  A closed horocycle $\mathcal{U}$ on $SL_N(\mathbb{Z}) \backslash
SL_N(\mathbb{R})/SO_N({\mathbb{R}})$ has many lifts to the universal cover
$SL_N(\mathbb{R})/SO_N({\mathbb{R}})$. Under some conditions on the horocycle,
we give a precise asymptotic count of its lifts of bounded distance away from a
given base point in the universal cover. This partially generalizes previous
work of Mohammadi--Golsefidy.
