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Asymptotic density and computably enumerable sets
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@MISC{Downey_asymptoticdensity,
author = {Rodney G. Downey and Carl G. Jockusch and Jr. and Paul E. Schupp},
title = {Asymptotic density and computably enumerable sets },
year = {}
}
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Abstract
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: (i) The degrees of such sets A are precisely the nonlow c.e. degrees. (ii) There is a c.e. set A of density 1 with no computable subset of nonzero density. (iii) 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.
Keyphrases
enumerable set computable subset asymptotic density tentative title classical asymptotic density different way nonzero density current paper unit interval upper density computational complexity computable enumerability classical smallness notion close connection small density high degree arithmetical complexity
