Finitary estimates for the distribution of lattice orbits in homogeneous spaces I: Riemannian metric

Let H < G both be noncompact connected semisimple real algebraic groups and < G be a lattice. Building on the work of Gorodnik-Weiss, we refine their techniques and obtain effective results. More precisely, we prove effective convergence of the distribution of dense -orbits in G/H to some limiting density on G/H assuming effective equidistribution of regions of maximal horospherical orbits under one-parameter diagonal flows inside a dense H-orbit in G. The significance of the effectivized argument is due to the recent effective equidistribution results of Lindenstrauss-Mohammadi-Wang for (SL₂ (R) ) < SL₂ (R) SL₂ (R) and SL₂ (R) < SL₂ (C) and arithmetic lattices, and future generalizations in that direction.