On the density of some sparse horocycles

Abstract

Let $\Gamma$ be a non-uniform lattice in $\mathrm{PSL}(2,\mathbb{R})$. 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 $p\in \mathrm{PSL}(2,\mathbb{R})/\Gamma$ which is not $u(t)$-periodic, the orbit $\{u(n^{1+\gamma })p:n\in \mathbb{N}\}$ is dense in $\mathrm{PSL}(2,\mathbb{R})/\Gamma$. We also prove that there exists $N\in \mathbb{N}$ such that for the set $\Omega (N)$ of $N$-almost primes, and for any $p\in \mathrm{PSL}(2,\mathbb{R})/\Gamma$ which is not $u(t)$-periodic, the orbit $\{u(x)p:x\in \Omega (N)\}$ is dense in $\mathrm{PSL}(2,\mathbb{R})/\Gamma$.

arXiv commentCorrect some arguments in the previous version