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## Fundamenta Mathematicae

1994 | 144 | 2 | 163-179
Tytuł artykułu

### Normal numbers and subsets of N with given densities

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
Kategorie tematyczne
Czasopismo
Rocznik
Tom
Numer
Strony
163-179
Opis fizyczny
Daty
wydano
1994
otrzymano
1992-12-16
poprawiono
1993-05-04
poprawiono
1993-09-02
Twórcy
autor
• Caltech 253-37, Pasadena, California 91125, U.S.A.
autor
• Department of Mathematics, Willamette University, 900 State Street, Salem, Oregon 97301, U.S.A.
Bibliografia
• [1] L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Wiley, New York, 1974.
• [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.
• [3] D. A. Martin, Borel determinacy, Ann. of Math. (2) 102 (1975), 363-371.
• [4] D. E. Miller, The invariant $Π^0_α$ 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.
• [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).
Typ dokumentu
Bibliografia
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