Separated distances in fixed dimension
A finite Euclidean point set is distance-separated if the lengths of any two distinct unordered edges differ by at least one. Counting gives a diameter lower bound of binom(n,2), whereas Erdős asked for a lower bound with leading term n^2. We give two constructions in fixed dimension. For every 0 = 4m0 there is a distance-separated n-point set in dimension 10m0-1 with diameter = 400000. The constructions use two orthogonal circle families to control distances within and between clusters separately. Their common geometric argument reduces to a uniform operator estimate for the first result…
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