The denominators of convergents for continued fractions

For any real number $x\in [0,1)$, we denote by $q_n(x)$ the denominator of the $n$-th convergent of the continued fraction expansion of $x$ $(n \in $\mathbb{N}$)$\mathrm{.}Itiswell-knownthattheLebesguemeasureofthesetofpoints$x \in [0,1)$forwhich$\log q_n(x)/n$deviatesawayfrom$\pi^2/(12\log2)$decaystozeroas$n$ tends to infinity. In this paper, we study the rate of this decay by giving an upper bound and a lower bound. What is interesting is that the upper bound is closely related to the Hausdorff dimensions of the level sets for $\log q_n(x)\mathrm{/}n$. As a consequence, we obtain a large deviation type result for $\log q_n(x)\mathrm{/}n$, which indicates that the rate of this decay is exponential.