Discrete Subsets of R 2 and the Associated Distance Sets

We prove that a well-distributed subset of R 2 can have a separated distance set only if the distance is induced by a polygon. Basic definitions Separated sets. We say that S ⊂ R d is separated if there exists c separation > 0 such that ||x − y|| ≥ c separation for every x, y ∈ S, x = y. Here and throughout the paper, ||x|| = x 2 1 + · · · + x 2 d is the standard Euclidean distance. Well-distributed sets. We say that S ⊂ R d is well-distributed if there exists a C density > 0 such that every cube of side-length C density contains at least one element of S. Notation: A B, with respect to the parameter R, means that there exists a positive constant C ǫ such that A ≤ C ǫ R ǫ B for any ǫ > 0. Similarly, A B means that there exists a C > 0 such that A ≤ CB, and A ≈ B means that A B and B A.