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2018 Three consecutive approximation coefficients: asymptotic frequencies in semi-regular cases
Jaap de Jonge, Cor Kraaikamp
Tohoku Math. J. (2) 70(2): 285-317 (2018). DOI: 10.2748/tmj/1527904823

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.

Citation

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Jaap de Jonge. Cor Kraaikamp. "Three consecutive approximation coefficients: asymptotic frequencies in semi-regular cases." Tohoku Math. J. (2) 70 (2) 285 - 317, 2018. https://doi.org/10.2748/tmj/1527904823

Information

Published: 2018
First available in Project Euclid: 2 June 2018

zbMATH: 06929336
MathSciNet: MR3810242
Digital Object Identifier: 10.2748/tmj/1527904823

Subjects:
Primary: 11J70
Secondary: 11K50

Rights: Copyright © 2018 Tohoku University

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Vol.70 • No. 2 • 2018
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