Range-Renewal Structure in Continued Fractions

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 $k\ge 1$, $${\displaystyle \underset{n\to \mathrm{\infty }}{\lim }\frac{R_n(\omega )}{\sqrt{n}}=\sqrt{\frac{\pi }{\log 2}},\underset{n\to \mathrm{\infty }}{\lim }\frac{R_{n,k}(\omega )}{R_n(\omega )}=\frac{C_{2k}^k}{(2k-1)\cdot 4^k},\underset{n\to \mathrm{\infty }}{\lim }\frac{R_{n,k}(\omega )}{R_{n,k+}(\omega )}=\frac{1}{2k}\mathrm{.}}$$ The Hausdorff dimensions of certain level sets about $R_n$ are discussed.