Asymptotic density and the coarse computability bound

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Hirschfeldt, Dennis
Jockusch, Carl
Schupp, Paul
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McNicholl, Timothy
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For r is an element of [0, 1] we say that a set A subset of omega 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 gamma (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 is an element of (0, 1] there are sets A(0), A(1) such that gamma(A(0)) = gamma(A(1)) = r where A(0) is coarsely computable at density r while A(1) is not coarsely computable at density r. We show that a real r is an element of [0, 1] is equal to gamma (A) for some c.e. set A if and only if r is left-Sigma(0)(3). A surprising result is that if G is a Delta(0)(2) 1-generic set, and A < = (T) G with gamma(A) = 1, then A is coarsely computable at density 1.


This is a manuscript of an article published as Hirschfeldt, Denis R., Carl G. Jockusch Jr, Timothy H. McNicholl, and Paul E. Schupp. "Asymptotic density and the coarse computability bound." Computability 5, no. 1 (2016): 13-27, doi:10.3233/COM-150035. Posted with permission.

Fri Jan 01 00:00:00 UTC 2016