Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-04T21:49:28.374Z Has data issue: false hasContentIssue false
  • English
  • Français

Metrical properties of best approximants

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

Published online by Cambridge University Press:  09 April 2009

Jos Blom
Jos Blom
Affiliation:
Harrelaers 11 1852 KT, Heiloo The, Netherlands
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A rational number is called a best approximant of the irrational number ζ if it lies closer to ζ than all rational numbers with a smaller denominator. Metrical properties of these best approximants are studied. The main tool is the two-dimensional ergodic system, underlying the continued fraction expansion.

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 53 , Issue 1 , August 1992 , pp. 78 - 91
Copyright
Copyright © Australian Mathematical Society 1992

References

[1]Billingsley, P., Ergodic theory and information (John Wiley, New York, 1965).Google Scholar
[2]Bosma, W., ‘Approximation by mediants’, Math. of Comp. 54 (1990), 421–434.Google Scholar
[3]Bosma, W., Jager, H. and Wiedijk, F., ‘Some metrical observations on the approximation by continued fractions’, Indag. Math. 45 (1983), 281–299.Google Scholar
[4]Erdös, P., ‘Some results on diophantine approximation’, Acta Arithmetica 5 (1959), 359–369.Google Scholar
[5]Jager, H., ‘Continued fractions and ergodic theory’, Transcendental Numbers and Related Topics, RIMS Kokyuroku 599, (Kyoto University, Kyoto, Japan 1986), pp. 55–59.Google Scholar
[6]Knuth, D. E., ‘The distribution of continued fraction approximations’, J. Number Theory 19 (1984), 443–448.Google Scholar
[7]Nakada, H., Ito, S. and Tanaka, S., ‘On the invariant measure for the transformations associated with some real continued fractions’, Keio Engineering Reports 30 (1977), 159–175.Google Scholar
[8]Perron, O., Die Lehre von den Kettenbrüchen, Dritte verbesserte Auflage, (B. G. Teubner, Stuttgart 1954).Google Scholar
[9]Schmidt, W. M., ‘A metrical theorem in Diophantine approximations’, Canad. J. Math. 11 (1960), 619–631.CrossRefGoogle Scholar