A shrinking target problem in homogeneous spaces of semisimple algebraic groups

Abstract

In this paper, we study a shrinking target problem with target at infinity in a homogeneous space of a semisimple algebraic group from the representation-theoretic point of view. Let $\ddot{I}:\mathbf{G}\to \mathbf{GL}(V)$ be an irreducible $\mathbb{Q}$-rational representation of a connected semisimple $\mathbb{Q}$-algebraic group $\mathbf{G}$ on a complex vector space $V$, $\{a_t{\}}_{t\in \mathbb{R}}$ a one-parameter subgroup in a $\mathbb{Q}$-split torus in $\mathbf{G}$ and $\ddot{I}ˆ:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ a positive function on ${\mathbb{R}}_{+}$. We define a subset $S_{\ddot{I}}(\ddot{I}ˆ)$ of $\ddot{I}ˆ$-Diophantine elements in $\mathbf{G}(\mathbb{R})$ in terms of the representation $\ddot{I}$ and $\{a_t{\}}_{t\in \mathbb{R}}$, and prove formulas for the Hausdorff dimension of the complement of $S_{\ddot{I}}(\ddot{I}ˆ)$. We also discuss the connections of our results to Diophantine approximation on flag varieties and rational approximation to linear subspaces in Grassmann varieties.