1 results
THE MULTIFRACTAL SPECTRUM OF CONTINUED FRACTIONS WITH NONDECREASING PARTIAL QUOTIENTS
- Part of
-
- Journal:
- Bulletin of the Australian Mathematical Society , First View
- Published online by Cambridge University Press:
- 25 November 2024, pp. 1-13
-
- Article
-
- You have access
- HTML
- Export citation
-
Let
$[a_1(x),a_2(x),\ldots ,a_n(x),\ldots ]$ be the continued fraction expansion of 
$x\in [0,1)$ and 
$q_n(x)$ be the denominator of its nth convergent. The irrationality exponent and Khintchine exponent of x are respectively defined by 
$$ \begin{align*} \overline{v}(x)=2+\limsup_{n\to\infty}\frac{\log a_{n+1}(x)}{\log q_n(x)} \quad \text{and}\quad \gamma(x)=\lim_{n\to\infty}\frac{1}{n}\sum_{i=1}^{n}\log a_i(x). \end{align*} $$We study the multifractal spectrum of the irrationality exponent and the Khintchine exponent for continued fractions with nondecreasing partial quotients. For any
$v>2$, we completely determine the Hausdorff dimensions of the sets 
$\{x\in [0,1): a_1(x)\leq a_2(x)\leq \cdots , \overline {v}(x)=v\}$ and 
$$ \begin{align*}\bigg\{x\in[0,1): a_1(x)\leq a_2(x)\leq\cdots, \lim\limits_{n\to\infty}\frac{\log a_1(x)+\log a_2(x)+\cdots+\log a_n(x)}{\psi(n)}=1\bigg\},\end{align*} $$where
$\psi :\mathbb {N}\rightarrow \mathbb {R}^+$ is a function satisfying 
$\psi (n)\to \infty $ as 
$n\to \infty $.
