Optimal discrete Riesz energy and discrepancy (Q2893517)

The presented paper is an overview of the problem of distribution of \(N\) points \(\mathbf x_1, \mathbf x_2,\dots, \mathbf x_N\) in the \(d\)-dimensional unit sphere \(\mathbb S^d\subset \mathbb R^{d+1}\) for which the Riesz \(s\)-energy \(\sum_{i\neq j =1}^N| \mathbf x_i-\mathbf x_j| ^{-s}\) is extremal. Here \(| \mathbf x-\mathbf y| \) is the usual Euclidean distance. If \(N\to \infty \), then this problem leads to finding extremes of the energy integral \(\iint | \mathbf x-\mathbf y| ^{-s}d\mu (\mathbf x)d\mu (\mathbf y)\) over the family of Borel probability measures \(\mu \) supported on \(\mathbb S^d\). In the first part the author discusses ``the long-standing open fundamental conjecture'': NEWLINE\[NEWLINE \sup_{\mathbf x_1,\dots ,\mathbf x_N}\sum_{i\neq j =1}^N| \mathbf x_i-\mathbf x_j| ^{-s}=\sup_{\mu}\iint | \mathbf x-\mathbf y| ^{-s}d\mu (\mathbf x)\,d\mu (\mathbf y)\,N^2+\Delta. NEWLINE\]NEWLINE Here the remainder term \(\Delta =\frac {C}{H}N^{1+s/d}+R\), \(C=C(s,d)\), \(H=H(\mathbb S^d)\) denotes the \(d\)-dimensional Hausdorff measure of \(\mathbb S^d\) and \(R/N^{1+s/d-\varepsilon}\to 0\) as \(N\to \infty \). In the parts 2 an 3 the author discusses the spherical cap discrepancy \(D_N\) and the spherical cap \(L_2\)-discrepancy \(D^{(2)}_N\) of the points \(\mathbf x_1,\dots ,\mathbf x_N\) (for definition see the \textit{M. Drmota} and \textit{R. F. Tichy} [Sequences, discrepancies and applications. Berlin: Springer (1997; Zbl 0877.11043)] Then he presents the \textit{K.~B. Stolarsky} [Proc. Am. Math. Soc. 41, 575--582 (1973; Zbl 0274.52012)] ``invariance principle'': NEWLINE\[NEWLINE \frac {1}{N^2}\sum_{i,j=1}^N| \mathbf x_i-\mathbf x_j| +\mathfrak {H(\mathbb S^d)}{H(\mathbb B^d)}(D^{(2)}_N)^2=\iint | \mathbf x-\mathbf x'| \,d\sigma (\mathbf x)\,d\sigma (\mathbf x') NEWLINE\]NEWLINE for any \(N\)-points \(\mathbf x_1,\dots ,\mathbf x_N\) on \(\mathbb S^d\). Here \(\sigma (\mathbf x)\) denotes the normalized surface area measure defined on caps \(C(\mathbf x,t)\). An extensive list of 48 references concludes the paper.