Über die \(<p,q>\)-Systeme in der euklidischen Ebene. (On \(<p,q>\)-systems in Euclidean plane) (Q1823473)

A point set \(S\subset {\mathbb{E}}^ 2\) is called a (p,q)-system (with p, q as positive integers), if for positive real numbers r, R each open circle of radius r contains at most p points of S and in every closed circle of radius R lie at least q points of S. Under these conditions and with \(r_ p=\sup r\), \(R_ q=\inf R\) the quotient \(r_ p/R_ q\) is named the (p,q)-width of S. For the case of lattice points, the maxima of the (p,q)-widths with \(1\leq p,q\leq 4\) (p\(\neq q)\) are given. Additionally, point lattices are described which are extremal in this sense.