Normal numbers and subsets of N with given densities

Ki, Haseo; Linton, Tom

Ki, Haseo ; Linton, Tom

Fundamenta Mathematicae, Tome 144 (1994), p. 163-179 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $D_{ξ} (Π_{α}^{0})$ iff $D_{X}$ is properly $D_{ξ} (Π_{1 + α}^{0})$ . We also show that for every nonempty set X ⊆[0,1], $D_{X}$ is $Π_{3}^{0}$ -hard. For each nonempty $Π_{2}^{0}$ set X ⊆ [0,1], in particular for X = x, $D_{X}$ is $Π_{3}^{0}$ -complete. For each n ≥ 2, the collection of real numbers that are normal or simply normal to base n is $Π_{3}^{0}$ -complete. Moreover, $D_{ℚ}$ , the subsets of ℕ with rational densities, is $D_{2} (Π_{3}^{0})$ -complete.

Publié le : 1994-01-01

Zbl 0809.04001

EUDML-ID : urn:eudml:doc:212021

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     author = {Haseo Ki and Tom Linton},
     title = {Normal numbers and subsets of N with given densities},
     journal = {Fundamenta Mathematicae},
     volume = {144},
     year = {1994},
     pages = {163-179},
     zbl = {0809.04001},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-fmv144i2p163bwm}
}

Ki, Haseo; Linton, Tom. Normal numbers and subsets of N with given densities. Fundamenta Mathematicae, Tome 144 (1994) pp. 163-179. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-fmv144i2p163bwm/

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