  We give a definition of thickness in $\mathbb{R}^d$ that is useful even for
totally disconnected sets, and prove a Gap Lemma type result. We also guarantee
an interval of distances in any direction in thick compact sets, relate thick
sets (for this definition of thickness) with winning sets, give a lower bound
for the Hausdorff dimension of the intersection of countably many of them, a
result guaranteeing the presence of large patterns, and lower bounds for the
Hausdorff dimension of a set in relationship with its thickness.
