On the abelian complexity of the Rudin-Shapiro sequence (Q2408601)

The abelian complexity \(\mathcal{\rho}_{\mathbf{w}}(n)\) of an infinite word \(\mathbf w\) counts the number of factors of length \(n\) which are pairwise abelian inequivalent. Two words are abelian equivalent if each alphabet symbol occurs in both of them the same number of times (they yield the same Parikh vector). The paper investigates the abelian complexity of the Rudin-Shapiro sequence \(\mathbf{r}=r(0)r(1)\cdots\in\{-1,1\}^{N}\) given by the recurrence relations \(r(0)=1\), \(r(2n)=r(n)\), \(r(2n+1)=(-1)^{n}r(n)\), \((n\geq0)\), and of the related sequence \(\mathbf{r}^{\prime}=r^{\prime}(0)r^{\prime}(1)\cdots \in\{-1,1\}^{N}\) where \(r^{\prime}(n)=(-1)^{n}r(n)\). The authors show that \(\rho_{\mathbf{r}}=\rho_{\mathbf{r}^{\prime}}\), thus it makes sense to denote this function by a common symbol \(\rho\). They prove that \(\rho\) is \(2\)-regular, i.e., the \(\mathbb{Z}\)-module generated by the kernel of \(\mathbf{r}\) -- the set of subsequences \(K_{2}(\mathbf{r}):=\{(r(2^{e} n+c))_{n\geq0}\mid e\geq0,0\leq c<2^{e}\}\) -- is finitely generated. Then they study the limit function \(\lambda(x):=\lim_{k\rightarrow\infty}\rho (4^{k}x)/\sqrt{4^{k}x}\), where \(\rho(x):=\rho(\left\lfloor x\right\rfloor )\) for every \(x>0\). They show that the box dimension of the graph of \(\lambda\left( x\right) \) on every subinterval of \(\left( 0,\infty\right) \) is \(3/2\). The box dimension of a non-empty bounded set \(F\) in \(\mathbb{R} ^{2}\) is the common value \(\lim\inf_{\delta\rightarrow\infty}\left( N_{\delta }\left( F\right) /-\log\delta\right) =\lim\sup_{\delta\rightarrow\infty }\left( N_{\delta}\left( F\right) /-\log\delta\right) \) (if the equality takes place), where \(N_{\delta}\) is the number of squares in the \(\left( \delta\times\delta\right) \)-square grid containing some point of \(F\).