  We investigate the asymptotic behavior, as $N$ grows, of the largest minimal
pairwise distance of $N$ points restricted to an arbitrary compact rectifiable
set embedded in Euclidean space, and we find the limit distribution of such
optimal configurations. For this purpose, we compare best-packing
configurations with minimal Riesz $s$-energy configurations and determine the
$s$-th root asymptotic behavior (as $s\to \infty)$ of the minimal energy
constants.
  We show that the upper and the lower dimension of a set defined through the
Riesz energy or best-packing coincides with the upper and lower Minkowski
dimension, respectively.
  For certain sets in ${\rm {\bf R}}^d$ of integer Hausdorff dimension, we show
that the limiting behavior of the best-packing distance as well as the minimal
$s$-energy for large $s$ is different for different subsequences of the
cardinalities of the configurations.
