The Minkowski grid has robustly many repeated distances

Abstract

We show that there exists a constant $δ> 0$ 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 $|A| \geq 2$, \[ \max_{λ>0} \#\{(a,b)\in A \times A: a\ne b,\ \lvert a-b\rvert=λ\} \gtrsim \frac{|A|^2}{n^{1-δ}}.\] 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 $n^{1+δ}$ 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 $\gtrsim n^{1-δ}$ must contain isosceles triangles, and (2) all subsets of $P$ of size $\gtrsim n^{1/2-δ}$ 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.

Disclosure

“discussions. We would also like to acknowledge the usage of AI in preparation of this manuscript. The collaboration and present paper started with the first author’s manuscript from [Lee26], who, independently of the other authors, used ChatGPT to discover the existence of a set of n points P in R2 with the property that every subset A ⊆ P of size |A| ≥ n1/2−δ determines a repeated distance. This established that fdist (n) ≲ n1/2−δ . The proof from [Lee26] combined the new local-”

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Substantial mathematical content or result generation

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10

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2026-08-24T00:34:09.244Z

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