Asymptotic Density and Computably Enumerable Sets

TL;DR

This paper explores the relationship between asymptotic density and computably enumerable sets, providing characterizations of degrees and densities, and connecting computational complexity with density as a real number.

Contribution

It offers new characterizations of non-low c.e. degrees, analyzes the complexity of densities, and links density notions with classical smallness properties.

Findings

Non-low c.e. degrees contain sets of density 1 with no computable density-1 subsets.

Every nonzero c.e. degree has a set that is generically computable but not coarsely computable.

Characterized the complexity of densities and introduced the concept of 'computable at density r'.

Abstract

We study connections between classical asymptotic density and c.e. sets. We prove that a c.e. Turing degree d is not low if and only if d contains a c.e. set A of density 1 which has no computable subsets of density 1, giving a natural characterization of non-low c.e. degrees. In contrast, we prove that every nonzero c.e. degree contains a set which is generically computable but not coarsely computable. There is a very close connection between the computational complexity of a set and the computational complexity of its density as a real number where we measure complexity of real numbers as the position of their left Dedekind cuts in the Arithmetic Hierarchy. We characterize the lower densities, upper densities and densities of both computable and computably enumerable sets. We also study "computable at density r" where r is an arbitrary real number in the unit interval. Finally, we study connections between density and classical smallness notions such as immunity and cohesiveness.