Weak mixing and sparse equidistribution

TL;DR

This paper extends the understanding of orbit equidistribution in dynamical systems by generalizing methods to higher dimensions and weakly mixing actions, addressing the behavior of all points on sparse subsets of integers.

Contribution

It generalizes Venkatesh's method to higher-dimensional actions and weak mixing, providing new insights into orbit behavior for all points on sparse integer subsets.

Findings

Established basic properties of weak mixing.

Proved weak mixing for the time 1-map of a weak mixing flow.

Extended Venkatesh's method to new settings.

Abstract

The celebrated Birkhoff Ergodic Theorem asserts that, for an ergodic map, orbits of almost every point equidistributes when sampled at integer times. This result was generalized by Bourgain to many natural sparse subsets of the integers. On the other hand, the behaviour of orbits of \textbf{all} points in a dynamical system is much less understood, especially for sparse subsets of the integers. We generalize a method introduced by A. Venkatesh to tackle this problem in two directions, general $\mathbb{R}^d$ actions instead of flows, and weak mixing, rather than mixing, actions. Along the way, we also establish some basic properties of weak mixing and show weak mixing for the time 1-map of a weak mixing flow.