On the limiting behaviour of arithmetic toral eigenfunctions

Abstract

We consider a wide class of families (Fm) m∈N of Gaussian fields on T d = R d /Z d defined bywhere the ζ λ 's are independent standard normals and Λm is the set of solutions λ ∈ Z d to the equation p(λ) = m for some fixed elliptic polynomial p with integer coefficients. The case p(x)amounts to considering a random Laplace eigenfunction whose law is sometimes called the arithmetic random wave and has been 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 the multiples of x 2 1 + x 2 2 + x 2 3 , and a wide family of polynomials in many variables.For these three classes of polynomials, we study the (d − 1)-dimensional volume Vm of the zero set of Fm. We compute the asymptotics, as m → +∞ along certain well chosen subsequences of integers, of the expectation and variance of Vm. Moreover, we prove that in the same limit,converges to a standard normal.As in previous analogous works on this topic for the arithmetic random wave, a very general method reduces the problem of these asymptotics to the study of certain arithmetic properties of the sets of solutions to p(λ) = m. More precisely, we need to study the number of such solutions for a fixed m, as well as the number of quadruples of solutions (λ, µ, ν, ι) satisfying λ + µ + ν + ι = 0, a.k.a. 4-correlations, and the rate of convergence of the (rescaled) counting measure of Λm towards a certain limiting measure on the hypersurface {p(x) = 1}. To this end, we use many previous results on this topic but also prove a new estimate on correlations which may be of independent interest.