Automatic sequences as good weights for ergodic theorems

TL;DR

This paper demonstrates that automatic sequences, which are generated by finite automata, serve as effective weights in various ergodic theorems, extending their applicability beyond traditional dynamical system weights.

Contribution

It introduces a new class of weights for ergodic theorems based on automatic sequences and establishes their effectiveness in both $L^2$ and $L^1$ settings.

Findings

Automatic sequences are good weights in $L^2$ for polynomial averages.

Totally balanced automatic sequences satisfy pointwise weighted ergodic theorems in $L^1$.

Invertible automatic sequences are good weights for pointwise polynomial ergodic theorems in $L^r$, $r>1$.

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 $L^r$, $r>1$.