The multiplicity of the two smallest distances among points Original Research Article

Let 1 = d1 < d2 < ⋯ < dk denote the distinct distances determined by a set of n points in the plane. The multiplicity of the two smallest distances is smaller than 6n and it is maximized by the triangular lattice, where d2 = √3. We partially answer a question of Erdös and Vesztergombi by proving that d2 ≠ √3 implies that the multiplicity of the two smallest distances is at most 4n unless d2 is (√5 + 1)/2 or 1/(2 sin 15). In the case d2 = (√5 + 1)/2, the multiplicity is at most 4.5n. We also show some extremal configurations for different values of d2.