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

Abstract

Consider $G=\mathrm{SL}_{ d }({\mathbb {R}})$ and $\Gamma =\mathrm{SL}_{ d }({\mathbb {Z}})$. It was recently shown by the second-named author (Shi, Pointwise equidistribution for one parameter diagonalizable group action on homogeneous space (preprint), arXiv:1405.2067, 2014) 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 $G/\Gamma$ are equidistributed with respect to continuous compactly supported functions $\varphi$ on $G/\Gamma$. 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 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.