Erdős Problem #661

Are there, for all large \(n\), some points \(x_1,\ldots,x_n,y_1,\ldots,y_n\in \mathbb{R}^2\) such that the number of distinct distances \(d(x_i,y_j)\) is \[o\left(\frac{n}{\sqrt{\log n}}\right)?\] We require all \(2n\) points to be distinct; without any distinctness one could take all points equal and get a single distance. Writing \(F(n)\) for the minimal number of such distances (`minimalCrossDistances ℝ² n`), the question asks whether \(F(n) = o(n / \sqrt{\log n})\). This is equivalent to the existence, for all large \(n\), of configurations whose count is at most \(g(n)\) for some fixed \(g(n) = o(n / \sqrt{\log n})\), since the infimum is attained. The source adds that one can also ask this for points in \(\mathbb{R}^3\). Taken literally, the question is easy there, since a cubic grid of \(2n\) points determines only \(O(n^{2/3})\) distinct distances, so we do not state it.