On the limiting behaviour of arithmetic toral eigenfunctions

We consider a wide class of families $(F_m)_{minmathbb{N}}$ of Gaussian fields on $mathbb{T}^d=mathbb{R}^d/mathbb{Z}^d$ defined by [F_m:xmapsto frac{1}{sqrt{|Lambda_m|}}sum_{lambdainLambda_m}zeta_lambda e^{2pi ilangle lambda,xrangle}] where the $zeta_lambda$s are independent std. normals and $Lambda_m$ is the set of solutions $lambdainmathbb{Z}^d$ to $p(lambda)=m$ for a fixed elliptic polynomial $p$ with integer coefficients. The case $p(x)=x_1^2+dots+x_d^2$ is a random Laplace eigenfunction whose law is sometimes called the $textit{arithmetic random wave}$, studied in the past by many authors. In contrast, we consider three classes of polynomials $p$: a certain family of positive definite quadratic forms in two variables, all positive definite quadratic forms in three variables except multiples of $x_1^2+x_2^2+x_3^2$, and a wide family of polynomials in many variables. For these classes of polynomials, we study the $(d-1)$-dimensional volume $mathcal{V}_m$ of the zero set of $F_m$. We compute the asymptotics, as $mto+infty$ along certain sequences of integers, of the expectation and variance of $mathcal{V}_m$. Moreover, we prove that in the same limit, $frac{mathcal{V}_m-mathbb{E}[mathcal{V}_m]}{sqrt{text{Var}(mathcal{V}_m)}}$ converges to a std. normal. As in previous works, one reduces the problem of these asymptotics to the study of certain arithmetic properties of the sets of solutions to $p(lambda)=m$. We need to study the number of such solutions for fixed $m$, the number of quadruples of solutions $(lambda,mu, u,iota)$ satisfying $lambda+mu+ u+iota=0$, ($4$-correlations), and the rate of convergence of the counting measure of $Lambda_m$ towards a certain limiting measure on the hypersurface ${p(x)=1}$. To this end, we use prior results on this topic but also prove a new estimate on correlations, of independent interest.