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

Abstract

Consider $G=\mathrm{SL}_{d}(\mathbb R)$ and $Γ=\mathrm{SL}_{d}(\mathbb Z)$. It was recently shown by the second-named author that for some diagonal subgroups $\{g_t\}\subset G$ and unipotent subgroups $U\subset G$, $g_t$-trajectories of almost all points on all $U$-orbits on $G/Γ$ are equidistributed with respect to continuous compactly supported functions $\varphi$ on $G/Γ$. In this paper we strengthen this result in two directions: by exhibiting an error rate of equidistribution when $\varphi$ is smooth and compactly supported, and by proving equidistribution with respect to certain unbounded functions, namely Siegel transforms of Riemann integrable functions on $\mathbb{R}^d$. For the first part we use a method based on effective double equidistribution of $g_t$-translates of $U$-orbits, which generalizes an earlier result of Kleinbock and Margulis. 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, are derived using the equidistribution result.