Discrepancy, separation and Riesz energy of finite point sets on compact connected Riemannian manifolds

Abstract

On a smooth compact connected $d$-dimensional Riemannian manifold $M$, if $0<s<d$ then an asymptotically equidistributed sequence of finite subsets of $M$ that is also well-separated yields a sequence of Riesz $s$-energies that converges to the energy double integral, with a rate of convergence depending on the geodesic ball discrepancy. This generalizes a known result for the sphere.

10 pages; based on work presented in 2012 at the 3rd Dolomites Workshop on Constructive Approximation and Applications; submitted December 2012 to Dolomites Research Notes on Approximation; revised January 2014; revised again after second referee's report May 2014; revised for publication June 2014

References in corpus (5)

• QMC designs: optimal order Quasi Monte Carlo Integration schemes on the sphere
• Beurling-Landau's density on compact manifolds
• Quasi-Monte Carlo rules for numerical integration over the unit sphere ${\mathbb{S}}^2$
• On Koksma-Hlawka inequality
• Quadrature rules and distribution of points on manifolds

Cited by in corpus (1)

• Diameter bounded equal measure partitions of Ahlfors regular metric measure spaces