Pointwise equidistribution with an error rate and with respect to unbounded functions

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Consider G=(R) and Γ=(Z). It was recently shown by the second-named author s that for some diagonal subgroups {gₜ}⊂ G and unipotent subgroups U⊂ G, gₜ-trajectories 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 ᵈ. For the first part we use a method based on effective double equidistribution of gₜ-translates of U-orbits, which generalizes the main result of km12. 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 agt1, are derived using the equidistribution result.