  For a closed subset $K$ of a compact metric space $A$ possessing an
$\alpha$-regular measure $\mu$ with $\mu(K)>0$, we prove that whenever
$s>\alpha$, any sequence of weighted minimal Riesz $s$-energy configurations
$\omega_N=\{x_{i,N}^{(s)}\}_{i=1}^N$ on $K$ (for `nice' weights) is
quasi-uniform in the sense that the ratios of its mesh norm to separation
distance remain bounded as $N$ grows large. Furthermore, if $K$ is an
$\alpha$-rectifiable compact subset of Euclidean space ($\alpha$ an integer)
with positive and finite $\alpha$-dimensional Hausdorff measure, it is possible
to generate such a quasi-uniform sequence of configurations that also has (as
$N\to \infty$) a prescribed positive continuous limit distribution with respect
to $\alpha$-dimensional Hausdorff measure. As a consequence of our energy
related results for the unweighted case, we deduce that if $A$ is a compact
$C^1$ manifold without boundary, then there exists a sequence of $N$-point
best-packing configurations on $A$ whose mesh-separation ratios have limit
superior (as $N\to \infty$) at most 2.
