Covering spheres with spheres (Q2471717)

The classical \textit{C. A. Rogers} bound [Mathematika, Lond. 4, 1--6 (1957; Zbl 0079.27203); Mathematika, Lond. 10, 157--164 (1963; Zbl 0158.19603)] states that for sufficiently large radius \(r\), both an \(n\)-dimensional ball and an \(n\)-dimensional sphere can be covered with density \[ \vartheta \;\leq \;\bigg(1+\frac{\ln\ln n}{\ln n} + \frac{5}{\ln n}\bigg) n\ln n. \] The present paper improves this upper bound on the density by approximately~\(\frac12\): For \(n\geq3\), unit balls can cover an \(n\)-dimensional sphere of radius~\(r>1\) with density \[ \vartheta(S_r^n) \;\leq \;\bigg(\frac12 + \frac{2\ln\ln n}{\ln n} + \frac{5}{\ln n}\bigg) n\ln n. \] As a corollary, for \(n\to\infty\), unit balls can cover the entire \(n\)-dimensional Euclidean space with density \[ \vartheta(\mathbb{R}^n) \;\leq \;\bigg(\frac12+o(1)\bigg)n\ln n. \] Randomly selecting some of the ball centers is an essential ingredient in constructing these new coverings.