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arXiv · 1710.08643

Automatic sequences as good weights for ergodic theorems

Tanja Eisner·Jakub Konieczny
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Abstract

We study correlation estimates of automatic sequences (that is, sequences computable by finite automata) with polynomial phases. As a consequence, we provide a new class of good weights for classical and polynomial ergodic theorems, not coming themselves from dynamical systems. We show that automatic sequences are good weights in $L^2$ for polynomial averages and totally ergodic systems. For totally balanced automatic sequences (i.e., sequences converging to zero in mean along arithmetic progressions) the pointwise weighted ergodic theorem in $L^1$ holds. Moreover, invertible automatic sequences are good weights for the pointwise polynomial ergodic theorem in $L^r$, $r>1$.

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Tanja Eisner, Jakub Konieczny. 2017-10-24. Automatic sequences as good weights for ergodic theorems. https://arxiv.org/abs/1710.08643

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