On the variance of the nodal volume of arithmetic random waves

Abstract

Rudnick and Wigman (Ann. Henri Poincaré, 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 $d\ge 4$, 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 $d\ge 5$ due to the proof of the ${\ell }^2$-decoupling conjecture by Bourgain and Demeter.

13 pages