  We prove global wellposedness of the Klein-Gordon equation with power
nonlinearity $|u|^{\alpha-1}u$, where $\alpha\in\left[1,\frac{d}{d-2}\right]$,
in dimension $d\geq3$ with initial data in $M_{p, p'}^{1}(\mathbb{R}^d)\times
M_{p,p'}(\mathbb{R}^d)$ for $p$ sufficiently close to $2$. The proof is an
application of the high-low method described by Bourgain [1] where the
Klein-Gordon equation is studied in one dimension with cubic nonlinearity for
initial data in Sobolev spaces.
