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2008 | 6 | 3 | 482-487
Tytuł artykułu

Rational points on the unit sphere

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Treść / Zawartość
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EN
Abstrakty
EN
It is known that the unit sphere, centered at the origin in ℝn, has a dense set of points with rational coordinates. We give an elementary proof of this fact that includes explicit bounds on the complexity of the coordinates: for every point ν on the unit sphere in ℝn, and every ν > 0; there is a point r = (r 1; r 2;…;r n) such that: ⊎ ‖r-v‖∞ < ε.⊎ r is also a point on the unit sphere; Σ r i 2 = 1.⊎ r has rational coordinates; $$ r_i = \frac{{a_i }} {{b_i }} $$ for some integers a i, b i.⊎ for all $$ i,0 \leqslant \left| {a_i } \right| \leqslant b_i \leqslant (\frac{{32^{1/2} \left\lceil {log_2 n} \right\rceil }} {\varepsilon })^{2\left\lceil {log_2 n} \right\rceil } $$ . One consequence of this result is a relatively simple and quantitative proof of the fact that the rational orthogonal group O(n;ℚ) is dense in O(n;ℝ) with the topology induced by Frobenius’ matrix norm. Unitary matrices in U(n;ℂ) can likewise be approximated by matrices in U(n;ℚ(i))
Wydawca
Czasopismo
Rocznik
Tom
6
Numer
3
Strony
482-487
Opis fizyczny
Daty
wydano
2008-09-01
online
2008-07-02
Twórcy
Bibliografia
  • [1] Beresnevich V.V., Bernik V.I., Kleinbock D.Y., Margulis G.A., Metric diophantine approximation: the Khintchine-Groshev theorem for nondegenerate manifolds, Mosc. Math. J., 2002, 2, 203–225
  • [2] Bernik V.I., Dodson M.M., Metric diophantine approximation on manifolds, Cambridge University Press, Cambridge, 1999
  • [3] Hardy G.H., Wright E.M., An introduction to the theory of numbers, 5th ed., Oxford University Press, Oxford, 1983
  • [4] Householder A., Unitary triangularization of a nonsymmetric matrix, J. ACM, 1958, 5, 339–342 http://dx.doi.org/10.1145/320941.320947
  • [5] Humke P.D., Krajewski L.L., A characterization of circles which contain rational points, Amer. Math. Monthly, 1979, 86, 287–290 http://dx.doi.org/10.2307/2320748
  • [6] Kleinbock D.Y., Margulis G.A., Flows on homogeneous spaces and diophantine approximation on manifolds, Ann. of Math.(2), 1998, 148, 339–360 http://dx.doi.org/10.2307/120997
  • [7] Margulis G.A., Some remarks on invariant means, Monatsh. Math., 1980, 90, 233–235 http://dx.doi.org/10.1007/BF01295368
  • [8] Mazur B., The topology of rational points, Experiment. Math., 1992, 1, 35–45
  • [9] Mazur B., Speculations about the topology of rational points: an update, Astérisque, 1995, 228, 165–182
  • [10] Milenkovic V.J., Milenkovic V., Rational orthogonal approximations to orthogonal matrices, Comput. Geom., 1997, 7, 25–35 http://dx.doi.org/10.1016/0925-7721(95)00048-8
  • [11] Platonov V.P., The problem of strong approximation and the Kneser-Tits hypothesis for algebraic groups, Izv. Akad. Nauk SSSR Ser. Mat., 1969, 33, 1211–1219 (in Russian)
  • [12] Platonov V.P., A supplement to the paper “The problem of strong approximation and the Kneser-Tits hypothesis for algebraic groups”, Izv. Akad. Nauk SSSR Ser. Mat., 1970, 34, 775–777 (in Russian)
  • [13] Platonov V.P., Rapinchuk A., Algebraic groups and number theory, Academic Press, Boston, 1994 http://dx.doi.org/10.1016/S0079-8169(08)62065-6
  • [14] Uhlig F., Constructive ways for generating (generalized) real orthogonal matrices as products of (generalized) symmetries, Linear Algebra Appl., 2001, 332/334, 459–467 http://dx.doi.org/10.1016/S0024-3795(01)00296-8
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-008-0038-4
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