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arXiv · 1606.06935

On the abelian complexity of the Rudin-Shapiro sequence

Xiaotao Lü·Jin Chen·Zhixiong Wen·Wen Wu
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Abstract

In this paper, we study the abelian complexity of the Rudin-Shapiro sequence and a related sequence. We show that these two sequences share the same complexity function $ρ(n)$ which satisfies certain recurrence relations. As a consequence, the abelian complexity function is $2$-regular. Further, we prove that the box dimension of the graph of the asymptotic function $λ(x)$ is $3/2$ where $λ(x)=\lim_{k\to\infty}ρ(4^{k}x)/\sqrt{4^{k}x}$ and $ρ(x)=ρ(\lfloor x\rfloor)$ for any $x> 0$.

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Xiaotao Lü, Jin Chen, Zhixiong Wen, Wen Wu. 2016-06-22. On the abelian complexity of the Rudin-Shapiro sequence. https://doi.org/10.1016/j.jmaa.2017.02.019

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