The exact convergence rate in the ergodic theorem of Lubotzky Phillips Sarnak

TL;DR

This paper precisely determines the convergence rates in ergodic theorems for free groups acting on spheres and tori, utilizing spectral bounds from algebraic geometry and number theory.

Contribution

It provides exact convergence rates for free groups acting on geometric spaces, extending previous spectral bounds and establishing optimal discrepancy bounds.

Findings

Exact convergence rate computed for free groups acting on spheres.

Finite rank free groups of automorphisms achieve minimal discrepancy.

Matching upper bounds on convergence rates are established.

Abstract

We compute exact convergence rates in von Neumann type ergodic theorems when the acting group of measure preserving transformations is free and the means are taken over spheres or over balls defined by a word metric. Relying on the upper bounds on the spectra of Koopman operators deduced by Lubozky, Phillips, and Sarnak from Deligne's work on the Weil conjecture, we compute the exact convergence rate for the free groups (of rank $(p+1)/2$ where $p\equiv 1\mod 4$ is prime) of isometries of the round sphere defined by Lipschitz quaternions. We also show that any finite rank free group of automorphisms of the torus realizes the lowest possible discrepancy and prove a matching upper bound on the convergence rate.