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Range-renewal structure in continued fractions

Published online by Cambridge University Press:  08 March 2016

Department of Mathematics, Huazhong University of Science and Technology, Wuhan 430074, PR China email
School of Mathematical Sciences, Fudan University, Shanghai 200433, PR China email


Let $\unicode[STIX]{x1D714}=[a_{1},a_{2},\ldots ]$ be the infinite expansion of a continued fraction for an irrational number $\unicode[STIX]{x1D714}\in (0,1)$, and let $R_{n}(\unicode[STIX]{x1D714})$ (respectively, $R_{n,k}(\unicode[STIX]{x1D714})$, $R_{n,k+}(\unicode[STIX]{x1D714})$) be the number of distinct partial quotients, each of which appears at least once (respectively, exactly $k$ times, at least $k$ times) in the sequence $a_{1},\ldots ,a_{n}$. In this paper, it is proved that, for Lebesgue almost all $\unicode[STIX]{x1D714}\in (0,1)$ and all $k\geq 1$,

$$\begin{eqnarray}\displaystyle \lim _{n\rightarrow \infty }\frac{R_{n}(\unicode[STIX]{x1D714})}{\sqrt{n}}=\sqrt{\frac{\unicode[STIX]{x1D70B}}{\log 2}},\quad \lim _{n\rightarrow \infty }\frac{R_{n,k}(\unicode[STIX]{x1D714})}{R_{n}(\unicode[STIX]{x1D714})}=\frac{C_{2k}^{k}}{(2k-1)\cdot 4^{k}},\quad \lim _{n\rightarrow \infty }\frac{R_{n,k}(\unicode[STIX]{x1D714})}{R_{n,k+}(\unicode[STIX]{x1D714})}=\frac{1}{2k}.\end{eqnarray}$$
The Hausdorff dimensions of certain level sets about $R_{n}$ are discussed.

Research Article
© Cambridge University Press, 2016 

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