Skip to main content
Log in

Rational points on the unit sphere

  • Research Article
  • Published:
Central European Journal of Mathematics


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))

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  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

    MATH  MathSciNet  Google Scholar 

  2. Bernik V.I., Dodson M.M., Metric diophantine approximation on manifolds, Cambridge University Press, Cambridge, 1999

    MATH  Google Scholar 

  3. Hardy G.H., Wright E.M., An introduction to the theory of numbers, 5th ed., Oxford University Press, Oxford, 1983

    Google Scholar 

  4. Householder A., Unitary triangularization of a nonsymmetric matrix, J. ACM, 1958, 5, 339–342

    Article  MATH  MathSciNet  Google Scholar 

  5. Humke P.D., Krajewski L.L., A characterization of circles which contain rational points, Amer. Math. Monthly, 1979, 86, 287–290

    Article  MATH  MathSciNet  Google Scholar 

  6. Kleinbock D.Y., Margulis G.A., Flows on homogeneous spaces and diophantine approximation on manifolds, Ann. of Math.(2), 1998, 148, 339–360

    Article  MATH  MathSciNet  Google Scholar 

  7. Margulis G.A., Some remarks on invariant means, Monatsh. Math., 1980, 90, 233–235

    Article  MATH  MathSciNet  Google Scholar 

  8. Mazur B., The topology of rational points, Experiment. Math., 1992, 1, 35–45

    MATH  MathSciNet  Google Scholar 

  9. Mazur B., Speculations about the topology of rational points: an update, Astérisque, 1995, 228, 165–182

    MathSciNet  Google Scholar 

  10. Milenkovic V.J., Milenkovic V., Rational orthogonal approximations to orthogonal matrices, Comput. Geom., 1997, 7, 25–35

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  13. Platonov V.P., Rapinchuk A., Algebraic groups and number theory, Academic Press, Boston, 1994

    Book  MATH  Google Scholar 

  14. Uhlig F., Constructive ways for generating (generalized) real orthogonal matrices as products of (generalized) symmetries, Linear Algebra Appl., 2001, 332/334, 459–467

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Eric Schmutz.

About this article

Cite this article

Schmutz, E. Rational points on the unit sphere. centr.eur.j.math. 6, 482–487 (2008).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: