Using random sets as oracles

Let R be a notion of algorithmic randomness for individual subsets of N. We say B is a base for R randomness if there is a Z >T B such that Z is R random relative to B. We show that the bases for 1-randomness are exactly the K-trivial sets and discuss several consequences of this result. We also show that the bases for computable randomness include every ∆2 set that is not diagonally noncomputable, but no set of PA-degree. As a consequence, we conclude that an n-c.e. set is a base for computable randomness iff it is Turing incomplete.