A note on Diophatine approximation in ${\mathrm{S}\mathrm{L}}_2(\mathbb{R})$

We prove a quantitative version of the following statement: the unipotent flow orbit of a typical lattice in ${\mathrm{S}\mathrm{L}}_2(\mathbb{R})\mathrm{/}{\mathrm{S}\mathrm{L}}_2(\mathbb{Z})$ is dense. Our quantitative result uses A. Weil’s bounds for Kloostermann sums.