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Mathematics of Computation

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ISSN 1088-6842 (online) ISSN 0025-5718 (print)

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Constructing multidimensional periodic continued fractions in the sense of Klein
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by O. N. Karpenkov PDF
Math. Comp. 78 (2009), 1687-1711 Request permission

Abstract:

We consider the geometric generalization of ordinary continued fractions to the multidimensional case introduced by F. Klein in 1895. A multidimensional periodic continued fraction is the union of sails with some special group acting freely on these sails. This group transposes the faces. In this article, we present a method of constructing “approximate” fundamental domains of algebraic multidimensional continued fractions and an algorithm testing whether this domain is indeed fundamental or not. We give some polynomial estimates on the number of the operations for the algorithm. In conclusion we present an example of a fundamental domain calculation for a two-dimensional series of two-dimensional periodic continued fractions.
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Additional Information
  • O. N. Karpenkov
  • Affiliation: Poncelet Laboratory (UMI 2615 of CNRS and Independent University of Moscow)
  • Email: karpenk@mccme.ru
  • Received by editor(s): May 12, 2005
  • Received by editor(s) in revised form: June 9, 2008
  • Published electronically: October 24, 2008
  • Additional Notes: The author was supported by SS-1972.2003.1 and RFBR-05-01-01012a grants.
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Math. Comp. 78 (2009), 1687-1711
  • MSC (2000): Primary 11J70; Secondary 11Y16
  • DOI: https://doi.org/10.1090/S0025-5718-08-02187-X
  • MathSciNet review: 2501070