Equidistribution of holomorphic cusp forms on thin sets
Qingfeng Sun, Qizhi Zhang
TL;DR
The paper investigates shrinking-scale equidistribution (rQUE) for holomorphic cusp forms by averaging over an orthonormal basis of weight‑k cusp forms and using the Bergman kernel as a core tool. By deriving precise bulk and elliptic-point asymptotics for the normalized Bergman kernel and applying a pre-trace formula, it establishes that the averaged measures converge to the ambient measure with the universal constant $3/π$, on vertical geodesics, horizontal geodesics, and the full modular surface. This approach avoids $L$-function techniques and leverages the group action of Γ on the upper half‑plane to obtain unconditional average equidistribution results. The work thereby confirms conjectured rQUE behavior in an averaged sense and provides explicit error terms tied to $k$, elliptic points, and geometric regions.
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}ψ(z) dμ_{R}(z)\to \frac{3}π\int_{R}ψ(z) dμ_{R}(z) \end{align*} where $R$ is some subset of $\mathbb{H}$, $ψ$ is a nice function relative to $R$, $dμ_{R}(z)$ is a suitable measure on $R$, and $\mathcal{B}_k$ is an orthonormal basis of the cusp forms for group $Γ$ with respect to weight $k$.