THE MULTIFRACTAL SPECTRUM OF CONTINUED FRACTIONS WITH NONDECREASING PARTIAL QUOTIENTS
THE MULTIFRACTAL SPECTRUM OF CONTINUED FRACTIONS WITH NONDECREASING PARTIAL QUOTIENTS
Abstract 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 n th 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 …