Asymptotic nodal length and log-integrability of toral eigenfunctions

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}.