Sparse Equidistribution of Unipotent Orbits in Finite-Volume Quotients of $\text{PSL}(2,\mathbb R)$, with appendices

Abstract: In this note, we consider the orbits ${pu(n^{1+\gamma})|n\in\mathbb N}$ in $\Gamma\backslash\text{PSL}(2,\mathbb R)$, where $\Gamma$ is a non-uniform lattice in $\text{PSL}(2,\mathbb R)$ and $u(t)$ is the standard unipotent group in $\text{PSL}(2,\mathbb R)$. Under a Diophantine condition on the intial point $p$, we can prove that ${pu(n^{1+\gamma})|n\in\mathbb N}$ is equidistributed in $\Gamma\backslash\text{PSL}(2,\mathbb R)$ for small $\gamma>0$, which generalizes a result of Venkatesh (Ann.of Math. 2010). We will compute Hausdorff dimensions of subsets of non-Diophantine points in Appendix A, using results of lattice counting problem. In Appendix B we will use a technique of Venkatesh (Ann.of Math. 2010) and an exponential mixing property to prove a weak version of a result of Str\"ombergsson (J Mod Dynam, 2013), which is about the effective equidistribution of horospherical orbits.