Solution to a Problem of Erdős Concerning Distances and Points
Benjamin Grayzel
TL;DR
This paper resolves Erdős's distance problem by constructing $n$-point sets in the plane with every $4$-point subset determining at least three distinct distances, while the total number of distinct distances is $O\left(\frac{n}{\sqrt{\log n}}\right)$. The construction uses an anisotropic lattice $L=\{(x,\sqrt{2}y):x,y\in\mathbb{Z}\}$ and its $m\times m$ box $P_m$, together with the representation of squared distances as $Q(u,v)=u^2+2v^2$ and Bernays's asymptotics for representations of primitive binary quadratic forms, yielding $|D(P_m)| \le B_Q(3m^2)$ with $B_Q(x)\sim C_{-8}\,x/\sqrt{\log x}$. The local constraint is verified by ruling out the six two-distance four-point configurations classified by Perucca, showing that none occur in $P_m$ due to elementary lattice obstructions and irrationality arguments in $\mathbb{Q}(\sqrt{2})$ and the golden ratio. This provides a constructive affirmative answer to Erdős's question and demonstrates how an anisotropic, distance-structure–aware lattice can simultaneously bound the global distinct-distance set and satisfy a strong local condition.
Abstract
In 1997, Erdős 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.