Klein polyhedron

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

Definition

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

Relation to continued fractions

The Klein continued fraction for α = φ {\displaystyle \textstyle \alpha =\varphi } {\displaystyle \textstyle \alpha =\varphi } (Golden Ratio) with the Klein polyhedra encoding the odd terms in blue and the Klein polyhedra encoding the even terms in red.

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

Graphs associated with the Klein polyhedron

Suppose C {\displaystyle \textstyle C} {\displaystyle \textstyle C} is generated by a basis ( a i ) {\displaystyle \textstyle (a_{i})} {\displaystyle \textstyle (a_{i})} of R n {\displaystyle \textstyle \mathbb {R} ^{n}} {\displaystyle \textstyle \mathbb {R} ^{n}} (so that C = { ∑ i λ i a i : ( ∀ i ) λ i ≥ 0 } {\displaystyle \textstyle C=\{\sum _{i}\lambda _{i}a_{i}:(\forall i)\;\lambda _{i}\geq 0\}} {\displaystyle \textstyle C=\{\sum _{i}\lambda _{i}a_{i}:(\forall i)\;\lambda _{i}\geq 0\}}), and let ( w i ) {\displaystyle \textstyle (w_{i})} {\displaystyle \textstyle (w_{i})} be the dual basis (so that C = { x : ( ∀ i ) ⟨ w i , x ⟩ ≥ 0 } {\displaystyle \textstyle C=\{x:(\forall i)\;\langle w_{i},x\rangle \geq 0\}} {\displaystyle \textstyle C=\{x:(\forall i)\;\langle w_{i},x\rangle \geq 0\}}). Write D ( x ) {\displaystyle \textstyle D(x)} {\displaystyle \textstyle D(x)} for the line generated by the vector x {\displaystyle \textstyle x} {\displaystyle \textstyle x}, and H ( x ) {\displaystyle \textstyle H(x)} {\displaystyle \textstyle H(x)} for the hyperplane orthogonal to x {\displaystyle \textstyle x} {\displaystyle \textstyle x}.

Call the vector x ∈ R n {\displaystyle \textstyle x\in \mathbb {R} ^{n}} {\displaystyle \textstyle x\in \mathbb {R} ^{n}} irrational if H ( x ) ∩ Q n = { 0 } {\displaystyle \textstyle H(x)\cap \mathbb {Q} ^{n}=\{0\}} {\displaystyle \textstyle H(x)\cap \mathbb {Q} ^{n}=\{0\}}; and call the cone C {\displaystyle \textstyle C} {\displaystyle \textstyle C} irrational if all the vectors a i {\displaystyle \textstyle a_{i}} {\displaystyle \textstyle a_{i}} and w i {\displaystyle \textstyle w_{i}} {\displaystyle \textstyle w_{i}} are irrational.

The boundary V {\displaystyle \textstyle V} {\displaystyle \textstyle V} of a Klein polyhedron is called a sail. Associated with the sail V {\displaystyle \textstyle V} {\displaystyle \textstyle V} of an irrational cone are two graphs:

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

Both of these graphs are structurally related to the directed graph Υ n {\displaystyle \textstyle \Upsilon _{n}} {\displaystyle \textstyle \Upsilon _{n}} whose set of vertices is G L n ( Q ) {\displaystyle \textstyle \mathrm {GL} _{n}(\mathbb {Q} )} {\displaystyle \textstyle \mathrm {GL} _{n}(\mathbb {Q} )}, where vertex A {\displaystyle \textstyle A} {\displaystyle \textstyle A} is joined to vertex B {\displaystyle \textstyle B} {\displaystyle \textstyle B} if and only if A − 1 B {\displaystyle \textstyle A^{-1}B} {\displaystyle \textstyle A^{-1}B} is of the form U W {\displaystyle \textstyle UW} {\displaystyle \textstyle UW} where

U = ( 1 ⋯ 0 c 1 ⋮ ⋱ ⋮ ⋮ 0 ⋯ 1 c n − 1 0 ⋯ 0 c n ) {\displaystyle 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)} {\displaystyle 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 c i ∈ Q {\displaystyle \textstyle c_{i}\in \mathbb {Q} } {\displaystyle \textstyle c_{i}\in \mathbb {Q} }, c n ≠ 0 {\displaystyle \textstyle c_{n}\neq 0} {\displaystyle \textstyle c_{n}\neq 0}) and W {\displaystyle \textstyle W} {\displaystyle \textstyle W} is a permutation matrix. Assuming that V {\displaystyle \textstyle V} {\displaystyle \textstyle V} has been triangulated, the vertices of each of the graphs Γ e ( V ) {\displaystyle \textstyle \Gamma _{\mathrm {e} }(V)} {\displaystyle \textstyle \Gamma _{\mathrm {e} }(V)} and Γ f ( V ) {\displaystyle \textstyle \Gamma _{\mathrm {f} }(V)} {\displaystyle \textstyle \Gamma _{\mathrm {f} }(V)} can be described in terms of the graph Υ n {\displaystyle \textstyle \Upsilon _{n}} {\displaystyle \textstyle \Upsilon _{n}}:

  • Given any path ( x 0 , x 1 , … ) {\displaystyle \textstyle (x_{0},x_{1},\ldots )} {\displaystyle \textstyle (x_{0},x_{1},\ldots )} in Γ e ( V ) {\displaystyle \textstyle \Gamma _{\mathrm {e} }(V)} {\displaystyle \textstyle \Gamma _{\mathrm {e} }(V)}, one can find a path ( A 0 , A 1 , … ) {\displaystyle \textstyle (A_{0},A_{1},\ldots )} {\displaystyle \textstyle (A_{0},A_{1},\ldots )} in Υ n {\displaystyle \textstyle \Upsilon _{n}} {\displaystyle \textstyle \Upsilon _{n}} such that x k = A k ( e ) {\displaystyle \textstyle x_{k}=A_{k}(e)} {\displaystyle \textstyle x_{k}=A_{k}(e)}, where e {\displaystyle \textstyle e} {\displaystyle \textstyle e} is the vector ( 1 , … , 1 ) ∈ R n {\displaystyle \textstyle (1,\ldots ,1)\in \mathbb {R} ^{n}} {\displaystyle \textstyle (1,\ldots ,1)\in \mathbb {R} ^{n}}.
  • Given any path ( σ 0 , σ 1 , … ) {\displaystyle \textstyle (\sigma _{0},\sigma _{1},\ldots )} {\displaystyle \textstyle (\sigma _{0},\sigma _{1},\ldots )} in Γ f ( V ) {\displaystyle \textstyle \Gamma _{\mathrm {f} }(V)} {\displaystyle \textstyle \Gamma _{\mathrm {f} }(V)}, one can find a path ( A 0 , A 1 , … ) {\displaystyle \textstyle (A_{0},A_{1},\ldots )} {\displaystyle \textstyle (A_{0},A_{1},\ldots )} in Υ n {\displaystyle \textstyle \Upsilon _{n}} {\displaystyle \textstyle \Upsilon _{n}} such that σ k = A k ( Δ ) {\displaystyle \textstyle \sigma _{k}=A_{k}(\Delta )} {\displaystyle \textstyle \sigma _{k}=A_{k}(\Delta )}, where Δ {\displaystyle \textstyle \Delta } {\displaystyle \textstyle \Delta } is the ( n − 1 ) {\displaystyle \textstyle (n-1)} {\displaystyle \textstyle (n-1)}-dimensional standard simplex in R n {\displaystyle \textstyle \mathbb {R} ^{n}} {\displaystyle \textstyle \mathbb {R} ^{n}}.

Generalization of Lagrange's theorem

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

Let K ⊆ R {\displaystyle \textstyle K\subseteq \mathbb {R} } {\displaystyle \textstyle K\subseteq \mathbb {R} } be a totally real algebraic number field of degree n {\displaystyle \textstyle n} {\displaystyle \textstyle n}, and let α i : K → R {\displaystyle \textstyle \alpha _{i}:K\to \mathbb {R} } {\displaystyle \textstyle \alpha _{i}:K\to \mathbb {R} } be the n {\displaystyle \textstyle n} {\displaystyle \textstyle n} real embeddings of K {\displaystyle \textstyle K} {\displaystyle \textstyle K}. The simplicial cone C {\displaystyle \textstyle C} {\displaystyle \textstyle C} is said to be split over K {\displaystyle \textstyle K} {\displaystyle \textstyle K} if C = { x ∈ R n : ( ∀ i ) α i ( ω 1 ) x 1 + … + α i ( ω n ) x n ≥ 0 } {\displaystyle \textstyle C=\{x\in \mathbb {R} ^{n}:(\forall i)\;\alpha _{i}(\omega _{1})x_{1}+\ldots +\alpha _{i}(\omega _{n})x_{n}\geq 0\}} {\displaystyle \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 ω 1 , … , ω n {\displaystyle \textstyle \omega _{1},\ldots ,\omega _{n}} {\displaystyle \textstyle \omega _{1},\ldots ,\omega _{n}} is a basis for K {\displaystyle \textstyle K} {\displaystyle \textstyle K} over Q {\displaystyle \textstyle \mathbb {Q} } {\displaystyle \textstyle \mathbb {Q} }.

Given a path ( A 0 , A 1 , … ) {\displaystyle \textstyle (A_{0},A_{1},\ldots )} {\displaystyle \textstyle (A_{0},A_{1},\ldots )} in Υ n {\displaystyle \textstyle \Upsilon _{n}} {\displaystyle \textstyle \Upsilon _{n}}, let R k = A k + 1 A k − 1 {\displaystyle \textstyle R_{k}=A_{k+1}A_{k}^{-1}} {\displaystyle \textstyle R_{k}=A_{k+1}A_{k}^{-1}}. The path is called periodic, with period m {\displaystyle \textstyle m} {\displaystyle \textstyle m}, if R k + q m = R k {\displaystyle \textstyle R_{k+qm}=R_{k}} {\displaystyle \textstyle R_{k+qm}=R_{k}} for all k , q ≥ 0 {\displaystyle \textstyle k,q\geq 0} {\displaystyle \textstyle k,q\geq 0}. The period matrix of such a path is defined to be A m A 0 − 1 {\displaystyle \textstyle A_{m}A_{0}^{-1}} {\displaystyle \textstyle A_{m}A_{0}^{-1}}. A path in Γ e ( V ) {\displaystyle \textstyle \Gamma _{\mathrm {e} }(V)} {\displaystyle \textstyle \Gamma _{\mathrm {e} }(V)} or Γ f ( V ) {\displaystyle \textstyle \Gamma _{\mathrm {f} }(V)} {\displaystyle \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 C ⊆ R n {\displaystyle \textstyle C\subseteq \mathbb {R} ^{n}} {\displaystyle \textstyle C\subseteq \mathbb {R} ^{n}}, with generators ( a i ) {\displaystyle \textstyle (a_{i})} {\displaystyle \textstyle (a_{i})} and ( w i ) {\displaystyle \textstyle (w_{i})} {\displaystyle \textstyle (w_{i})} as above and with sail V {\displaystyle \textstyle V} {\displaystyle \textstyle V}, the following three conditions are equivalent:

  • C {\displaystyle \textstyle C} {\displaystyle \textstyle C} is split over some totally real algebraic number field of degree n {\displaystyle \textstyle n} {\displaystyle \textstyle n}.
  • For each of the a i {\displaystyle \textstyle a_{i}} {\displaystyle \textstyle a_{i}} there is periodic path of vertices x 0 , x 1 , … {\displaystyle \textstyle x_{0},x_{1},\ldots } {\displaystyle \textstyle x_{0},x_{1},\ldots } in Γ e ( V ) {\displaystyle \textstyle \Gamma _{\mathrm {e} }(V)} {\displaystyle \textstyle \Gamma _{\mathrm {e} }(V)} such that the x k {\displaystyle \textstyle x_{k}} {\displaystyle \textstyle x_{k}} asymptotically approach the line D ( a i ) {\displaystyle \textstyle D(a_{i})} {\displaystyle \textstyle D(a_{i})}; and the period matrices of these paths all commute.
  • For each of the w i {\displaystyle \textstyle w_{i}} {\displaystyle \textstyle w_{i}} there is periodic path of chambers σ 0 , σ 1 , … {\displaystyle \textstyle \sigma _{0},\sigma _{1},\ldots } {\displaystyle \textstyle \sigma _{0},\sigma _{1},\ldots } in Γ f ( V ) {\displaystyle \textstyle \Gamma _{\mathrm {f} }(V)} {\displaystyle \textstyle \Gamma _{\mathrm {f} }(V)} such that the σ k {\displaystyle \textstyle \sigma _{k}} {\displaystyle \textstyle \sigma _{k}} asymptotically approach the hyperplane H ( w i ) {\displaystyle \textstyle H(w_{i})} {\displaystyle \textstyle H(w_{i})}; and the period matrices of these paths all commute.

Example

Take n = 2 {\displaystyle \textstyle n=2} {\displaystyle \textstyle n=2} and K = Q ( 2 ) {\displaystyle \textstyle K=\mathbb {Q} ({\sqrt {2}})} {\displaystyle \textstyle K=\mathbb {Q} ({\sqrt {2}})}. Then the simplicial cone { ( x , y ) : x ≥ 0 , | y | ≤ x / 2 } {\displaystyle \textstyle \{(x,y):x\geq 0,\vert y\vert \leq x/{\sqrt {2}}\}} {\displaystyle \textstyle \{(x,y):x\geq 0,\vert y\vert \leq x/{\sqrt {2}}\}} is split over K {\displaystyle \textstyle K} {\displaystyle \textstyle K}. The vertices of the sail are the points ( p k , ± q k ) {\displaystyle \textstyle (p_{k},\pm q_{k})} {\displaystyle \textstyle (p_{k},\pm q_{k})} corresponding to the even convergents p k / q k {\displaystyle \textstyle p_{k}/q_{k}} {\displaystyle \textstyle p_{k}/q_{k}} of the continued fraction for 2 {\displaystyle \textstyle {\sqrt {2}}} {\displaystyle \textstyle {\sqrt {2}}}. The path of vertices ( x k ) {\displaystyle \textstyle (x_{k})} {\displaystyle \textstyle (x_{k})} in the positive quadrant starting at ( 1 , 0 ) {\displaystyle \textstyle (1,0)} {\displaystyle \textstyle (1,0)} and proceeding in a positive direction is ( ( 1 , 0 ) , ( 3 , 2 ) , ( 17 , 12 ) , ( 99 , 70 ) , … ) {\displaystyle \textstyle ((1,0),(3,2),(17,12),(99,70),\ldots )} {\displaystyle \textstyle ((1,0),(3,2),(17,12),(99,70),\ldots )}. Let σ k {\displaystyle \textstyle \sigma _{k}} {\displaystyle \textstyle \sigma _{k}} be the line segment joining x k {\displaystyle \textstyle x_{k}} {\displaystyle \textstyle x_{k}} to x k + 1 {\displaystyle \textstyle x_{k+1}} {\displaystyle \textstyle x_{k+1}}. Write x ¯ k {\displaystyle \textstyle {\bar {x}}_{k}} {\displaystyle \textstyle {\bar {x}}_{k}} and σ ¯ k {\displaystyle \textstyle {\bar {\sigma }}_{k}} {\displaystyle \textstyle {\bar {\sigma }}_{k}} for the reflections of x k {\displaystyle \textstyle x_{k}} {\displaystyle \textstyle x_{k}} and σ k {\displaystyle \textstyle \sigma _{k}} {\displaystyle \textstyle \sigma _{k}} in the x {\displaystyle \textstyle x} {\displaystyle \textstyle x}-axis. Let T = ( 3 4 2 3 ) {\displaystyle \textstyle T=\left({\begin{array}{cc}3&4\\2&3\end{array}}\right)} {\displaystyle \textstyle T=\left({\begin{array}{cc}3&4\\2&3\end{array}}\right)}, so that x k + 1 = T x k {\displaystyle \textstyle x_{k+1}=Tx_{k}} {\displaystyle \textstyle x_{k+1}=Tx_{k}}, and let R = ( 6 1 − 1 0 ) = ( 1 6 0 − 1 ) ( 0 1 1 0 ) {\displaystyle \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)} {\displaystyle \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 M e = ( 1 2 1 2 1 4 − 1 4 ) {\displaystyle \textstyle M_{\mathrm {e} }=\left({\begin{array}{cc}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{4}}&-{\frac {1}{4}}\end{array}}\right)} {\displaystyle \textstyle M_{\mathrm {e} }=\left({\begin{array}{cc}{\frac {1}{2}}&{\frac {1}{2}}\\{\frac {1}{4}}&-{\frac {1}{4}}\end{array}}\right)}, M ¯ e = ( 1 2 1 2 − 1 4 1 4 ) {\displaystyle \textstyle {\bar {M}}_{\mathrm {e} }=\left({\begin{array}{cc}{\frac {1}{2}}&{\frac {1}{2}}\\-{\frac {1}{4}}&{\frac {1}{4}}\end{array}}\right)} {\displaystyle \textstyle {\bar {M}}_{\mathrm {e} }=\left({\begin{array}{cc}{\frac {1}{2}}&{\frac {1}{2}}\\-{\frac {1}{4}}&{\frac {1}{4}}\end{array}}\right)}, M f = ( 3 1 2 0 ) {\displaystyle \textstyle M_{\mathrm {f} }=\left({\begin{array}{cc}3&1\\2&0\end{array}}\right)} {\displaystyle \textstyle M_{\mathrm {f} }=\left({\begin{array}{cc}3&1\\2&0\end{array}}\right)}, and M ¯ f = ( 3 1 − 2 0 ) {\displaystyle \textstyle {\bar {M}}_{\mathrm {f} }=\left({\begin{array}{cc}3&1\\-2&0\end{array}}\right)} {\displaystyle \textstyle {\bar {M}}_{\mathrm {f} }=\left({\begin{array}{cc}3&1\\-2&0\end{array}}\right)}.

  • The paths ( M e R k ) {\displaystyle \textstyle (M_{\mathrm {e} }R^{k})} {\displaystyle \textstyle (M_{\mathrm {e} }R^{k})} and ( M ¯ e R k ) {\displaystyle \textstyle ({\bar {M}}_{\mathrm {e} }R^{k})} {\displaystyle \textstyle ({\bar {M}}_{\mathrm {e} }R^{k})} are periodic (with period one) in Υ 2 {\displaystyle \textstyle \Upsilon _{2}} {\displaystyle \textstyle \Upsilon _{2}}, with period matrices M e R M e − 1 = T {\displaystyle \textstyle M_{\mathrm {e} }RM_{\mathrm {e} }^{-1}=T} {\displaystyle \textstyle M_{\mathrm {e} }RM_{\mathrm {e} }^{-1}=T} and M ¯ e R M ¯ e − 1 = T − 1 {\displaystyle \textstyle {\bar {M}}_{\mathrm {e} }R{\bar {M}}_{\mathrm {e} }^{-1}=T^{-1}} {\displaystyle \textstyle {\bar {M}}_{\mathrm {e} }R{\bar {M}}_{\mathrm {e} }^{-1}=T^{-1}}. We have x k = M e R k ( e ) {\displaystyle \textstyle x_{k}=M_{\mathrm {e} }R^{k}(e)} {\displaystyle \textstyle x_{k}=M_{\mathrm {e} }R^{k}(e)} and x ¯ k = M ¯ e R k ( e ) {\displaystyle \textstyle {\bar {x}}_{k}={\bar {M}}_{\mathrm {e} }R^{k}(e)} {\displaystyle \textstyle {\bar {x}}_{k}={\bar {M}}_{\mathrm {e} }R^{k}(e)}.
  • The paths ( M f R k ) {\displaystyle \textstyle (M_{\mathrm {f} }R^{k})} {\displaystyle \textstyle (M_{\mathrm {f} }R^{k})} and ( M ¯ f R k ) {\displaystyle \textstyle ({\bar {M}}_{\mathrm {f} }R^{k})} {\displaystyle \textstyle ({\bar {M}}_{\mathrm {f} }R^{k})} are periodic (with period one) in Υ 2 {\displaystyle \textstyle \Upsilon _{2}} {\displaystyle \textstyle \Upsilon _{2}}, with period matrices M f R M f − 1 = T {\displaystyle \textstyle M_{\mathrm {f} }RM_{\mathrm {f} }^{-1}=T} {\displaystyle \textstyle M_{\mathrm {f} }RM_{\mathrm {f} }^{-1}=T} and M ¯ f R M ¯ f − 1 = T − 1 {\displaystyle \textstyle {\bar {M}}_{\mathrm {f} }R{\bar {M}}_{\mathrm {f} }^{-1}=T^{-1}} {\displaystyle \textstyle {\bar {M}}_{\mathrm {f} }R{\bar {M}}_{\mathrm {f} }^{-1}=T^{-1}}. We have σ k = M f R k ( Δ ) {\displaystyle \textstyle \sigma _{k}=M_{\mathrm {f} }R^{k}(\Delta )} {\displaystyle \textstyle \sigma _{k}=M_{\mathrm {f} }R^{k}(\Delta )} and σ ¯ k = M ¯ f R k ( Δ ) {\displaystyle \textstyle {\bar {\sigma }}_{k}={\bar {M}}_{\mathrm {f} }R^{k}(\Delta )} {\displaystyle \textstyle {\bar {\sigma }}_{k}={\bar {M}}_{\mathrm {f} }R^{k}(\Delta )}.

Generalization of approximability

A real number α > 0 {\displaystyle \textstyle \alpha >0} {\displaystyle \textstyle \alpha >0} is called badly approximable if { q ( p α − q ) : p , q ∈ Z , q > 0 } {\displaystyle \textstyle \{q(p\alpha -q):p,q\in \mathbb {Z} ,q>0\}} {\displaystyle \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.[1] This fact admits of a generalization in terms of Klein polyhedra.

Given a simplicial cone C = { x : ( ∀ i ) ⟨ w i , x ⟩ ≥ 0 } {\displaystyle \textstyle C=\{x:(\forall i)\;\langle w_{i},x\rangle \geq 0\}} {\displaystyle \textstyle C=\{x:(\forall i)\;\langle w_{i},x\rangle \geq 0\}} in R n {\displaystyle \textstyle \mathbb {R} ^{n}} {\displaystyle \textstyle \mathbb {R} ^{n}}, where ⟨ w i , w i ⟩ = 1 {\displaystyle \textstyle \langle w_{i},w_{i}\rangle =1} {\displaystyle \textstyle \langle w_{i},w_{i}\rangle =1}, define the norm minimum of C {\displaystyle \textstyle C} {\displaystyle \textstyle C} as N ( C ) = inf { ∏ i ⟨ w i , x ⟩ : x ∈ Z n ∩ C ∖ { 0 } } {\displaystyle \textstyle N(C)=\inf\{\prod _{i}\langle w_{i},x\rangle :x\in \mathbb {Z} ^{n}\cap C\setminus \{0\}\}} {\displaystyle \textstyle N(C)=\inf\{\prod _{i}\langle w_{i},x\rangle :x\in \mathbb {Z} ^{n}\cap C\setminus \{0\}\}}.

Given vectors v 1 , … , v m ∈ Z n {\displaystyle \textstyle \mathbf {v} _{1},\ldots ,\mathbf {v} _{m}\in \mathbb {Z} ^{n}} {\displaystyle \textstyle \mathbf {v} _{1},\ldots ,\mathbf {v} _{m}\in \mathbb {Z} ^{n}}, let [ v 1 , … , v m ] = ∑ i 1 < ⋯ < i n | det ( v i 1 ⋯ v i n ) | {\displaystyle \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 } {\displaystyle \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 { ∑ i λ i v i : ( ∀ i ) 0 ≤ λ i ≤ 1 } {\displaystyle \textstyle \{\sum _{i}\lambda _{i}\mathbf {v} _{i}:(\forall i)\;0\leq \lambda _{i}\leq 1\}} {\displaystyle \textstyle \{\sum _{i}\lambda _{i}\mathbf {v} _{i}:(\forall i)\;0\leq \lambda _{i}\leq 1\}}.

Let V {\displaystyle \textstyle V} {\displaystyle \textstyle V} be the sail of an irrational simplicial cone C {\displaystyle \textstyle C} {\displaystyle \textstyle C}.

  • For a vertex x {\displaystyle \textstyle x} {\displaystyle \textstyle x} of Γ e ( V ) {\displaystyle \textstyle \Gamma _{\mathrm {e} }(V)} {\displaystyle \textstyle \Gamma _{\mathrm {e} }(V)}, define [ x ] = [ v 1 , … , v m ] {\displaystyle \textstyle [x]=[\mathbf {v} _{1},\ldots ,\mathbf {v} _{m}]} {\displaystyle \textstyle [x]=[\mathbf {v} _{1},\ldots ,\mathbf {v} _{m}]} where v 1 , … , v m {\displaystyle \textstyle \mathbf {v} _{1},\ldots ,\mathbf {v} _{m}} {\displaystyle \textstyle \mathbf {v} _{1},\ldots ,\mathbf {v} _{m}} are primitive vectors in Z n {\displaystyle \textstyle \mathbb {Z} ^{n}} {\displaystyle \textstyle \mathbb {Z} ^{n}} generating the edges emanating from x {\displaystyle \textstyle x} {\displaystyle \textstyle x}.
  • For a vertex σ {\displaystyle \textstyle \sigma } {\displaystyle \textstyle \sigma } of Γ f ( V ) {\displaystyle \textstyle \Gamma _{\mathrm {f} }(V)} {\displaystyle \textstyle \Gamma _{\mathrm {f} }(V)}, define [ σ ] = [ v 1 , … , v m ] {\displaystyle \textstyle [\sigma ]=[\mathbf {v} _{1},\ldots ,\mathbf {v} _{m}]} {\displaystyle \textstyle [\sigma ]=[\mathbf {v} _{1},\ldots ,\mathbf {v} _{m}]} where v 1 , … , v m {\displaystyle \textstyle \mathbf {v} _{1},\ldots ,\mathbf {v} _{m}} {\displaystyle \textstyle \mathbf {v} _{1},\ldots ,\mathbf {v} _{m}} are the extreme points of σ {\displaystyle \textstyle \sigma } {\displaystyle \textstyle \sigma }.

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

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

See also

References

  1. Bugeaud, Yann (2012). Distribution modulo one and Diophantine approximation. Cambridge Tracts in Mathematics. Vol. 193. Cambridge: Cambridge University Press. p. 245. ISBN 978-0-521-11169-0. Zbl 1260.11001.
  • 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.