Uniform distribution in nilmanifolds along functions from a Hardy field

Abstract

We study equidistribution properties of translations on nilmanifolds along functions of polynomial growth from a Hardy field. More precisely, if $X=G/\Gamma$ is a nilmanifold, $a_1,\dots ,a_k\in G$ are commuting nilrotations, and $f_1,\dots ,f_k$ are functions of polynomial growth from a Hardy field then we show that $\bullet$ the distribution of the sequence $a_1^{f_1(n)}\cdot \dots \cdot a_k^{f_k(n)}\Gamma$ is governed by its projection onto the maximal factor torus, which extends Leibman's Equidistribution Criterion form polynomials to a much wider range of functions; and $\bullet$ the orbit closure of $a_1^{f_1(n)}\cdot \dots \cdot a_k^{f_k(n)}\Gamma$ is always a finite union of sub-nilmanifolds, which extends some of the previous work of Leibman and Frantzikinakis on this topic.

49 pages, 2nd revised version after referee comments

References in corpus (1)

• Multiple ergodic averages along functions from a Hardy field: convergence, recurrence and combinatorial applications

Cited by in corpus (2)

• Multiple ergodic averages along functions from a Hardy field: convergence, recurrence and combinatorial applications
• Pointwise convergence in nilmanifolds along smooth functions of polynomial growth