The denominators of convergents for continued fractions

TL;DR

This paper investigates the decay rate of the measure of points in [0,1) where the normalized logarithm of the denominators of continued fraction convergents deviates from its typical value, establishing bounds and a large deviation result.

Contribution

It provides new bounds on the decay rate of deviation probabilities and links these bounds to Hausdorff dimensions, offering a large deviation principle for continued fraction denominators.

Findings

Established upper and lower bounds for decay rates.

Linked decay bounds to Hausdorff dimensions of level sets.

Proved exponential decay rate for deviation probabilities.

Abstract

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.