On Visibility Problems with an Infinite Discrete Set of Obstacles

This paper deals with visibility problems in Euclidean spaces where the set of obstacles Y is an infinite discrete point set. We prove five independent results. Consider the following problem. Given ε> 0 , imagine a forest whose trees have radius ε and their locations are given by the set Y. Suppose that a light source is at infinity, and that there are no arbitrarily large clearings in the forest. Then, are there always dark points (namely, points that do not see infinity)? We answer the above question positively. We also examine other visibility problems. In particular we show that there exists a relatively dense subset Y of Z^d such that every point in R^d has a ray to infinity with positive distance from Y. In addition, we derive a number of other results clarifying how the sizes of the sets of obstacles may affect the sets of points that are visible from infinity. We also present a geometric Ramsey type result concerning finding patterns in uniformly separated subsets of the plane, whose growth is faster than linear.