On equidistribution of polynomial sequences in quotients of $PS{L}_2(\mathbb{R})$

In this paper, it is shown that for every lattice $\mathrm{\Gamma }\subset PS{L}_2(\mathbb{R})$ there exists a $c>0$ such that for any $0\le \gamma <c$ the sequence $ph(n^{1+\gamma })$ equidistributes for any $p\in \mathrm{\Gamma }\mathrm{\backslash }PS{L}_2(\mathbb{R})$, where $h$ is the horocycle flow. This makes modest progress towards a conjecture of Shah and generalizes a result of Venkatesh (arXiv:math/0506224), who established the same equidistribution for co-compact lattices. The proof utilizes a dichotomy between good equidistribution estimates and approximability of $\{ph(t),t\le T\}$ by closed horocycles of small period.