  We introduce a definition of thickness in $\mathbb{R}^d$ and obtain a lower
bound for the Hausdorff dimension of the intersection of finitely or countably
many thick compact sets using a variant of Schmidt's game. As an application we
prove that given any compact set in $\mathbb{R}^d$ with thickness $\tau$, there
is a number $N(\tau)$ such that the set contains a translate of all
sufficiently small similar copies of every set in $\mathbb{R}^d$ with at most
$N(\tau)$ elements; indeed the set of such translations has positive Hausdorff
dimension. We also prove a gap lemma and bounds relating Hausdorff dimension
and thickness.
