How far can you see in a forest?

TL;DR

This paper constructs a forest with finite density in any dimension where visibility from any point in any direction is bounded by a specific polynomial rate, using Fourier analysis and exponential sum estimates.

Contribution

It provides a new construction of forests with controlled visibility bounds in arbitrary dimensions, advancing understanding of geometric visibility problems.

Findings

Visibility bound: V(ε) = O(ε^{-2d-η}) for any η > 0

Construction of forests with finite density in any dimension

Application of Fourier analysis and exponential sums in geometric problems

Abstract

We address a visibility problem posed by Solomon & Weiss. More precisely, in any dimension $n := d + 1 \ge 2$, we construct a forest $\F$ with finite density satisfying the following condition : if $\e > 0$ denotes the radius common to all the trees in $\F$, then the visibility $\V$ therein satisfies the estimate $\V(\e) = O(\e^{-2d-\eta})$ for any $\eta > 0$, no matter where we stand and what direction we look in. The proof involves Fourier analysis and sharp estimates of exponential sums.