Three consecutive approximation coefficients: asymptotic frequencies in semi-regular cases
The authors consider the continued fraction convergents \(p_n/q_n\ (n\geq 1)\) of a real irrational number \(x\) in the form of a so called \textit{semi-regular continued fraction} (SRCF), denoted by \[\text{SRCF}:\quad [a_0;\varepsilon_1a_1,\varepsilon_2a_2,\varepsilon_3a_3,\ldots],\] where \(a_n\geq 1\ (n\geq 1),\varepsilon_{n+1}+a_n\geq 1\ (n\geq 1),\ \varepsilon_{n+1}+a_n\geq 2\) infinitely often. Specifically the \textit{approximation coefficients} are studied \[\theta_n(x)=q_n^ 2\Big|x-\frac{p_n}{q_n}\Big|\ (n\geq 1).\] Special types of these SRCF are the regular-, optimal- and nearest-integer continued fraction \[ \begin{matrix} \text{RCF}:& \quad \varepsilon_n=1,\ n\geq 1 \\ \text{OCF}:& \quad q_n\hbox{ maximal},\ \sup\theta_n\hbox{ minimal},\\ \text{NICF}:& \quad \varepsilon_n \in \{-1,1\}\ (n\geq 1),\ a_0\in\mathbb{Z},\ a_n\geq 2\ (n\geq 1),\ \varepsilon_{n+1}+a_n\geq 2\ (n\geq 1). \end{matrix} \] The aim is to calculate the asymptotic frequencies of the following six patterns: \[ \begin{matrix} \mathcal{A}:\ \theta_{n-1}<\theta_n<\theta_{n+1}, & \mathcal{B}:\ \theta_{n-1}<\theta_{n+1}<\theta_n, & \mathcal{C}:\ \theta_n<\theta_{n-1}<\theta_{n+1}, \\ \mathcal{D}:\ \theta_n<\theta_{n+1}<\theta_{n-1}, & \mathcal{E}:\ \theta_{n+1}<\theta_{n-1}<\theta_n, & \mathcal{F}:\ \theta_{n+1}<\theta_n<\theta_{n-1}. \end{matrix}\] This asymptotic frequency s defined by for instance \[AF(\mathcal{A})=\lim_{N\rightarrow\infty}\,\frac{1}{N}\#\{n\in\mathbb{N}\,|\,2\leq n\leq N,\ \theta_{n-1}<\theta_n<\theta_{n+1}\}.\] For the RCF these asymptotic values were given by \textit{H. Jager} and \textit{J. de Jonge} [Indag. Math., New Ser. 25, No. 4, 816--824 (2014; Zbl 1297.11083)]. In the paper under review, the authors give the explicit values for the OCF and the NICF: Theorem 4. For the optimal continued fraction, the asymptotic frequencies of the six patterns of three consecutive approximation constants are given by \[ AF_O(\mathcal{A})=AF_O(\mathcal{F})=\frac{3}{2}+\frac{1}{\log{G}} \left(\sum_{a=2}^{\infty}\, \log{\frac{(1+\sqrt{1-\frac{4}{a^ 2}})(1+\sqrt{1+\frac{4}{a^ 2}})}{2(1+\sqrt{1+\frac{1}{a^ 4}})}}\right) \approx 0.160377\ldots \] \begin{multline*} AF_O(\mathcal{B})=AF_O(\mathcal{C})=AF_O(\mathcal{D})=AF_O(\mathcal{E}) \\ =-\frac{1}{2}+\frac{1}{2\log{G}}\left(\sum_{a=2}^{\infty}\, \log{\frac{2\big(1+\sqrt{1+\frac{1}{a^ 4}}\big)}{\big(1+\sqrt{1-\frac{4}{a^ 2}}\big)\big(1+\sqrt{1+\frac{4}{a^ 2}}\big)}}\right) \approx 0.169811\ldots \end{multline*} Theorem 5. For the nearest integer continued fraction, the asymptotic frequencies of the six patterns of three consecutive approximation constants are given by \[ \begin{matrix} AF_N(\mathcal{A})=0.168790\ldots & AF_N(\mathcal{B})=0.162629\ldots & AF_N(\mathcal{C})=0.167112\ldots \\ AF_N(\mathcal{D})=0.172509\ldots & AF_N(\mathcal{E})=0.176993\ldots & AF_N(\mathcal{F})=0.151962\ldots \\ \end{matrix} \] A paper that is a pleasure to read. The proof, using ergodic systems connected with the continued fraction algorithm, is a ``sight for sore eyes''.
scientific article; zbMATH DE number 6929336
scientific article