Some visibility problems in point lattices

[摘要] NOTE: Text or symbols not renderable in plain ASCII are indicated by [...]. Abstract is included in .pdf document.We say that one lattice point is visible from another if no third lattice point lies on the line joining them.A lattice point visible from the origin is called a visible point.We study the manner in which the visible points are distributed throughout the lattice and show that, in a k-dimensional lattice, the fraction of such points in an expanding region "usually" tends to [...]. On the other hand there exist arbitrarily large "gaps" containing no visible points.The following is a typical theorem: The maximum number of lattice points mutually visible in pairs is [...], and if [...], the "density" of points visible from each of a fixed set of n points, themselves mutually visible in pairs, is [...].The last section is devoted to a study of the function [...], which is defined to be the number of distinct solutions of the congruence [...] having [...]. A special case of this function arises in connection with a certain visibility problem.A typical result is that [...].

[发布机构] University:California Institute of Technology;Department:Physics, Mathematics and Astronomy

[关键词] Mathematics