On equidistribution of polynomial sequences in quotients of PSL 2 ( ℝ )
Lauritz Streck · Ergodic Theory and Dynamical Systems · 2025
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 , 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 [Sparse equidistribution problems, period bounds, and subconvexity. Ann. of Math. (2) 172 (2) (2010), 989–1094], who established the same equidistribution for co-compact lattices. The proof uses a dichotomy between good equidistribution estimates and approximability of $\{p h(t), t \leq T \}$ by closed horocycles of small period.