Asymptotic Nodal Length and Log-Integrability of Toral Eigenfunctions

We study the nodal set of Laplace eigenfunctions on flat 2d torus $${\mathbb {T}}^2$$ . prove an asymptotic law for length such eigenfunctions, under some growth assumptions their Fourier coefficients. Moreover, we show that is asymptotically equidistributed The proofs are based Bourgain’s de-randomisation technique and main new ingredient, which might be independent interest, integrability arbitrarily large powers doubling index , work Nazarov (Algebra Anal 5:3–66, 1993; Summability logarithm classic lacunary series its simplest consequences https://users.math.msu.edu/users/fedja/prepr.html 1995).