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 $\{pu(n^{1+γ})|n\in\mathbb N\}$ in $Γ\backslash\text{PSL}(2,\mathbb R)$, where $Γ$ 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 $\{pu(n^{1+γ})|n\in\mathbb N\}$ is equidistributed in $Γ\backslash\text{PSL}(2,\mathbb R)$ for small $γ>0$, 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 Strömbergsson (J Mod Dynam, 2013), which is about the effective equidistribution of horospherical orbits.

33 pages. The main part of the paper (without appendices) will appear in Journal of Modern Dynamics