On the density of some sparse horocycles

Abstract

Let $\Gamma$ be a non-uniform lattice in ${\text {PSL}}(2,\mathbb R)$. Here, we show that there exists a constant $\gamma _0>0$ such that for any $0<\gamma <\gamma _0$, any one-parameter unipotent subgroup $\{u(t)\}_{t\in {\mathbb {R}}}$ and any $p\in {\text {PSL}}(2,\mathbb R)/\Gamma$ which is not u(t)-periodic, the orbit $\{u(n^{1+\gamma })p:n\in {\mathbb {N}}\}$ is dense in ${\text {PSL}}(2,\mathbb R)/\Gamma$. We also prove that there exists $N\in {\mathbb {N}}$ such that for the set $\Omega (N)$ of N-almost primes, and for any $p\in {\text {PSL}}(2,{\mathbb {R}})/\Gamma$ which is not u(t)-periodic, the orbit $\{u(x)p:x\in \Omega (N)\}$ is dense in ${\text {PSL}}(2,\mathbb R)/\Gamma$.