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Uniformly discrete forests with poor visibility. (English) Zbl 1397.51003

A dense planar forest is a set \(F \subset {\mathbb R}^2\) for which there exists a function \(f : (0, 1) \rightarrow {\mathbb R}^+\) such that for any \(\varepsilon\in (0, 1)\) and any line segment \({\mathcal l}\) in the plane of length at least \(f(\varepsilon)\), there is a point \(x \in F\) whose distance from \({\mathcal l}\) is at most \(\varepsilon\). Such a function \(f\) is called a visibility function for \(F\). The set \(F\), to be referred to as a forest, is uniformly discrete if there is a positive \(r\) so that the distance between any two points of \(F\) is at least \(r\). The forest has finite density if there exists a finite \(C\) such that for every \(t\geq 1\), the number of points of \(F\) in the ball of radius \(t\) centred at the origin is at most \(Ct^2\). Uniform discreteness is a strictly stronger requirement than that of finite density.
Y. Solomon and B. Weiss [Ann. Sci. Éc. Norm. Supér. (4) 49, No. 5, 1053–1074 (2016; Zbl 1379.52020)], while proving the existence of a uniformly discrete dense forest, using homogeneous dynamics and relying on a theorem of Ratner, provided no explicit bound for the visibility function. On the other hand, F. Adiceam [“How far can you see in a forest?”, IMRN, No. 16, 4867–4881 (2016)] proved the existence of a dense forest, which is finitely dense but not uniformly discrete, with visibility function \(c(\delta) \varepsilon^{ -2-\delta}\), for any fixed \(\delta > 0\).
The result in this paper significantly improves both of the above results by showing that:
There are absolute positive constants \(r\) and \(C\), so that there exists a uniformly discrete planar forest in which the distance between any two points is at least \(r\), and the function \[ f(\varepsilon)=\frac{2^{C\sqrt{\log(1/\varepsilon)}}}{\varepsilon} \] is a visibility function.
However, unlike the forests in the above papers, which are described explicitely, the proof here is a probabilitic one, showing only the existence of such a forest. The proof combines “some geometric considerations with the Lovász local lemma and a compactness argument.”

MSC:

51F99 Metric geometry
52C99 Discrete geometry

Citations:

Zbl 1379.52020
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References:

[1] Adiceam, F. (2016) How far can you see in a forest? IMRN 4867-4881. · Zbl 1404.52009
[2] Alon, N., A non-linear lower bound for planar epsilon-nets, Discrete Comput. Geom., 47, 235-244, (2012) · Zbl 1232.68161 · doi:10.1007/s00454-010-9323-7
[3] Alon, N. and Spencer, J. H. (2008) The Probabilistic Method, third edition, Wiley. · Zbl 1148.05001
[4] Bishop, C. J., A set containing rectifiable arcs QC-locally but not QC-globally, Pure Appl. Math. Quart., 1, 121-138, (2011) · Zbl 1250.30022
[5] Pólya, G. (1976) Problems and Theorems in Analysis, Vol. 2, Springer. · Zbl 0338.00001
[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. · Zbl 1405.37015
[7] Solomon, Y. and Weiss, B. (2014) Dense forests and Danzer sets. Ann. Sci. Éc. Norm. Supér491053-1074. · Zbl 1379.52020
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