On the variance of the nodal volume of arithmetic random waves

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