Asymptotic density and the coarse computability bound

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 left-Σ0 3. A surprising result is that if G is a ∆0 2 1-generic set, and A T G with γ(A) = 1, then A is coarsely

• Type: Journal
• Year: 2016
• Where: COMPUTABILITY
• Authors: Denis R. Hirschfeldt, Carl G. Jockusch Jr., Timothy H. McNicholl, Paul E. Schupp
• Keywords: COMPUTABILITY 2016