Israel Journal of Mathematics

, Volume 219, Issue 1, pp 479–505 | Cite as

Invariant measures for solvable groups and Diophantine approximation



We show that if \(\mathcal{L}\) is a line in the plane containing a badly approximable vector, then almost every point in \(\mathcal{L}\) does not admit an improvement in Dirichlet’s theorem. Our proof relies on a measure classification result for certain measures invariant under a nonabelian two-dimensional group on the homogeneous space SL3(ℝ)/SL3(ℤ). Using the measure classification theorem, we reprove a result of Shah about planar nondegenerate curves (which are not necessarily analytic), and prove analogous results for the framework of Diophantine approximation with weights. We also show that there are line segments in ℝ3 which do contain badly approximable points, and for which all points do admit an improvement in Dirichlet’s theorem.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    J. An, V. Beresnevich and S. Velani, Badly approximable points on planar curves and winning, preprint, (2014).Google Scholar
  2. [2]
    D. Badziahin and S. Velani, Badly approximable points on planar curves and a problem of Davenport, Mathematische Annalen 359 (2014), 969–1023.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    S. G. Dani, Divergent trajectories of flows on homogeneous spaces and Diophantine approximation, Journal für die Reine und Angewandte Mathematk 359 (1985), 55–89.MathSciNetMATHGoogle Scholar
  4. [4]
    S. G. Dani and G. A. Margulis, Limit distributions of orbits of unipotent flows and values of quadratic forms, Advances in Soviet Mathematics 16 (1993), 91–137.MathSciNetMATHGoogle Scholar
  5. [5]
    H. Davenport and W. M. Schmidt, Dirichlet’s theorem on Diophantine approximation, in Symposia Mathematica, Vol. IV (INDAM, Rome, 1968/69), Academic Press, London, 1970, pp. 113–132.Google Scholar
  6. [6]
    D. Kleinbock, Flows on homogeneous spaces and Diophantine properties of matrices, Duke Mathematical Journal 95 (1998), 107–124.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    D. Kleinbock, An ‘almost all versus no’ dichotomy in homogeneous dynamics and Diophantine approximation, Geometriae Dedicata 149 (2010), 205–218.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    D. Kleinbock and G. A. Margulis, Flows on homogeneous spaces and Diophantine approximation on manifolds, Annals of Mathematics 148 (1998), 339–360.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    D. Kleinbock and B. Weiss, Dirichlet’s theorem on Diophantine approximation and homogeneous flows, Journal of Modern Dynamics 4 (2008), 43–62.MathSciNetMATHGoogle Scholar
  10. [10]
    E. Lindenstrauss and B. Weiss, On sets invariant under the action of the diagonal group, Ergodic Theory and Dynamical Systems 21 (2001), 1481–1500.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    G. A. Margulis and G. M. Tomanov, Measure rigidity for almost linear groups and its applications, Journal d’Analyse Mathématique 69 (1996), 25–54.MathSciNetCrossRefMATHGoogle Scholar
  12. [12]
    S. Mozes, Epimorphic subgroups and invariant measures, Ergodic Theory and Dynamical Systems 15 (1995), 1207–1210.MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    M. Ratner, On Raghunathan’s measure conjecture, Annals of Mathematics 134 (1991), 545–607.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    N. A. Shah, Equidistribution of expanding translates of curves and Dirichlet’s theorem on Diophantine approximation, Inventiones Mathematicae 177 (2009), 509–532.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    N. A. Shah, Expanding translates of curves and Dirichlet-Minkowski theorem on linear forms, Journal of the American Mathematical Society 23 (2010), 563–589.MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    R. Shi, Convergence of measures under diagonal actions on homogeneous spaces, Advances in Mathematics 229 (2012), 1417–1434.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    R. Shi, Equidistribution of expanding measures with local maximal dimension and Diophantine Approximation, Monatshefte für Mathematik 165 (2012), 513–541.MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    B. Weiss, Divergent trajectories on noncompact parameter spaces, Geometric and Functional Analysis 14 (2004), 94–149.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 2017

Authors and Affiliations

  1. 1.School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael
  2. 2.School of Mathematical SciencesXiamen UniversityXiamenPR China

Personalised recommendations