I classes, LR degrees and Turing degrees

We say that A ≤LR B if every B-random set is A-random with respect to Martin-Löf randomness. We study this reducibility and its interactions with the Turing reducibility, Π1 classes, hyperimmunity and other recursion theoretic notions. A natural variant of the Turing reducibility from the point of view of Martin-Löf randomness is the LR reducibility which was introduced in [12]. We say that A ≤LR B for two sets A,B if every B-random real is A-random (throughout this paper randomness means Martin-Löf randomness). Intuitively this means that whenever A can derandomize a real, B also has this ability. This reducibility naturally induces an equivalence relation ≡LR which defines a partition of Cantor space into the LR degrees. Two reals A,B belong to the same LR degree iff the A-random reals and B-random reals coincide. The LR degrees were first introduced by André Nies [12] and were further studied by Barmpalias, Lewis, Soskova [2] and Simpson [15]. In this paper we study ≤LR and its interactions with ≤T . In Section 1 we lay out the basic framework and facts which are used throughout the rest of the paper. In Section 2 we study Π1 classes of sets ≤LR ∅′ and as an application we apply a basis theorem to deduce that there is some A which is not low for random but is LR reducible to an A-random set. This contrasts the situation in ≤T . In Section 3 we show that there is a hyperimmune-free Turing degree ≤LR ∅′ (again via a basis theorem) and prove more results about hyperimmunity in relation to ≤LR. We also construct a superlow r.e. set A whose lower cone with respect to ≤LR contains a perfect Π1 class. In section 4 we study the Turing degrees inside an LR degree (globally). In Section 5 we look at recursively enumerable LR degrees and the r.e. Turing degrees inside them. We also prove a weak density result for the recursively enumerable LR degrees. In the last section we show that every jump traceable set in the REA hierarchy is superlow, thus extending (in one direction) the result of Nies that jump traceability and superlowness coincide in the r.e. sets. In the following, we use r.e. sets of strings to generate subclasses of the Cantor space. In particular, we never use the relations ⊂, ⊆, ⊃ and ⊇, the measure μ and the operations ∩ and ∪ for sets U of strings; these relations and operations always refer to the class S(U) = {α ∈ {0, 1} | ∃n(α(0)α(1) . . . α(n) ∈ U)} Date: Last updated on 20 May 2007.