The density of planar sets avoiding unit distances
Metric Geometry
2024-06-25 v3
Abstract
By improving upon previous estimates on a problem posed by L. Moser, we prove a conjecture of Erd\H{o}s that the density of any measurable planar set avoiding unit distances cannot exceed . Our argument implies the upper bound of .
Keywords
Cite
@article{arxiv.2207.14179,
title = {The density of planar sets avoiding unit distances},
author = {Gergely Ambrus and Adrián Csiszárik and Máté Matolcsi and Dániel Varga and Pál Zsámboki},
journal= {arXiv preprint arXiv:2207.14179},
year = {2024}
}
Comments
24 pages, 6 figures. Final version, to appear in Mathematical Programming