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On the density of sets of the Euclidean plane avoiding distance 1
release_xo62dw64fbgzblxjifsikc4nee
by
Thomas Bellitto, Arnaud Pêcher, Antoine Sédillot
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2020
Abstract
A subset A ⊂ℝ^2 is said to avoid distance 1 if: ∀
x,y ∈ A, x-y _2 ≠ 1. In this paper we study the number
m_1(ℝ^2) which is the supremum of the upper densities of measurable
sets avoiding distance 1 in the Euclidean plane. Intuitively, m_1(ℝ^2) represents the highest proportion of the plane that can be filled by a
set avoiding distance 1. This parameter is related to the fractional chromatic
number χ_f(ℝ^2) of the plane.
We establish that m_1(ℝ^2) ≤ 0.25647 and χ_f(ℝ^2)
≥ 3.8991.
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1810.00960v2
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