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Type of Document	Dissertation
Author	Rearick, David Francis
URN	etd-06232006-133908
Persistent URL	http://resolver.caltech.edu/CaltechETD:etd-06232006-133908
Title	Some visibility problems in point lattices
Degree	PhD
Option	Mathematics
Advisory Committee	Advisor Name Title Tom M. Apostol Committee Chair
Keywords	none
Date of Defense	1960-01-01
Availability	unrestricted
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 [...].
Files	Filename       Size       Approximate Download Time (Hours:Minutes:Seconds)     28.8 Modem   56K Modem   ISDN (64 Kb)   ISDN (128 Kb)   Higher-speed Access    Rearick_df_1960.pdf 2.03 Mb 00:09:22 00:04:49 00:04:13 00:02:06 00:00:10