Israel Journal of Mathematics

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

Invariant measures for solvable groups and Diophantine approximation

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Abstract

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.

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© 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

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