On the variance of the nodal volume of arithmetic random waves

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 $Etoinfty$, 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 $dgeq 4$, 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 $dgeq 5$ due to the proof of the $ell^{2}$-decoupling conjecture by Bourgain and Demeter.