Pointwise Convergence for Subsequences of Weighted Averages

TL;DR

This paper establishes conditions under which subsequences of weighted averages satisfy a pointwise ergodic theorem, linking Fourier decay properties to convergence behavior in ergodic theory.

Contribution

It proves that certain Fourier decay conditions imply the existence of subsequences satisfying pointwise ergodic theorems in $L^1$, and explores the influence of growth rates on these subsequences.

Findings

Fourier decay implies subsequential pointwise convergence.

Growth rate of $ ho$ affects the existence of good subsequences.

Connections between Fourier decay and ergodic theorems are established.

Abstract

We prove that if $\mu_n$ are probability measures on $Z$ such that $\hat \mu_n$ converges to 0 uniformly on every compact subset of $(0,1)$, then there exists a subsequence $\{n_k\}$ such that the weighted ergodic averages corresponding to $\mu_{n_k}$ satisfy a pointwise ergodic theorem in $L^1$. We further discuss the relationship between Fourier decay and pointwise ergodic theorems for subsequences, considering in particular the averages along $n^2+ \lfloor \rho(n)\rfloor$ for a slowly growing function $\rho$. Under some monotonicity assumptions, the rate of growth of $\rho'(x)$ determines the existence of a "good" subsequence of these averages.