On the density of sets of the Euclidean plane avoiding distance 1Article
Authors: Thomas Bellitto ; Arnaud Pêcher ; Antoine Sédillot
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Thomas Bellitto;Arnaud Pêcher;Antoine Sédillot
A subset $A \subset \mathbb R^2$ is said to avoid distance $1$ if: $\forall
x,y \in A, \left\| x-y \right\|_2 \neq 1.$ In this paper we study the number
$m_1(\mathbb R^2)$ which is the supremum of the upper densities of measurable
sets avoiding distance 1 in the Euclidean plane. Intuitively, $m_1(\mathbb
R^2)$ 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 $\chi_f(\mathbb R^2)$ of the plane.
We establish that $m_1(\mathbb R^2) \leq 0.25647$ and $\chi_f(\mathbb R^2)
\geq 3.8991$.