Quasi-uniformity of minimal weighted energy points on compact metric spaces

For a closed subset K of a compact metric space A possessing an α-regular measure μ with μ(K)>0, we prove that whenever s>α, any sequence of weighted minimal Riesz s-energy configurations ωN={xi,N(s)}i=1N 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 α-rectifiable compact subset of Euclidean space (α an integer) with positive and finite α-dimensional Hausdorff measure, it is possible to generate such a quasi-uniform sequence of configurations that also has (as N→∞) a prescribed positive continuous limit distribution with respect to α-dimensional Hausdorff measure. ► Weighted Riesz s-energy minimal configurations are quasi-uniform for s large. ► Weight can be chosen such that minimal configurations approach a given limiting density. ► Quasi-uniformity for best-packing configurations is deduced from energy configurations.