Some properties of stationary determinantal point processes on $\mathbb{Z}$

We study properties of stationary determinantal point processes $\text{X}$ on $\mathbb{Z}$ from different points of views. It is proved that $\text{X}\cap \mathbb{N}$ is almost surely Bohr-dense and good universal for almost everywhere convergence in $L^1$, and that $\text{X}$ is not syndetic but $\text{X}+\text{X}=\mathbb{Z}$. 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 ${\mathbb{Z}}^d$.