Erdős Problem #957

Let \(A\subset \mathbb{R}^2\) be a set of size \(n\) and let \(\{d_1<\ldots<d_k\}\) be the set of distinct distances determined by \(A\). Let \(f(d)\) be the number of times the distance \(d\) is determined. Is it true that \[f(d_1)f(d_k) \leq (\tfrac{9}{8}+o(1))n^2?\] A question of Erdős and Pach [ErPa90]. The answer is yes, proved by Dumitrescu [Du19]. Here \(f(d)\) counts unordered pairs of points of \(A\) at distance \(d\) (`distanceMultiplicity`), and \(o(1)\) is uniform over all \(A\) of size \(n\).