Tohoku Mathematical Journal

Three consecutive approximation coefficients: asymptotic frequencies in semi-regular cases

Jaap de Jonge and Cor Kraaikamp

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Abstract

Denote by $p_n/q_n, n=1,2,3,\ldots,$ the sequence of continued fraction convergents of a real irrational number $x$. Define the sequence of approximation coefficients by $\theta_n(x):=q_n\left|q_nx-p_n\right|, n=1,2,3,\ldots$. In the case of regular continued fractions the six possible patterns of three consecutive approximation coefficients, such as $\theta_{n-1}<\theta_n<\theta_{n+1}$, occur for almost all $x$ with only two different asymptotic frequencies. In this paper it is shown how these asymptotic frequencies can be determined for two other semi-regular cases. It appears that the optimal continued fraction has a similar distribution of only two asymptotic frequencies, albeit with different values. The six different values that are found in the case of the nearest integer continued fraction will show to be closely related to those of the optimal continued fraction.

Article information

Source
Tohoku Math. J. (2), Volume 70, Number 2 (2018), 285-317.

Dates
First available in Project Euclid: 2 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1527904823

Digital Object Identifier
doi:10.2748/tmj/1527904823

Mathematical Reviews number (MathSciNet)
MR3810242

Zentralblatt MATH identifier
06929336

Subjects
Primary: 11J70: Continued fractions and generalizations [See also 11A55, 11K50]
Secondary: 11K50: Metric theory of continued fractions [See also 11A55, 11J70]

Keywords
Continued fractions metric theory

Citation

Jonge, Jaap de; Kraaikamp, Cor. Three consecutive approximation coefficients: asymptotic frequencies in semi-regular cases. Tohoku Math. J. (2) 70 (2018), no. 2, 285--317. doi:10.2748/tmj/1527904823. https://projecteuclid.org/euclid.tmj/1527904823


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