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Weighted Birkhoff ergodic theorem with oscillating weights

Published online by Cambridge University Press:  25 September 2017

AI-HUA FAN
AI-HUA FAN*
Affiliation:
School of Mathematics and Statistics, Central China Normal University, 430079, Wuhan, China LAMFA, UMR 7352 CNRS, University of Picardie, 33 rue Saint Leu, 80039 Amiens, France email [email protected]
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Abstract

We consider sequences of Davenport type or Gelfond type and prove that sequences of Davenport exponent larger than $\frac{1}{2}$ are good sequences of weights for the ergodic theorem, and that the ergodic sums weighted by a sequence of strong Gelfond property are well controlled almost everywhere. We prove that for any $q$-multiplicative sequence, the Gelfond property implies the strong Gelfond property and that sequences realized by dynamical systems can be fully oscillating and have the Gelfond property.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 5 , May 2019 , pp. 1275 - 1289
Copyright
© Cambridge University Press, 2017 

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