We prove that degrees that are low for Kurtz randomness cannot be diagonally non-recursive. Together with the work of Stephan and Yu , 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.
"Lowness for Kurtz randomness." J. Symbolic Logic 74 (2) 665 - 678, June 2009. https://doi.org/10.2178/jsl/1243948333