The Minkowski grid has robustly many repeated distances

Abstract

We show that there exists a constant $\hat{I}´>0$ 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 $|A|\ge 2$, \[ \max_{λ>0} \#\{(a,b)\in A \times A: a\ne b,\ \lvert a-b\rvert=λ\} \gtrsim \frac{|A|^2}{n^{1-δ}}.\] 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 Erdős unit distance problem and the Elekes-Rónyai problem. Taking $A=P$, the inequality above gives a distance occurring $n^{1+\hat{I}´}$ 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 $\gtrsim n^{1-\hat{I}´}$ must contain isosceles triangles, and (2) all subsets of $P$ of size $\gtrsim n^{1/2-\hat{I}´}$ must contain repeated distances. These features give polynomially improved estimates for old problems of Erdős. The existence of a set satisfying property (1) confirms a conjecture of Erdős from 1980, whereas the existence of a set with property (2) answers a question of Conlon-Fox-Gasarch-Harris-Ulrich-Zbarsky in the negative.

8 pages