The Minkowski grid has robustly many repeated distances

Abstract

We show that there exists a constant $\delta > 0$ such that for any positive integer $n$ there exists a set of $n$ points $P \subset \mathbb{R}^2$ with the following property: for every subset $A \subseteq P$ of size $|A| \geq 2$, \[ \max_{\lambda>0} \#\{(a,b)\in A \times A: a\ne b,\ \lvert a-b\rvert=\lambda\} \gtrsim \frac{|A|^2}{n^{1-\delta}}.\] Our result is a vertical amplification of a robust Ramanujan estimate recently established by Croot-Mao-Pohoata-Sheffer-Yip for arbitrary subsets of the ordinary square grid, and is inspired by recent constructions for the Erd\H{o}s unit distance problem and the Elekes-R\'onyai problem. Taking $A=P$, the inequality above gives a distance occurring $n^{1+\delta}$ times in $P$; thereby a scaled copy of $P$ is a counterexample for the unit-distance conjecture. In addition, the same inequality shows that (1) all subsets of $P$ of size $\gtrsim n^{1-\delta}$ must contain isosceles triangles, and (2) all subsets of $P$ of size $\gtrsim n^{1/2-\delta}$ must contain repeated distances. These features give polynomially improved estimates for old problems of Erd\H{o}s. The existence of a set satisfying property (1) confirms a conjecture of Erd\H{o}s from 1980, whereas the existence of a set with property (2) answers a question of Conlon-Fox-Gasarch-Harris-Ulrich-Zbarsky in the negative.