Normal numbers and subsets of N with given densities

Volume 144 / 1994

Haseo Ki, Tom Linton Fundamenta Mathematicae 144 (1994), 163-179 DOI: 10.4064/fm-144-2-163-179


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 $D_ξ(Π^0_α)$ iff $D_X$ is properly $D_ξ(Π^0_{1+α})$. We also show that for every nonempty set X ⊆[0,1], $D_X$ is $Π^0_3$-hard. For each nonempty $Π^0_2$ set X ⊆ [0,1], in particular for X = {x}, $D_X$ is $Π^0_3$-complete. For each n ≥ 2, the collection of real numbers that are normal or simply normal to base n is $Π^0_3$-complete. Moreover, $D_ℚ$, the subsets of ℕ with rational densities, is $D_2(Π^0_3)$-complete.


  • Haseo KiCaltech 253–37
    Pasadena, California 91125
  • Tom LintonDepartment of Mathematics
    Willamette University
    900 State Street
    Salem, Oregon 97301

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