The density of planar sets avoiding unit distances

Abstract

By improving upon previous estimates on a problem posed by L. Moser, we prove a conjecture of Erdős that the density of any measurable planar set avoiding unit distances cannot exceed $1/4$. Our argument implies the upper bound of $0.2470$.

24 pages, 6 figures. Final version, to appear in Mathematical Programming

References in corpus (1)

• Array Programming with NumPy

Cited by in corpus (2)

• Optimization of trigonometric polynomials with crystallographic symmetry and spectral bounds for set avoiding graphs
• On lower bounds of the density of planar periodic sets without unit distances