Asymptotic density and the Ershov hierarchy

We classify the asymptotic densities of the Δ 2 0 sets according to their level in the Ershov hierarchy. In particular, it is shown that for n ≥ 2 , a real r ∈ [ 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 2 0 sets, and there are ω‐c.e. sets whose density is not the density of an n ‐c.e. set for any n ∈ ω .