On the density of some sparse horocycles

Abstract: Let $\Gamma$ be a non-uniform lattice in $\operatorname{PSL}(2,\mathbb R)$. In this note, we show that there exists a constant $\gamma_0>0$ such that for any $0<\gamma<\gamma_0$, any one-parametrer unipotent subgroup ${u(t)}_{t\in\mathbb R}$ and any $p\in\operatorname{PSL}(2,\mathbb R)/\Gamma$ which is not $u(t)$-periodic, the orbit ${u(n^{1+\gamma})p:n\in\mathbb N}$ is dense in $\operatorname{PSL}(2,\mathbb R)/\Gamma$. We also prove that there exists $N\in\mathbb N$ such that for the set $\Omega(N)$ of $N$-almost primes, and for any $p\in\operatorname{PSL}(2,\mathbb R)/\Gamma$ which is not $u(t)$-periodic, the orbit ${u(x)p:x\in\Omega(N)}$ is dense in $\operatorname{PSL}(2,\mathbb R)/\Gamma$.