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

TL;DR

This paper investigates the shrinking-target problem in the space of unimodular lattices in three dimensions, providing an explicit formula for the Hausdorff dimension of points with orbits approaching the cusp at a certain rate.

Contribution

It establishes a precise Hausdorff dimension formula for points with orbits approaching infinity in the space of unimodular lattices under a diagonal flow.

Findings

Derived an explicit Hausdorff dimension formula for the shrinking-target set.

Analyzed the behavior of orbits under diagonal flows in SL(3,R)/SL(3,Z).

Connected orbit behavior to geometric properties of the 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 $\operatorname{SL}_3(\mathbb R)/\operatorname{SL}_3(\mathbb Z)$. 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 $\gamma\geq0$.