### Lowness for Kurtz randomness

Noam Greenberg and Joseph S. Miller
Source: J. Symbolic Logic Volume 74, Issue 2 (2009), 665-678.

#### Abstract

We prove that degrees that are low for Kurtz randomness cannot be diagonally non-recursive. Together with the work of Stephan and Yu [16], this proves that they coincide with the hyperimmune-free non-DNR degrees, which are also exactly the degrees that are low for weak 1-genericity.

We also consider Low(ℳ,Kurtz), the class of degrees a such that every element of ℳ is a-Kurtz random. These are characterised when ℳ is the class of Martin—Löf random, computably random, or Schnorr random reals. We show that Low(ML,Kurtz) coincides with the non-DNR degrees, while both Low(CR,Kurtz) and Low(Schnorr,Kurtz) are exactly the non-high, non-DNR degrees.

First Page:
Primary Subjects: 03D80
Secondary Subjects: 68Q30
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Permanent link to this document: http://projecteuclid.org/euclid.jsl/1243948333
Digital Object Identifier: doi:10.2178/jsl/1243948333
Zentralblatt MATH identifier: 05561772
Mathematical Reviews number (MathSciNet): MR2518817

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