  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}.

Spectral Theory;Analysis of PDEs;Number Theory