The density of planar sets avoiding unit distances

TL;DR

This paper proves that the maximum density of a measurable planar set avoiding any pair of points at distance one is at most 1/4, improving previous bounds and confirming a conjecture of Erdős.

Contribution

The authors establish a new upper bound of 0.2470 for the density, confirming Erdős's conjecture and improving upon earlier estimates.

Findings

Maximum density of such sets is at most 1/4

Improved upper bound of 0.2470 on density

Confirmed Erdős's conjecture

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 $1/4$. Our argument implies the upper bound of $0.2470$.