Clustering in typical unit-distance avoiding sets

2024/07/06 by Cohen, Alex, Mani, Nitya
#Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2407.05071

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 ℝ². 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.

Related

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• Measurable sets with excluded distances
• On the unit distance problem
• Clustering Based on Pairwise Distances When the Data is of Mixed Dimensions
• Online unit clustering in higher dimensions