Relative Randomness and Cardinality

A set B ⊆ N is called low for Martin-Löf random if every Martin-Löf random set is also Martin-Löf random relative to B. We show that a ∆2 set B is low for Martin-Löf random iff the class of oracles which compress less efficiently than B, namely the class C = {A | ∀n K(n) ≤ K(n)} is countable (where K denotes the prefix free complexity and ≤ denotes inequality modulo a constant). It follows that ∆2 is the largest arithmetical class with this property and if C is uncountable, it contains a perfect Π1 set of reals. The proof introduces a new method for constructing non-trivial reals below a ∆2 set which is not low for Martin-Löf random.