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In this paper, we study the shrinking-target problem with target at infinityinduced by the injectivity radius function under the action of a regulardiagonalizable flow on $\operatorname{SL}_3(\mathbbR)/\operatorname{SL}_3(\mathbb Z)$. In particular, we establish an explicitformula for the Hausdorff dimension of the subset of points $p$ whose orbitapproaches the cusp infinitely often with a rate $\gamma\geq0$.

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