Ben Lund and I were sitting in a small Korean restaurant and discussing an old conjecture of Erdős. Erdős suggested that if a set of points in spans distinct distances, then the vertices of this set must span either a square or an equilateral triangle. Presumably, this conjecture was made because the two best-known examples of sets with distinct distances are subsets of the integer lattice and of the triangular lattice. (By the triangular lattice, I mean the vertices of a tiling of the plane with equilateral triangles.)

* A subset of the integer lattice and a subset of the triangular lattice.*

A subset of the integer lattice spans many squares and no equilateral triangles, while a subset of the triangular lattice spans many equilateral triangles and no squares. We can create various similar configurations by taking one of these two configurations and then applying rotations, uniform scaling, and taking subsets. Such configurations still contain squares or equilateral triangles. Ben and I were wondering whether we can find additional configurations, which are not “related” to the integer lattice or to the triangular lattice. Surprisingly, we managed to find a large family of other configurations. Moreover, many of these configurations do not span any squares or triangles, disproving Erdős’ conjecture. We then discovered that these configurations were already known, and appeared in a paper by Moree and Osburn. I guess that this is another case of lack of communication between different mathematical communities…

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