Clustering in typical unit-distance avoiding sets
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10474_2025_1556_ReferencePDF.pdf
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2.58 MB
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239dc53d7fc54fba2abd66756226b4ef
Author(s)
Cohen, A.
Mani, N.
Date Issued
September 22, 2025
Journal
Acta Mathematica Hungarica
Publisher
Springer International Publishing
Citation
Cohen, A., Mani, N. Clustering in typical unit-distance avoiding sets. Acta Math. Hungar. 176, 473–497 (2025).
Version
Author's final manuscript
Abstract
In the 1960s Moser asked how dense a subset of R d can be if no pairs of points in the subset are exactly distance 1 apart. There has been a long line of work showing upper bounds on this density. One curious feature of dense unit distance avoiding sets is that they appear to be ''clumpy,'' i.e. forbidding unit distances comes hand in hand with having more than the expected number distance ≈ 2 pairs. In this work we rigorously establish this phenomenon in R 2 . We show that dense unit distance avoiding sets have over-represented distance ≈ 2 pairs, and that this clustering extends to typical unit distance avoiding sets. To do so, we build off of the linear programming approach used previously to prove upper bounds on the density of unit distance avoiding sets.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s10474-025-01556-w
