Effective mass equidistribution conjecture for holomorphic Hecke forms

Let $\Gamma\subseteq \operatorname{SL}_2(\operatorname{\mathbb{Z}})$ be a Fuchsian group of the first kind, let $f\in S_k(\Gamma)$ be a normalized Hecke cusp form, and let $\mu_f$ be its associated probability measure. Let $\widetilde{\nu}$ denote the uniform probability measure on $\Gamma\backslash\operatorname{\mathbb{H}}$, and let $\lVert\cdot\rVert_{W^{T,2}}$ be the Sobolev norm on that surface. Effective mass equidistribution conjecture. There exist $\delta>0$ and $T<\infty$ such that, for every $g\in C_0^\infty(\Gamma\backslash\operatorname{\mathbb{H}})$,

$$\left\lvert\mu_f(g)-\widetilde{\nu}(g)\right\rvert\ll k^{-\delta}\lVert g\rVert_{W^{T,2}}.$$

This strengthens mass equidistribution by requiring a polynomial rate of convergence in the weight. The paper proves such an effective statement for almost all Hecke forms in suitable short weight intervals, with any exponent $\delta<1/2$, but not for every form.