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 $p(n)=n^{1+\epsilon }$, $0<\epsilon <1$. We prove that, almost surely, for every measure-preserving system $(X,T)$ and every $f\in L^1(X)$ the modulated, random averages \[ \frac{1}{N} \sum_{n = 1}^N e(p(n)) T^{a_n(ω)} f\] converge to $0$ pointwise almost everywhere.

v2: corrected the chain of approximations, see (2.8) in v2; small corrections following referee report

References in corpus (1)

• (Uniform) Convergence of Twisted Ergodic Averages

Cited by in corpus (1)

• (Uniform) Convergence of Twisted Ergodic Averages