Ergodic averages with deterministic weights
Annales de l'Institut Fourier, Volume 52 (2002) no. 2, pp. 561-583.

We study the convergence of the ergodic averages 1 N k=0 N-1 θ(k)fT u k where (θ(k)) k is a bounded sequence and (u k ) k a strictly increasing sequence of integers such that Sup α | k=0 N-1 θ(k) exp (2iπαu k )|=O(N δ ) for some δ<1. Moreover we give explicit such sequences θ and u and we investigate in particular the case where θ is a q-multiplicative sequence.

Nous étudions la convergence de moyennes ergodiques 1 N k=0 N-1 θ(k)fT u k (θ(k)) k est une suite bornée et (u k ) k une suite d’entiers strictement croissante tels que Sup α | k=0 N-1 θ(k) exp (2iπαu k )|=O(N δ ) avec δ<1. De plus nous donnons des exemples explicites de telles suites θ et u puis, nous nous intéressons aux cas où θ est une suite q- multiplicative.

DOI: 10.5802/aif.1894
Classification: 37A05,  28D05,  11K99
Keywords: weighted ergodic averages, central limit theorem, almost sure convergence, q-multiplicative sequences, substitutive sequences, generalized Thue-Morse sequences
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     author = {Durand, Fabien and Schneider, Dominique},
     title = {Ergodic averages with deterministic weights},
     journal = {Annales de l'Institut Fourier},
     pages = {561--583},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {52},
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Durand, Fabien; Schneider, Dominique. Ergodic averages with deterministic weights. Annales de l'Institut Fourier, Volume 52 (2002) no. 2, pp. 561-583. doi : 10.5802/aif.1894. https://aif.centre-mersenne.org/articles/10.5802/aif.1894/

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