Optimal logarithmic energy points on the unit sphere

Abstract:

We study minimum energy point charges on the unit sphere $\mathbb {S}^{d}$ in $\mathbb {R}^{d+1}$, $d\geq 3$, that interact according to the logarithmic potential $\log (1/r)$, where $r$ is the Euclidean distance between points. Such optimal $N$-point configurations are uniformly distributed as $N\to \infty$. We quantify this result by estimating the spherical cap discrepancy of optimal energy configurations. The estimate is of order $\mathcal {O}(N^{-1/(d+2)})$. Essential is an improvement of the lower bound of the optimal logarithmic energy which yields the second term $(1/d)(\log N)/N$ in the asymptotical expansion of the optimal energy. Previously, this was known for the unit sphere in $\mathbb {R}^{3}$ only. Furthermore, we present an upper bound for the error of integration for an equally-weighted numerical integration rule $Q_{N}$ with the $N$ nodes forming an optimal logarithmic energy configuration. For polynomials $p$ of degree at most $n$ this bound is $C_{d} ( N^{1/d} / n )^{-d/2} \| p\|_{\infty }$ as $n/N^{1/d}\to 0$. For continuous functions $f$ of $\mathbb {S}^{d}$ satisfying a Lipschitz condition with constant $C_{f}$ the bound is $(12dC_{f} + C_{d}’ \|f\|_{\infty }) \mathcal {O}(N^{-1/(d+2)})$ as $N\to \infty$.