Pointwise Convergence for Subsequences of Weighted Averages

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 ${\rho }^{\prime }(x)$ determines the existence of a "good" subsequence of these averages.

LaTeX, 11 pages; corrected from previous version (which included an erroneous minor result)