On maximal autocorrelations of Rudin-Shapiro sequences

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.