Equidistribution of holomorphic cusp forms on thin sets

TL;DR

This paper establishes equidistribution results for holomorphic cusp forms on certain subsets of the upper half-plane, contributing to the understanding of restriction quantum unique ergodicity for these forms.

Contribution

It provides new equidistribution results for holomorphic cusp forms on thin sets, advancing the study of quantum ergodicity in this context.

Findings

Average of weighted cusp form squares converges to a measure integral

Results connect to restriction quantum unique ergodicity

Shows equidistribution on specific subsets of the upper half-plane

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 $k$.