Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-23T02:56:41.711Z Has data issue: false hasContentIssue false
  • English
  • Français

On the continued fraction algorithm

Published online by Cambridge University Press:  17 April 2009

J. M. Mack
J. M. Mack
Affiliation:
University of Sydney, Sydney, New South Wales.
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.

The fact that continued fractions can be described in terms of Farey sections is used to obtain a generalised continued fraction algorithm. Geometrically, the algorithm transfers the continued fraction process from the real line R to an arbitrary rational line l in Rn. Arithmetically, the algorithm provides a sequence of simultaneous rational approximations to a set of n real numbers θ1, …, θn in the extreme case where all of the numbers are rationally dependent on 1 and (say) θ1. All but a finite number of best approximations are given by the algorithm.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 3 , Issue 3 , December 1970 , pp. 413 - 422
Copyright
Copyright © Australian Mathematical Society 1970

References

[1]Cassels, J.W.S., Ledermann, W. and Mahler, K., “Farey section in k(i) and k(p)”, philos. Trans. Roy. Soc. London Ser. A 243 (1951), 585–628.Google Scholar
[2]Hardy, G.H. and Wright, E.M., An introduction to the theory of numbers 4th ed., (Clarendon Press, Oxford, 1960).Google Scholar
[3]Hurwitz, A., “Ueber die ahgenäherte Darstellung der Zahlen durch rationale Brüche”, Math. Ann. 44 (1894), 417–436.Google Scholar
[4]Mack, J.M., “A note on simultaneous approximation”, Bull. Austral. Math. Soc. 3 (1970), 81–83.Google Scholar
[5]Mahler, K., “Farey sections in the field of Gauss and Eisenstein”, Proc. Internat. Congress of Mathematicians, Cambridge, Mass. 1 (1950), 281–285. (Amer. Math. Soc., Providence, Rhode Island, 1952.)Google Scholar
[6]Szekeres, G., “Multidimensional continued fractions”, Ann. Univ. Sci. Budapest Eötvös. Sect. Math. (to appear).Google Scholar