How far can you see in a forest?

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 $\V(\e) = O(\e^{-2d-η})$ 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.

arXiv commentThis is an extended version of a paper to appear. Minor typos have been corrected