Pointwise convergence in nilmanifolds along smooth functions of polynomial growth

TL;DR

This paper investigates the distribution of certain orbits in nilmanifolds generated by smooth functions with polynomial growth, establishing conditions for their uniform distribution and applying these results to ergodic averages.

Contribution

It introduces new conditions on smooth functions of polynomial growth for equidistribution in nilmanifolds and applies these to prove convergence of multiple ergodic averages.

Findings

Orbits are uniformly distributed on subnilmanifolds under specific growth conditions.

Established a norm convergence result for multiple ergodic averages involving Hardy field functions.

Utilized Green-Tao equidistribution results to analyze polynomial orbits in nilmanifolds.

Abstract

We study the equidistribution of orbits of the form $b_1^{a_1(n)}... b_k^{a_k(n)}\Gamma$ in a nilmanifold $X$, where the sequences $a_i(n)$ arise from smooth functions of polynomial growth belonging to a Hardy field. We show that under certain assumptions on the growth rates of the functions $a_1,...,a_k$, these orbits are uniformly distributed on some subnilmanifold of the space $X$. As an application of these results and in combination with the Host-Kra structure theorem for measure preserving systems, as well as some recent seminorm estimates of the author for ergodic averages concerning Hardy field functions, we deduce a norm convergence result for multiple ergodic averages. Our method mainly relies on an equidistribution result of Green-Tao on finite polynomial orbits of a nilmanifold.