Asymptotic Nodal Length and Log-Integrability of Toral Eigenfunctions

We study the nodal set of Laplace eigenfunctions on the flat 2d torus 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 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 T-2, based on the work of Nazarov (Algebra Anal 5:3-66, 1993; Summability of large powers of logarithm of classic lacunary series and its simplest consequences https://users.math.msu.edu/users/fedja/prepr.html, 1995).