Geodesic
On the density of sets of the Euclidean plane avoiding distance 1
Discrete mathematics & theoretical computer science, Tome 23 (2021-2022) no. 1.

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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 $\chi_f(\mathbb R^2) \geq 3.8991$.
DOI : 10.46298/dmtcs.5153
Classification : 05C12, 05C15, 52C20, 52C26 
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Bellitto, Thomas; Pêcher, Arnaud; Sédillot, Antoine. On the density of sets of the Euclidean plane avoiding distance 1. Discrete mathematics & theoretical computer science, Tome 23 (2021-2022) no. 1. doi : 10.46298/dmtcs.5153. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.5153/

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