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Fabien Durand; Dominique Schneider
Ergodic averages with deterministic weights
(Moyennes ergodiques pondérées par des poids déterministes)
Annales de l'institut Fourier, 52 no. 2 (2002), p. 561-583, doi: 10.5802/aif.1894
Article PDF | Reviews MR 1906483 | Zbl 1054.37003
Class. Math.: 37A05, 28D05, 11K99
Keywords: weighted ergodic averages, central limit theorem, almost sure convergence, $q$-multiplicative sequences, substitutive sequences, generalized Thue-Morse sequences

Résumé - Abstract

We study the convergence of the ergodic averages ${1\over N}\sum^{N-1}_{k=0}\theta (k) f\circ T^{u_k}$ where $(\theta (k))_{k\in{\Bbb N}}$ is a bounded sequence and $(u_k)_{k\in{\Bbb N}}$ a strictly increasing sequence of integers such that ${\rm Sup}_{\alpha\in{\Bbb R}}\vert\sum^{N-1}_{k=0}\theta (k){\rm exp}(2i\pi\alpha u_k)\vert =O(N^\delta)$ for some $\delta<1$. Moreover we give explicit such sequences $\theta$ and $u$ and we investigate in particular the case where $\theta$ is a $q$-multiplicative sequence.

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