Hausdorff dimension and countable Borel equivalence relations

We show that if $E$ is a countable Borel equivalence relation on $\mathbb{R}^n$, then there is a closed subset $A \subset [0,1]^n$ of Hausdorff dimension $n$ so that $E \restriction A$ is smooth. More generally, if $\leq_Q$ is a locally countable Borel quasi-order on $2^{\omega}$ and $g$ is any gauge function of lower order than the identity, then there is a closed set $A$ so that $A$ is an antichain in $\leq_Q$ and $H^g(A) > 0$.