Sets of visible points
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...
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| Format: | Others |
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1961
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| Online Access: | https://thesis.library.caltech.edu/6275/1/Rumsey_h_1961.pdf Rumsey, Howard Calvin (1961) Sets of visible points. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/9Y69-0F82. https://resolver.caltech.edu/CaltechTHESIS:03282011-140809241 <https://resolver.caltech.edu/CaltechTHESIS:03282011-140809241> |
| Summary: | 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.
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