Lowness for Kurtz randomness

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.