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Uniformly Discrete Forests with Poor Visibility

Part of: Discrete geometry Metric geometry

Published online by Cambridge University Press:  09 October 2017

NOGA ALON
NOGA ALON*
Affiliation:
Sackler School of Mathematics and Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, USA (e-mail: nogaa@tau.ac.il)
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Abstract

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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.

MSC classification

Primary: 51F99: None of the above, but in this section
Secondary: 52C99: None of the above, but in this section
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 27 , Special Issue 4: Special Issue: Oberwolfach Workshop , July 2018 , pp. 442 - 448
Copyright
Copyright © Cambridge University Press 2017 

References

[1] Adiceam, F. (2016) How far can you see in a forest? IMRN 4867–4881.CrossRefGoogle Scholar
[2] Alon, N. (2012) A non-linear lower bound for planar epsilon-nets. Discrete Comput. Geom. 47 235–244.CrossRefGoogle Scholar
[3] Alon, N. and Spencer, J. H. (2008) The Probabilistic Method, third edition, Wiley.CrossRefGoogle Scholar
[4] Bishop, C. J. (2011) A set containing rectifiable arcs QC-locally but not QC-globally. Pure Appl. Math. Quart. 1 121–138.CrossRefGoogle Scholar
[5] Pólya, G. (1976) Problems and Theorems in Analysis, Vol. 2, Springer.CrossRefGoogle Scholar
[6] Solan, O., Solomon, Y. and Weiss, B. On problems of Danzer and Gowers and dynamics on the space of closed subsets of ℝd. IMRN to appear.Google Scholar
[7] Solomon, Y. and Weiss, B. (2014) Dense forests and Danzer sets. Ann. Sci. Éc. Norm. Supér 49 1053–1074.CrossRefGoogle Scholar