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 $E\upharpoonright A$ is smooth. More generally, if ${\le }_Q$ 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 ${\le }_Q$ and $H^g(A)>0$.