Solution to a Problem of Erd\H{o}s Concerning Distances and Points

Abstract

In 1997, Erd\H{o}s asked whether for arbitrarily large $n$ there exists a set of $n$ points in $\mathbb{R}^2$ that determines $O(\frac{n}{\sqrt{\log n}})$ distinct distances while satisfying the local constraint that every 4-point subset determines at least 3 distinct pairwise distances. We construct $n$-point sets from an $m\times m$ box of the lattice $L = \{(x,\sqrt{2}y):x,y \in \mathbb{Z}\} \subset \mathbb{R}^2.$ The distinct distance bound follows from applying Bernays' theorem to the number of integers represented by the binary quadratic form $u^2 + 2v^2$. The local 4-point constraint is verified through Perucca's similarity classification of the six similarity types determining exactly two distances.