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arXiv · 2203.11609

Pointwise convergence in nilmanifolds along smooth functions of polynomial growth

Konstantinos Tsinas
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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.

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Konstantinos Tsinas. 2022-03-22. Pointwise convergence in nilmanifolds along smooth functions of polynomial growth. https://doi.org/10.1017/etds.2023.68

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