Bounds for spectral projectors on the three-dimensional torus

We study $L^2$ to $L^p$ operator norms of spectral projectors for the Euclidean Laplacian on the torus in the case where the spectral window is narrow. With a window of constant size this is a classical result of Sogge; in the small-window limit we are left with $L^p$ norms of eigenfunctions of the Laplacian, as considered for instance by Bourgain. For the three-dimensional torus we prove new cases of a previous conjecture of the first two authors concerning the size of these norms; we also refine certain prior results to remove $\epsilon$-losses in all dimensions. We use methods from number theory: the geometry of numbers, the circle method and exponential sum bounds due to Guo. We complement these techniques with height splitting and a bilinear argument to prove sharp results. We exposit on the various techniques used and their limitations.