CLT for Planck scale mass distribution of toral Laplace eigenfunctions

We study the fine scale $L^2$-mass distribution of toral Laplace eigenfunctions with respect to random position, in 2 and 3 dimensions. In 2d, under certain flatness assumptions on the Fourier coefficients and generic restrictions on energy levels, both the asymptotic shape of the variance is determined and the limiting Gaussian law is established, in the optimal Planck-scale regime. In 3d the asymptotic behaviour of the variance is analysed in a more restrictive scenario ("Bourgain's eigenfunctions"). Other than the said precise results, lower and upper bounds are proved for the variance, under more general flatness assumptions on the Fourier coefficients.