A random pointwise ergodic theorem with Hardy field weights

Abstract

Let $a_{n}$ be the random increasing sequence of natural numbers which takes each value independently with probability $n^{-a}$, $0<a<1/2$, and let $p(n)=n^{1+\varepsilon}$, $0<\varepsilon<1$. We prove that, almost surely, for every measure-preserving system $(X,T)$ and every $f\in L^{1}(X)$ the modulated, random averages

\[\frac{1}{N}\sum_{n=1}^{N}e(p(n))T^{a_{n}(\omega)}f\] converge to 0 pointwise almost everywhere.