### Normal numbers and subsets of N with given densities

Haseo Ki, Tom Linton (1994)

Fundamenta Mathematicae

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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,...