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 $\Gamma$.
Categories: math.NT math.DS