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