# Normal numbers and subsets of N with given densities

Fundamenta Mathematicae (1994)

- Volume: 144, Issue: 2, page 163-179
- ISSN: 0016-2736

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topKi, Haseo, and Linton, Tom. "Normal numbers and subsets of N with given densities." Fundamenta Mathematicae 144.2 (1994): 163-179. <http://eudml.org/doc/212021>.

@article{Ki1994,

abstract = {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.},

author = {Ki, Haseo, Linton, Tom},

journal = {Fundamenta Mathematicae},

keywords = {Borel hierarchy; Borel class; densities},

language = {eng},

number = {2},

pages = {163-179},

title = {Normal numbers and subsets of N with given densities},

url = {http://eudml.org/doc/212021},

volume = {144},

year = {1994},

}

TY - JOUR

AU - Ki, Haseo

AU - Linton, Tom

TI - Normal numbers and subsets of N with given densities

JO - Fundamenta Mathematicae

PY - 1994

VL - 144

IS - 2

SP - 163

EP - 179

AB - 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.

LA - eng

KW - Borel hierarchy; Borel class; densities

UR - http://eudml.org/doc/212021

ER -

## References

top- [1] L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Wiley, New York, 1974. Zbl0281.10001
- [2] A. Louveau and J. Saint-Raymond, Borel classes and closed games: Wadge-type and Hurewicz-type results, Trans. Amer. Math. Soc. 304 (1987), 431-467. Zbl0655.04001
- [3] D. A. Martin, Borel determinacy, Ann. of Math. (2) 102 (1975), 363-371.
- [4] D. E. Miller, The invariant ${\Pi}_{\alpha}^{0}$ separation principle, Trans. Amer. Math. Soc. 242 (1978), 185-204.
- [5] I. Niven, Irrational Numbers, The Carus Math. Monographs 11, Math. Assoc. America, Quinn and Boden, Rahway, N.J., 1956.
- [6] W. Schmidt, On normal numbers, Pacific J. Math. 10 (1960), 661-672. Zbl0093.05401
- [7] W. Wadge, Degrees of complexity of subsets of the Baire space, Notices Amer. Math. Soc. 19 (1972), A-714-A-715 (abstract 72T-E91).

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