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Sets of visible points


Rumsey, Howard Calvin (1961) Sets of visible points. Dissertation (Ph.D.), California Institute of Technology.


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:
Default Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:6275
Deposited By: Benjamin Perez
Deposited On:28 Mar 2011 21:26
Last Modified:26 Dec 2012 04:33

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