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.
翻译:在本文中,我们提出了一个替代证据,表明卢丁-沙皮罗(Rudin-Shapiro)第2x-4美元实际根值为$x{%3}+ x ⁇ 2} - 2xx-4美元的最高周期自动关系。这一结果最初由Alloche、Choi、Denise、Erd\'elyi(2019年)和Saffari(20202020年)用将问题翻译成线性代数来证明。我们的方法简化了线性代数翻译并提供另一种方法处理崔英杰给出的计算。此外,我们证明鲁丁-沙皮罗(Rudin-Shapiro)序列的最大周期自动调节结果相似。我们最后讨论了所提供的证据与共同光谱半径理论之间的联系,以及几条关于自动调节为最高值的预测。