Asymptotics of $d$-Dimensional Visibility

Abstract

We consider the space $[0,n]^3$, imagined as a three dimensional, axis-aligned grid world partitioned into $n^3$ $1\times 1 \times 1$ unit cubes. Each cube is either considered to be empty, in which case a line of sight can pass through it, or obstructing, in which case no line of sight can pass through it. From a given position, some of these obstructing cubes block one's view of other obstructing cubes, leading to the following extremal problem: What is the largest number of obstructing cubes that can be simultaneously visible from the surface of an observer cube, over all possible choices of which cubes of $[0,n]^3$ are obstructing? We construct an example of a configuration in which $\Omega\big(n^\frac{8}{3}\big)$ obstructing cubes are visible, and generalize this to an example with $\Omega\big(n^{d-\frac{1}{d}}\big)$ visible obstructing hypercubes for dimension $d>3$. Using Fourier analytic techniques, we prove an $O\big(n^{d-\frac{1}{d}}\log n\big)$ upper bound in a reduced visibility setting.