## Pointwise convergence for subsequences of weighted averages

### Volume 124 / 2011

Colloquium Mathematicum 124 (2011), 157-168
MSC: Primary 42B25; Secondary 37A30.
DOI: 10.4064/cm124-2-2

#### Abstract

We prove that if $\mu_n$ are probability measures on $\mathbb Z$ 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.