Klein polyhedron explained

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

styleC

be a closed simplicial cone in Euclidean space

styleRⁿ

. The Klein polyhedron of

styleC

is the convex hull of the non-zero points of

styleC\capZⁿ

Relation to continued fractions

Suppose

style\alpha>0

is an irrational number. In

styleR²

, 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

styleZ^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

styleC

is generated by a basis

style(a_i)

styleRⁿ

(so that

styleC=\{\sum_iλ_ia_i:(\foralli) λ_i\geq0\}

), and let

style(w_i)

be the dual basis (so that

styleC=\{x:(\foralli) \langlew_i,x\rangle\geq0\}

). Write

styleD(x)

for the line generated by the vector

stylex

, and

styleH(x)

for the hyperplane orthogonal to

stylex

Call the vector

stylex\inRⁿ

irrational if

styleH(x)\capQⁿ=\{0\}

; and call the cone

styleC

irrational if all the vectors

stylea_i

and

stylew_i

are irrational.

The boundary

styleV

of a Klein polyhedron is called a sail. Associated with the sail

styleV

of an irrational cone are two graphs:

the graph

style\Gamma_e(V)

whose vertices are vertices of

styleV

, two vertices being joined if they are endpoints of a (one-dimensional) edge of

styleV

;

the graph

style\Gamma_f(V)

whose vertices are

style(n-1)

-dimensional faces (chambers) of

styleV

, 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

styleGL_n(Q)

, where vertex

styleA

is joined to vertex

styleB

if and only if

styleA^-1B

is of the form

styleUW

where

U=\left(\begin{array}{cccc}1& … &0&c₁\ \vdots&\ddots&\vdots&\vdots\ 0& … &1&c_n-1\ 0& … &0&c_n\end{array}\right)

(with

stylec_i\inQ

stylec_n ≠ 0

) and

styleW

is a permutation matrix. Assuming that

styleV

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)

style\Gamma_e(V)

, one can find a path

style(A_0,A_1,\ldots)

style\Upsilon_n

such that

stylex_k=A_k(e)

, where

stylee

is the vector

style(1,\ldots,1)\inRⁿ

Given any path

style(\sigma_0,\sigma_1,\ldots)

style\Gamma_f(V)

, one can find a path

style(A_0,A_1,\ldots)

style\Upsilon_n

such that

style\sigma_k=A_k(\Delta)

, where

style\Delta

is the

style(n-1)

-dimensional standard simplex in

styleRⁿ

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

styleK\subseteqR

be a totally real algebraic number field of degree

stylen

, and let

style\alpha_i:K\toR

be the

stylen

real embeddings of

styleK

. The simplicial cone

styleC

is said to be split over

styleK

styleC=\{x\inRⁿ:(\foralli) \alpha_i(\omega₁₎x₁+\ldots+\alpha_i(\omega_n)x_n\geq0\}

where

style\omega_1,\ldots,\omega_n

is a basis for

styleK

over

styleQ

Given a path

style(A_0,A_1,\ldots)

style\Upsilon_n

, let

styleR_k=A_k+1

	-1
A
	k

. The path is called periodic, with period

stylem

, if

styleR_k+qm=R_k

for all

stylek,q\geq0

. The period matrix of such a path is defined to be

styleA_m

	-1
A
	0

. A path in

style\Gamma_e(V)

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

styleC\subseteqRⁿ

, with generators

style(a_i)

and

style(w_i)

as above and with sail

styleV

, the following three conditions are equivalent:

styleC

is split over some totally real algebraic number field of degree

stylen

For each of the

stylea_i

there is periodic path of vertices

stylex_0,x_1,\ldots

style\Gamma_e(V)

such that the

stylex_k

asymptotically approach the line

styleD(a_i)

; and the period matrices of these paths all commute.

For each of the

stylew_i

there is periodic path of chambers

style\sigma_0,\sigma_1,\ldots

style\Gamma_f(V)

such that the

style\sigma_k

asymptotically approach the hyperplane

styleH(w_i)

; and the period matrices of these paths all commute.

Example

Take

stylen=2

and

styleK=Q(\sqrt{2})

. Then the simplicial cone

style\{(x,y):x\geq0,\verty\vert\leqx/\sqrt{2}\}

is split over

styleK

. The vertices of the sail are the points

style(p_k,\pmq_k)

corresponding to the even convergents

stylep_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

stylex_k

stylex_k+1

. Write

style\bar{x}_k

and

style\bar{\sigma}_k

for the reflections of

stylex_k

and

style\sigma_k

in the

stylex

-axis. Let

styleT=\left(\begin{array}{cc}3&4\ 2&3\end{array}\right)

, so that

stylex_k+1=Tx_k

, and let

styleR=\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

styleM_e=\left(\begin{array}{cc}

	12
	&

\	14
	&

	14
	\end{array}

\right)

style\bar{M}_e=\left(\begin{array}{cc}

	12
	&

\ -	14
	&

	14
	\end{array}

\right)

styleM_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_eR^k)

and

style(\bar{M}_eR^k)

are periodic (with period one) in

style\Upsilon₂

, with period matrices

styleM_eR

	-1
M
	e

and

style\bar{M}_eR

	-1
\bar{M}
	e

=T^-1

. We have

stylex_k=M_eR^k(e)

and

style\bar{x}_k=\bar{M}_eR^k(e)

The paths

style(M_fR^k)

and

style(\bar{M}_fR^k)

are periodic (with period one) in

style\Upsilon₂

, with period matrices

styleM_fR

	-1
M
	f

and

style\bar{M}_fR

	-1
\bar{M}
	f

=T^-1

. We have

style\sigma_k=M_fR^k(\Delta)

and

style\bar{\sigma}_k=\bar{M}_fR^k(\Delta)

Generalization of approximability

A real number

style\alpha>0

is called badly approximable if

style\{q(p\alpha-q):p,q\inZ,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

styleC=\{x:(\foralli) \langlew_i,x\rangle\geq0\}

styleRⁿ

, where

style\langlew_i,w_i\rangle=1

, define the norm minimum of

styleC

styleN(C)=inf\{\prod_i\langlew_i,x\rangle:x\inZⁿ\capC\setminus\{0\}\}

Given vectors

stylev_1,\ldots,v_m\inZⁿ

, let

style[v_1,\ldots,v_m]=

\sum
	i₁< … <i_n

\vert

\det(v
	i₁

…

v
	i_n

)\vert

. This is the Euclidean volume of

style\{\sum_iλ_iv_i:(\foralli) 0\leqλ_i\leq1\}

Let

styleV

be the sail of an irrational simplicial cone

styleC

For a vertex

stylex

style\Gamma_e(V)

, define

style[x]=[v_1,\ldots,v_m]

where

stylev_1,\ldots,v_m

are primitive vectors in

styleZⁿ

generating the edges emanating from

stylex

For a vertex

style\sigma

style\Gamma_f(V)

, define

style[\sigma]=[v_1,\ldots,v_m]

where

stylev_1,\ldots,v_m

are the extreme points of

style\sigma

Then

styleN(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

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

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

Klein polyhedron explained

Definition

Relation to continued fractions

Graphs associated with the Klein polyhedron

Generalization of Lagrange's theorem

Example

Generalization of approximability

See also

References

Notes and References