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In this paper we develope a quantitative non-divergence theorem fortranslates of horospherical orbits, using the technique of Margulis' inequalityas developed by Eskin-Margulis-Mozes and Eskin-Margulis. As we use theMargulis' inequality, our results do not depend on the spectral gap of theaction. As an application of our techniques, we show that given a horosphericalflow over the space of lattices, the horospherical orbit of every latticedefined over a number field, not contained in a proper rational parabolicsubgroup is equidistributed with an effective rate.

Margulis' Inequality for translates of horospherical orbits and applications to equidistribution