On the density of some sparse horocycles

TL;DR

This paper proves the density of certain sparse horocycle orbits in non-uniform lattices of PSL(2,R), extending classical results to sparser sequences and almost primes.

Contribution

It establishes new density results for horocycle orbits along polynomial and almost prime sequences in non-uniform lattices.

Findings

Sparse polynomial sequences yield dense horocycle orbits.

Orbits along almost primes are dense in the homogeneous space.

Results extend classical horocycle density to sparser and arithmetic sequences.

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 $\operatorname{PSL}(2,\mathbb R)/\Gamma$.