Asymptotic nodal length and log-integrability of toral eigenfunctions

TL;DR

This paper investigates the asymptotic behavior of the nodal length and distribution of Laplace eigenfunctions on the 2D torus, revealing equidistribution and integrability properties under certain Fourier coefficient growth conditions.

Contribution

It establishes an asymptotic law for the nodal length and proves equidistribution of the nodal set, introducing new integrability results for the doubling index of eigenfunctions.

Findings

Nodal length follows an asymptotic law

Nodal set is asymptotically equidistributed

Large powers of the doubling index are integrable

Abstract

We study the nodal set of Laplace eigenfunctions on the flat $2d$ torus $\mathbb{T}^2$. We prove an asymptotic law for the nodal length of such eigenfunctions, under some growth assumptions on their Fourier coefficients. Moreover, we show that their nodal set is asymptotically equidistributed on $\mathbb{T}^2$. The proofs are based on Bourgain's de-randomisation technique and the main new ingredient, which might be of independent interest, is the integrability of arbitrarily large powers of the doubling index of Laplace eigenfunctions on $\mathbb{T}^2$, based on the work of Nazarov \cite{N93,Nun}.