Normal numbers and completeness results for difference sets

Abstract

We consider some natural sets of real numbers arising in ergodic theory and show that they are, respectively, complete in the classes $\mathcal D_2 (\mathbf\Pi^0_3)$ and $\mathcal D_\omega (\mathbf \Pi^0_3)$, that is, the class of sets which are 2-differences (respectively, $\omega$-differences) of $\mathbf \Pi^0_3$ sets.