THE COMPUTATIONAL CONTENT OF INTRINSIC DENSITY

In a previous article, 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 (a′ ≥T ∅″) or compute a diagonally noncomputable function. By contrast, a classic construction of an immune set in every noncomputable degree actually yields a set with intrinsic lower density 0 in every noncomputable degree. We also show that the former result holds in the sense of reverse mathematics, in that (over RCA₀) the existence of a dominating or diagonally noncomputable function is equivalent to the existence of a set with intrinsic density 0.