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Consider $G=\SL_{ d }(\mathbb R)$ and $ \Gamma=\SL_{ d }(\mathbb Z)$. It wasrecently shown by the second-named author \cite{s} that for some diagonalsubgroups $\{g_t\}\subset G$ and unipotent subgroups $U\subset G$,$g_t$-trajectories of almost all points on all $U$-orbits on $G/\Gamma$ areequidistributed with respect to continuous compactly supported functions$\varphi$ on $G/\Gamma$. In this paper we strengthen this result in twodirections: by exhibiting an error rate of equidistribution when $\varphi$ issmooth and compactly supported, and by proving equidistribution with respect tocertain unbounded functions, namely Siegel transforms of Riemann integrablefunctions on $\R^d$. For the first part we use a method based on effectivedouble equidistribution of $g_t$-translates of $U$-orbits, which generalizesthe main result of \cite{km12}. The second part is based on Schmidt's resultson counting of lattice points. Number-theoretic consequences involvingspiraling of lattice approximations, extending recent work of Athreya, Ghoshand Tseng \cite{agt1}, are derived using the equidistribution result.

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