The descriptive complexity of the set of Poisson generic numbers

Abstract

Let b ≥ 2 be an integer. We show that the set of real numbers that are Poisson generic in base b is Π0 3-complete in the Borel hierarchy of subsets of the real line. Furthermore, the set of real numbers that are Borel normal in base b and not Poisson generic in base b is complete for the class given by the differences between Π0 3 sets. We also show that the effective versions of these results hold in the effective Borel hierarchy.