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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

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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 [...].

Item Type:Thesis (Dissertation (Ph.D.))
Degree Grantor:California Institute of Technology
Division:Physics, Mathematics and Astronomy
Major Option:Mathematics
Thesis Availability:Public (worldwide access)
Research Advisor(s):
  • Apostol, Tom M.
Thesis Committee:
  • Apostol, Tom M. (chair)
  • Rumsey, Howard C.
Defense Date:1 January 1960
Record Number:CaltechETD:etd-06232006-133908
Persistent URL:http://resolver.caltech.edu/CaltechETD:etd-06232006-133908
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:2705
Collection:CaltechTHESIS
Deposited By: Imported from ETD-db
Deposited On:30 Jun 2006
Last Modified:26 Dec 2012 02:53

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