Discrepancy of minimal Riesz energy points (Q2067507)

This paper is concerned with upper bounds for the spherical cap discrepancy of the set of minimizers of the Riesz \(s\)-energy and the logarithmic energy on the sphere \({\mathbb S}^d\). Recall, that a spherical cap \(D_r(x)\) is the set \( \{y \in{\mathbb S}^d\ :\ |x-y|<r\}, \) where \(x\in {\mathbb S}^d \), \(r\in[0,2]\), and the spherical cap discrepancy of a point set \(X\subset {\mathbb S}^d\) is \[ \sup_{r,x}\Big|\frac{\#\big(X\cap D_r(x)\big)}{\# X}-\tilde{\sigma}(D_r(x))\Big|, \] where \(\tilde{\sigma}\) is the uniform probability measure on \({\mathbb S}^d\). The Riesz/logarithmic energy of a set \(V\subset {\mathbb S}^d\) is given by \[ E_s(V)=\sum_{\substack{x,y\in V \\ x\neq y}}R_s(x,y),\hspace{0.4cm}\mbox{ where }\hspace{0.4cm} R_s(x,y)=\begin{cases} |x-y|^{-s}, &\mbox{for }0<s<d\\ -\log(|x-y|),&\mbox{for }s=0.\end{cases} \] Let \(V_N\) denote all subsets of \({\mathbb S}^d\) with \(N\)-elements, where \(N\in {\mathbb N}\). Since the kernel \(R_s(x,y)\) is lower semi-continuous and \({\mathbb S}^d\) is compact, there exists a minimizing configuration \(\omega^s_N=\{x_1,\ldots,x_N\}\in V_N\) for the energy, for every \(N\in {\mathbb N}\). Let \(\chi_A\) denote the characteristic function of the set \(A\), i.e., \(\chi_A(x)=1\) if \(x\in A\) and \(\chi_A(x)=0\) else. The authors show the following upper bound: Theorem 1.1. Let \(\omega^s_N=\{x_1,\ldots,x_N\}\) be the \(N\)-point minimizer of the Riesz or logarithmic energy, then \[ c_{s,d}\cdot\sup_{r,x}\Big|\frac{\#\big(\omega^s_N\cap D_r(x)\big)}{N}-\tilde{\sigma}(D_r(x))\Big|\ \leq\ \chi_{[0,d-2]}(s)\cdot N^{-\frac{2}{d(d-s+1)}}+\chi_{(d-2,d)}(s)\cdot N^{-\frac{2(d-s)}{d(d-s+4)}}, \] with a constant \(c_{s,d}\) that depends on \(d,s\) only. This theorem is derived by relating the spherical cap discrepancy with a notion of discrepancy involving Sobolev norms (Theorem 1.5 and Proposition 5.2). The behavior of following integral operator on \(L^2\big({\mathbb S}^d\big)\) is investigated (Section 2 and Section 3): \[ R_s(f)=\int_{{\mathbb S}^d}R_s(x,y)f(y)d\tilde{\sigma}(y), \] where, among other things, it is shown (in Proposition 2.2) that \(R_s\) diagonalizes in the standard basis of spherical harmonics, and its eigenvalues are computed as well as their asymptotic behavior. Further, some formal identities were derived (in Remark 2.3) for \(R_s(x,y)\) in terms of Gegenbauer, also known as ultraspherical polynomials, which might be of independent interest. The authors derive a differential equation to relate \(R_s(x,x_0)\) to \(R_{s+2}(x,x_0)\) for \(x\neq x_0\) (in Lemma 2.5) via the Laplace-Beltrami operator, which is later used to show superharmonicity of \(R_s(x,x_0)\) near \(x_0\) (in Lemma 3.1), and to obtain an upper bound (in Corollary 3.7) of \[ \frac{1}{N^2}\sum_{j\neq k}\int_{D_j}\int_{D_k}R_s(x,y)d\tilde{\sigma}(x)d\tilde{\sigma}(y), \] where each \(D_j\) is a small disc around \(x_j\in\omega^s_N \). This result is then applied to derive the upper bound for the Sobolev discrepancy (in Theorem 1.5). This paper generalizes an unpublished result due to \textit{T. Wolff} [``Fekete points on spheres'', Preprint]. The conclusion of Theorem 1.1 (partially) improves upon results, obtained by Kleiner, Sjögren, Götz and Brauchart. Some typos were found on page 479, where the average of a function \(f\) over a disc \(D\) should be integrated over \(D\); on page 499 we find \(R_s(x-y)\) instead of \(R_s(x,y)\).