Normal numbers and subsets of N with given densities
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.
@article{10_4064_fm_144_2_163_179,
author = {Haseo Ki and Tom Linton},
title = {Normal numbers and subsets of {N} with given densities},
journal = {Fundamenta Mathematicae},
pages = {163--179},
publisher = {mathdoc},
volume = {144},
number = {2},
year = {1994},
doi = {10.4064/fm-144-2-163-179},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/fm-144-2-163-179/}
}
TY - JOUR AU - Haseo Ki AU - Tom Linton TI - Normal numbers and subsets of N with given densities JO - Fundamenta Mathematicae PY - 1994 SP - 163 EP - 179 VL - 144 IS - 2 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4064/fm-144-2-163-179/ DO - 10.4064/fm-144-2-163-179 LA - en ID - 10_4064_fm_144_2_163_179 ER -
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
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