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Weighted Birkhoff ergodic theorem with oscillating weights

  • AI-HUA FAN (a1) (a2)

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.

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