On Non-Uniformly Discrete Orbits

We study the property of uniform discreteness within discrete orbits of non-uniform lattices in $SL_2(\mathbb{R})$, acting on $\mathbb{R}^2$ by linear transformations. We provide quantitative consequences of previous results by using Diophantine properties. We give a partial result toward a conjecture of Leli\`evre regarding the set of long cylinder holonomy vectors of the "golden L" translation surface: for any $\epsilon>0$, three points of this set can be found on a horizontal line within a distance of $\epsilon$ of each other.