Sets of visible points

• Main Author: Rumsey, Howard Calvin
• Format: Others
• Published: 1961
• 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>

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.