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ЧЕБЫШЕВСКИЙ СБОРНИК Том 11 Выпуск 1 (2010)
Труды VII Международной конференции Алгебра и теория чисел: современные проблемы и приложения, посвященной памяти профессора Анатолия Алексеевича Карацубы
STRUCTURE of the BEST DIOPHANTINE APPROXIMATIONS and MULTIDIMENSIONAL GENERALIZATIONS of the CONTINUED FRACTION 1
A. D. Bruno (t. Moscow)
Аннотация
Let in a three-dimensional real space two forms be given: a linear form and a quadratic one which is a product of two complex conjugate linear forms.
Their root sets are a plane and a straight line correspondingly. We assume that the line does not lie in the plane. Voronoi (1896) and author (2006) proposed two different algorithms for computation of integer points giving the best approximations to roots of these two forms. The both algorithms are one-way: the Voronoi algorithms is directed to the plane and the authors algorithms is directed to the line.
Here we propose an algorithm, which works in both directions. We give also a survey of results on such approach to simultaneous Diophantine approximations in any dimensions.
1. Statement of the problem
Let in R3 with eoordinates X = (xi;x2,x3) the linear form
1l(X) = (J, X) =f jiXi + j2X2 + 33x3 and the quadratic form /2(X) = {K,X){K,X) be given. Product
f (X ) = 1i(X )l2(X).
The root set f(X) = 0 is L1 U L2 where the piane L1 = {X : (J, X) = 0}, the line
L2 = {X : (R.K, X) = (QK,X) = 0}. Put
шг(X) = |1i(X)|, i = 1, 2; M(X) = (mi(X),m2(X)).
The integer point X G Z3 is the best approximation to Li U L2 if a point Y G Z3, Y = 0 with
M(Y)^M(X), ||М(У)|| < || АфОИ
1 Supported by RFBR, grants 08-01-0082 and 09-01-00291
is absent.
Problem. To find algorithm for computation of best approximations.
Xow there are two algorithms solving the problem: by G, Voronoi (1896) |1|, it tends to Li, and by A. Bruno and V. Parusnikov (2006) [2], it tends to L2, Here we propose new algorithm working in both directions: to Li and to L2,
2. The principal construction
The vector-function
M(X) = (mi(X)) = (|1i(X)|, |fe(X)|)
maps R3 into the first quadrant S+ of the plane S = R2 9 (mi,m2^^et Z be the image of Z3 except X = 0:
Z = M(Z3 \ 0).
M is the convex hull of Z, and 5M is ^^e boundary of M,
In consists of vertices and edges. All its vertices are images of the integer points X 6 Z3, Some such images can be at edges. All points of SM n Z are images of the best approximations. So computing the polygon SM is enough for solving our Problem,
For two points U = (ui)u2^d V = (vi,v2) 6 S+ the function
V2 - U2
Zi(u, v )
U2(Ui — Vi)
is the value m- i in the point (mi, 0) 6 S+ of the axis mi where the axis intersects
UV
3. Algorithm
Let we have the basis Bi; B2, B3 6 Z3, det(Bg, B2g, Bg) = ±1, ordered in some manner. We compute the next basis Bi; B2, B3 bv the linear transformation
where a, b, c 6 Z; |a|, |b|, |c| ^ Ki, differences M(B2) — M(B3), M(B3) — M(B3) belong either to the III quadrant or to I and IV quadrants (Fig, 2);
M(B3)
Puc, 2: Places for images of new basis points.
B2 : Zi(M(B3), M(B2)) = max(i(M№),M(B2));
a
B3 : Zi(M(Bs),M(B3)) = max(1(M(Bs),M(B3)).
b,c
Among all possible values of B2 and B3 we choose those giving biggest inclinations for lines going through points M(B2), M(B3) and the point M(B3),
Here к = 3 ■ 2l is the movable boundarv for |a|, |b|, |c|: if оте of |a|, |b|, |c| reaches k, then we rep lace к by к1+1; and repeat computation of B2 and B3, To the basis B 1, B2, B3 we again apply the described algorithm and so on (Fig, 1), The algorithm works in both directions: to L1 and to L2, enough to permute forms /1(X) and /2(X),
4. Algebraic case
Let the polynomial P(A) = A3 + a\2 + /ЗА + 7 have integral coefficients and negative discriminant. It has 3 roots Ai Є R, Л2 = Л3 Є C, Put J = (1, Аь A2), K = (1, A2, Al), h(X) = (J,X), l2(X) = {K,X)(K,X). According to the Diriehlet Theorem, the field Q(A1) has one fundamental unity which corresponds to the unimodular substitution X = DY, The substitution is the automorphism of |f (X)| = |11(X)||12(X)| and of open polygon dM, Thus, the polygon SM is periodic.
So our algorithm allows to find the minimal period of SM and the corresponding fundamental unit of the field.
The algorithm was implemented and was checked on a lot of polynomials [3].
5. Three linear forms and positive discriminant
The similar approach in the case of three linear forms leads to consideration
of a polyhedral surface dM in the 3-dimensional first octant (m1,m2,m3) ^ 0. In the algebraic case the polynomial P(A) must have the positive discriminant then three real roots of P(A) give 3 linear forms. The surface SM has 2 independent periods corresponding to 2 fundamental units. The algorithm was described early (2005-2007 [4, 5, 6]).
6. General situation
In R™ there are given / linear forms /j(X), i = 1, ..., / and k quadratic forms fj (X), j = / + 1 ...,/ + k and / + 2k = n. The map
mi = |/i(X)|, i = 1,..., / + k
transforms Z™ \ 0 in the set Z C R++fc, M C R++fc is convex hull of Z, Boundary
dM has dimens ion / + k — 1,
In algebraic case dM has / + k — 1 independent periods [7], They can be found bv computation of SM bv a similar algorithm,
7. Comparison with other approaches
In 1895-1896 F, Klein [8], H, Minkowski [9] and G, Voronoi [1] proposed 3 different approaches to generalization of the continued fraction for the case of 3
linear forms in R3, Our approach is nearer to the Voronoi’s one, but different from it.
The Klein’s approach was proposed again independently by B, Scubenko (1988) [10] and by V, Arnold (1998) [11]. The term “Klein’s polyhedra” I introduced (1994) [12] as reaction on the term “Arnold’s polyhedra” introduced by G. Lachand
(1993) [13], We found (1994-2002) [12, 14, 15, 16, 17] that Klein’s polyhedra cannot give a background for an algorithm generalizing the continued fraction. So I proposed (2005) [4] one polyhedron M which is in a sense the convex hull of 8 Klein’s polyhedra. Nevertheless now several groups in different countries study the Klein’s polyhedra.
The Minkowski’s approach was developed for n linear forms in Rn bv J, Lagarias
(1994) [18], But his algorithm is essentially more complicated than algorithm proposed by the author,
СПИСОК ЦИТИРОВАННОЙ ЛИТЕРАТУРЫ
[1] G, F, Voronoi, On Generalization of the Algorithm of Continued Fractions // Warszawa University, 1896, Also in: Collected Works in 3 Volumes, Vol. 1, Izdat, Akad, Nauk USSR, Kiev, 1952 (in Russian),
[2] A, D, Bruno, V.I. Parusnikov, Further generalization of the continued fraction // Dokladv Akademii Nauk 410:1 (2006) 12-16 (R) = Dokladv Mathematics 74:2 (2006) 628-632 (E).
[3] A, D, Bruno, V, I, Parusnikov, Two-way generalization of the continued fraction // Dokladv Akademii Nauk 429:6 (2009) 727-730 (R) = Dokladv Mathematics 80:3 (2009) 887-890 (E).
[4] A, D, Bruno, Structure of best Diophantine approximations // Dokladv Akademii Nauk 402:4 (2005) 439-444 (R) = Dokladv Mathematics 71:3 (2005) 396-400 (E)
[5] A, D, Bruno, Generalizied continued fraction algorithm // Ibid, 402:6 (2005) 732-736 (R) = Ibid. 71:3 (2005) 446-450 (E)
[6] A. D. Bruno, Generalizations of continued fraction // Chebvshevskii sbornik, 7:3 (2006) 4-71. (R)
[7] A. D. Bruno, Structure of multidimensional Diophantine approximations // Dokladv Akademii Nauk, 433:5 (2010) (R) = Dokladv Mathematics 82:1 (2010)
(E)
[8] F. Klein, Uber eine geometrische Auffassung der gewohnlichen Kettenbru-chentwicklung // Nachr. Ges. Wiss. Gottingen Math.-Phvs. Kl. 1895. N 3. S. 357-359.
[9] H, Minkowski, Generalisation de le theorie des fractions continues // Ann, Sci, Ec, Norm, Super, ser III, 1896, t, 13, p. 41-60, Also in: Gesamm, Abh, I, S, 278-292.
[10] B.F. Skubenko, Minimum of a decomposable cubic form of three variables // J. Sov. Math. 53, No. 3, 302-321 (1991).
[11] V.I. Arnold, Higher dimensional continued fractions // Regular and Chaotic Dynamics, 1998, v, 3, no. 3, p. 10-17.
[12] A.D, Bruno, V.I, Parusnikov, Klein polvhedrals for two cubic Davenport forms // Mat. Zametki 56:4 (1994) 9-27 (R) = Math. Notes 56:3-4 (1994) 994-1007
(E)
[13] G. Lachaud, Polvedre d’Arnol’d et voile d’un cone simplicial: analogues du theoreme de Lagrange // C.R. Acad. Sci. Ser. 1. 1993. V. 317. P. 711-716.
[14] A. D. Bruno, V. I. Parusnikov, Comparison of various generalizations of continued fractions // Matem. Zametki 61:3 (1997) 339-348 (R) = Mathem. Notes 61:3 (1997) 278-286 (E).
[15] V.I. Parusnikov, Klein polvhedra for complete decomposable forms // Number theory. Diophantine, Computational and Algebraic Aspects. (Editors: K. Gvorv, A. Petho and V.T. Sos.) De Gruvter, Berlin, New York. 1998, p. 453-463.
[16] V.I, Parusnikov, Klein polvhedra for the fourth extremal cubic form // Matem, Zametki 67:1 (2000) 110-128 (R) = Math. Notes 67:1 (2000) 87-102 (E).
[17] V.I. Parusnikov, Klein’s polvhedra for three extremal forms // Matem. Zametki 77:4 (2005) 566-583 (Russian) = Math. Notes 77:4 (2005) 523-538 (English).
[18] J.C. Lagarias, Geodesic multidimensional continued fractions // Proc. London Math. Soc. (3) 69 (1994) 464-488.
Институт прикладной математики им. М. В. Келдыша РАН
Получено 25.05.2010
