How far can you see in a forest?

TY - JOUR

T1 - How far can you see in a forest?

AU - Adiceam, Faustin

N1 - 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.

PY - 2016

Y1 - 2016

N2 - 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.

AB - 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.

KW - math.NT

KW - 11J71, 11J25, 11L03, 11L07, 11Z05, 51N20

U2 - 10.1093/imrn/rnv292

DO - 10.1093/imrn/rnv292

M3 - Article

SN - 1687-0247

VL - 16

SP - 4867

EP - 4881

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

ER -