On the density of sets of the Euclidean plane avoiding distance 1 [article]

Thomas Bellitto, Arnaud Pêcher, Antoine Sédillot
2021 arXiv   pre-print
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.
arXiv:1810.00960v3 fatcat:4tt224lztvbz7epak3jpfgcqfq