Equidistribution of sparse sequences on nilmanifolds

TL;DR

This paper investigates the equidistribution of nil-orbits along sparse sequences, including polynomial, Hardy field, and random sequences, extending previous results to more general sparse settings on nilmanifolds.

Contribution

It proves that certain sparse sequences, such as non-integer powers and Hardy field functions, produce equidistributed nil-orbits, broadening understanding of distribution properties on nilmanifolds.

Findings

Sequences of the form (b^{[n^c]}x) are equidistributed for ergodic nilrotations.

Hardy field functions with polynomial growth and sub-polynomial deviation also produce equidistributed sequences.

Random sequences with sub-exponential growth exhibit equidistribution in nilmanifolds.

Abstract

We study equidistribution properties of nil-orbits $(b^nx)_{n\in\N}$ when the parameter $n$ is restricted to the range of some sparse sequence that is not necessarily polynomial. For example, we show that if $X=G/\Gamma$ is a nilmanifold, $b\in G$ is an ergodic nilrotation, and $c\in \R\setminus \Z$ is positive, then the sequence $(b^{[n^c]}x)_{n\in\N}$ is equidistributed in $X$ for every $x\in X$. This is also the case when $n^c$ is replaced with $a(n)$, where $a(t)$ is a function that belongs to some Hardy field, has polynomial growth, and stays logarithmically away from polynomials, and when it is replaced with a random sequence of integers with sub-exponential growth. Similar results have been established by Boshernitzan when $X$ is the circle.