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Uniformly Discrete Forests with Poor Visibility

  • NOGA ALON (a1) (a2)
Abstract

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.

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References
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[1] Adiceam F. (2016) How far can you see in a forest? IMRN 4867–4881.
[2] Alon N. (2012) A non-linear lower bound for planar epsilon-nets. Discrete Comput. Geom. 47 235244.
[3] Alon N. and Spencer J. H. (2008) The Probabilistic Method, third edition, Wiley.
[4] Bishop C. J. (2011) A set containing rectifiable arcs QC-locally but not QC-globally. Pure Appl. Math. Quart. 1 121138.
[5] Pólya G. (1976) Problems and Theorems in Analysis, Vol. 2, Springer.
[6] Solan O., Solomon Y. and Weiss B. On problems of Danzer and Gowers and dynamics on the space of closed subsets of ℝ d . IMRN to appear.
[7] Solomon Y. and Weiss B. (2014) Dense forests and Danzer sets. Ann. Sci. Éc. Norm. Supér 49 10531074.
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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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