  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.
