Correlation functions of the Rudin-Shapiro sequence

TL;DR

This paper investigates the correlation functions of the Rudin-Shapiro sequence, revealing that odd-point correlations vanish and even-point correlations depend on a single parameter, with detailed analysis of four-point functions and their symmetries.

Contribution

It provides a comprehensive analysis of correlation functions of the Rudin-Shapiro sequence, including explicit formulas and symmetry properties, advancing understanding of its arithmetic and spectral structure.

Findings

All odd-point correlation functions vanish.

Even-point correlations depend on a single parameter.

Four-point correlations exhibit specific arithmetic structures.

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.