Klein polyhedron explained

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

Let

style C be a closed simplicial cone in Euclidean space style R ^{n} . The Klein polyhedron of style C is the convex hull of the non-zero points of style C \cap Z ^{n} .

Relation to continued fractions

Suppose

style \alpha > 0 is an irrational number. In style R ^{2} , the cones generated by style \{( 1 , \alpha), ( 1 , 0 )\} and by style \{( 1 , \alpha), ( 0 , 1 )\} 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 style Z ^{2.} Then the integer lengths of the edges of these two Klein polyhedra encode the continued-fraction expansion of style \alpha , one matching the even terms and the other matching the odd terms.

Graphs associated with the Klein polyhedron

Suppose

style C is generated by a basis style (a _{i)} of style R ^{n} (so that style C = \{ \sum _{i} λ _{i} a _{i} : (\forall i) λ _{i} \geq 0 \} ), and let style (w _{i)} be the dual basis (so that style C = \{ x : (\forall i) \langle w _{i,} x \rangle \geq 0 \} ). Write style D(x) for the line generated by the vector style x , and style H(x) for the hyperplane orthogonal to style x .

Call the vector

style x \in R ^{n} irrational if style H(x) \cap Q ^{n} = \{ 0 \} ; and call the cone style C irrational if all the vectors style a _{i} and style w _{i} are irrational.

The boundary

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

• the graph

style \Gamma _{e} (V) whose vertices are vertices of style V , two vertices being joined if they are endpoints of a (one-dimensional) edge of style V ;

• the graph

style \Gamma _{f} (V) whose vertices are style (n -1 ) -dimensional faces (chambers) of style V , two chambers being joined if they share an style (n -2 ) -dimensional face.

Both of these graphs are structurally related to the directed graph

style \Upsilon _{n} whose set of vertices is style GL _{n(Q)} , where vertex style A is joined to vertex style B if and only if style A ^{-1} B is of the form style UW whereU = \left( \begin{array}{cccc} 1 & … & 0 & c _{1} \ \vdots & \ddots & \vdots & \vdots \ 0 & … & 1 & c _{n-1} \ 0 & … & 0 & c _{n} \end{array} \right)

(with

style c _{i} \in Q , style c _{n} ≠ 0 ) and style W is a permutation matrix. Assuming that style V has been triangulated, the vertices of each of the graphs style \Gamma _{e} (V) and style \Gamma _{f} (V) can be described in terms of the graph style \Upsilon _{n} :

• Given any path

style (x _{0,} x _{1,} \ldots) in style \Gamma _{e} (V) , one can find a path style (A _{0,} A _{1,} \ldots) in style \Upsilon _{n} such that style x _{k} = A _{k} (e) , where style e is the vector style ( 1 , \ldots, 1 ) \in R ^{n} .

• Given any path

style (\sigma _{0,} \sigma _{1,} \ldots) in style \Gamma _{f} (V) , one can find a path style (A _{0,} A _{1,} \ldots) in style \Upsilon _{n} such that style \sigma _{k} = A _{k} (\Delta) , where style \Delta is the style (n -1 ) -dimensional standard simplex in style R ^{n} .

Generalization of Lagrange's theorem

Lagrange proved that for an irrational real number

style \alpha , the continued-fraction expansion of style \alpha is periodic if and only if style \alpha is a quadratic irrational. Klein polyhedra allow us to generalize this result.

Let

style K \subseteq R be a totally real algebraic number field of degree style n , and let style \alpha _{i} : K \to R be the style n real embeddings of style K . The simplicial cone style C is said to be split over style K if style C = \{ x \in R ^{n} : (\forall i) \alpha _{i(\omega} _{1)} x _{1} + \ldots + \alpha _{i(\omega} _{n)} x _{n} \geq 0 \} where style \omega _{1,} \ldots, \omega _{n} is a basis for style K over style Q .

Given a path

style (A _{0,} A _{1,} \ldots) in style \Upsilon _{n} , let style R _{k} = A _{k+1} A ^{-1} _{k} . The path is called periodic, with period style m , if style R _{k+qm} = R _{k} for all style k, q \geq 0 . The period matrix of such a path is defined to be style A _{m} A ^{-1} _{0} . A path in style \Gamma _{e} (V) or style \Gamma _{f} (V) 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

style C \subseteq R ^{n} , with generators style (a _{i)} and style (w _{i)} as above and with sail style V , the following three conditions are equivalent:style C is split over some totally real algebraic number field of degree style n .

• For each of the

style a _{i} there is periodic path of vertices style x _{0,} x _{1,} \ldots in style \Gamma _{e} (V) such that the style x _{k} asymptotically approach the line style D(a _{i)} ; and the period matrices of these paths all commute.

• For each of the

style w _{i} there is periodic path of chambers style \sigma _{0,} \sigma _{1,} \ldots in style \Gamma _{f} (V) such that the style \sigma _{k} asymptotically approach the hyperplane style H(w _{i)} ; and the period matrices of these paths all commute.

Example

Take

style n = 2 and style K = Q (\sqrt{ 2 }) . Then the simplicial cone style \{(x,y) : x \geq 0 , \vert y \vert \leq x / \sqrt{ 2 }\} is split over style K . The vertices of the sail are the points style (p _{k,} \pm q _{k)} corresponding to the even convergents style p _{k} / q _{k} of the continued fraction for style \sqrt{ 2 } . The path of vertices style (x _{k)} in the positive quadrant starting at style ( 1 , 0 ) and proceeding in a positive direction is style (( 1 , 0 ), ( 3 , 2 ), ( 17 , 12 ), ( 99 , 70 ), \ldots) . Let style \sigma _{k} be the line segment joining style x _{k} to style x _{k+1} . Write style \bar{x} _{k} and style \bar{\sigma} _{k} for the reflections of style x _{k} and style \sigma _{k} in the style x -axis. Let style T = \left( \begin{array}{cc} 3 & 4 \ 2 & 3 \end{array} \right) , so that style x _{k+1} = T x _{k} , and let style R = \left( \begin{array}{cc} 6 & 1 \ -1 & 0 \end{array} \right) = \left( \begin{array}{cc} 1 & 6 \ 0 & -1 \end{array} \right) \left( \begin{array}{cc} 0 & 1 \ 1 & 0 \end{array} \right) .

Let

style M _{e} = \left( \begin{array}{cc} (12)/(&) (12)/(- \ (14)/(&)) (14)/(\end{array}) \right) , style \bar{M} _{e} = \left( \begin{array}{cc} (12)/(&) (12)/(\ - (14)/(&)) (14)/(\end{array}) \right) , style M _{f} = \left( \begin{array}{cc} 3 & 1 \ 2 & 0 \end{array} \right) , and style \bar{M} _{f} = \left( \begin{array}{cc} 3 & 1 \ -2 & 0 \end{array} \right) .

• The paths

style (M _{e} R ^{k)} and style (\bar{M} _{e} R ^{k)} are periodic (with period one) in style \Upsilon _{2} , with period matrices style M _{e} R M ^{-1} _{e} = T and style \bar{M} _{e} R \bar{M} ^{-1} _{e} = T ^{-1} . We have style x _{k} = M _{e} R ^{k} (e) and style \bar{x} _{k} = \bar{M} _{e} R ^{k} (e) .

• The paths

style (M _{f} R ^{k)} and style (\bar{M} _{f} R ^{k)} are periodic (with period one) in style \Upsilon _{2} , with period matrices style M _{f} R M ^{-1} _{f} = T and style \bar{M} _{f} R \bar{M} ^{-1} _{f} = T ^{-1} . We have style \sigma _{k} = M _{f} R ^{k} (\Delta) and style \bar{\sigma} _{k} = \bar{M} _{f} R ^{k} (\Delta) .

Generalization of approximability

A real number

style \alpha > 0 is called badly approximable if style \{ q (p \alpha - q) : p,q \in Z , q > 0 \} is bounded away from zero. An irrational number is badly approximable if and only if the partial quotients of its continued fraction are bounded.^[1] This fact admits of a generalization in terms of Klein polyhedra.

Given a simplicial cone

style C = \{ x : (\forall i) \langle w _{i,} x \rangle \geq 0 \} in style R ^{n} , where style \langle w _{i,} w _{i} \rangle = 1 , define the norm minimum of style C as style N(C) = inf \{ \prod _{i} \langle w _{i,} x \rangle : x \in Z ^{n} \cap C \setminus \{ 0 \} \} .

Given vectors

style v _{1,} \ldots, v _{m} \in Z ^{n} , let style [ v _{1,} \ldots, v _{m]} = \sum _{i _{1} < … < i _{n}} \vert \det( v _{i _{1}} … v _{i _{n}} ) \vert . This is the Euclidean volume of style \{ \sum _{i} λ _{i} v _{i} : (\forall i) 0 \leq λ _{i} \leq 1 \} .

Let

style V be the sail of an irrational simplicial cone style C .

• For a vertex

style x of style \Gamma _{e} (V) , define style [x] = [ v _{1,} \ldots, v _{m]} where style v _{1,} \ldots, v _{m} are primitive vectors in style Z ^{n} generating the edges emanating from style x .

• For a vertex

style \sigma of style \Gamma _{f} (V) , define style [\sigma] = [ v _{1,} \ldots, v _{m]} where style v _{1,} \ldots, v _{m} are the extreme points of style \sigma .

Then

style N(C) > 0 if and only if style \{ [x] : x \in \Gamma _{e} (V) \} and style \{ [\sigma] : \sigma \in \Gamma _{f} (V) \} are both bounded.

The quantities

style [x] and style [\sigma] are called determinants. In two dimensions, with the cone generated by style \{( 1 , \alpha), ( 1 , 0 )\} , they are just the partial quotients of the continued fraction of style \alpha .

See also

• Building (mathematics)

References

• 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.

Notes and References

1. Book: Bugeaud, Yann . Distribution modulo one and Diophantine approximation . Cambridge Tracts in Mathematics . 193 . Cambridge . . 2012 . 978-0-521-11169-0 . 1260.11001 . 245 .