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

TL;DR

This paper proves that minimal weighted Riesz energy points on certain compact metric spaces become quasi-uniform as their number grows, and constructs configurations with prescribed distributions, with implications for best-packing on manifolds.

Contribution

It establishes quasi-uniformity of minimal energy points on compact metric spaces and constructs configurations with prescribed limit distributions, extending understanding of energy minimization and point distributions.

Findings

Weighted minimal Riesz energy points are quasi-uniform for s > alpha.

Constructed configurations with prescribed limit distributions on rectifiable sets.

Existence of point sequences with bounded mesh-separation ratios on manifolds.

Abstract

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) 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. 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.