Continued fraction and decimal expansions of an irrational number
Let \(k_n(x)\) be the exact number of partial quotients in the continued fraction expansion of the irrational number \(x\) that can be computed from the first \(n\) decimal digits of \(x\). It was proven by \textit{G. Lochs} [Abh. Math. Semin. Univ. Hamb. 27, 142--144 (1964; Zbl 0124.28003)] that, for almost all \(x\), \(\lim_{n\to+\infty} {k_n(x)\over n}= {6\log 2\log 10\over \pi^2}\). Define \(\beta_*(x)\), \(\beta^*(x)\), and \(\beta(x)\) by \(\beta_*(x)= \varliminf_{n\to+\infty} {\log q_n(x)\over n}\), \(\beta^*(x)= \varlimsup_{n\to+\infty}{\log q_n(x)\over n}\), where \(q_n(x)\) is the denominator of the \(n\)th convergent of \(x\); furthermore if \(\beta_*(x)= \beta^*(x)\) (say \(\beta(x)\)) this common value is called the Lévy constant of \(x\). \textit{C. Faivre} proved in [Acta Sci. Math. 67, No. 3--4, 505--519 (2001; Zbl 1017.11040)] that if \(\beta^*= \beta_*\) and if the \(n\)th partial quotient of \(x\) is \(O(\alpha^n)\) for all \(\alpha> 1\), then \(\lim_{n\to+\infty} {k_n(x)\over n}= {\log 10\over 2\beta(x)}\). The author of the paper under review proves that for any irrational \(x\) \[ \varliminf{k_n(x)\over n}= {\log 10\over 2\beta_*(x)}\quad\text{and}\quad \varlimsup{k_n(x)\over x}= {\log 10\over 2\beta^*(x)}. \] Furthermore if \(\beta_*(x)< \beta^*(x)\) any value inbetween can be obtained by letting \(n\) tend to infinity along some subsequence of \(k_n(x)/n\). Please note that [13] appeared [see \textit{J. Wu}, Monatsh. Math. 147, No. 3, 259--264 (2006; Zbl 1091.11028)]. Also note the recent relevant reference by the same author [Monatsh. Math. 153, No. 1, 83--87 (2008; Zbl 1136.11050)].
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