The problem of investigation of the simplest n-dimensional continued fraction
in the sense of Klein for n>2 was posed by V.Arnold. The answer for the case of
n=2 can be found in the works of E.Korkina and G.Lachaud. In present work we
study the case of n=3

Karpenkov, Oleg

English

arXiv

Numérisation de Documents Anciens Mathématiques

C. R. Acad. Sci. Paris, Ser. I 343 (2006) 5–7http://france.elsevier.com/direct/CRASS1/Number Theory/GeometryThree examples of three-dimensional continued fractionsin the sense of KleinOleg Karpenkov 1CEREMADE – UMR 7534, université Paris-Dauphine France, 75775 Paris cedex 16, FranceReceived 20 January 2006; accepted 24 April 2006Available online 9 June 2006Presented by Vladimir Arnol’dAbstractThe problem of the investigation of the simplest n-dimensional continued fraction in the sense of Klein for n  2 was posed byV. Arnold. The answer for the case n = 2 can be found in the works of E. Korkina (1995) and G. Lachaud (1995). In present Notewe study the case n = 3. To cite this article: O. Karpenkov, C. R. Acad. Sci. Paris, Ser. I 343 (2006).© 2006 Académie des sciences. Published by Elsevier SAS. All rights reserved.RésuméTrois exemples des fractions continues trois-dimensional en sens de Klein. Le problème de l’étude les plus simple fractionscontinues n-dimensional en sens de Klein pour n  2 a été poser de V. Arnold. Le solution pour la case de n = 2 a presentédans les articles de E. Korkina (1995) et G. Lachaud (1995). Dans la Note présente, on étude la case de n = 3. Pour citer cetarticle : O. Karpenkov, C. R. Acad. Sci. Paris, Ser. I 343 (2006).© 2006 Académie des sciences. Published by Elsevier SAS. All rights reserved.1. DefinitionsA point of Rn+1 is called integer if all its coordinates are integers. A hyperplane is called integer if all its integervectors generate an n-dimensional sublattice of integer lattice. Consider some integer hyperplane and an integer pointin the complement to this plane. Let the Euclidean distance from the given point to the given plane equal l. Theminimal value of nonzero Euclidean distances from integer points of the space Rn+1 to the plane is denoted by l0. Theratio l/ l0 is said to be the integer distance from the given integer point to the given integer hyperplane.2. Definition of multidimensional continued fraction in the sense of KleinConsider arbitrary n+1 hyperplanes in Rn+1 that intersect at the unique point: at the origin. Assume also that allthe given planes do not contain any integer point different to the origin. The complement to these hyperplanes consistsE-mail address: karpenk@ceremade.dauphine.fr (O. Karpenkov).1 Partially supported by RFBR grant SS-1972.2003.1 and by RFBR grant 05-01-01012a.1631-073X/$ – see front matter © 2006 Académie des sciences. Published by Elsevier SAS. All rights reserved.doi:10.1016/j.crma.2006.04.0236 O. Karpenkov / C. R. Acad. Sci. Paris, Ser. I 343 (2006) 5–7of 2n+1 open orthants. Consider one of these orthants. The boundary of the convex hull of all integer points except theorigin in the closure of the orthant is called the sail of the orthant. The set of all 2n+1 sails is called the n-dimensionalcontinued fraction constructed accordingly to the given n+1 hyperplanes. Two n-dimensional continued fractions aresaid to be equivalent if there exists a linear lattice preserving transformation of Rn+1 taking all sails of one continuedfraction to the sails of the other continued faction.We associate to any hyperbolic irreducible operator A of SL(n+1,Z) an n-dimensional continued fraction con-structed according to the set of all n+1 eigen-hyperplanes for A. Any sail of such continued fraction is homeomorphicto Rn. From Dirichlet unity theorem it follows that the group of all SL(n+1,Z)-operators commuting with A and pre-serving the sails is homeomorphic to Zn and its action is free (we denote this group by Ξ(A)). A fundamental domainof the sail with respect to the action of the group Ξ(A) is a face union that contains exactly one face of the sail fromeach orbit. (For more information see [1–5].)3. The examplesDenote by Aa,b,c,d the following integer operator⎛⎜⎝0 1 0 00 0 1 00 0 0 1a b c d⎞⎟⎠ .Example 1. Consider the operator A1 = A1,−3,0,4. The group Ξ(A1) is generated by the operators B11 = A−21 ,B12 = (A1−E)2A−21 , and B13 = (A1−E)2(A1+E)A−21 . Let us enumerate all three-dimensional faces for one ofthe fundamental domains of the sail containing the vertex (0,0,0,1). Let V10 = (−3,−2,−1,1), V1,4i+2j+k =Bi11Bj12Bk13(V10) for i, j, k ∈ {0,1}. One of the fundamental domains of the sail contains the following three-dimensional faces: T11 = V10V12V14V15, T12 = V12V14V15V16, T13 = V12V15V16V17, T14 = V12V13V15V17, T15 =V10V12V13V15, T16 = V10V11V13V15, and T17 = V10V11V12V13 (see Fig. 1 (left)). All listed tetrahedra are taken bysome integer affine transformations to the unit basis tetrahedron. The integer distance from the origin to the planescontaining the faces T11, . . . , T17 equal 4, 3, 2, 4, 3, 2, and 1, respectively.Statement 1. The continued fraction constructed for any hyperbolic matrix of SL(4,Z) with irreducible characteristicpolynomial over rationals and with the sum of absolute values of the elements smaller than 8 is equivalent to thecontinued fraction of Example 1.Statement 2. The symmetry (not commuting with A1) defined by the matrix⎛⎜⎝4 −16 17 −33 −11 11 −23 −8 6 −16 −8 −2 1⎞⎟⎠acts on the sail of Example 1. This symmetry permutes the equivalence classes (with respect to the action of Ξ(A1))of tetrahedra T11 and T14, T12 and T15, T13 and T16, and takes the class of T17 to itself.Example 2. Let us consider the operator A2 = A1,−4,1,4. The group Ξ(A2) is generated by the operators B21 = A−22 ,B22 = (A2−E)2A−22 , and B23 = (A2+E)A−12 . Let us enumerate all three-dimensional faces for one of the fundamen-tal domains of the sail containing the vertex (0,0,0,1). Let V20 = (−4,−3,−2,0), V2,4i+2j+k = Bi21Bj22Bk23(V20)for i, j, k ∈ {0,1}. One of the fundamental domains of the sail contains the following three-dimensional faces:T21 = V20V21V23V24, T22 = V21V23V24V25, T23 = V20V22V23V24, T24 = V22V23V24V26, T25 = V23V24V25V27, andT26 = V23V24V26V27 (see Fig. 1 (middle)). All listed tetrahedra are taken by some integer affine transformationsto the unit basis tetrahedron. The integer distance from the origin to the planes containing the faces T21, . . . , T26 equal1, 2, 2, 4, 8, and 13, respectively.O. Karpenkov / C. R. Acad. Sci. Paris, Ser. I 343 (2006) 5–7 7Fig. 1. Gluing faces; see text for details.Example 3. Consider the operator A3 = A−1,−3,1,3. The group Ξ(A3) is generated by the operators B31 = A−23 ,B32 = (A3−E)A−13 , and B33 = A3+E. Any fundamental domain of the sail with (0,0,0,1) as a vertex contains aunique three-dimensional face. The polyhedron V30V31V32V33V34V35V36V37 shown on Fig. 1 (right) is an example ofsuch face, here V30 = (−1,−1,−1,0), V31 = B33(V30), V32 = B32B33(V30), V33 = B31B−132 (V30), V34 = B−132 (V30),V35 = B31B233(V30), V36 = B31B33(V30), V37 = B31B−132 B33(V30). The described face is contained in the plane on theunit distance from the origin. The integer volume of the face equals 8.Example 3 provides the negative answer to the following question for the case of n = 3: is it true, that any n-periodic n-dimensional sail contains an n-dimensional face in some hyperplane on integer distance to the origingreater than one? The answers for n = 2,4,5,6, . . . are unknown. The answer to the following question is alsounknown to the author: is it true, that any n-periodic n-dimensional sail contains an n-dimensional face in somehyperplane on unit integer distance to the origin?We show with dotted lines (Fig. 1) how to glue the faces to obtain the combinatorial scheme of the describedfundamental domains.AcknowledgementsThe author is grateful to V. Arnold for constant attention to this work and useful remarks and Université Paris-Dauphine — CEREMADE for the hospitality and excellent working conditions.References[1] V.I. Arnold, Continued Fractions, Moscow Center of Continuous Mathematical Education, Moscow, 2002.[2] O.N. Karpenkov, On tori decompositions associated with two-dimensional continued fractions of cubic irrationalities, Funct. Anal. Appl. 38 (2)(2004) 28–37.[3] E.I. Korkina, Two-dimensional continued fractions. The simplest examples, Proc. Steklov Inst. Math. 209 (1995) 143–166.[4] G. Lachaud, Voiles et Polyèdres de Klein, preprint no. 95-22, Laboratoire de Mathématiques Discrètes du C.N.R.S., Luminy, 1995.[5] J.-O. Moussafir, Voiles et Polyédres de Klein: Geometrie, Algorithmes et Statistiques, docteur en sciences thése, Université Paris IX-Dauphine,2000.

Three examples of three-dimensional continued fractions in the sense of Klein

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