Asymptotic density and the Ershov hierarchy

Abstract

We classify the asymptotic densities of the sets according to their level in the Ershov hierarchy. In particular, it is shown that for , a real is the density of an n‐c.e. set if and only if it is a difference of left‐ reals. Further, we show that the densities of the ω‐c.e. sets coincide with the densities of the sets, and there are ω‐c.e. sets whose density is not the density of an n‐c.e. set for any .