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Sets of visible points

Citation

Rumsey, Howard Calvin (1961) Sets of visible points. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechTHESIS:03282011-140809241

Abstract

We say that two lattice points are visible from one another if there is no lattice point on the open line segment joining them. If Q is a subset of the n-dimensional integer lattice L^n, we write VQ for the set of points which can see every point of Q, and we call a set S a set of visible points if S = VQ for some set Q. In the first section we study the elementary properties of the operator V and of certain associated operators. A typical result is that Q is a set of visible points if and only if Q = V(VQ). In the second and third sections we study sets of visible points in greater detail. In particular we show that if Q is a finite subset of L^n, then VQ has a "density" which is given by the Euler product ^π_p (1 – r_p(Q)/p_n) where the numbers r_p (Q) are certain integers determined by the set Q and the primes p. And if Q is an infinite subset of L^ n, we give necessary and sufficient conditions on the set Q such that VQ has a density which is given by this or other related products. In the final section we compute the average values of a certain class of functions defined on L^n, and we show that the resulting formula may be used to compute the density of a set of visible points VQ generated by a finite set Q.

Item Type:Thesis (Dissertation (Ph.D.))
Subject Keywords:Mathematics
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:
  • Unknown, Unknown
Defense Date:1 January 1961
Record Number:CaltechTHESIS:03282011-140809241
Persistent URL:http://resolver.caltech.edu/CaltechTHESIS:03282011-140809241
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:6275
Collection:CaltechTHESIS
Deposited By: Benjamin Perez
Deposited On:28 Mar 2011 21:26
Last Modified:26 Dec 2012 04:33

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