  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.
