A shrinking-target problem in the space of unimodular lattices in the three dimensional Euclidean space

Abstract

In this paper, we study the shrinking-target problem with target at infinity induced by the injectivity radius function under the action of a regular diagonalizable flow on ${\mathrm{SL}}_3(\mathbb{R})/{\mathrm{SL}}_3(\mathbb{Z})$. In particular, we establish an explicit formula for the Hausdorff dimension of the subset of points $p$ whose orbit approaches the cusp infinitely often with a rate $\gamma \ge 0$.