Asymptotic density and the Ershov hierarchy

We classify the asymptotic densities of the $Δ^0_2$ 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-$Π_2^0$ reals. Further, we show that the densities of the $ω$-c.e.\ sets coincide with the densities of the $Δ^0_2$ sets, and there are $ω$-c.e.\ sets whose density is not the density of an $n$-c.e. set for any $n \in ω$.

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