The denominators of convergents for continued fractions

Abstract

For any real number x ∈ [0, 1), we denote by qn(x) the denominator of the n-th convergent of the continued fraction expansion of x (n ∈ N). It is well-known that the Lebesgue measure of the set of points x ∈ [0, 1) for which log qn(x)/n deviates away from π 2 /(12 log 2) 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 qn(x)/n. As a consequence, we obtain a large deviation type result for log qn(x)/n, which indicates that the rate of this decay is exponential.