The computational content of intrinsic density

TL;DR

This paper explores the concept of intrinsic density in computability theory, showing that sets with well-defined intrinsic density exist only in certain high Turing degrees and linking this to reverse mathematics principles.

Contribution

It establishes the conditions under which sets with intrinsic density exist and connects these conditions to the existence of diagonally non-computable functions and high degrees.

Findings

Sets with intrinsic density 0 exist only in high or diagonally non-computable degrees.

A classic immune set construction yields a set with intrinsic lower density 0 in every non-computable degree.

Over RCA_0, the existence of a dominating or diagonally non-computable function is equivalent to the existence of a set with intrinsic density 0.

Abstract

In a previous paper, the author introduced the idea of intrinsic density --- a restriction of asymptotic density to sets whose density is invariant under computable permutation. We prove that sets with well-defined intrinsic density (and particularly intrinsic density 0) exist only in Turing degrees that are either high ($\mathbf{a}'\ge_{\rm T}\emptyset"$) or compute a diagonally non-computable function. By contrast, a classic construction of an immune set in every non-computable degree actually yields a set with intrinsic lower density 0 in every non-computable degree. We also show that the former result holds in the sense of reverse mathematics, in that (over $\mathsf{RCA}_0$) the existence of a dominating or diagonally non-computable function is equivalent to the existence of a set with intrinsic density 0.