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 $Nto infty$) 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 $Nto infty$) at most 2.
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