On the variance of the nodal volume of arithmetic random waves

Abstract

Rudnick and Wigman (Ann. Henri Poincar\'{e}, 2008; arXiv:math-ph/0702081) conjectured that the variance of the volume of the nodal set of arithmetic random waves on the $d$-dimensional torus is $O(E/\mathcal{N})$, as $E\to\infty$, where $E$ is the energy and $\mathcal{N}$ is the dimension of the eigenspace corresponding to $E$. Previous results have established this with stronger asymptotics when $d=2$ and $d=3$. In this brief note we prove an upper bound of the form $O(E/\mathcal{N}^{1+\alpha(d)-\epsilon})$, for any $\epsilon>0$ and $d\geq 4$, where $\alpha(d)$ is positive and tends to zero with $d$. The power saving is the best possible with the current method (up to $\epsilon$) when $d\geq 5$ due to the proof of the $\ell^{2}$-decoupling conjecture by Bourgain and Demeter.