Weighted Birkhoff ergodic theorem with oscillating weights
Résumé
We consider sequences of Davenport type or Gelfond type and prove that sequences of Davenport exponent larger than 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.