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We study properties of stationary determinantal point processes   X   on   Z   from different points of views. It is proved that    X  ∩  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  =  Z   . For the associated centered random field, we obtain a sub‐Gaussian property, a Salem–Littlewood inequality and a Khintchine–Kahane inequality.

Some properties of stationary determinantal point processes on Z