The Journal of Symbolic Logic Volume 00, Number 0, XXX 0000 THE COMPUTATIONAL CONTENT OF INTRINSIC DENSITY ERIC P. ASTOR 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 (a ′ ≥T ∅ ′′ ) 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 RCA0) the existence of a dominating or diagonally non-computable function is equivalent to the existence of a set with intrinsic density 0. §1. Introduction. Shortly after the launch of the field of computability, practitioners began exploring the connections between the computability of a set and the scarcity of its elements. Post, seeking a non-computable c.e. set with less computational power than the halting problem, based his approach on the idea of creating sets with increasingly thin complement, making their comple- ments computationally difficult to distinguish from finite. Those following his program developed the classical “immunity hierarchy” of thinness properties, from immune sets to cohesive sets. (See Figure 1.) Though none of these exhib- ited the strict upper bound Post sought, lower bounds on the Turing degrees of levels of this hierarchy describe useful dividing lines in computational content. For example, immune sets exist precisely in every non-computable degree, while the hyperimmune-free degrees are those that contain only computably-bounded functions. Moving upwards in the hierarchy, cohesiveness and other forms of immunity force co-c.e. sets to be high (a ′ ≥ T ∅ ′′ ), while revealing much more complex patterns outside of the Δ 0 2 degrees. [13] More recently, Jockusch and Schupp [12] (inspired by work of Kapovich, Myas- nikov, Schupp, and Shpilrain [14] on decidability in group theory) constructed new notions of near-computation, considering computations modulo sets with as- ymptotic density 0. This approach of computability modulo sparse sets, joined by other researchers (including Downey [5, 4], Dzhafarov [6], Hirschfeldt [2, 7, 8], Igusa [6, 10], and McNicholl [4, 8]), has uncovered still more connections between thinness and computation. However, since asymptotic density is not invariant Received by the editors January 5, 2018. 2000 Mathematics Subject Classification. 03D28. Key words and phrases. intrinsic density, Turing degrees, reverse mathematics. c  0000, Association for Symbolic Logic 0022-4812/00/0000-0000/$00.00 1