Correlations of the Thue--Morse sequence

Michael Baake; Michael Coons

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arXiv · 2209.07102

Correlations of the Thue--Morse sequence

Michael Baake·Michael Coons

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Abstract

The pair correlations of the Thue--Morse sequence and system are revisited, with focus on asymptotic results on various means. First, it is shown that all higher-order correlations of the Thue--Morse sequence with general real weights are effectively determined by a single value of the balanced $2$-point correlation. As a consequence, we show that all odd-order correlations of the balanced Thue--Morse sequence vanish, and that, for any even $n$, the $n$-point correlations of the balanced Thue--Morse sequence have mean value zero, as do their absolute values, raised to an arbitrary positive power. All these results also apply to the entire Thue--Morse system. We finish by showing how the correlations of the Thue--Morse system with general real weights can be derived from the balanced $2$-point correlations.

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Michael Baake, Michael Coons. 2022-09-15. Correlations of the Thue--Morse sequence. https://arxiv.org/abs/2209.07102

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