The density of planar sets avoiding unit distances

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.

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Bibliographic Details
Published in:Mathematical programming 2024-09, Vol.207 (1-2), p.303-327
Main Authors: Ambrus, Gergely, Csiszárik, Adrián, Matolcsi, Máté, Varga, Dániel, Zsámboki, Pál
Format: Article
Language:English
Subjects:
Calculus of Variations and Optimal Control
Optimization
Combinatorics
Density
Full Length Paper
Mathematical and Computational Physics
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Mathematics of Computing
Numerical Analysis
Theoretical
Upper bounds
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Description
Summary: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.
ISSN:0025-5610
1436-4646
DOI:10.1007/s10107-023-02012-9