Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-24T06:37:32.039Z Has data issue: false hasContentIssue false
  • English
  • Français

The unreasonable effectualness of continued function expansions

Part of: Diophantine approximation, transcendental number theory Convergence and divergence of infinite limiting processes

Published online by Cambridge University Press:  09 April 2009

Greg Martin
Greg Martin
Affiliation:
Department of Mathematics, University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, BC V6T 1Z2, Canada e-mail: [email protected]
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.

Many generalizations of continued fractions, where the reciprocal function has been replaced by a more general function, have been studied, and it is often asked whether such generalized expansions can have nice properties. For instance, we might ask that algebraic numbers of a given degree have periodic expansions, just as quadratic irrationals have periodic continued fractions; or we might ask that familiar transcendental constants such as e or π have periodic or terminating expansions. In this paper, we show that there exist such generalized continued function expansions with essentially any desired behaviour.

MSC classification

Secondary: 11J70: Continued fractions and generalizations 40A15: Convergence and divergence of continued fractions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 77 , Issue 3 , December 2004 , pp. 305 - 320
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Bissinger, B. H., ‘A generalization of continued fractions’, Bull. Amer. Math. Soc. 50 (1944), 868–876.CrossRefGoogle Scholar
[2]Schweiger, F., Ergodic theory of fibred systems and metric number theory (The Clarendon Press Oxford University Press, New York, 1995).Google Scholar
[3]Viader, P., Paradís, J. and Bibiloni, L., ‘A new light on Minkowski's ?(x) function’, J. Number Theory 73 (1998), 212–227.CrossRefGoogle Scholar
[4]Voronoi, G. F., On a generalization of the algorithm of continued fractions (Ph.D. Thesis, Warsaw, 1896), (Russian).Google Scholar
[5]Wigner, E. P., ‘The unreasonable effectiveness of mathematics in the natural sciences [Comm. Pure Appl. Math. 13 (1960), 1–14]’,CrossRefGoogle Scholar
Mathematical analysis of physical systems (Van Nostrand Reinhold, New York, 1985) pp. 1–14.Google Scholar