A point set everywhere dense in the plane
Let \(0_ 1\), \(0_ 2\) be two points in the plane having mutual distance at most 2. Put \(S_ 1=\{0_ 1,0_ 2\}\). If \(S_ i\), \(i\geq 1\), has already been determined, let \(S_{i+1}\) be the union of \(S_ i\) and the set of all intersection points of unit circles whose centers are in \(S_ i.\) The authors prove the following conjecture of L. Fejes-Tóth: The set \(\cup^{\infty}_{i=1}S_ i\) is either identical with the set of vertices of a regular triangle-lattice of side 1, or it is everywhere dense in the plane. It is shown, that this result is valid for Minkowski planes. Similar assertions can be proved for hyperbolic planes and spheres.
point sets everywhere dense in the plane
scientific article; zbMATH DE number 3915104