Uniform Discreteness of Discrete Orbits of Non-Uniform Lattices in $SL_2(\mathbb{R})$

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 a new proof of the conditions under which the orbit of a non-uniform lattice in $SL_2(\mathbb{R})$ is uniformly discrete, by using Diophantine properties. Our results include a detailed analysis of the asymptotic behavior of the error terms. Focusing on a specific group $\Gamma$ and a discrete orbit of it, $S$, the main result of this paper is that for any $\epsilon>0$, three points in $S$ can be found on a horizontal line within distance $\epsilon$ of each other. This gives a partial result toward a conjecture of Leli\`evre. The set $S$ and group $\Gamma$ are respectively the set of long cylinder holonomy vectors, and Veech group, of the "golden L" translation surface.