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Correlation functions of the Rudin-Shapiro sequence

Jan Mazáč

TL;DR

This paper analyzes higher-order correlation functions of the binary Rudin--Shapiro sequence. By employing renormalisation techniques, symmetry properties, and matrix representations, it proves that all odd-order correlations vanish and that even-order correlations are determined by a single baseline value, with explicit results for the 2-point case. It then provides a detailed treatment of 4-point correlations, revealing arithmetic structures and infinite constant-level sets, yet shows that the mean absolute correlations vanish asymptotically, highlighting a nuanced mix of order and randomness. The findings advance understanding of long-range order in RS sequences and distinguish their higher-order statistics from random and Bernoulli models, while connecting to topological factors with maximal pure point spectrum. The methods offer a framework for examining higher-order statistics in substitution sequences with absolutely continuous spectra.

Abstract

In this paper, we show that all odd-point correlation functions of the balanced Rudin--Shapiro sequence vanish and that all even-point correlation functions depend only on a single number, which holds for any weighted correlation function as well. For the four-point correlation functions, we provide a more detailed exposition which reveals some arithmetic structures and symmetries. In particular, we show that one can obtain the autocorrelation coefficients of its topological factor with maximal pure point spectrum among them.

Correlation functions of the Rudin-Shapiro sequence

TL;DR

This paper analyzes higher-order correlation functions of the binary Rudin--Shapiro sequence. By employing renormalisation techniques, symmetry properties, and matrix representations, it proves that all odd-order correlations vanish and that even-order correlations are determined by a single baseline value, with explicit results for the 2-point case. It then provides a detailed treatment of 4-point correlations, revealing arithmetic structures and infinite constant-level sets, yet shows that the mean absolute correlations vanish asymptotically, highlighting a nuanced mix of order and randomness. The findings advance understanding of long-range order in RS sequences and distinguish their higher-order statistics from random and Bernoulli models, while connecting to topological factors with maximal pure point spectrum. The methods offer a framework for examining higher-order statistics in substitution sequences with absolutely continuous spectra.

Abstract

In this paper, we show that all odd-point correlation functions of the balanced Rudin--Shapiro sequence vanish and that all even-point correlation functions depend only on a single number, which holds for any weighted correlation function as well. For the four-point correlation functions, we provide a more detailed exposition which reveals some arithmetic structures and symmetries. In particular, we show that one can obtain the autocorrelation coefficients of its topological factor with maximal pure point spectrum among them.
Paper Structure (6 sections, 18 theorems, 88 equations)

This paper contains 6 sections, 18 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Rudin--Shapiro sequence
  3. Correlation functions
  4. Renormalisation relations
  5. Matrix representation
  6. 4-point correlation functions

Key Result

Lemma 2.1

TAO The four bi-infinite binary Rudin--Shapiro sequences are fixed under the substitution rules where $\varrho_{\mathrm{even}}$ and $\varrho_{\mathrm{odd}}$ have to be applied to letters at even and odd positions, respectively. In particular, this rule is non-local, as one needs a reference point to apply the rule. ∎

Theorems & Definitions (29)

  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Remark 2.4
  • Proposition 3.1
  • Remark 3.2
  • Lemma 4.1
  • Proposition 4.2
  • proof
  • ...and 19 more