Quantitative equidistribution properties of toral eigenfunctions

TL;DR

This paper establishes that eigenfunctions of the Laplacian on the rational torus become uniformly distributed at a polynomial rate, with implications for symbols supported in shrinking regions.

Contribution

It provides the first quantitative polynomial decay rates for equidistribution of toral eigenfunctions and extends results to symbols supported in shrinking balls.

Findings

Eigenfunctions exhibit polynomial decay in equidistribution rate.

Equidistribution holds for symbols supported in shrinking balls at polynomial rates.

Quantitative bounds improve understanding of eigenfunction distribution on tori.

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.