On the limiting behaviour of arithmetic toral eigenfunctions

Abstract: We consider a wide class of families $(F_m){m\in\mathbb{N}}$ of Gaussian fields on $\mathbb{T}^d=\mathbb{R}^d/\mathbb{Z}^d$ defined by [F_m:x\mapsto \frac{1}{\sqrt{|\Lambda_m|}}\sum{\lambda\in\Lambda_m}\zeta_\lambda e^{2\pi i\langle \lambda,x\rangle}] where the $\zeta_\lambda$'s are independent std. normals and $\Lambda_m$ is the set of solutions $\lambda\in\mathbb{Z}^d$ 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 $m\to+\infty$ 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 $(\lambda,\mu,\nu,\iota)$ satisfying $\lambda+\mu+\nu+\iota=0$, ($4$-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.