Asymptotic density and the Ershov hierarchy

We classify the asymptotic densities of the $\Delta^0_2$ sets according to their level in the Ershov hierarchy. In particular, it is shown that for $n \geq 2$, a real $r \in [0,1]$ is the density of an $n$-c.e.\ set if and only if it is a difference of left-$\Pi_2^0$ reals. Further, we show that the densities of the $\omega$-c.e.\ sets coincide with the densities of the $\Delta^0_2$ sets, and there are $\omega$-c.e.\ sets whose density is not the density of an $n$-c.e. set for any $n \in \omega$.Comment: To appear in Mathematical Logic Quarterl

Asymptotic density and the coarse computability bound

For $r \in [0,1]$ we say that a set $A \subseteq \omega$ is \emph{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 $\gamma(A) = \sup \{r : A \hbox{ is coarsely computable at density } r\}$. We study the interactions of these concepts with Turing reducibility. For example, we show that if $r \in (0,1]$ 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 $r \in [0,1]$ 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 $A \leq\sub{T} G$ with $\gamma(A) = 1$, then $A$ is coarsely computable at density $1$

Asymptotic density and the Ershov hierarchy

We classify the asymptotic densities of the Delta(0)(2) sets according to their level in the Ershov hierarchy. In particular, it is shown that for n2, a real r[0,1] is the density of an n-c.e. set if and only if it is a difference of left-Delta(0)(2) reals. Further, we show that the densities of the w-c.e. sets coincide with the densities of the Delta(0)(2) sets, and there are n-c.e. sets whose density is not the density of an n-c.e. set for any n epsilon omega.This is the pre-peer reviewed version of the following article: Downey, Rod, Carl Jockusch, Timothy H. McNicholl, and Paul Schupp. "Asymptotic density and the Ershov hierarchy." Mathematical Logic Quarterly 61, no. 3 (2015): 189-195, which has been published in final form at doi:10.1002/malq.201300081. This article may be used for non-commercial purposes in accordance With Wiley Terms and Conditions for self-archiving. Posted with permission.</p