Range-Renewal Structure in Continued Fractions

Abstract

Let $ω=[a_1, a_2, \cdots]$ be the infinite expansion of continued fraction for an irrational number $ω\in (0,1)$; let $R_n (ω)$ (resp. $R_{n, \, k} (ω)$, $R_{n, \, k+} (ω)$) 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 $ω\in (0,1)$ and all $k \geq 1$, $$ \displaystyle \lim_{n \to \infty} \frac{R_n (ω)}{\sqrt{n}}=\sqrt{\fracπ{\log 2}}, \quad \lim_{n \to \infty} \frac{R_{n, \, k} (ω)}{R_n (ω)}=\frac{C_{2 k}^k}{(2k-1) \cdot 4^k}, \quad \lim_{n \to \infty} \frac{R_{n, \, k} (ω)}{R_{n, \, k+} (ω)}=\frac{1}{2k}. $$ The Hausdorff dimensions of certain level sets about $R_n$ are discussed.