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On Construction of Multidimensional Continued Fractions

  • Oleg Karpenkov
Chapter
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Part of the Algorithms and Computation in Mathematics book series (AACIM, volume 26)

Abstract

In the first part of this book we saw that the LLS sequences completely determine all possible sails of integer angles in the one-dimensional case. The situation in the multidimensional case is much more complicated. Of course, the convex hull algorithms can compute all the vertices and faces of sails for finite continued fractions, but it is not clear how to construct (or to describe) vertices of infinite sails of multidimensional continued fractions in general. What integer-combinatorial structures could the infinite sails have? There is no single example in the case of aperiodic infinite continued fractions of dimension greater than one. The situation is better with periodic algebraic sails, where each sail is characterized by its fundamental domain and the group of period shifts (i.e., the positive Dirichlet group).

In this chapter we show the main algorithms that are used to construct examples of multidimensional continued fractions (finite, periodic, or finite parts of arbitrary sails). We begin with some definitions and background. Further, we discuss one inductive and two deductive algorithms to construct continued fractions. Finally, we demonstrate one of the deductive algorithms on a particular example.

Keywords

Convex Hull Continue Fraction Fundamental Domain Integer Point Inductive Algorithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Oleg Karpenkov
    • 1
  1. 1.Dept. of Mathematical SciencesUniversity of LiverpoolLiverpoolUK

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