Sets of visible points

[摘要] We say that two lattice points are visible from one another ifthere is no lattice point on the open line segment joining them. IfQ is a subset of the n-dimensional integer lattice L^n, we write VQfor the set of points which can see every point of Q, and we call aset S a set of visible points if S = VQ for some set Q.In the first section we study the elementary properties of theoperator V and of certain associated operators. A typical result isthat Q is a set of visible points if and only if Q = V(VQ). In thesecond and third sections we study sets of visible points in greaterdetail. 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 Qand the primes p. And if Q is an infinite subset of L^ n, we givenecessary and sufficient conditions on the set Q such that VQ hasa density which is given by this or other related products.In the final section we compute the average values of a certainclass of functions defined on L^n, and we show that the resultingformula may be used to compute the density of a set of visible pointsVQ generated by a finite set Q.

[发布机构] University:California Institute of Technology;Department:Physics, Mathematics and Astronomy

[关键词] Mathematics