Interesting Esoterica

The denominators of convergents for continued fractions

Article by Lulu Fang and Min Wu and Bing Li
  • Published in 2016
  • Added on
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.

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BibTeX entry

@article{Thedenominatorsofconvergentsforcontinuedfractions,
	title = {The denominators of convergents for continued fractions},
	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.},
	url = {http://arxiv.org/abs/1608.01246v1 http://arxiv.org/pdf/1608.01246v1},
	author = {Lulu Fang and Min Wu and Bing Li},
	comment = {},
	urldate = {2016-08-06},
	archivePrefix = {arXiv},
	eprint = {1608.01246},
	primaryClass = {math.NT},
	year = 2016
}