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 = {x i,N} N i=1 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 quasiuniform sequence of configurations that also has (as N →∞) a prescribed positive continuous limit distribution with respect to α-dimensional Hausdorff measure.