Dimension estimates for the set of points with non-dense orbit in homogeneous spaces

TL;DR

This paper provides upper bounds on the Hausdorff dimension of points with non-dense orbits in homogeneous spaces, extending previous results and applying to Diophantine approximation problems.

Contribution

It introduces new dimension estimates for non-dense orbit sets in homogeneous spaces using exponential mixing, generalizing recent work and applying to weighted badly approximable systems.

Findings

Upper bounds for Hausdorff dimension of non-dense orbit sets

Extension of Kadyrov's results to broader homogeneous spaces

Applications to Diophantine approximation, including weighted badly approximable systems

Abstract

Let $X = G/\Gamma$, where $G$ is a Lie group and $\Gamma$ is a lattice in $G$, and let $U$ be a subset of $X$ whose complement is compact. We use the exponential mixing results for diagonalizable flows on $X$ to give upper estimates for the Hausdorff dimension of the set of points whose trajectories miss $U$. This extends a recent result of Kadyrov and produces new applications to Diophantine approximation, such as an upper bound for the Hausdorff dimension of the set of weighted uniformly badly approximable systems of linear forms, generalizing an estimate due to Broderick and Kleinbock.