Density of orbits of horocycle flows at sub-quadratic polynomial times

Let $\mathrm{\Gamma }\subset PSL(2,\mathbb{R})$ be such that the space $X=\mathrm{\Gamma }\mathrm{\backslash }PSL(2,\mathbb{R})$ is not compact. Let $(h_t)$ be the horocycle flow acting on $X$. We show that for every $x\in X$ that is not periodic for $(h_t)$ and for every $\delta \in (0,1)$ the orbit $\{h_{n^{2-\delta }}x{\}}_{n\in \mathbb{N}}$ is dense in $X$. Assuming additionally the Hardy-Littlewood conjecture we show that for every non-periodic $x\in X$, $\{h_{p}x{\}}_{p-\text{prime}}$ is dense in $X$. Finally we show that for $\mathrm{\Gamma }=PSL(2,\mathbb{Z})$, $\{h_{n^2}{y}_q{\}}_{n<q}$ equidistribute, as $q\to \mathrm{\infty }$ along primes congruent to $1(\mathrm{m}\mathrm{o}\mathrm{d}4)$, towards Haar measure, where $\{y_q\}$ is a sequence of periodic points of period $q$.