Pointwise convergence in nilmanifolds along smooth functions of polynomial growth

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.

34 pages, Several corrections, Referee suggestions incorporated, To appear in Ergodic Theory and Dynamical Systems

References in corpus (2)

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