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ASYMPTOTIC DENSITY AND THE COARSE COMPUTABILITY BOUND
"... Abstract. For r ∈ [0, 1] we say that a set A ⊆ ω is coarsely computable at density r if there is a computable set C such that {n: C(n) = A(n)} has lower density at least r. Let γ(A) = sup{r: A is coarsely computable at density r}. We study the interactions of these concepts with Turing reducibilit ..."
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Abstract. For r ∈ [0, 1] we say that a set A ⊆ ω is coarsely computable at density r if there is a computable set C such that {n: C(n) = A(n)} has lower density at least r. Let γ(A) = sup{r: A is coarsely computable at density r}. We study the interactions of these concepts with Turing reducibility. For example, we show that if r ∈ (0, 1] there are sets A0, A1 such that γ(A0) = γ(A1) = r where A0 is coarsely computable at density r while A1 is not coarsely computable at density r. We show that a real r ∈ [0, 1] is equal to γ(A) for some c.e. set A if and only if r is leftΣ03. A surprising result is that if G is a ∆02 1generic set, and A 6T G with γ(A) = 1, then A is coarsely computable at density 1. 1.