Klein polyhedron

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

File:Klein continued fraction.svg) with the Klein polyhedra encoding the odd terms in blue and the Klein polyhedra encoding the even terms in red.]]

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

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

Call the vector $\backslash textstyle\; x\; \backslash in\; \backslash mathbb\{R\}^n$ irrational if $\backslash textstyle\; H(x)\; \backslash cap\; \backslash mathbb\{Q\}^n\; =\; \backslash \{0\backslash \}$; and call the cone $\backslash textstyle\; C$ irrational if all the vectors $\backslash textstyle\; a\_i$ and $\backslash textstyle\; w\_i$ are irrational.

The boundary $\backslash textstyle\; V$ of a Klein polyhedron is called a sail. Associated with the sail $\backslash textstyle\; V$ of an irrational cone are two graphs:

• the graph $\backslash textstyle\; \backslash Gamma\_\{\backslash mathrm\; e\}(V)$ whose vertices are vertices of $\backslash textstyle\; V$, two vertices being joined if they are endpoints of a (one-dimensional) edge of $\backslash textstyle\; V$;
• the graph $\backslash textstyle\; \backslash Gamma\_\{\backslash mathrm\; f\}(V)$ whose vertices are $\backslash textstyle\; (n-1)$-dimensional faces (chambers) of $\backslash textstyle\; V$, two chambers being joined if they share an $\backslash textstyle\; (n-2)$-dimensional face.

Both of these graphs are structurally related to the directed graph $\backslash textstyle\; \backslash Upsilon\_n$ whose set of vertices is $\backslash textstyle\; \backslash mathrm\{GL\}\_n(\backslash mathbb\{Q\})$, where vertex $\backslash textstyle\; A$ is joined to vertex $\backslash textstyle\; B$ if and only if $\backslash textstyle\; A^\{-1\}B$ is of the form $\backslash textstyle\; UW$ where

:

(with $\backslash textstyle\; c\_i\; \backslash in\; \backslash mathbb\{Q\}$, $\backslash textstyle\; c\_n\; \backslash neq\; 0$) and $\backslash textstyle\; W$ is a permutation matrix. Assuming that $\backslash textstyle\; V$ has been triangulated, the vertices of each of the graphs $\backslash textstyle\; \backslash Gamma\_\{\backslash mathrm\; e\}(V)$ and $\backslash textstyle\; \backslash Gamma\_\{\backslash mathrm\; f\}(V)$ can be described in terms of the graph $\backslash textstyle\; \backslash Upsilon\_n$:

• Given any path $\backslash textstyle\; (x\_0,\; x\_1,\; \backslash ldots)$ in $\backslash textstyle\; \backslash Gamma\_\{\backslash mathrm\; e\}(V)$, one can find a path $\backslash textstyle\; (A\_0,\; A\_1,\; \backslash ldots)$ in $\backslash textstyle\; \backslash Upsilon\_n$ such that $\backslash textstyle\; x\_k\; =\; A\_k\; (e)$, where $\backslash textstyle\; e$ is the vector $\backslash textstyle\; (1,\; \backslash ldots,\; 1)\; \backslash in\; \backslash mathbb\{R\}^n$.
• Given any path $\backslash textstyle\; (\backslash sigma\_0,\; \backslash sigma\_1,\; \backslash ldots)$ in $\backslash textstyle\; \backslash Gamma\_\{\backslash mathrm\; f\}(V)$, one can find a path $\backslash textstyle\; (A\_0,\; A\_1,\; \backslash ldots)$ in $\backslash textstyle\; \backslash Upsilon\_n$ such that $\backslash textstyle\; \backslash sigma\_k\; =\; A\_k\; (\backslash Delta)$, where $\backslash textstyle\; \backslash Delta$ is the $\backslash textstyle\; (n-1)$-dimensional standard simplex in $\backslash textstyle\; \backslash mathbb\{R\}^n$.

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

Let $\backslash textstyle\; K\; \backslash subseteq\; \backslash mathbb\{R\}$ be a totally real algebraic number field of degree $\backslash textstyle\; n$, and let $\backslash textstyle\; \backslash alpha\_i\; :\; K\; \backslash to\; \backslash mathbb\{R\}$ be the $\backslash textstyle\; n$ real embeddings of $\backslash textstyle\; K$. The simplicial cone $\backslash textstyle\; C$ is said to be split over $\backslash textstyle\; K$ if $\backslash textstyle\; C\; =\; \backslash \{\; x\; \backslash in\; \backslash mathbb\{R\}^n\; :\; (\backslash forall\; i)\; \backslash ;\; \backslash alpha\_i(\backslash omega\_1)\; x\_1\; +\; \backslash ldots\; +\; \backslash alpha\_i(\backslash omega\_n)\; x\_n\; \backslash geq\; 0\; \backslash \}$ where $\backslash textstyle\; \backslash omega\_1,\; \backslash ldots,\; \backslash omega\_n$ is a basis for $\backslash textstyle\; K$ over $\backslash textstyle\; \backslash mathbb\{Q\}$.

Given a path $\backslash textstyle\; (A\_0,\; A\_1,\; \backslash ldots)$ in $\backslash textstyle\; \backslash Upsilon\_n$, let $\backslash textstyle\; R\_k\; =\; A\_\{k+1\}\; A\_k^\{-1\}$. The path is called periodic, with period $\backslash textstyle\; m$, if $\backslash textstyle\; R\_\{k+qm\}\; =\; R\_k$ for all $\backslash textstyle\; k,\; q\; \backslash geq\; 0$. The period matrix of such a path is defined to be $\backslash textstyle\; A\_m\; A\_0^\{-1\}$. A path in $\backslash textstyle\; \backslash Gamma\_\{\backslash mathrm\; e\}(V)$ or $\backslash textstyle\; \backslash Gamma\_\{\backslash mathrm\; f\}(V)$ 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 $\backslash textstyle\; C\; \backslash subseteq\; \backslash mathbb\{R\}^n$, with generators $\backslash textstyle\; (a\_i)$ and $\backslash textstyle\; (w\_i)$ as above and with sail $\backslash textstyle\; V$, the following three conditions are equivalent:

• $\backslash textstyle\; C$ is split over some totally real algebraic number field of degree $\backslash textstyle\; n$.
• For each of the $\backslash textstyle\; a\_i$ there is periodic path of vertices $\backslash textstyle\; x\_0,\; x\_1,\; \backslash ldots$ in $\backslash textstyle\; \backslash Gamma\_\{\backslash mathrm\; e\}(V)$ such that the $\backslash textstyle\; x\_k$ asymptotically approach the line $\backslash textstyle\; D(a\_i)$; and the period matrices of these paths all commute.
• For each of the $\backslash textstyle\; w\_i$ there is periodic path of chambers $\backslash textstyle\; \backslash sigma\_0,\; \backslash sigma\_1,\; \backslash ldots$ in $\backslash textstyle\; \backslash Gamma\_\{\backslash mathrm\; f\}(V)$ such that the $\backslash textstyle\; \backslash sigma\_k$ asymptotically approach the hyperplane $\backslash textstyle\; H(w\_i)$; and the period matrices of these paths all commute.

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

Let , , , and .

• The paths $\backslash textstyle\; (M\_\{\backslash mathrm\; e\}\; R^k)$ and $\backslash textstyle\; (\backslash bar\{M\}\_\{\backslash mathrm\; e\}\; R^k)$ are periodic (with period one) in $\backslash textstyle\; \backslash Upsilon\_2$, with period matrices $\backslash textstyle\; M\_\{\backslash mathrm\; e\}\; R\; M\_\{\backslash mathrm\; e\}^\{-1\}\; =\; T$ and $\backslash textstyle\; \backslash bar\{M\}\_\{\backslash mathrm\; e\}\; R\; \backslash bar\{M\}\_\{\backslash mathrm\; e\}^\{-1\}\; =\; T^\{-1\}$. We have $\backslash textstyle\; x\_k\; =\; M\_\{\backslash mathrm\; e\}\; R^k\; (e)$ and $\backslash textstyle\; \backslash bar\{x\}\_k\; =\; \backslash bar\{M\}\_\{\backslash mathrm\; e\}\; R^k\; (e)$.
• The paths $\backslash textstyle\; (M\_\{\backslash mathrm\; f\}\; R^k)$ and $\backslash textstyle\; (\backslash bar\{M\}\_\{\backslash mathrm\; f\}\; R^k)$ are periodic (with period one) in $\backslash textstyle\; \backslash Upsilon\_2$, with period matrices $\backslash textstyle\; M\_\{\backslash mathrm\; f\}\; R\; M\_\{\backslash mathrm\; f\}^\{-1\}\; =\; T$ and $\backslash textstyle\; \backslash bar\{M\}\_\{\backslash mathrm\; f\}\; R\; \backslash bar\{M\}\_\{\backslash mathrm\; f\}^\{-1\}\; =\; T^\{-1\}$. We have $\backslash textstyle\; \backslash sigma\_k\; =\; M\_\{\backslash mathrm\; f\}\; R^k\; (\backslash Delta)$ and $\backslash textstyle\; \backslash bar\{\backslash sigma\}\_k\; =\; \backslash bar\{M\}\_\{\backslash mathrm\; f\}\; R^k\; (\backslash Delta)$.

A real number $\backslash textstyle\; \backslash alpha\; >\; 0$ is called badly approximable if $\backslash textstyle\; \backslash \{\; q\; (p\; \backslash alpha\; -\; q)\; :\; p,q\; \backslash in\; \backslash mathbb\{Z\},\; q\; >\; 0\backslash \}$ is bounded away from zero. An irrational number is badly approximable if and only if the partial quotients of its continued fraction are bounded.{{cite book | last=Bugeaud | first=Yann | title=Distribution modulo one and Diophantine approximation | series=Cambridge Tracts in Mathematics | volume=193 | location=Cambridge | publisher=Cambridge University Press | year=2012 | isbn=978-0-521-11169-0 | zbl=1260.11001 | page=245 }} This fact admits of a generalization in terms of Klein polyhedra.

Given a simplicial cone $\backslash textstyle\; C\; =\; \backslash \{\; x\; :\; (\backslash forall\; i)\; \backslash ;\; \backslash langle\; w\_i,\; x\; \backslash rangle\; \backslash geq\; 0\backslash \}$ in $\backslash textstyle\; \backslash mathbb\{R\}^n$, where $\backslash textstyle\; \backslash langle\; w\_i,\; w\_i\; \backslash rangle\; =\; 1$, define the norm minimum of $\backslash textstyle\; C$ as $\backslash textstyle\; N(C)\; =\; \backslash inf\; \backslash \{\; \backslash prod\_i\; \backslash langle\; w\_i,\; x\; \backslash rangle\; :\; x\; \backslash in\; \backslash mathbb\{Z\}^n\; \backslash cap\; C\; \backslash setminus\; \backslash \{0\backslash \}\; \backslash \}$.

Given vectors $\backslash textstyle\; \backslash mathbf\{v\}\_1,\; \backslash ldots,\; \backslash mathbf\{v\}\_m\; \backslash in\; \backslash mathbb\{Z\}^n$, let . This is the Euclidean volume of $\backslash textstyle\; \backslash \{\; \backslash sum\_i\; \backslash lambda\_i\; \backslash mathbf\{v\}\_i\; :\; (\backslash forall\; i)\; \backslash ;\; 0\; \backslash leq\; \backslash lambda\_i\; \backslash leq\; 1\; \backslash \}$.

Let $\backslash textstyle\; V$ be the sail of an irrational simplicial cone $\backslash textstyle\; C$.

• For a vertex $\backslash textstyle\; x$ of $\backslash textstyle\; \backslash Gamma\_\{\backslash mathrm\; e\}(V)$, define $\backslash textstyle\; [x]\; =\; [\backslash mathbf\{v\}\_1,\; \backslash ldots,\; \backslash mathbf\{v\}\_m]$ where $\backslash textstyle\; \backslash mathbf\{v\}\_1,\; \backslash ldots,\; \backslash mathbf\{v\}\_m$ are primitive vectors in $\backslash textstyle\; \backslash mathbb\{Z\}^n$ generating the edges emanating from $\backslash textstyle\; x$.
• For a vertex $\backslash textstyle\; \backslash sigma$ of $\backslash textstyle\; \backslash Gamma\_\{\backslash mathrm\; f\}(V)$, define $\backslash textstyle\; [\backslash sigma]\; =\; [\backslash mathbf\{v\}\_1,\; \backslash ldots,\; \backslash mathbf\{v\}\_m]$ where $\backslash textstyle\; \backslash mathbf\{v\}\_1,\; \backslash ldots,\; \backslash mathbf\{v\}\_m$ are the extreme points of $\backslash textstyle\; \backslash sigma$.

Then $\backslash textstyle\; N(C)\; >\; 0$ if and only if $\backslash textstyle\; \backslash \{\; [x]\; :\; x\; \backslash in\; \backslash Gamma\_\{\backslash mathrm\; e\}(V)\; \backslash \}$ and $\backslash textstyle\; \backslash \{\; [\backslash sigma]\; :\; \backslash sigma\; \backslash in\; \backslash Gamma\_\{\backslash mathrm\; f\}(V)\; \backslash \}$ are both bounded.

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

• Building (mathematics)

{{reflist}}

• 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.

Category:Geometry of numbers