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In the geometry of numbers, the Klein polyhedron, named after Felix Klein, is used to generalize the concept of continued fractions to higher dimensions.

Definition

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Let   be a closed simplicial cone in Euclidean space  . The Klein polyhedron of   is is the convex hull of the non-zero points of  .

Relation to continued fractions

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Suppose   is an irrational number. In  , the cones generated by   and by   give rise to two Klein polyhedra, each of which is bounded by a sequence of adjoining line segments. Define the integer length of a line segment to be one less than the size of its intersection with  . Then the integer lengths of the edges of these two Klein polyhedra encode the continued-fraction expansion of  , one matching the even terms and the other matching the odd terms.

Graphs associated with the Klein polyhedron

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Suppose   is generated by a basis   of   (so that  ), and let   be the dual basis (so that  ). Write   for the line generated by the vector  , and   for the hyperplane orthogonal to  .

Call the vector   irrational if  ; and call the cone   irrational if all the vectors   and   are irrational.

The boundary   of a Klein polyhedron is called a sail. Associated with the sail   of an irrational cone are two graphs:

  • the graph   whose vertices are vertices of  , two vertices being joined if they are endpoints of a (one-dimensional) edge of  ;
  • the graph   whose vertices are  -dimensional faces (chambers) of  , two chambers being joined if they share an  -dimensional face.

Both of these graphs are structurally related to the directed graph   whose set of vertices is  , where vertex   is joined to vertex   if and only if   is of the form   where

 

(with  ,  ) and   is a permutation matrix. Assuming that   has been triangulated, the vertices of each of the graphs   and   can be described in terms of the graph  :

  • Given any path   in  , one can find a path   in   such that  , where   is the vector  .
  • Given any path   in  , one can find a path   in   such that  , where   is the  -dimensional standard simplex in  .

Generalization of Lagrange's theorem

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Lagrange proved that for an irrational real number  , the continued-fraction expansion of   is periodic if and only if   is a quadratic irrational. Klein polyhedra allow us to generalize this result.

Let   be a totally-real algebraic number field of degree  , and let   be the   real embeddings of  . The simplicial cone   is said to be split over   if   where   is a basis for   over  .

Given a path   in  , let  . The path is called periodic, with period  , if   for all  . The period matrix of such a path is defined to be  . A path in   or   associated with such a path is also said to be periodic, with the same period matrix.

The generalized Lagrange theorem states that for an irrational simplicial cone  , with generators   and   as above and with sail  , the following three conditions are equivalent:

  •   is split over some totally-real algebraic number field of degree  .
  • For each of the   there is periodic path of vertices   in   such that the   asymptotically approach the line  ; and the period matrices of these paths all commute.
  • For each of the   there is periodic path of chambers   in   such that the   asymptotically approach the hyperplane  ; and the period matrices of these paths all commute.

Example

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Take   and  . Then the simplicial cone   is split over  . The vertices of the sail are the points   corresponding to the even convergents   of the continued fraction for  . The path of vertices   in the positive quadrant starting at   and proceeding in a positive direction is  . Let   be the line segment joining   to  . Write   and   for the reflections of   and   in the  -axis. Let  , so that  , and let  .

Let  ,  ,  , and  .

  • The paths   and   are periodic (with period one) in  , with period matrices   and  . We have   and  .
  • The paths   and   are periodic (with period one) in  , with period matrices   and  . We have   and  .

Generalization of approximability

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A real number   is called badly approximable if   is bounded away from zero. An irrational number is badly approximable if and only if the partial quotients of its continued fraction are bounded. This fact admits of a generalization in terms of Klein polyhedra.

Given a simplicial cone   in  , where  , define the norm minimum of   as  .

Given vectors  , let  . This is the Euclidean volume of  .

Let   be the sail of an irrational simplicial cone  .

  • For a vertex   of  , define   where   are primitive vectors in   generating the edges emanating from  .
  • For a vertex   of  , define   where   are the extreme points of  .

Then   if and only if   and   are both bounded.

The quantities   and   are called determinants. In two dimensions, with the cone generated by  , they are just the partial quotients of the continued fraction of  .

See also

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References

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  • O. N. German, 2007, "Klein polyhedra and lattices with positive norm minima". Journal de Théorie des Nombres de Bordeaux 19: 175--190.
  • E. I. Korkina, 1995, "Two-dimensional continued fractions. The simplest examples". Proc. Steklov Institute of Mathematics 209: 124–144.
  • G. Lachaud, 1998, "Sails and Klein polyhedra" in Contemporary Mathematics 210. American Mathematical Society: 373–385.