How far can you see in a forest?

2015/09/14 by Faustin Adiceam, Adiceam, Faustin
Mathematics · #11J25 #11J71 #11L03 #11L07 #11Z05 #51N20 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11J25 #msc:11J71 #msc:11L03 #msc:11L07 #msc:11Z05 #msc:51N20

paper · pdf · doi:10.48550/arxiv.1509.04188

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

arxiv created 2015/09/27 · arxiv updated 2015/09/29

Abstract

We address a visibility problem posed by Solomon & Weiss. More precisely, in any dimension n := d + 1 ≥ 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 η> 0, no matter where we stand and what direction we look in. The proof involves Fourier analysis and sharp estimates of exponential sums.

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