Relative Randomness and Cardinality

Abstract

A set $B\subseteq \mathbb{N}$ 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 ${\mathrm{\Delta }}_2^0$ set B is low for Martin-Löf random if and only if the class of oracles which compress less efficiently than B, namely, the class ${𝒞}^B=\{A|\mathrm{\forall }n{K}^B(n){\le }^{+}{K}^A(n)\}$ is countable (where K denotes the prefix-free complexity and ${\le }^{+}$ denotes inequality modulo a constant. It follows that ${\mathrm{\Delta }}_2^0$ is the largest arithmetical class with this property and if ${𝒞}^B$ is uncountable, it contains a perfect ${\mathrm{\Pi }}_1^0$ set of reals. The proof introduces a new method for constructing nontrivial reals below a ${\mathrm{\Delta }}_2^0$ set which is not low for Martin-Löf random.