The density of planar sets avoiding unit distances

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Gergely Ambrus, Adrián Csiszárik, Máté Matolcsi, Dániel Varga, Pál Zsámboki

Published: 01 Jan 2024, Last Modified: 15 May 2025Math. Program. 2024EveryoneRevisionsBibTeXCC BY-SA 4.0
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 is less than 1/4. Our argument implies the upper bound of 0.2470.