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arXiv · 0911.3927

Pointwise Convergence for Subsequences of Weighted Averages

Patrick LaVictoire
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Abstract

We prove that if $μ_n$ are probability measures on $Z$ such that $\hat μ_n$ 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 $μ_{n_k}$ 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 $n^2+ \lfloor ρ(n)\rfloor$ for a slowly growing function $ρ$. Under some monotonicity assumptions, the rate of growth of $ρ'(x)$ determines the existence of a "good" subsequence of these averages.

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Patrick LaVictoire. 2011-09-13. Pointwise Convergence for Subsequences of Weighted Averages. https://arxiv.org/abs/0911.3927

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