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For X ⊆ [0,1], let $D_X$ 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 $D_X$. For α ≥ 3, X is properly $D_{\xi }\left({\Pi }_{\alpha }^0\right)$ iff $D_X$ is properly $D_{\xi }\left({\Pi }_{1+\alpha }^0\right)$. We also show that for every nonempty set X ⊆[0,1], $D_X$ is ${\Pi }_3^0$-hard. For each nonempty ${\Pi }_2^0$ set X ⊆ [0,1], in particular for X = x, $D_X$ is ${\Pi }_3^0$-complete. For each n ≥ 2, the collection of real numbers that are normal or simply normal to base n is ${\Pi }_3^0$-complete. Moreover, $D_{\mathbb{Q}}$, the...