Cone avoidance and randomness preservation

Abstract

Let X be an infinite sequence of 0's and 1's. Let f be a computable function. Recall that X is strongly f-random if and only if the a priori Kolmogorov complexity of each finite initial segment τ of X is bounded below by f(τ) minus a constant. We study the problem of finding a PA-complete Turing oracle which preserves the strong f-randomness of X while avoiding a Turing cone. In the context of this problem, we prove that the cones which cannot always be avoided are precisely the K-trivial ones. We also prove: (1) If f is convex and X is strongly f-random and Y is Martin-Löf random relative to X, then X is strongly f-random relative to Y. (2) X is complex relative to some oracle if and only if X is random with respect to some continuous probability measure.