Pointwise Equidistribution with an Error Rate and with Respect to Unbounded Functions

Consider G = SLd(R) and Γ = SLd(Z). It was recently shown by the second-named author [21] that for some diagonal subgroups {gt} ⊂ G and unipotent subgroups U ⊂ G, gttrajectories of almost all points on all U -orbits on G/Γ are equidistributed with respect to continuous compactly supported functions φ on G/Γ. In this paper we strengthen this result in two directions: by exhibiting an error rate of equidistribution when φ is smooth and compactly supported, and by proving equidistribution with respect to certain unbounded functions, namely Siegel transforms of Riemann integrable functions on R. For the first part we use a method based on effective double equidistribution of gttranslates of U -orbits, which generalizes the main result of [13]. 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, Ghosh and Tseng [1], are derived using the equidistribution result.