We prove that if μₙ are probability measures on ℤ such that μ̂ₙ converges to 0 uniformly on every compact subset of (0,1), then there exists a subsequence such that the weighted ergodic averages corresponding to satisfy a pointwise ergodic theorem in L¹. We further discuss the relationship between Fourier decay and pointwise ergodic theorems for subsequences, considering in particular the averages along n² + ⌊ρ(n)⌋ 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm124-2-2,
author = {Patrick LaVictoire},
title = {Pointwise convergence for subsequences of weighted averages},
journal = {Colloquium Mathematicae},
volume = {122},
year = {2011},
pages = {157-168},
zbl = {1228.42024},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm124-2-2}
}
Patrick LaVictoire. Pointwise convergence for subsequences of weighted averages. Colloquium Mathematicae, Tome 122 (2011) pp. 157-168. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm124-2-2/