Multidimensional Continued Fractions in the Sense of Klein

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


In 1839, C. Hermite posed the problem of generalizing ordinary continued fractions to the higher-dimensional case. Since then, there have been many different definitions generalizing different properties of ordinary continued fractions. In this book we focus on the geometric generalization proposed by F. Klein. We start this chapter with an introduction of general notions and definitions. Further we consider the case of finite continued fractions in more details. As an additional tool we will use multidimensional Kronecker’s approximation theory, which we formulate and prove it in this chapter. After that we discuss homeomorphic and polyhedral structure of the sails in general. Finally we classify all two-dimensional faces with integer distance to the origin greater then one and say a few words about the two-dimensional faces with integer distance to the origin equals one.


General Position Continue Fraction Integer Point Simplicial Cone Polyhedral Structure 
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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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