The descriptive complexity of the set of Poisson generic numbers

Let [Formula: see text] be an integer. We show that the set of real numbers that are Poisson generic in base [Formula: see text] is [Formula: see text]-complete in the Borel hierarchy of subsets of the real line. Furthermore, the set of real numbers that are Borel normal in base [Formula: see text] and not Poisson generic in base [Formula: see text] is complete for the class given by the differences between [Formula: see text] sets. We also show that the effective versions of these results hold in the effective Borel hierarchy.