Geodesic
Normal numbers and subsets of N with given densities
Haseo Ki1 ; Tom Linton2
1 Caltech 253–37 Pasadena, California 91125 U.S.A.
2 Department of Mathematics Willamette University 900 State Street Salem, Oregon 97301 U.S.A.
Fundamenta Mathematicae, Tome 144 (1994) no. 2, pp. 163-179.

Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences

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.
DOI : 10.4064/fm-144-2-163-179

Haseo Ki 1 ; Tom Linton 2

1 Caltech 253–37 Pasadena, California 91125 U.S.A.
2 Department of Mathematics Willamette University 900 State Street Salem, Oregon 97301 U.S.A. 
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Haseo Ki; Tom Linton. Normal numbers and subsets of N with given densities. Fundamenta Mathematicae, Tome 144 (1994) no. 2, pp. 163-179. doi : 10.4064/fm-144-2-163-179. http://geodesic.mathdoc.fr/articles/10.4064/fm-144-2-163-179/

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