On Visibility Problems with an Infinite Discrete, set of Obstacles

Abstract

This paper studies visibility problems in Euclidean spaces $\mathbb{R}^d$ where the obstacles are the points of infinite discrete sets $Y\subseteq\mathbb{R}^d$. A point $x\in\mathbb{R}^d$ is called $\varepsilon$-visible for $Y$ (notation: $x\in\mathbf{vis}(Y, \varepsilon))$ if there exists a ray $L\subseteq\mathbb{R}^d$ emanating from $x$ such that $||y-z||\geq\varepsilon$, for all $y\in Y\setminus\{x\}$ and $z\in L$. A point $x\in\mathbb{R}^d$ is called visible for $Y$ (notation: $x\in\mathbf{vis}(Y))$ if $x\in\mathbf{vis}(Y, \varepsilon))$, for some $\varepsilon>0$.\\ Our main result is the following. For every $\varepsilon>0$ and every relatively dense set $Y\subseteq\mathbb{R}^2$, $\mathbf{vis}(Y, \varepsilon))\neq\mathbb{R}^2$. 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 $Y\subseteq \mathbb{Z}^d$ such that $\mathbf{vis}(Y)=\mathbb{R}^d$. (One easily verifies that $\mathbf{vis}(\mathbb{Z}^d)=\mathbb{R}^d\setminus\mathbb{Z}^d$, for all $d\geq 2$). We derive a number of other results clarifying how the size of a sets $Y\subseteq\mathbb{R}^d$ may affect the sets $\mathbf{vis}(Y)$ and $\mathbf{vis}(Y,\varepsilon)$. We present a Ramsey type result concerning uniformly separated subsets of $\mathbb{R}^2$ whose growth is faster than linear.