Range-Renewal Structure in Continued Fractions

TL;DR

This paper investigates the asymptotic behavior of the number of distinct partial quotients in continued fraction expansions for almost all irrationals, revealing precise limit laws and discussing the fractal dimensions of related level sets.

Contribution

It establishes new limit theorems for the growth and distribution of partial quotients in continued fractions, including explicit constants and ratios, and explores the Hausdorff dimensions of specific level sets.

Findings

The number of distinct partial quotients grows proportionally to the square root of n with a specific constant.

Ratios of counts of partial quotients appearing exactly k times to at least k times converge to explicit constants.

Hausdorff dimensions of certain level sets related to partial quotient counts are analyzed.

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.