Asymptotic density and the Ershov hierarchy

TL;DR

This paper classifies the asymptotic densities of $ ext{Delta}^0_2$ sets based on their Ershov hierarchy levels, revealing precise characterizations for $n$-c.e. ess and the densities of $ ext{omega}$-c.e. ess.

Contribution

It provides a complete characterization of densities of $n$-c.e. ess and $ ext{omega}$-c.e. ess in terms of left-$ ext{Pi}_2^0$ reals, advancing understanding of their asymptotic properties.

Findings

A real $r$ is the density of an $n$-c.e. ess iff it is a difference of left-$ ext{Pi}_2^0$ reals for $n \\geq 2$.

Densities of $ ext{omega}$-c.e. ess coincide with those of $ ext{Delta}^0_2$ sets.

There exist $ ext{omega}$-c.e. ess sets with densities not corresponding to any $n$-c.e. ess set.

Abstract

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 $n \in \omega$.