Clustering in typical unit-distance avoiding sets

TL;DR

This paper proves that dense sets avoiding unit distances in the plane tend to cluster around pairs separated by approximately 2 units, revealing a 'clumpy' structure in such sets.

Contribution

It rigorously establishes the clustering phenomenon in dense unit distance avoiding sets in the plane, extending previous upper bound methods.

Findings

Dense unit distance avoiding sets have over-represented pairs at distance ~2.

Clustering phenomenon extends to typical such sets.

Builds on linear programming methods used for upper bounds.

Abstract

In the 1960s Moser asked how dense a subset of $\mathbb{R}^d$ 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 $\approx 2$ pairs. In this work we rigorously establish this phenomenon in $\mathbb{R}^2$. We show that dense unit distance avoiding sets have over-represented distance $\approx 2$ 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.