Correlations of the Thue--Morse sequence

TL;DR

This paper investigates the pair and higher-order correlations of the Thue--Morse sequence, revealing that all odd correlations vanish and even correlations have mean zero, with results applicable to the entire system.

Contribution

It demonstrates that all higher-order correlations are determined by a single 2-point correlation and extends these results to the entire Thue--Morse system.

Findings

All odd-order correlations of the balanced Thue--Morse sequence vanish.

Even-order correlations have mean value zero.

Correlations with real weights can be derived from the 2-point correlations.

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.