Sharp Eigenfunction Bounds on the Torus for large $p$

TL;DR

This paper establishes optimal $L^p$ bounds for Laplacian eigenfunctions on the torus for large $p$, improving previous results and applying circle method refinements.

Contribution

It proves the discrete restriction conjecture with no loss for $p > 2d/(d-4)$ on the torus, providing the first sharp bounds since Cooke and Zygmund.

Findings

Proves sharp $L^p$ bounds for eigenfunctions on the torus for large $p$

Improves upon Bourgain and Demeter's results

Provides applications to spectral projectors and lattice point energy

Abstract

We prove the discrete restriction conjecture holds with no loss when $p>\frac{2d}{d-4}$ and $d\geq 5$. 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 $p$.