Quantitative equidistribution properties of toral eigenfunctions

Abstract

We prove quantitative equidistribution properties for orthonormal bases of eigenfunctions of the Laplacian on the rational $d$-torus. We show that the rate of equidistribution of such eigenfunctions is of polynomial decay. We also prove that equidistribution of eigenfunctions holds for symbols supported in balls with a radius shrinking at a polynomial rate.

This article is based on the appendix of our previous preprint: arXiv:1411.4078. We have included improvements and have simplified the proofs (no semiclassical/microlocal techniques are necessary)