Schnorr trivial reals: a construction

Abstract

A real is Martin-Lof (Schnorr) random if it does not belong to any effectively presented null $${\Sigma^0_1}$$ (recursive) class of reals. Although these randomness notions are very closely related, the set of Turing degrees containing reals that are K-trivial has very different properties from the set of Turing degrees that are Schnorr trivial. Nies proved in (Adv Math 197(1):274–305, 2005) that all K-trivial reals are low. In this paper, we prove that if $${{\bf h'} \geq_T {\bf 0''}}$$ , then h contains a Schnorr trivial real. Since this concept appears to separate computational complexity from computational strength, it suggests that Schnorr trivial reals should be considered in a structure other than the Turing degrees.