# Rational points on the unit sphere

Open Mathematics (2008)

• Volume: 6, Issue: 3, page 482-487
• ISSN: 2391-5455

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## Abstract

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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 $i,0⩽\left|{a}_{i}\right|⩽{b}_{i}⩽{\left(\frac{{32}^{1/2}⌈lo{g}_{2}n⌉}{\epsilon }\right)}^{2⌈lo{g}_{2}n⌉}$ . 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))

## How to cite

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Eric Schmutz. "Rational points on the unit sphere." Open Mathematics 6.3 (2008): 482-487. <http://eudml.org/doc/269530>.

@article{EricSchmutz2008,
abstract = {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))},
author = {Eric Schmutz},
journal = {Open Mathematics},
keywords = {Diophantine approximation; orthogonal group; unitary group; rational points; unit sphere},
language = {eng},
number = {3},
pages = {482-487},
title = {Rational points on the unit sphere},
url = {http://eudml.org/doc/269530},
volume = {6},
year = {2008},
}

TY - JOUR
AU - Eric Schmutz
TI - Rational points on the unit sphere
JO - Open Mathematics
PY - 2008
VL - 6
IS - 3
SP - 482
EP - 487
AB - 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))
LA - eng
KW - Diophantine approximation; orthogonal group; unitary group; rational points; unit sphere
UR - http://eudml.org/doc/269530
ER -

## References

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1. [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 Zbl1013.11039
2. [2] Bernik V.I., Dodson M.M., Metric diophantine approximation on manifolds, Cambridge University Press, Cambridge, 1999 Zbl0933.11040
3. [3] Hardy G.H., Wright E.M., An introduction to the theory of numbers, 5th ed., Oxford University Press, Oxford, 1983 Zbl0020.29201
4. [4] Householder A., Unitary triangularization of a nonsymmetric matrix, J. ACM, 1958, 5, 339–342 http://dx.doi.org/10.1145/320941.320947 Zbl0121.33802
5. [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 Zbl0404.10007
6. [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 Zbl0922.11061
7. [7] Margulis G.A., Some remarks on invariant means, Monatsh. Math., 1980, 90, 233–235 http://dx.doi.org/10.1007/BF01295368 Zbl0425.43001
8. [8] Mazur B., The topology of rational points, Experiment. Math., 1992, 1, 35–45 Zbl0784.14012
9. [9] Mazur B., Speculations about the topology of rational points: an update, Astérisque, 1995, 228, 165–182 Zbl0851.14009
10. [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 Zbl0869.68105
11. [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. [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. [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. [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 Zbl0982.65049

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