Abstract
Consider and . It was recently shown by the second-named author (Shi, Pointwise equidistribution for one parameter diagonalizable group action on homogeneous space (preprint), 2014) that for some diagonal subgroups and unipotent subgroups , -trajectories of almost all points on all U-orbits on are equidistributed with respect to continuous compactly supported functions on . In this paper we strengthen this result in two directions: by exhibiting an error rate of equidistribution when is smooth and compactly supported, and by proving equidistribution with respect to certain unbounded functions, namely Siegel transforms of Riemann integrable functions on . For the first part we use a method based on effective double equidistribution of -translates of U-orbits, which generalizes the main result of Kleinbock and Margulis (On effective equidistribution of expanding translates of certain orbits in the space of lattices, Number theory, analysis and geometry 385-396, 2012). The second part is based on Schmidt's results on counting of lattice points. Number-theoretic consequences involving spiraling of lattice approximations, extending recent work of Athreya et al. (J Lond Math Soc 91(2):383-404, 2015), are derived using the equidistribution result.