Pointwise Convergence of Ergodic Averages Along Hardy Field Sequences

TL;DR

This paper proves pointwise convergence of ergodic averages along Hardy field sequences with distinct growth rates, establishing maximal and variational inequalities, and providing quantitative bounds on exponential sums.

Contribution

It introduces new convergence results for ergodic averages along non-polynomial Hardy field functions with distinct growth rates, including variational inequalities and explicit convergence rates.

Findings

Pointwise convergence of ergodic averages for Hardy field sequences.

Maximal inequalities for the averages.

Quantitative bounds on exponential sums.

Abstract

Let $(X,\mu)$ be an arbitrary measure space equipped with a family of pairwise commuting measure preserving transformations $T_1, \dotsc, T_m$. We prove that the ergodic averages \[ A_{N;X}^{P_1, \dotsc, P_m}f = \frac{1}{N} \sum_{n=1}^N T_1^{\lfloor P_1(n) \rfloor} \dotsm T_m^{\lfloor P_m(n) \rfloor} f \] converge pointwise $\mu$-almost everywhere as $N \to \infty$ for any $f \in L^p(X)$ with $p>1$, where $P_1, \dotsc, P_m$ are Hardy field functions which are "non-polynomial" and have distinct growth rates. To establish pointwise convergence we will prove a long-variational inequality, which will in turn prove that a maximal inequality holds for our averages. Additionally, by restricting the class of Hardy field functions to those with the same growth rate as $t^c$ for $c>0$ non-integer, we also prove full variational estimates. We are therefore able to provide quantitative bounds on the rate of convergence of exponential sums of the form \[ \frac{1}{N} \sum_{n=1}^N e(\xi_1 \lfloor n^{c_1} \rfloor + \dotsb + \lfloor n^{c_m} \rfloor) \] where $0<c_1<\dotsb<c_m$ are non-integer.