Invariant measures for solvable groups and diophantine approximation

TL;DR

This paper investigates the measure-theoretic properties of points on lines in the plane related to Diophantine approximation, showing that almost all points on certain lines do not improve Dirichlet's theorem, using invariant measure classification.

Contribution

It introduces a measure classification approach for non-abelian group actions on homogeneous spaces to analyze Diophantine approximation on lines and curves.

Findings

Almost every point on certain lines containing badly approximable vectors does not admit an improvement in Dirichlet's theorem.

Reproves Shah's result on planar nondegenerate curves using measure classification.

Identifies line segments in b2^3 with badly approximable points where all points admit improvements.

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 non-abelian two dimensional group on the homogeneous space $\mathrm{SL}_3(\mathbb{R})/\mathrm{SL}_3(\mathbb{Z})$. 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 $\mathbb{R}^3$, which do contain badly approximable points, and for which all points do admit an improvement in Dirichlet's theorem.