Erdős Problem #95

Let \(x_1,\ldots,x_n\in\mathbb{R}^2\) determine the set of distances \(\{u_1,\ldots,u_t\}\). Suppose \(u_i\) appears as the distance between \(f(u_i)\) many pairs of points. Then for all \(\epsilon>0\) \[\sum_i f(u_i)^2 \ll_\epsilon n^{3+\epsilon}.\] The case when the points determine a convex polygon was solved by Altman [Al63]. Note it is trivial that \(\sum f(u_i)=\binom{n}{2}\). Solved by Guth and Katz [GuKa15] who proved the upper bound \[\sum_i f(u_i)^2 \ll n^3\log n.\] See also [94](https://www.erdosproblems.com/94).