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Asymptotic density, computable traceability, and 1-randomness

Uri Andrews, Mingzhong Cai, David Diamondstone, Carl Jockusch, Steffen Lempp (2016)

Fundamenta Mathematicae

Let r ∈ [0,1]. A set A ⊆ ω is said to be coarsely computable at density r if there is a computable function f such that {n | f(n) = A(n)} has lower density at least r. Our main results are that A is coarsely computable at density 1/2 if A is computably traceable or truth-table reducible to a 1-random set. In the other direction, we show that if a degree a is hyperimmune or PA, then there is an a-computable set which is not coarsely computable at any positive density.

Coherent randomness tests and computing the K -trivial sets

Laurent Bienvenu, Noam Greenberg, Antonín Kučera, André Nies, Dan Turetsky (2016)

Journal of the European Mathematical Society

We introduce Oberwolfach randomness, a notion within Demuth’s framework of statistical tests with moving components; here the components’ movement has to be coherent across levels. We show that a ML-random set computes all K -trivial sets if and only if it is not Oberwolfach random, and indeed that there is a K -trivial set which is not computable from any Oberwolfach random set. We show that Oberwolfach random sets satisfy effective versions of almost-everywhere theorems of analysis, such as the...

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