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Automatic sequences as good weights for ergodic theorems

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  • 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. 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$.

    Mathematics Subject Classification: Primary: 11B85, 37A30; Secondary: 47A35, 68Q45.


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