Asymptotic Density and Computability

Abstract

We prove that a set is bi-immune if and only if its images under computable permutations are not generically computable or effectively densely computable sets. An example of a coarsely computable bi-immune set is constructed. It is also proved that for any set there is a permutation from the same Turing degree such that its image under this permutation is an effectively densely computable set. Upper densities of weakly 1-generic sets are studied.