The computational content of intrinsic density

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.