On maximal autocorrelations of Rudin-Shapiro sequences

TL;DR

This paper offers a simplified proof for the order of maximal autocorrelation in Rudin-Shapiro sequences, connecting it to spectral radius theory and proposing conjectures on autocorrelation maxima.

Contribution

It introduces an alternative, simplified proof method for autocorrelation bounds in Rudin-Shapiro sequences and explores their spectral radius connections.

Findings

Maximal aperiodic autocorrelation grows as λ^m, with λ being a specific root.

Analogous results are established for maximal periodic autocorrelation.

Discussion links proofs to joint spectral radius theory and suggests new conjectures.

Abstract

In this paper, we present an alternative proof showing that the maximal aperiodic autocorrelation of the $m$-th Rudin-Shapiro sequence is of the same order as $\lambda^{m}$, where $\lambda$ is the real root of $x^{3} + x^{2} - 2x - 4$. This result was originally proven by Allouche, Choi, Denise, Erd\'elyi, and Saffari (2019) and Choi (2020) using a translation of the problem into linear algebra. Our approach simplifies this linear algebraic translation and provides another method of dealing with the computations given by Choi. Additionally, we prove an analogous result for the maximal periodic autocorrelation of the $m$-th Rudin-Shapiro sequence. We conclude with a discussion on the connection between the proofs given and joint spectral radius theory, as well as a couple of conjectures on which autocorrelations are maximal.