Correlations of the Thue–Morse sequence

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.