[Paper Review] Optimal Discrete Riesz Energy and Discrepancy

Johann S. Brauchart|arXiv (Cornell University)|Mar 16, 2011

Mathematical Approximation and Integration25 references3 citations

TL;DR

This paper investigates the asymptotic behavior of optimal Riesz $s$-energy configurations on the unit sphere $\mathbb{S}^d$ and their connection to spherical cap $\mathbb{L}_2$-discrepancy via Stolarsky's invariance principle. It proposes a conjecture for the leading terms of the energy expansion and derives the corresponding discrepancy bound, showing that optimal energy configurations yield near-optimal discrepancy for $s = -1$, with explicit asymptotic expansions involving zeta and Bernoulli functions.

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 $s o0$ 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 $N o \infty$.

Motivation & Objective

• To establish a conjectured asymptotic expansion for the optimal Riesz $s$-energy of $N$ points on $\mathbb{S}^d$ for $-2 < s < d+2$, $s \neq 0,d$.
• To link the $\mathbb{L}_2$-discrepancy of spherical cap distributions to the Riesz $s$-energy via Stolarsky's invariance principle.
• To derive the leading-order asymptotic behavior of the spherical cap discrepancy for $s = -1$ using known energy expansions.
• To analyze the discrepancy properties of optimal energy configurations, particularly in relation to known bounds and conjectures such as Korevaar’s.

Proposed method

• Utilizes Stolarsky’s invariance principle to relate the $\mathbb{L}_2$-discrepancy of spherical cap distributions to the Riesz $s$-energy for $s = -1$.
• Applies the fundamental conjecture on the asymptotic expansion of $\mathcal{E}_s(\mathbb{S}^d; N)$, with leading terms involving $N^2$ and $N^{1+s/d}$, to deduce discrepancy behavior.
• Employs known exact expansions of Riesz energy for $N$th roots of unity on $\mathbb{S}^1$ to derive the discrepancy expansion for $s = -1$, using zeta and Bernoulli number identities.
• Uses generating functions for coefficients $\alpha_n(s)$ via $\left(\frac{\sin \pi z}{\pi z}\right)^{-s} = \sum_{n=0}^\infty \alpha_n(s) z^{2n}$ to express higher-order terms in the energy expansion.
• Applies LeVeque-type and Erdős–Turán-type inequalities to estimate discrepancy in terms of spherical harmonic sums and Weyl sums.
• Leverages results from potential theory and geometric measure theory, including the use of generalized transfinite diameters and equilibrium measures, to justify the conjectured energy asymptotics.

Experimental results

Research questions

• RQ1 What is the asymptotic expansion of the optimal Riesz $s$-energy $\mathcal{E}_s(\mathbb{S}^d; N)$ as $N \to \infty$ for $-2 < s < d+2$, $s \neq 0,d$?
• RQ2 How does the spherical cap $\mathbb{L}_2$-discrepancy relate to the Riesz $s$-energy through Stolarsky’s invariance principle?
• RQ3 What is the leading-order asymptotic behavior of the $\mathbb{L}_2$-discrepancy for $s = -1$ configurations on $\mathbb{S}^d$?
• RQ4 Do optimal Riesz $s$-energy configurations achieve near-optimal discrepancy, and how do they compare to known bounds like $\mathcal{O}(N^{-1/d})$?
• RQ5 Can the discrepancy be bounded using energy-based estimates derived from spherical harmonic and Weyl sum techniques?

Key findings

• The conjectured asymptotic expansion of $\mathcal{E}_s(\mathbb{S}^d; N)$ is $V_s(\mathbb{S}^d)N^2 + \frac{C_{s,d}}{[\mathcal{H}_d(\mathbb{S}^d)]^{s/d}} N^{1+s/d} + \mathcal{R}_s(\mathbb{S}^d; N)$, with $\mathcal{R}_s(\mathbb{S}^d; N)/N^{1+s/d - \varepsilon} \to 0$ as $N \to \infty$.
• For $s = -1$, the $\mathbb{L}_2$-discrepancy satisfies $\frac{1}{\pi} \left[ D_{\mathrm{C}}^{\mathbb{L}_2}(X_N) \right]^2 = \frac{2[-\zeta(-1)]}{(2\pi)^{-1}} N^{-2} + \mathcal{O}(N^{-4})$, with $[-\zeta(-1)] = \frac{1}{12}$, yielding $\left[ D_{\mathrm{C}}^{\mathbb{L}_2}(X_N) \right]^2 \sim \frac{1}{3\pi} N^{-2}$.
• The discrepancy expansion for $s = -1$ is derived from the exact energy expansion of $N$th roots of unity, which includes terms involving $\zeta(s)$ and $\alpha_n(s)$, with $\alpha_n(-1)\zeta(-1-2n) < 0$ for $n \geq 1$.
• The spherical cap discrepancy is bounded via LeVeque-type inequalities involving spherical harmonic sums, with coefficients $a_\ell \asymp \ell^{-d-1}$.
• The discrepancy of optimal $s$-energy configurations is not necessarily optimal; for $s = d-1$, the discrepancy is conjectured to be $\mathcal{O}(N^{-1/d})$, and this bound is sharp for $d=2$.
• Numerical evidence suggests that digital nets lifted to $\mathbb{S}^2$ achieve $\mathcal{O}(N^{-1/2})$ discrepancy, but with a constant in the range $[0.50, 0.59]$, indicating suboptimal performance compared to theoretical bounds.