Weighted Birkhoff ergodic theorem with oscillating weights

TL;DR

This paper extends the weighted Birkhoff ergodic theorem by analyzing sequences with Davenport and Gelfond properties, showing their effectiveness as weights for ergodic averages and exploring their oscillatory behavior.

Contribution

It establishes that sequences with Davenport exponent > 1/2 are effective weights for the ergodic theorem and links Gelfond properties to oscillation in dynamical systems.

Findings

Sequences with Davenport exponent > 1/2 are good weights for ergodic theorems.

Sequences with strong Gelfond property are well-controlled almost everywhere.

Gelfond property implies strong Gelfond property for q-multiplicative sequences.

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 is 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.