Skip to main content
SpringerLink
Log in

Badly approximable vectors on fractals

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

For a large class of closed subsetsC of ℝn, we show that the intersection ofC with the set of badly approximable vectors has the same Hausdorff dimension asC. The sets are described in terms of measures they support. Examples include (but are not limited to) self-similar sets such as Cantor’s ternary sets or attractors for irreducible systems of similarities satisfying Hutchinson’s open set condition.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. T. Bedford,Hausdorff dimension and box dimension in self-similar sets, inTopology and Measure, V (Binz, 1987), Ernst-Moritz-Arndt University, Greifswald, 1988, pp. 17–26.

    Google Scholar 

  2. V. Bernik, D. Kleinbock and G. A. Margulis,Khintchine-type theorems on manifolds: convergence case for standard and multiplicative versions, International Mathematics Research Notices no. 9 (2001), 453–486.

  3. S. G. Dani,Divergent trajectories of flows on homogeneous spaces and diophantine approximation, Journal für die reine und angewandte Mathematik359 (1985), 55–89.

    Article  MATH  MathSciNet  Google Scholar 

  4. S. G. Dani,On orbits of unipotent flows on homogeneous spaces. II, Ergodic Theory and Dynamical Systems6 (1986), 167–182.

    MATH  MathSciNet  Google Scholar 

  5. J. E. Hutchinson,Fractals and self-similarity, Indiana University Mathematics Journal30 (1981), 713–747.

    Article  MATH  MathSciNet  Google Scholar 

  6. D. Kleinbock,Flows on homogeneous spaces and Diophantine properties of matrices, Duke Mathematical Journal95 (1998), 107–124.

    Article  MATH  MathSciNet  Google Scholar 

  7. D. Kleinbock,Bounded orbits conjecture and Diophantine approximation, inLie Groups and Ergodic Theory (Mumbai, 1996), Tata Institute of Fundamental Research, Bombay, 1998, pp. 119–130.

    Google Scholar 

  8. D. Kleinbock,Badly approximable systems of affine forms, Journal of Number Theory79 (1999), 83–102.

    Article  MATH  MathSciNet  Google Scholar 

  9. D. Kleinbock,Some applications of homogeneous dynamics to number theory, Proceedings of Symposia in Pure Mathematics69 (2001), 639–660.

    MathSciNet  Google Scholar 

  10. D. Kleinbock, E. Lindenstrauss and B. Weiss,On fractal measures and diophantine approximation, Selecta Mathematica10 (2004), 479–523.

    Article  MATH  MathSciNet  Google Scholar 

  11. D. Kleinbock and G. A. Margulis,Bounded orbits of nonquasiunipotent flows on homogeneous spaces, American Mathematical Society Translations171 (1996), 141–172.

    MathSciNet  Google Scholar 

  12. D. Kleinbock and G. A. Margulis,Flows on homogeneous spaces and Diophantine approximation on manifolds, Annals of Mathematics148 (1998), 339–360.

    Article  MATH  MathSciNet  Google Scholar 

  13. D. Kleinbock and G. Tomanov,Flows on S-arithmetic homogeneous spaces and applications to metric Diophantine approximation, Max Planck Institute Preprint 2003-65, 2003.

  14. D. Kleinbock and B. Weiss,Bounded geodesics in moduli space, International Mathematics Research Notices no. 30 (2004), 1551–1560.

  15. S. Kristensen, R. Thorn, and S. Velani,Diophantine approximation and badly approximable sets, preprint (2004).

  16. G. A. Margulis,On the action of unipotent groups in the space of lattices, inLie Groups and Their Representations (Proc. Summer School, Bolyai, János Math. Soc., Budapest, 1971), Halsted, New York, 1975, pp. 365–370.

    Google Scholar 

  17. P. Mattila,Geometry of Sets and Measures in Euclidean Space. Fractals and Rectifiability, Cambridge Studies in Advanced Mathematics, Vol. 44, Cambridge University Press, Cambridge, 1995.

    Google Scholar 

  18. D. Mauldin and M. Urbański,Dimensions and measures in infinite iterated function systems, Proceedings of the London Mathematical Society (3)73 (1996), 105–154.

    Article  MATH  MathSciNet  Google Scholar 

  19. C. McMullen,Area and Hausdorff dimension of Julia sets of entire functions, Transactions of the American Mathematical Society300 (1987), 329–342.

    Article  MATH  MathSciNet  Google Scholar 

  20. Y. N. Minsky and B. Weiss,Nondivergence of horocyclic flows on moduli space, Journal für die reine und angewandte Mathematik552 (2002), 131–177.

    Article  MATH  MathSciNet  Google Scholar 

  21. Y. Pesin,Dimension Theory in Dynamical Systems. Contemporary Views and Applications, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1997.

    Google Scholar 

  22. A. Pollington and S. Velani,On simultaneously badly approximable numbers, Journal of the London Mathematical Society (2)66 (2002), 29–40.

    Article  MATH  MathSciNet  Google Scholar 

  23. A. Pollington and S. Velani,Metric Diophantine approximation and ‘absolutely friendly’ measures, preprint (2003).

  24. M. S. Raghunathan,Discrete Subgroups of Lie Groups, Springer-Verlag, New York-Heidelberg, 1972.

    MATH  Google Scholar 

  25. W. M. Schmidt,A metrical theorem in diophantine approximation, Canadian Journal of Mathematics12 (1960), 619–631.

    MATH  Google Scholar 

  26. W. M. Schmidt,Badly approximable systems of linear forms, Journal of Number Theory1 (1969), 139–154.

    Article  MATH  MathSciNet  Google Scholar 

  27. W. M. Schmidt,Diophantine Approximation, Lecture Notes in Mathematics, Vol. 785, Springer-Verlag, Berlin, 1980.

    MATH  Google Scholar 

  28. W. M. Schmidt,Open problems in Diophantine approximation, inDiophantine Approximations and Transcendental Numbers (Luminy, 1982), Progress in Mathematics31, Birkhäuser, Boston, 1983, pp. 271–287.

    Google Scholar 

  29. M. Urbanski,The Hausdorff dimension of the set of points with non-dense orbit under a hyperbolic dynamical system, Nonlinearity4 (1991), 385–397.

    Article  MATH  MathSciNet  Google Scholar 

  30. M. Urbanski,Diophantine approximation for conformal measure of one-dimensional iterated function systems, Compositio Mathematica, to appear.

  31. M. Urbanski,Diophantine approximation and self-conformal measures, Journal of Number Theory, to appear.

  32. W. A. Veech,Measures supported on the set of uniquely ergodic directions of an arbitrary holomorphic 1-form, Ergodic Theory and Dynamical Systems19 (1999), 1093–1109.

    Article  MATH  MathSciNet  Google Scholar 

  33. B. Weiss,Almost no points on a Cantor set are very well approximable Proceedings of the Royal Society of London, Series A457 (2001), 949–952.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Brandeis University, 02454-9110, Waltham, MA, USA

    Dmitry Kleinbock

  2. Department of Mathematics, Ben Gurion University, 84105, Be’er Sheva, Israel

    Barak Weiss

Authors
  1. Dmitry Kleinbock
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Barak Weiss
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dmitry Kleinbock.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kleinbock, D., Weiss, B. Badly approximable vectors on fractals. Isr. J. Math. 149, 137–170 (2005). https://doi.org/10.1007/BF02772538

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02772538

Keywords

Search

Navigation