Equidistribution of holomorphic cusp forms on thin sets

Abstract

We find some equidistribution results connected to restriction quantum unique ergodicity problem in this paper. We shows that \begin{align*} \frac{1}{|\mathcal{B}_k|}\sum_{f\in \mathcal{B}_k} \int_{R}y^{k}|f(z)|^{2}\psi(z) d\mu_{R}(z)\to \frac{3}{\pi}\int_{R}\psi(z) d\mu_{R}(z) \end{align*} where $R$ is some subset of $\mathbb{H}$, $\psi$ is a nice function relative to $R$, $d\mu_{R}(z)$ is a suitable measure on $R$, and $\mathcal{B}_k$ is an orthonormal basis of the cusp forms for group $\Gamma$ with respect to weight $k$.