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

Abstract

For a closed subset $K$ of a compact metric space $A$ possessing an $\alpha$-regular measure $\mu$ with $\mu \left(K\right)>0$, we prove that whenever $s>\alpha$, any sequence of weighted minimal Riesz $s$-energy configurations ${\omega }_N={\left\{x_{i,N}^{\left(s\right)}\right\}}_{i=1}^N$ 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.