Hausdorff Dimension of Non-Diophantine Points in Finite-Volume Quotients of Rank One Semisimple Lie Groups

Cheng Zheng

Published 2016-10-06Version 1

In this note, we give a definition of Diophantine points of type $\gamma$ for $\gamma\geq0$ in a homogeneous space $G/\Gamma$ and we compute Hausdorff dimensions of subsets of points which are not Diophantine of type $\gamma$ when $G$ is a semisimple Lie group of real rank one. As a corollary, we will give a description of Hausdorff dimensions of non-Diophantine points of various types in Heisenberg groups, which could be considered as a Jarnik-Besicovitch Theorem on Diophantine approximation in Heisenberg groups.