On Visibility Problems with an Infinite Discrete, set of Obstacles
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This paper studies visibility problems in Euclidean spaces R^d where the obstacles are the points of infinite discrete sets Y⊆R^d. A point x∈R^d is called ε-visible for Y (notation: x∈vis(Y, ε)) if there exists a ray L⊆R^d emanating from x such that ||y-z||≥ε, for all y∈ Y∖{x} and z∈ L. A point x∈R^d is called visible for Y (notation: x∈vis(Y)) if x∈vis(Y, ε)), for some ε>0. Our main result is the following. For every ε>0 and every relatively dense set Y⊆R^2, vis(Y, ε))≠R^2. This result generalizes a theorem of Dumitrescu and Jiang, which settled Mitchell's dark forest conjecture. On the other hand, we show that there exists a relatively dense subset Y⊆Z^d such that vis(Y)=R^d. (One easily verifies that vis(Z^d)=R^d∖Z^d, for all d≥ 2). We derive a number of other results clarifying how the size of a sets Y⊆R^d may affect the sets vis(Y) and vis(Y,ε). We present a Ramsey type result concerning uniformly separated subsets of R^2 whose growth is faster than linear.
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