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Constructing multidimensional periodic continued fractions in the sense of Klein


Author: O. N. Karpenkov
Journal: Math. Comp. 78 (2009), 1687-1711
MSC (2000): Primary 11J70; Secondary 11Y16
DOI: https://doi.org/10.1090/S0025-5718-08-02187-X
Published electronically: October 24, 2008
MathSciNet review: 2501070
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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

DOI: https://doi.org/10.1090/S0025-5718-08-02187-X
Keywords: Multidimensional continued fractions, convex polygons, integer lattices
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.
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.