Optimal Discrete Riesz Energy and Discrepancy

Abstract

The Riesz $s$-energy of an $N$-point configuration in the Euclidean space ${\mathbb{R}}^p$ is defined as the sum of reciprocal $s$-powers of all mutual distances in this system. In the limit $s\to 0$ the Riesz $s$-potential $1/r^s$ ($r$ the Euclidean distance) governing the point interaction is replaced with the logarithmic potential $\log (1/r)$. In particular, we present a conjecture for the leading term of the asymptotic expansion of the optimal $\IL_2$-discrepancy with respect to spherical caps on the unit sphere in ${\mathbb{R}}^{d+1}$ which follows from Stolarsky's invariance principle [Proc. Amer. Math. Soc. 41 (1973)] and the fundamental conjecture for the first two terms of the asymptotic expansion of the optimal Riesz $s$-energy of $N$ points as $N\to \infty$.

8 pages

References in corpus (1)

• Quasi-Monte Carlo rules for numerical integration over the unit sphere ${\mathbb{S}}^2$

Cited by in corpus (2)

• Distributing many points on spheres: minimal energy and designs
• Generation of off-critical zeros for hypercubic Epstein zeta-functions