Invariant measures for solvable groups and diophantine approximation

2014/08/30 by Ronggang Shi, Shi, Ronggang, Barak Weiss +1
Mathematics · #Dynamical Systems (math.DS) #FOS: Mathematics #Geometric and Algebraic Topology #Geometry and complex manifolds #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1409.0097

openalex publication_date 2014/08/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 non-abelian 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.

Citations

• Badly approximable points on planar curves and winning

Related

• Diophantine approximation and badly approximable sets
• Bad is null
• Inhomogeneous Diophantine approximation on curves and Hausdorff dimension
• Khintchine's theorem and Diophantine approximation on manifolds
• Diophantine approximation on curves