Translate of horospheres and counting problems (Q2927716)

Let \(G\) be a higher rank semisimple Lie group and \(\Gamma\) be a nonuniform lattice. In this paper, the authors give necessary and sufficient conditions for a sequence of translations of a closed horospherical orbit in \(G/\Gamma\) to be equidistributed (in the limit) in a homogeneous closed subset. They further use Ratner's theorem to prove similar properties for translations of a measure which is absolutely continuous with respect to a horospherical measure. Finally, they obtain two different types of applications. One is a geometric result on locally symmetric finite volume orbifolds generalizing a result of \textit{A. Eskin} and \textit{C. McMullen} [Duke Math. J. 71, No. 1, 181--209 (1993; Zbl 0798.11025)]. The other application is on counting rational points on a flag variety with respect to any metrized line bundle.