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 $α$-regular measure $μ$ with $μ(K)>0$, we prove that whenever $s>α$, any sequence of weighted minimal Riesz $s$-energy configurations $ω_N=\{x_{i,N}^{(s)}\}_{i=1}^N$ 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 $α$-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 $N\to \infty$) a prescribed positive continuous limit distribution with respect to $α$-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 $N\to \infty$) at most 2.