On multiplicities of interpoint distances
Felix Christian Clemen, Adrian Dumitrescu, D. Liu · Acta Mathematica Academiae Scientiarum Hungaricae · 2025
Abstract Given a set $$X\subseteq\mathbb{R}^2$$ X ⊆ R 2 of $$n$$ n points and a distance $$d>0$$ d > 0 , the multiplicity of $$d$$ d is the number of times the distance $$d$$ d appears between points in $$X$$ X . Let $$a_1(X) \geq a_2(X) \geq \cdots \geq a_m(X)$$ a 1 ( X ) ≥ a 2 ( X ) ≥ ⋯ ≥ a m ( X ) denote the multiplicities of the $$m$$ m distances determined by $$X$$ X and let $$a(X)=(a_1(X),\dots,a_m(X))$$ a ( X ) = ( a 1 ( X ) , ⋯ , a m ( X ) ) . In this paper, we study several questions from Erdős’s time regarding distance multiplicities. Among other results, we show that: (1) If $$X$$ X is convex or “not too convex”, then there exists a distance other than the diameter that has multiplicity at most $$n$$ n . (2) There exists a set $$X\subseteq\mathbb{R}^2$$ X ⊆ R