Linnik problem for Maass--Hecke cuspforms and effective multiplicity one theorem

We investigate two related problems concerning the dimension of joint eigenspaces of the Laplace--Beltrami operator and a finite set of Hecke operators on $\mathbb{X}=\mathrm{PGL}_2(\mathbb{Z})\backslash \mathbb{H}$. First, we consider Linnik problem for Maass--Hecke cuspforms. We prove that the dimension of such a joint eigenspace, for Maass--Hecke cuspforms with eigenparameter in $[T, T+1]$, associated to Hecke operators $T_p$ with $p < (\log T)^\alpha$ is $O_\epsilon (T^{{\frac{4}{\alpha}} + \epsilon})$. For this, we prove a new form of spectral large sieve inequality for symmetric-squares of Maass--Hecke cuspforms, by exploiting the fact that the forms under consideration are unramified at every non-archimedean place. Second, we consider the effective multiplicity one problem, determining the minimal number of Hecke eigenvalues needed to distinguish two Maass--Hecke cusp forms with the same Laplace eigenvalue. We prove that for any fixed $\eta>0$, if two Maass--Hecke cuspforms, with eigenparameter $t$, share Hecke eigenvalues $\lambda_{\phi_1}(n) = \lambda_{\phi_2}(n)$ for all $n < \eta t$, and $t$ is sufficiently large, then the forms are proportional. This improves the previously known best bound due to Huntley in 1991. Key ingredient for the improvement is the result by Brook and Lindenstrauss that classifies quantum limits of a joint eigenfunction of a Hecke operator and the Laplace--Beltrami operator on arithmetic hyperbolic surfaces. We also discuss generalizations of these results to Maass--Hecke cuspforms on $\mathrm{PGL}_2$ over arbitrary number field.