  In this paper we develope a quantitative non-divergence theorem for
translates of horospherical orbits, using the technique of Margulis' inequality
as developed by Eskin-Margulis-Mozes and Eskin-Margulis. As we use the
Margulis' inequality, our results do not depend on the spectral gap of the
action. As an application of our techniques, we show that given a horospherical
flow over the space of lattices, the horospherical orbit of every lattice
defined over a number field, not contained in a proper rational parabolic
subgroup is equidistributed with an effective rate.
