On the density of sets of the Euclidean plane avoiding distance 1

Abstract

A subset $A\subset {\mathbb{R}}^2$ is said to avoid distance $1$ if: $\forall x,y\in A,{\Vert x-y\Vert }_2\ne 1.$ 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 $m_1({\mathbb{R}}^2)\le 0.25647$ and ${\chi }_f({\mathbb{R}}^2)\ge 3.8991$.

13 pages, 8 figures

References in corpus (1)

• Better bounds for planar sets avoiding unit distances

Cited by in corpus (1)

• The density of planar sets avoiding unit distances