Uniformly Discrete Forests with Poor Visibility

We prove that there is a set F in the plane so that the distance between any two points of F is at least 1, and for any positive ϵ < 1, and every line segment in the plane of length at least ϵ−1−o(1), there is a point of F within distance ϵ of the segment. This is tight up to the o(1)-term in the...

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Published in:Combinatorics, probability & computing probability & computing, 2018-07, Vol.27 (4), p.442-448
Main Author: ALON, NOGA
Format: Article
Language:English
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Summary:We prove that there is a set F in the plane so that the distance between any two points of F is at least 1, and for any positive ϵ < 1, and every line segment in the plane of length at least ϵ−1−o(1), there is a point of F within distance ϵ of the segment. This is tight up to the o(1)-term in the exponent, improving earlier estimates of Peres, of Solomon and Weiss, and of Adiceam.
ISSN:0963-5483
1469-2163
DOI:10.1017/S0963548317000505