Given a set \(X\subseteq\mathbb{R}^2\) of \(n\) points and a distance \(d>0\) , the multiplicity of \(d\) is thenumber of times the distance \(d\) appears between points in \(X\) . Let \(a_1(X) \geq a_2(X) \geq \cdots \geq a_m(X)\) denote the multiplicities of the \(m\) distances determined by \(X\) and let \(a(X)=(a_1(X),\dots,a_m(X))\) . In this paper, we study severalquestions from Erdős’s time regarding distance multiplicities. Among other results, we show that: (1) If \(X\) is convex or “not too convex”, then there exists a distance other than the diameter that has multiplicity at most \(n\) .
(2) There exists a set \(X\subseteq\mathbb{R}^2\) of \(n\) points, such that many distances occur with high multiplicity. In particular, at least \(n^{\Omega(1/\log\log{n})}\) distances have superlinear multiplicity in \(n\) .
(3) For any (not necessarily fixed) integer \(1\leq k\leq\log{n}\) , there exists \( {X\subseteq\mathbb{R}^2 } \) of \(n\) points, such that the difference between the \(k^{\text{th}}\) and \((k+1)^{\text{th}}\) largest multiplicities is at least \(\Omega(\frac{n\log{n}}{k})\) . Moreover, the distances in \(X\) with the largest \(k\) multiplicities can be prescribed.
(4) For every \(n\in N\) , there exists \(X\subseteq\mathbb{R}^2\) of \(n\) points, not all collinear or cocircular, such that \(a(X)= (n-1,n-2,\ldots,1)\) . There also exists \(X\subseteq\mathbb{R}^2\) of \(n\) points with pairwise distinct distance multiplicities and \(a(Y) \neq (n-1,n-2,\ldots,1)\) .
