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

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 \({\text {SL}}_3({\mathbb {R}})/{\text {SL}}_3({\mathbb {Z}})\) ${\text{SL}}_3(\mathbb{R})/{\text{SL}}_3(\mathbb{Z})$ . 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 \(\gamma \ge 0\) $\gamma \ge 0$ .