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On Construction of Multidimensional Continued Fractions

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Geometry of Continued Fractions

Part of the book series: Algorithms and Computation in Mathematics ((AACIM,volume 26))

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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.

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Authors and Affiliations

  1. Dept. of Mathematical Sciences, University of Liverpool, Liverpool, UK

    Oleg Karpenkov

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  1. Oleg Karpenkov
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Karpenkov, O. (2013). On Construction of Multidimensional Continued Fractions. In: Geometry of Continued Fractions. Algorithms and Computation in Mathematics, vol 26. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39368-6_20

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