Non-Concentration of Primes in $\Gamma \backslash PSL_2(\mathbb{R})$

This paper generalizes the result of Sarnak and Ubis \cite{sarnak-ubis} about non-concentration of primes in horocycle orbits on $PSL_2(\mathbb{Z}) \backslash PSL_2(\mathbb{R})$ to any lattice in $PSL_2(\mathbb{R})$. The proof combines the asymptotic result of Str\"ombergsson \parencite{strombergsson} and Venkatesh's method \parencite{venkatesh} with the approach of Sarnak and Ubis of approximating horocycle pieces with periodic horocycles. The key step is to establish a dichotomy between $\{\xi h(t), t \in [0, T] \}$ having good equidistribution in $\Gamma \backslash PSL_2(\mathbb{R})$ and it being approximable by closed horocycle pieces with small period. In a follow-up paper, a similar approach will be used to show equidistribution of $\xi h(n^{1+\gamma})$ for small $\gamma>0$, generalizing Venkatesh's result \parencite{venkatesh} to non-compact $\Gamma$.