Minimal degrees recursive in 1-generic degrees

Abstract

The problem on the existence of a set of minimal Turing degree recursive in an n-generic set is of considerable interest. Jockusch showed that if n≥2, then there is no set of minimal degree recursive in an n-generic set (or, equivalently, no n-generic degree bounds a minimal degree). Here we prove the next theorem: there is a 1-generic degree bounding a minimal degree below 0'. Furthermore, the 1-generic degree may be chosen to lie below 0''. Follows from the main result of this paper, as stated in the following theorem: Let M⊂ω. If there is no infinite r.e. sequence that is Σ 1 -dense in M, then M is recursive in a 1-generic set