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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-uniformlattice in $\text{PSL}(2,\mathbb R)$ and $u(t)$ is the standard unipotent groupin $\text{PSL}(2,\mathbb R)$. Under a Diophantine condition on the intial point$p$, we can prove that $\{pu(n^{1+\gamma})|n\in\mathbb N\}$ is equidistributedin $\Gamma\backslash\text{PSL}(2,\mathbb R)$ for small $\gamma>0$, whichgeneralizes a result of Venkatesh (Ann.of Math. 2010). We will computeHausdorff dimensions of subsets of non-Diophantine points in Appendix A, usingresults of lattice counting problem. In Appendix B we will use a technique ofVenkatesh (Ann.of Math. 2010) and an exponential mixing property to prove aweak version of a result of Str\"ombergsson (J Mod Dynam, 2013), which is aboutthe effective equidistribution of horospherical orbits.

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