On a problem of L. Fejes Toth (Q1144797)

Let \(0\leq x_1\leq x_2\leq x_3\leq\dots\) be a sequence of real numbers, \(\lim x_i = +\infty\). The authors prove that if \(\sum_il/x_i^{n-k}=+\infty\) then there exists a point-system \(P=\{z_1,z_2,\dots\}\) in the \(n\)-dimensional space \(\mathbb E^n\), for which \(|z_i|=x_i\) holds (\(i=1,2,\dots\)), and any \(k\)-dimensional plane comes arbitrarily near to \(P\). this result is best possible in the sense that if \(P\{z_1,z_2,\dots\}\) is a point-system satisfying \(\sum_il/|z_i|^{n-k}< +\infty\) then for every \(C> 0\) there exists a \(k\)-dimensional plane in \(BbbE^n\), whose distance from all members of \(P\) is at least \(C\). A generalization is also proved. This settles a problem of \textit{L. Fejes Tóth} [Mat. Lapok 25 (1974), 13-20 (1976; Zbl 0359.52010)].