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 Γ < 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 Γ-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 Γ G. The significance of the effectivized argument is due to the recent effective equidistribution results of Lindenstrauss-Mohammadi-Wang for Δ(SL₂( R)) < SL₂( R) × SL₂( R) and SL₂( R) < SL₂( C) and arithmetic lattices Γ, and future generalizations in that direction.