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 $\Z^d$.