  We study properties of stationary determinantal point processes $\X$ on $\Z$
from different points of views. It is proved that $\X\cap \N$ is almost surely
Bohr-dense and good universal for almost everywhere convergence in $L^1$, and
that $\X$ is not syndetic but $\X +\X = \mathbb{Z}$. For the associated
centered random field, we obtain a sub-Gaussian property, a Salem-Littlewood
inequality and a Khintchine-Kahane inequality. Results can be generalized to
$\Z^d$.
