Normal numbers and subsets of N with given densities

For X ⊆ [0,1], let ${\displaystyle D_X}$ denote the collection of subsets of ℕ whose densities lie in X. Given the exact location of X in the Borel or difference hierarchy, we exhibit the exact location of ${\displaystyle D_X}$. For α ≥ 3, X is properly ${\displaystyle D_{\xi }({\Pi }^0\_\alpha )}$ iff ${\displaystyle D_X}$ is properly ${\displaystyle D_{\xi }({\Pi }^0\_\{1+\alpha \})}$. We also show that for every nonempty set X ⊆[0,1], ${\displaystyle D_X}$ is ${\displaystyle {\Pi }^0\_3}$-hard. For each nonempty ${\displaystyle {\Pi }^0\_2}$ set X ⊆ [0,1], in particular for X = {x}, ${\displaystyle D_X}$ is ${\displaystyle {\Pi }^0\_3}$-complete. For each n ≥ 2, the collection of real numbers that are normal or simply normal to base n is ${\displaystyle {\Pi }^0\_3}$-complete. Moreover, ${\displaystyle D_{\mathbb{Q}}}$, the subsets of ℕ with rational densities, is ${\displaystyle D_2({\Pi }^0\_3)}$-complete.