Normal numbers and subsets of N with given densities

• Autorzy: Haseo Ki , Tom Linton
• 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.

Kategorie tematyczne

• 11K16: Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.
• 26A21: Classification of real functions; Baire classification of sets and functions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 1994
• Tom: 144
• Numer: 2
• Strony: 163-179

Daty

wydano

1994

otrzymano

1992-12-16

poprawiono

1993-05-04

poprawiono

1993-09-02

Twórcy

autor

Haseo Ki

• Caltech 253-37, Pasadena, California 91125, U.S.A.

autor

Tom Linton

• Department of Mathematics, Willamette University, 900 State Street, Salem, Oregon 97301, U.S.A.

Bibliografia

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• [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).