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-\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.Comment: This is an extended version of a paper to appear. Minor typos have been correcte