Let C be a closed simplicial cone in Euclidean space R n . The Klein polyhedron of C is the convex hull of the non-zero points of C ∩ Z n .
Suppose α > 0 is an irrational number. In R 2 , the cones generated by { ( 1 , α ) , ( 1 , 0 ) } and by { ( 1 , α ) , ( 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 n . Then the integer lengths of the edges of these two Klein polyhedra encode the continued-fraction expansion of α , one matching the even terms and the other matching the odd terms.
Suppose C is generated by a basis ( a i ) of R n (so that C = { ∑ i λ i a i : ( ∀ i ) λ i ≥ 0 } ), and let ( w i ) be the dual basis (so that C = { x : ( ∀ i ) ⟨ w i , x ⟩ ≥ 0 } ). Write D ( x ) for the line generated by the vector x , and H ( x ) for the hyperplane orthogonal to x .
Call the vector x ∈ R n irrational if H ( x ) ∩ Q n = { 0 } ; and call the cone C irrational if all the vectors a i and w i are irrational.
The boundary V of a Klein polyhedron is called a sail. Associated with the sail V of an irrational cone are two graphs:
the graph Γ e ( V ) whose vertices are vertices of V , two vertices being joined if they are endpoints of a (one-dimensional) edge of V ;the graph Γ f ( V ) whose vertices are ( n − 1 ) -dimensional faces (chambers) of V , two chambers being joined if they share an ( n − 2 ) -dimensional face.Both of these graphs are structurally related to the directed graph Υ n whose set of vertices is G L n ( Q ) , where vertex A is joined to vertex B if and only if A − 1 B is of the form U W where
U = ( 1 ⋯ 0 c 1 ⋮ ⋱ ⋮ ⋮ 0 ⋯ 1 c n − 1 0 ⋯ 0 c n ) (with c i ∈ Q , c n ≠ 0 ) and W is a permutation matrix. Assuming that V has been triangulated, the vertices of each of the graphs Γ e ( V ) and Γ f ( V ) can be described in terms of the graph Υ n :
Given any path ( x 0 , x 1 , … ) in Γ e ( V ) , one can find a path ( A 0 , A 1 , … ) in Υ n such that x k = A k ( e ) , where e is the vector ( 1 , … , 1 ) ∈ R n .Given any path ( σ 0 , σ 1 , … ) in Γ f ( V ) , one can find a path ( A 0 , A 1 , … ) in Υ n such that σ k = A k ( Δ ) , where Δ is the ( n − 1 ) -dimensional standard simplex in R n .Lagrange proved that for an irrational real number α , the continued-fraction expansion of α is periodic if and only if α is a quadratic irrational. Klein polyhedra allow us to generalize this result.
Let K ⊆ R be a totally real algebraic number field of degree n , and let α i : K → R be the n real embeddings of K . The simplicial cone C is said to be split over K if C = { x ∈ R n : ( ∀ i ) α i ( ω 1 ) x 1 + … + α i ( ω n ) x n ≥ 0 } where ω 1 , … , ω n is a basis for K over Q .
Given a path ( A 0 , A 1 , … ) in Υ n , let R k = A k + 1 A k − 1 . The path is called periodic, with period m , if R k + q m = R k for all k , q ≥ 0 . The period matrix of such a path is defined to be A m A 0 − 1 . A path in Γ e ( V ) or Γ 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 , with generators ( a i ) and ( w i ) as above and with sail V , the following three conditions are equivalent:
C is split over some totally real algebraic number field of degree n .For each of the a i there is periodic path of vertices x 0 , x 1 , … in Γ e ( V ) such that the x k asymptotically approach the line D ( a i ) ; and the period matrices of these paths all commute.For each of the w i there is periodic path of chambers σ 0 , σ 1 , … in Γ f ( V ) such that the σ k asymptotically approach the hyperplane H ( w i ) ; and the period matrices of these paths all commute.Take n = 2 and K = Q ( 2 ) . Then the simplicial cone { ( x , y ) : x ≥ 0 , | y | ≤ x / 2 } is split over K . The vertices of the sail are the points ( p k , ± q k ) corresponding to the even convergents p k / q k of the continued fraction for 2 . The path of vertices ( x k ) in the positive quadrant starting at ( 1 , 0 ) and proceeding in a positive direction is ( ( 1 , 0 ) , ( 3 , 2 ) , ( 17 , 12 ) , ( 99 , 70 ) , … ) . Let σ k be the line segment joining x k to x k + 1 . Write x ¯ k and σ ¯ k for the reflections of x k and σ k in the x -axis. Let T = ( 3 4 2 3 ) , so that x k + 1 = T x k , and let R = ( 6 1 − 1 0 ) = ( 1 6 0 − 1 ) ( 0 1 1 0 ) .
Let M e = ( 1 2 1 2 1 4 − 1 4 ) , M ¯ e = ( 1 2 1 2 − 1 4 1 4 ) , M f = ( 3 1 2 0 ) , and M ¯ f = ( 3 1 − 2 0 ) .
The paths ( M e R k ) and ( M ¯ e R k ) are periodic (with period one) in Υ 2 , with period matrices M e R M e − 1 = T and M ¯ e R M ¯ e − 1 = T − 1 . We have x k = M e R k ( e ) and x ¯ k = M ¯ e R k ( e ) .The paths ( M f R k ) and ( M ¯ f R k ) are periodic (with period one) in Υ 2 , with period matrices M f R M f − 1 = T and M ¯ f R M ¯ f − 1 = T − 1 . We have σ k = M f R k ( Δ ) and σ ¯ k = M ¯ f R k ( Δ ) .A real number α > 0 is called badly approximable if { q ( p α − q ) : p , q ∈ 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 C = { x : ( ∀ i ) ⟨ w i , x ⟩ ≥ 0 } in R n , where ⟨ w i , w i ⟩ = 1 , define the norm minimum of C as N ( C ) = inf { ∏ i ⟨ w i , x ⟩ : x ∈ Z n ∩ C ∖ { 0 } } .
Given vectors v 1 , … , v m ∈ Z n , let [ v 1 , … , v m ] = ∑ i 1 < ⋯ < i n | det ( v i 1 ⋯ v i n ) | . This is the Euclidean volume of { ∑ i λ i v i : ( ∀ i ) 0 ≤ λ i ≤ 1 } .
Let V be the sail of an irrational simplicial cone C .
For a vertex x of Γ e ( V ) , define [ x ] = [ v 1 , … , v m ] where v 1 , … , v m are primitive vectors in Z n generating the edges emanating from x .For a vertex σ of Γ f ( V ) , define [ σ ] = [ v 1 , … , v m ] where v 1 , … , v m are the extreme points of σ .Then N ( C ) > 0 if and only if { [ x ] : x ∈ Γ e ( V ) } and { [ σ ] : σ ∈ Γ f ( V ) } are both bounded.
The quantities [ x ] and [ σ ] are called determinants. In two dimensions, with the cone generated by { ( 1 , α ) , ( 1 , 0 ) } , they are just the partial quotients of the continued fraction of α .
