Distinct distances in finite planar sets

Abstract

For each n ⩾ 3 let Fn denote the set of all integer vectors f = (f1, f2, …, fn) with 1 ⩽ f1 ⩽ f2 ⩽ ··· ⩽ fn ⩽ n − 1 for which there exists a set x1, x2, …, xn of n points in the plane such that each xi has exactly fi different distances to the other n − 1 points. Thus, F3 = n0 ⊮(1, 1, 1), (1, 2, 2), (2, 2, 2) . We determine all n-point configurations for n ⩽ 7 that minimize ∑fi over Fn and show that min Larr; bEfi: f ∈ F8gsim; = 24. We note that every small n except n ∈ 8, 9 has a subset of the triangular lattice amont its sum-minimizing configurations, and conjecture that subsets of the lattice minimize ∑fi for all large n.