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# On the denseness of some sparse horocycles

Cornell University - arXiv
Aug 2021

Let $\Gamma$ be a non-uniform lattice in $\textrm{PSL}(2,\mathbb R)$. 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 $p\in\textrm{PSL}(2,\mathbb R)/\Gamma$ which is not $u(t)$-periodic, the orbit $\{u(n^{1+\gamma})p:n\in\mathbb N\}$ is dense in $\textrm{PSL}(2,\mathbb R)/\Gamma$. We also prove that there exists $N\in\mathbb N$ such that for the set $\Omega$ of $N$-almost primes, the orbit $\{u(x)p:x\in\Omega\}$ is dense in $\textrm{PSL}(2,\mathbb R)/\Gamma$. Moreover, for a one-parameter unipotent flow $u(t)$ on $\textrm{PSL}(2,\mathbb R)/\textrm{PSL}(2,\mathbb Z)$, we construct points $x$ with the property that the orbit $\{u(n^2)x:n\in\mathbb N\}$ is dense in $\textrm{PSL}(2,\mathbb R)/\textrm{PSL}(2,\mathbb Z)$.

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