Schnorr Trivial Reals: A construction

Abstract

A real is Martin-Löf (Schnorr) random if it does not belong to any effectively presented null Σ10 (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 [11] that all K-trivial reals are low. In this paper, we prove that if h is a high degree, then every degree a⩾Th 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.