How far can you see in a forest?

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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.
Original languageEnglish
Pages (from-to)4867-4881
Number of pages13
JournalInternational Mathematics Research Notices
Early online date14 Oct 2015
Publication statusPublished - 2016

Bibliographical note

This is an extended version of a paper to appear. Minor typos have been corrected. © The Author(s) 2015. Published by Oxford University Press. All rights reserved.


  • math.NT
  • 11J71, 11J25, 11L03, 11L07, 11Z05, 51N20

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