Planck-Scale Mass Equidistribution of Toral Laplace Eigenfunctions

We study the small scale distribution of the L 2 -mass of eigenfunctions of the Laplacian on the two-dimensional flat torus. Given an orthonormal basis of eigenfunctions, Lester and Rudnick (Commun. Math. Phys. 350(1):279–300, 2017 ) showed the existence of a density one subsequence whose L 2 -mass equidistributes more-or-less down to the Planck scale. We give a more precise version of their result showing equidistribution holds down to a small power of log above Planck scale, and also showing that the L 2 -mass fails to equidistribute at a slightly smaller power of log above the Planck scale. This article rests on a number of results about the proximity of lattice points on circles, much of it based on foundational work of Javier Cilleruelo.