Normal numbers and completeness results for difference sets

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

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