On the abelian complexity of the Rudin-Shapiro sequence

TL;DR

This paper investigates the abelian complexity of the Rudin-Shapiro sequence, revealing its 2-regularity and analyzing the fractal dimension of an associated asymptotic function's graph.

Contribution

It establishes the 2-regularity of the abelian complexity function and analyzes the fractal dimension of the graph of a related asymptotic function.

Findings

The abelian complexity function is 2-regular.

The graph of the asymptotic function has box dimension 3/2.

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 $\rho(n)$ 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 $\lambda(x)$ is $3/2$ where $\lambda(x)=\lim_{k\to\infty}\rho(4^{k}x)/\sqrt{4^{k}x}$ and $\rho(x)=\rho(\lfloor x\rfloor)$ for any $x> 0$.