Klein polyhedron

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

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

Graphs associated with the Klein polyhedron
Suppose $$\textstyle C$$ is generated by a basis $$\textstyle (a_i)$$ of $$\textstyle \mathbb{R}^n$$ (so that $$\textstyle C = \{ \sum_i \lambda_i a_i : (\forall i) \; \lambda_i \geq 0 \}$$), and let $$\textstyle (w_i)$$ be the dual basis (so that $$\textstyle C = \{ x : (\forall i) \; \langle w_i, x \rangle \geq 0\}$$). Write $$\textstyle D(x)$$ for the line generated by the vector $$\textstyle x$$, and $$\textstyle H(x)$$ for the hyperplane orthogonal to $$\textstyle x$$.

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

The boundary $$\textstyle V$$ of a Klein polyhedron is called a sail. Associated with the sail $$\textstyle V$$ of an irrational cone are two graphs:
 * the graph $$\textstyle \Gamma_{\mathrm e}(V)$$ whose vertices are vertices of $$\textstyle V$$, two vertices being joined if they are endpoints of a (one-dimensional) edge of $$\textstyle V$$;
 * the graph $$\textstyle \Gamma_{\mathrm f}(V)$$ whose vertices are $$\textstyle (n-1)$$-dimensional faces (chambers) of $$\textstyle V$$, two chambers being joined if they share an $$\textstyle (n-2)$$-dimensional face.

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


 * $$U = \left( \begin{array}{cccc} 1 & \cdots & 0 & c_1 \\ \vdots & \ddots & \vdots & \vdots \\ 0 & \cdots & 1 & c_{n-1} \\ 0 & \cdots & 0 & c_n \end{array} \right)$$

(with $$\textstyle c_i \in \mathbb{Q}$$, $$\textstyle c_n \neq 0$$) and $$\textstyle W$$ is a permutation matrix. Assuming that $$\textstyle V$$ has been triangulated, the vertices of each of the graphs $$\textstyle \Gamma_{\mathrm e}(V)$$ and $$\textstyle \Gamma_{\mathrm f}(V)$$ can be described in terms of the graph $$\textstyle \Upsilon_n$$:
 * Given any path $$\textstyle (x_0, x_1, \ldots)$$ in $$\textstyle \Gamma_{\mathrm e}(V)$$, one can find a path $$\textstyle (A_0, A_1, \ldots)$$ in $$\textstyle \Upsilon_n$$ such that $$\textstyle x_k = A_k (e)$$, where $$\textstyle e$$ is the vector $$\textstyle (1, \ldots, 1) \in \mathbb{R}^n$$.
 * Given any path $$\textstyle (\sigma_0, \sigma_1, \ldots)$$ in $$\textstyle \Gamma_{\mathrm f}(V)$$, one can find a path $$\textstyle (A_0, A_1, \ldots)$$ in $$\textstyle \Upsilon_n$$ such that $$\textstyle \sigma_k = A_k (\Delta)$$, where $$\textstyle \Delta$$ is the $$\textstyle (n-1)$$-dimensional standard simplex in $$\textstyle \mathbb{R}^n$$.

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

Let $$\textstyle K \subseteq \mathbb{R}$$ be a totally real algebraic number field of degree $$\textstyle n$$, and let $$\textstyle \alpha_i : K \to \mathbb{R}$$ be the $$\textstyle n$$ real embeddings of $$\textstyle K$$. The simplicial cone $$\textstyle C$$ is said to be split over $$\textstyle K$$ if $$\textstyle C = \{ x \in \mathbb{R}^n : (\forall i) \; \alpha_i(\omega_1) x_1 + \ldots + \alpha_i(\omega_n) x_n \geq 0 \}$$ where $$\textstyle \omega_1, \ldots, \omega_n$$ is a basis for $$\textstyle K$$ over $$\textstyle \mathbb{Q}$$.

Given a path $$\textstyle (A_0, A_1, \ldots)$$ in $$\textstyle \Upsilon_n$$, let $$\textstyle R_k = A_{k+1} A_k^{-1}$$. The path is called periodic, with period $$\textstyle m$$, if $$\textstyle R_{k+qm} = R_k$$ for all $$\textstyle k, q \geq 0$$. The period matrix of such a path is defined to be $$\textstyle A_m A_0^{-1}$$. A path in $$\textstyle \Gamma_{\mathrm e}(V)$$ or $$\textstyle \Gamma_{\mathrm f}(V)$$ 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 $$\textstyle C \subseteq \mathbb{R}^n$$, with generators $$\textstyle (a_i)$$ and $$\textstyle (w_i)$$ as above and with sail $$\textstyle V$$, the following three conditions are equivalent:
 * $$\textstyle C$$ is split over some totally real algebraic number field of degree $$\textstyle n$$.
 * For each of the $$\textstyle a_i$$ there is periodic path of vertices $$\textstyle x_0, x_1, \ldots$$ in $$\textstyle \Gamma_{\mathrm e}(V)$$ such that the $$\textstyle x_k$$ asymptotically approach the line $$\textstyle D(a_i)$$; and the period matrices of these paths all commute.
 * For each of the $$\textstyle w_i$$ there is periodic path of chambers $$\textstyle \sigma_0, \sigma_1, \ldots$$ in $$\textstyle \Gamma_{\mathrm f}(V)$$ such that the $$\textstyle \sigma_k$$ asymptotically approach the hyperplane $$\textstyle H(w_i)$$; and the period matrices of these paths all commute.

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

Let $$\textstyle M_{\mathrm e} = \left( \begin{array}{cc} \frac12 & \frac12 \\ \frac14 & -\frac14 \end{array} \right)$$, $$\textstyle \bar{M}_{\mathrm e} = \left( \begin{array}{cc} \frac12 & \frac12 \\ -\frac14 & \frac14 \end{array} \right)$$, $$\textstyle M_{\mathrm f} = \left( \begin{array}{cc} 3 & 1 \\ 2 & 0 \end{array} \right)$$, and $$\textstyle \bar{M}_{\mathrm f} = \left( \begin{array}{cc} 3 & 1 \\ -2 & 0 \end{array} \right)$$.


 * The paths $$\textstyle (M_{\mathrm e} R^k)$$ and $$\textstyle (\bar{M}_{\mathrm e} R^k)$$ are periodic (with period one) in $$\textstyle \Upsilon_2$$, with period matrices $$\textstyle M_{\mathrm e} R M_{\mathrm e}^{-1} = T$$ and $$\textstyle \bar{M}_{\mathrm e} R \bar{M}_{\mathrm e}^{-1} = T^{-1}$$. We have $$\textstyle x_k = M_{\mathrm e} R^k (e)$$ and $$\textstyle \bar{x}_k = \bar{M}_{\mathrm e} R^k (e)$$.
 * The paths $$\textstyle (M_{\mathrm f} R^k)$$ and $$\textstyle (\bar{M}_{\mathrm f} R^k)$$ are periodic (with period one) in $$\textstyle \Upsilon_2$$, with period matrices $$\textstyle M_{\mathrm f} R M_{\mathrm f}^{-1} = T$$ and $$\textstyle \bar{M}_{\mathrm f} R \bar{M}_{\mathrm f}^{-1} = T^{-1}$$. We have $$\textstyle \sigma_k = M_{\mathrm f} R^k (\Delta)$$ and $$\textstyle \bar{\sigma}_k = \bar{M}_{\mathrm f} R^k (\Delta)$$.

Generalization of approximability
A real number $$\textstyle \alpha > 0$$ is called badly approximable if $$\textstyle \{ q (p \alpha - q) : p,q \in \mathbb{Z}, q > 0\}$$ is bounded away from zero. An irrational number is badly approximable if and only if the partial quotients of its continued fraction are bounded. This fact admits of a generalization in terms of Klein polyhedra.

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

Given vectors $$\textstyle \mathbf{v}_1, \ldots, \mathbf{v}_m \in \mathbb{Z}^n$$, let $$\textstyle [\mathbf{v}_1, \ldots, \mathbf{v}_m] = \sum_{i_1 < \cdots < i_n} \vert \det(\mathbf{v}_{i_1} \cdots \mathbf{v}_{i_n}) \vert$$. This is the Euclidean volume of $$\textstyle \{ \sum_i \lambda_i \mathbf{v}_i : (\forall i) \; 0 \leq \lambda_i \leq 1 \}$$.

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


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

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

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