  We establish sharp time decay estimates for the the Klein-Gordon equation on
the cubic lattice in dimensions $d=2,3,4$. The $\ell^1\to\ell^{\infty}$
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.
