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 \[F_m:x\mapsto \frac{1}{\sqrt{|Λ_m|}}\sum_{λ\inΛ_m}ζ_λe^{2πi\langle λ,x\rangle}\] 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 $p(x)=x_1^2+\cdots +x_d^2$ is a random Laplace eigenfunction whose law is sometimes called the $\text{arithmetic random wave}$, 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.

References in corpus (1)

• The angular distribution of integral ideal numbers with a fixed norm in quadratic extensions