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Asymptotic Density and Computability

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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.

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Funding

This work is supported by the Russian Science Foundation (project no. 18-11-00028).

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Authors and Affiliations

  1. Kazan Federal University, 18 Kremlyovskaya str., 420008, Kazan, Russia

    I. I. Batyrshin

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  1. I. I. Batyrshin
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Correspondence to I. I. Batyrshin.

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Russian Text © The Author(s), 2021, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2021, No. 10, pp. 3–14.

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Batyrshin, I.I. Asymptotic Density and Computability. Russ Math. 65, 1–9 (2021). https://doi.org/10.3103/S1066369X21100017

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