A Note on the Abelian Complexity of the Rudin-Shapiro Sequence

Mathematics

Pengju Han¹  Xiaotao Lü¹ 

[1] College of Science, Huazhong Agricultural University, Wuhan 430070, China;

关键词: Rudin-Shapiro sequence;    abelian complexity;    growth order;    dense property;   

DOI  :  10.3390/math10020221

来源: DOAJ

【 摘 要 】

Let ${\left\{r\left(n\right)\right\}}_{n\ge 0}$ be the Rudin-Shapiro sequence, and let $\rho \left(n\right):=max\{{\sum }_{j=i}^{i+n-1}r\left(j\right)\mid i\ge 0\}+1$ be the abelian complexity function of the Rudin-Shapiro sequence. In this note, we show that the function $\rho \left(n\right)$ has many similarities with the classical summatory function $S_{\mathbf{r}}\left(n\right):={\sum }_{i=0}^{n}r\left(i\right)$. In particular, we prove that for every positive integer n, $\sqrt{3}\le \frac{\rho \left(n\right)}{\sqrt{n}}\le 3$. Moreover, the point set $\{\frac{\rho \left(n\right)}{\sqrt{n}}:n\ge 1\}$ is dense in $[\sqrt{3},3]$.

【 授权许可】

Unknown