On mixing and sparse ergodic theorems

TL;DR

This paper investigates the impact of mixing properties on ergodic theorems, specifically for horocyclic flows, providing bounds on exceptional sets and supporting conjectures on equidistribution in homogeneous spaces.

Contribution

It establishes new bounds on the Hausdorff dimension of exceptional sets in ergodic theorems under mixing conditions, independent of spectral gaps.

Findings

Bound the Hausdorff dimension of exceptional sets from above

Provide evidence supporting conjectures on equidistribution

Establish a uniform upper bound independent of spectral gap

Abstract

We consider Bourgain's ergodic theorem regarding arithmetic averages in the cases where quantitative mixing is present in the dynamical system. Focusing on the case of the horocyclic flow, those estimates allows us to bound from above the Hausdorff dimension of the exceptional set, providing evidence towards conjectures by Margulis,Shah and Sarnak regarding equidistribution of arithmetic averages in homogeneous spaces. We also prove the existence of a uniform upper bound for the Hausdorff dimension of the exceptional set which is independent from the spectral gap.