  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 $x$ $(n \in
\mathbb{N})$. It is well-known that the Lebesgue measure of the set of points
$x \in [0,1)$ for which $\log q_n(x)/n$ deviates away from $\pi^2/(12\log2)$
decays to zero as $n$ 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 $\log q_n(x)/n$. As a consequence, we obtain a large deviation
type result for $\log q_n(x)/n$, which indicates that the rate of this decay is
exponential.
