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Optimal approximation by continued fractions

Part of: Diophantine approximation, transcendental number theory Probabilistic theory: distribution modulo $1$; metric theory of algorithms

Published online by Cambridge University Press:  09 April 2009

Wieb Bosma  and
Cor Kraaikamp
Wieb Bosma
Affiliation:
Department of Pure Mathematics University of SydneySydney, NSW 2006, Australia
Cor Kraaikamp
Affiliation:
Fakulteit Wiskunde en Informatika Universiteit van AmsterdamPlantage Muidergracht 24 1018 TV Amsterdam, The Netherlands
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Abstract

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Among all possible semiregular continued fraction expansions of an irrational number the one with the best approximation properties, in a well-defined and natural sense, is determined. Some properties of this so called optimal continued fraction expansion are described.

Keywords

continued fractionsdiophantine approximation

MSC classification

Secondary: 11J70: Continued fractions and generalizations 11K50: Metric theory of continued fractions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 50 , Issue 3 , June 1991 , pp. 481 - 504
Copyright
Copyright © Australian Mathematical Society 1991

References

[1]Bosma, W., ‘Optimal continued fractions’, Indag. Math. 49 (1987), 353–379.CrossRefGoogle Scholar
[2]Bosma, W., Jager, H. and Wiedijk, F., ‘Some metrical observations on the approximation by continued fractions’, Indag. Math. 45 (1983), 281–299.CrossRefGoogle Scholar
[3]Bosma, W. and Kraaikamp, C., ‘Metrical theory for optimal continued fractions’, J. Number Theory 34 (1990), 251–270.CrossRefGoogle Scholar
[4]Jager, H. and Kraaikamp, C., ‘On the approximation by continued fractions’, Indag. Math. 51 (1989), 289–307.CrossRefGoogle Scholar
[5]Keller, O.-H., ‘Eine Bemerkung zu den verschiedenen Möglichkeiten eine Zahl in einen Kettenbruch zu entwickeln’, Math. Ann. 116 (1939), 733–741.Google Scholar
[5]Koksma, F., Diophantische Approximation, Springer, Berlin, 1936.Google Scholar
[7]Kraaikamp, C., ‘Statistic and ergodic properties of Minkowski's diagonal continued fraction’, Theoret. Comp. Sci. 65 (1989), 197–212.CrossRefGoogle Scholar
[8]Perron, O., Die Lehre von den Kettenbrüchen, Chelsea, New York, 1929.Google Scholar
[9]Rieger, G. J., ‘Mischung und Ergodizität bei Kettenbrüchen nach nächsten Ganzen’, J. Reine Angew. Math. 310 (1979), 171–181.Google Scholar
[10]Selenius, C.-O., ‘Konstruktion und Theorie halbregelmässiger Kettenbrüche mit idealer relativer Approximation’, Acta Acad. Aboensis Math. et Phys. XXII.2 (1960), 1–75.Google Scholar
[11]Selenius, C.-O., ‘Tafel der Kettenbrüche mit idealer Approximation für Quadratwurzeln aus natürlichen Zahlen’, Acta Acad. Aboensis Math. et Phys. XXII.10 (1960), 1–37.Google Scholar
[12]Tong, J., ‘The conjugate property of the Borel theorem on Diophantine approximation’, Math. Z. 184 (1983), 151–153.Google Scholar