A Shrinking Target Problem with Target at Infinity in Rank One Homogeneous Spaces

Abstract

In this paper, we give a definition of Diophantine points of type $\gamma$ for $\gamma \ge 0$ in a homogeneous space $G/\Gamma$, and compute the Hausdorff dimension of the subset of points which are not Diophantine of type $\gamma$ when $G$ is a semisimple Lie group of real rank one. We also deduce a Jarník-Besicovitch Theorem on Diophantine approximation in Heisenberg groups.

Acknowledgements updated. To appear in Monatsh. Math

References in corpus (2)

• Sparse Equidistribution of Unipotent Orbits in Finite-Volume Quotients of $\text{PSL}(2,\mathbb{R})$, with appendices
• Non-Divergence of Unipotent Flows on Quotients of Rank One Semisimple Groups

Cited by in corpus (2)

• Diophantine approximations, large intersections and geodesics in negative curvature
• Hausdorff dimension of limsup sets of rectangles in the Heisenberg group