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Let $b\ge 2$ be an integer. We show that the set of real numbers that arePoisson generic in base $b$ is $\boldsymbol{\Pi}^0_3$-complete in the Borelhierarchy of subsets of the real line. Furthermore, the set of real numbersthat are Borel normal in base $b$ and not Poisson generic in base $b$ iscomplete for the class given by the differences between $\boldsymbol{\Pi}^0_3$sets. We also show that the effective versions of these results hold in theeffective Borel hierarchy.

The descriptive complexity of the set of Poisson generic numbers