Bounded trajectories of quasi-rays on homogeneous spaces and Diophantine approximation with weight functions

Abstract

Let $G$ be a connected semisimple real Lie group, $Γ$ an irreducible lattice in $G$ and $X = G/Γ$. Let $F = \{g_t: t\ge 0\}$ be a non-quasiunipotent one-parameter subsemigroup of $G$. Then it is known that the set of points in $X$ with bounded $F$-trajectories has full Hausdorff dimension. In addition, if $U$ is the expanding horospherical subgroup relative to $g_1$, then for any $x \in X$ the set of points $u \in U$ such that the $F$-trajectory of $ux$ is bounded has full Hausdorff dimension. In this paper we take $U$ to be a horospherical subgroup of $G$ and apply Shi's equidistribution theorem for elements of the expanding cone with respect to $U$ to describe a class of subsets $F$ in $G$, not presupposing the group structure, for which the above full Hausdorff dimension statements also hold. As an application, we prove that the set of badly approximable matrices in the set-up of Diophantine approximations with quasimultiplicative weight functions has full Hausdorff dimension.