Quantum ergodicity for shrinking balls in arithmetic hyperbolic manifolds

We study a refinement of the quantum unique ergodicity conjecture for shrinking balls on arithmetic hyperbolic manifolds, with a focus on dimensions $2$ and $3$. For the Eisenstein series for the modular surface $\mathrm{PSL}_2( {\mathbb Z}) \backslash \mathbb{H}^2$ we prove failure of quantum unique ergodicity close to the Planck-scale and an improved bound for its quantum variance. For arithmetic $3$-manifolds we show that quantum unique ergodicity of Hecke-Maa{\ss} forms fails on shrinking balls centered on an arithmetic point and radius $R \asymp t_j^{-\delta}$ with $\delta > 3/4$. For $\mathrm{PSL}_2(\mathcal{O}_K) \setminus \mathbb{H}^3$ with $\mathcal{O}_K$ being the ring of integers of an imaginary quadratic number field of class number one, we prove, conditionally on the generalized Lindel\"of hypothesis, that equidistribution holds for Hecke-Maa{ss} forms if $\delta < 2/5$. Furthermore, we prove that equidistribution holds unconditionally for the Eisenstein series if $\delta < (1-2\theta)/(34+4\theta)$ where $\theta$ is the exponent towards the Ramanujan-Petersson conjecture. For $\mathrm{PSL}_2(\mathbb{Z}[i])$ we improve the last exponent to $\delta < (1-2\theta)/(27+2\theta)$. Studying mean Lindel\"of estimates for $L$-functions of Hecke-Maa{\ss} forms we improve the last exponent on average to $\delta < 2/5$. Finally, we study massive irregularities for Laplace eigenfunctions on $n$-dimensional compact arithmetic hyperbolic manifolds for $n \geq 4$. We observe that quantum unique ergodicity fails on shrinking balls of radii $R \asymp t^{-\delta_n+\epsilon}$ away from the Planck-scale, with $\delta_n = 5/(n+1)$ for $n \geq 5$.