Geometriae Dedicata

, Volume 167, Issue 1, pp 295–307 | Cite as

A quantitative version of Steinhaus’ theorem for compact, connected, rank-one symmetric spaces

  • Fernando Mário de Oliveira Filho
  • Frank Vallentin
Original Paper


Let \(d_1,\,d_2\), ... be a sequence of positive numbers that converges to zero. A generalization of Steinhaus’ theorem due to Weil implies that, if a subset of a homogeneous Riemannian manifold has no pair of points at distances \(d_1,\,d_2\), ... from each other, then it has to have measure zero. We present a quantitative version of this result for compact, connected, rank-one symmetric spaces, by showing how to choose distances so that the measure of a subset not containing pairs of points at these distances decays exponentially in the number of distances.


Steinhaus’ theorem Geometric Ramsey theory Linear programming Orthogonal polynomials 

Mathematics Subject Classification (1991)

42B05 90C05 



We would like to thank Roderick Wong for his comments and observations concerning the results of [22]. We thank the referee for valuable comments.


  1. 1.
    Andrews, G.E., Askey, R., Roy, R.: Special Functions, Encyclopedia of Mathematics and its Applications, vol. 71. Cambridge University Press, Cambridge (1999)Google Scholar
  2. 2.
    Alon, N., Kahale, N.: Approximating the independence number via the \(\theta \)-function. Math. Program. 80, 253–264 (1998)Google Scholar
  3. 3.
    Askey, R.: Orthogonal Polynomials and Special Functions. Society for Industrial and Applied Mathematics, Philadelphia (1975)CrossRefGoogle Scholar
  4. 4.
    Bachoc, C., Nebe, G., de Oliveira Filho, F.M., Vallentin, F.: Lower bounds for measurable chromatic numbers. Geom. Funct. Anal. 19, 645–661 (2009)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bukh, B.: Measurable sets with excluded distances. Geom. Funct. Anal. 18, 668–697 (2008)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Croft, H.T., Falconer, K.J., Guy, R.K.: Unsolved Problems in Geometry. Springer, New York (1991)CrossRefMATHGoogle Scholar
  7. 7.
    de Oliveira Filho, F.M., Vallentin, F.: Fourier analysis, linear programming, and densities of distance avoiding sets in \({\mathbb{R}}^n\). J. Eur. Math. Soc. 12, 1417–1428 (2010)CrossRefMATHGoogle Scholar
  8. 8.
    Falconer, K.J.: The realization of small distances in plane sets of positive measure. Bull. London Math. Soc. 18, 475–477 (1986)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Frenzen, C.L., Wong, R.: A uniform asymptotic expansion of the Jacobi polynomials with error bounds. Can. J. Math. 37, 979–1007 (1985)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Halmos, P.R.: Measure Theory. Springer, New York (1974)MATHGoogle Scholar
  11. 11.
    Kolountzakis, M.N.: Distance sets corresponding to convex bodies. Geom. Funct. Anal. 14, 734–744 (2004)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Landau, L.J.: Bessel functions: monotonicity and bounds. J. London Math. Soc. 61, 197–215 (2000)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Lovász, L.: On the Shannon capacity of a graph. IEEE Trans. Inf. Theory IT-25, 1–7 (1979)Google Scholar
  14. 14.
    Steinhaus, H.: Sur les distances des points dans les ensembles de mesure positive. Fund. Math. 1, 93–104 (1920)MATHGoogle Scholar
  15. 15.
    Stromberg, K.: An elementary proof of Steinhaus’s theorem. Proc. Am. Math. Soc. 36, 308 (1972)MathSciNetMATHGoogle Scholar
  16. 16.
    Szegö, G.: Orthogonal polynomials. In: American Mathematical Society Colloquium Publications Volume XXIII, 4th edn. American Mathematical Society, Providence (1975)Google Scholar
  17. 17.
    Székely, L.A.: Remarks on the chromatic number of geometric graphs, in: Graphs and other combinatorial topics, M. Fiedler, ed., Teubner-Texte zur Mathematik 59, Teubner, Leipzig, 1983, pp. 312–315Google Scholar
  18. 18.
    Tao, T.: Structure and Randomness: Pages from Year One of a Mathematical Blog. American Mathematical Society, Providence (2008)Google Scholar
  19. 19.
    Wang, H.C.: Two-point homogeneous spaces. Ann. Math. 55, 177–191 (1952)CrossRefMATHGoogle Scholar
  20. 20.
    Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge (1922)MATHGoogle Scholar
  21. 21.
    Weil, A.: L’intégration dans les groupes topologiques et ses applications, Publications de l’Institut de Mathématique de l’Université de Strasbourg IV, deuxième édition. Hermann, Paris (1965)Google Scholar
  22. 22.
    Wong, R., Zhang, J.-M.: Asymptotic monotonicity of the relative extrema of Jacobi polynomials. Can. J. Math. 46, 1318–1337 (1994)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  • Fernando Mário de Oliveira Filho
    • 1
  • Frank Vallentin
    • 2
  1. 1.Institute of Mathematics, FU BerlinBerlinGermany
  2. 2.Delft Institute of Applied MathematicsTechnical University of DelftDelftThe Netherlands

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