On equidistribution of polynomial sequences in quotients of $PSL_2(\mathbb{R})$

Abstract

In this paper, it is shown that for every lattice $\Gamma \subset PSL_2(\mathbb{R})$ there exists a $c>0$ such that for any $0 \leq \gamma<c$ the sequence $p h(n^{1+\gamma})$ equidistributes for any $p \in \Gamma \backslash PSL_2(\mathbb{R})$, 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 $\{p h(t), t \leq T \}$ by closed horocycles of small period.