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