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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) isquasi-uniform in the sense that the ratios of its mesh norm to separationdistance 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 possibleto generate such a quasi-uniform sequence of configurations that also has (as$N\to \infty$) a prescribed positive continuous limit distribution with respectto $\alpha$-dimensional Hausdorff measure. As a consequence of our energyrelated results for the unweighted case, we deduce that if $A$ is a compact$C^1$ manifold without boundary, then there exists a sequence of $N$-pointbest-packing configurations on $A$ whose mesh-separation ratios have limitsuperior (as $N\to \infty$) at most 2.

Quasi-uniformity of Minimal Weighted Energy Points on Compact Metric Spaces