Some visibility problems in point lattices

Citation

Rearick, David Francis (1960) Some visibility problems in point lattices. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechETD:etd-06232006-133908

Abstract

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 [...].

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