Sharp Eigenfunction Bounds on the Torus for large $p$

We prove the discrete restriction conjecture holds with no loss when $p>\frac{2d}{d-4}$ and $d\ge 5$. That is, we show optimal $L^p$ bounds for eigenfunctions of the Laplacian on the square torus for large values of $p$. This improves the results of Bourgain and Demeter. Our proof method is a refinement of the circle method approach previously used to establish results with a subpolynomial loss. This represents the first sharp $L^p$ bounds for eigenfunctions on the torus since the work of Cooke and Zygmund. We present applications to bounds for spectral projectors and the additive energy of integer lattice points on higher dimensional spheres. These results are similarly sharp. We also prove results with a logarithmic loss that hold in a wider range of $p$.