Planck-scale mass equidistribution of toral Laplace eigenfunctions

TL;DR

This paper refines the understanding of how Laplace eigenfunctions on a 2D torus distribute their mass at very small scales, showing precise thresholds for equidistribution and its failure near the Planck scale.

Contribution

It provides a more precise characterization of the scale at which eigenfunction mass equidistributes or fails on the torus, improving previous results by Lester and Rudnick.

Findings

Equidistribution holds down to a small power of log above Planck scale.

Mass fails to equidistribute at a slightly smaller power of log above Planck scale.

Results rely on properties of lattice points on circles, based on Javier Cilleruelo's work.

Abstract

We study the small scale distribution of the $L^2$-mass of eigenfunctions of the Laplacian on the the two-dimensional flat torus. Given an orthonormal basis of eigenfunctions, Lester and Rudnick 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.