We study connections between classical asymptotic density, computability and computable enumerability. In an earlier paper, the second two authors proved that there is a computably enumerable set A of density 1 with no computable subset of density 1. In the current paper, we extend this result in three different ways: The degrees of such sets A are precisely the nonlow c.e. degrees. There is a c.e. set A of density 1 with no computable subset of nonzero density. There is a (...) c.e. set A of density 1 such that every subset of A of density 1 is of high degree. We also study the extent to which c.e. sets A can be approximated by their computable subsets B in the sense that A\B has small density. There is a very close connection between the computational complexity of a set and the arithmetical complexity of its density and we characterize the lower densities, upper densities and densities of both computable and computably enumerable sets. We also study the notion of "computable at density r" where r is a real in the unit interval. Finally, we study connections between density and classical smallness notions such as immunity, hyperimmunity, and cohesiveness. (shrink)

We consider the question “Is every nonzero generic degree a density-1-bounding generic degree?” By previous results [8] either resolution of this question would answer an open question concerning the structure of the generic degrees: A positive result would prove that there are no minimal generic degrees, and a negative result would prove that there exist minimal pairs in the generic degrees.We consider several techniques for showing that the answer might be positive, and use those techniques to prove that a wide (...) class of assumptions is sufficient to prove density-1-bounding.We also consider a historic difficulty in constructing a potential counterexample: By previous results [7] any generic degree that is not density-1-bounding must be quasiminimal, so in particular, any construction of a non-density-1-bounding generic degree must use a method that is able to construct a quasiminimal generic degree. However, all previously known examples of quasiminimal sets are also density-1, and so trivially density-1-bounding. We provide several examples of non-density-1 sets that are quasiminimal.Using cofinite and mod-finite reducibility, we extend our results to the uniform coarse degrees, and to the nonuniform generic degrees. We define all of the above terms, and we provide independent motivation for the study of each of them.Combined with a concurrently written paper of Hirschfeldt, Jockusch, Kuyper, and Schupp [4], this paper provides a characterization of the level of randomness required to ensure quasiminimality in the uniform and nonuniform coarse and generic degrees. (shrink)