Sharp time decay estimates for the discrete Klein-Gordon equation

We establish sharp time decay estimates for the the Klein-Gordon equation on the cubic lattice in dimensions $d = 2, 3, 4$. The $l^1\mapsto l^\infty$ dispersive decay rate is $|t|-3/4$ for $d = 2$, $|t|-7/6$ for $d = 3$ and $|t|-3/2 \log |t|$ for $d = 4$. These decay rates are faster than conjectured by Kevrekidis and Stefanov (2005). The proof relies on oscillatory integral estimates and proceeds by a detailed analysis of the the singularities of the associated phase function. We also prove new Strichartz estimates and discuss applications to nonlinear PDEs and spectral theory.