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Correlations of the Thue--Morse sequence

Michael Baake, Michael Coons

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.

Correlations of the Thue--Morse sequence

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 -point correlation. As a consequence, we show that all odd-order correlations of the balanced Thue--Morse sequence vanish, and that, for any even , the -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 -point correlations.
Paper Structure (5 sections, 12 theorems, 72 equations)

This paper contains 5 sections, 12 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Pair correlations for balanced weights
  4. General correlations for balanced weights
  5. Correlations for general weights

Key Result

Lemma 2.1

For $m\in\mathbb{N}_0$, we have $(t_0,t_1,\ldots,t_{2^m-1})= (1, -1)^{\otimes m}$.

Theorems & Definitions (27)

  • Lemma 2.1
  • proof
  • Remark 3.1
  • Remark 3.2
  • Proposition 3.3
  • Corollary 3.4
  • Theorem 3.5
  • proof
  • Remark 3.6
  • Theorem 3.7
  • ...and 17 more