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

TL;DR

This paper studies the size and properties of points with specific Diophantine approximation characteristics in rank one homogeneous spaces, extending classical theorems to new geometric contexts.

Contribution

It introduces a new definition of Diophantine points of type γ in homogeneous spaces and computes their Hausdorff dimension, also extending Jarník-Besicovitch theorem to Heisenberg groups.

Findings

Hausdorff dimension of non-Diophantine points computed

Established Diophantine approximation results in rank one spaces

Extended classical theorems to Heisenberg groups

Abstract

In this paper, we give a definition of Diophantine points of type $\gamma$ for $\gamma\geq0$ 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 Jarn\'ik-Besicovitch Theorem on Diophantine approximation in Heisenberg groups.