Invariant measures for solvable groups and Diophantine approximation

Ronggang Shi, Barak Weiss

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

We show that if ℒ is a line in the plane containing a badly approximable vector, then almost every point in 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.

Original languageEnglish
Pages (from-to)479-505
Number of pages27
JournalIsrael Journal of Mathematics
Volume219
Issue number1
DOIs
StatePublished - 1 Apr 2017
Externally publishedYes

ASJC Scopus subject areas

  • Mathematics (all)

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