Range-Renewal Structure in Continued Fractions

Abstract

Let $\omega=[a_1, a_2, \cdots]$ be the infinite expansion of continued fraction for an irrational number $\omega \in (0,1)$; let $R_n (\omega)$ (resp. $R_{n, \, k} (\omega)$, $R_{n, \, k+} (\omega)$) be the number of distinct partial quotients each of which appears at least once (resp. exactly $k$ times, at least $k$ times) in the sequence $a_1, \cdots, a_n$. In this paper it is proved that for Lebesgue almost all $\omega \in (0,1)$ and all $k \geq 1$, $$ \displaystyle \lim_{n \to \infty} \frac{R_n (\omega)}{\sqrt{n}}=\sqrt{\frac{\pi}{\log 2}}, \quad \lim_{n \to \infty} \frac{R_{n, \, k} (\omega)}{R_n (\omega)}=\frac{C_{2 k}^k}{(2k-1) \cdot 4^k}, \quad \lim_{n \to \infty} \frac{R_{n, \, k} (\omega)}{R_{n, \, k+} (\omega)}=\frac{1}{2k}. $$ The Hausdorff dimensions of certain level sets about $R_n$ are discussed.