorem of Lubotzky–Phillips–Sarnak and a universal lower bound on discrepancies

Abstract

We establish a new result about the equidistribution of points on the two-dimensional round sphere. More precisely, we improve an upper bound of Lubotzky–Phillips–Sarnak on the discrepancies of some finite symmetric sets of isometries of the sphere defined with the help of Lipschitz quaternions. We show that a simple application of the spectral theorem leads to the best possible upper bound. Our proof relies on the deep result of Lubotzky–Phillips– Sarnak about spectral properties of some special free groups of isometries of the sphere. It leads to an upper bound on the discrepancy which matches exactly a general lower bound suspected by Lubotzky. We confirm Lubotzky’s guess by proving a universal lower bound on the discrepancies of actions on atomless probability spaces. We also mention some facts relating discrepancies to standard deviations, spectral gaps, amenability, Kazdhan pairs and $ϵ$-good sets. In an appendix, we emphasize the relation between the computed discrepancies and the values of the Harish-Chandra function of a free group.