Crescent configurations in normed spaces

TL;DR

This paper extends the study of crescent configurations, originally posed in Euclidean spaces, to general normed spaces, constructing specific configurations and analyzing their structural properties, especially in the Chebyshev norm.

Contribution

It introduces the concept of strong crescent configurations in normed spaces, constructs examples in various norms, and characterizes the structure of line-like crescent configurations in the Chebyshev norm.

Findings

Constructed strong crescent configurations of size 4 in arbitrary norms.

Built larger configurations in Euclidean, taxicab, and Chebyshev norms.

Proved structural rigidity of line-like crescent configurations in the Chebyshev norm for sizes n ≥ 7.

Abstract

We study the problem of crescent configurations, posed by Erd\H{o}s in 1989. A crescent configuration is a set of $n$ points in the plane such that: 1) no three points lie on a common line, 2) no four points lie on a common circle, 3) for each $1 \leq i \leq n - 1$, there exists a distance which occurs exactly $i$ times. Constructions of sizes $n \leq 8$ have been provided by Liu, Pal\'{a}sti, and Pomerance. Erd\H{o}s conjectured that there exists some $N$ for which there do not exist crescent configurations of size $n$ for all $n \geq N$. We extend the problem of crescent configurations to general normed spaces $(\mathbb{R}^2, \| \cdot \|)$ by studying strong crescent configurations in $\| \cdot \|$. In an arbitrary norm $\|\cdot \|$, we construct a strong crescent configuration of size 4. We also construct larger strong crescent configurations in the Euclidean, taxicab, and Chebyshev norms, of sizes $n \leq 6$, $n \leq 8$, and $n \leq 8$ respectively. When defining strong crescent configurations, we introduce the notion of line-like configurations in $\|\cdot \|$. A line-like configuration in $\|\cdot \|$ is a set of points whose distance graph is isomorphic to the distance graph of equally spaced points on a line. In a broad class of norms, we construct line-like configurations of arbitrary size. Our main result is a crescent-type result about line-like configurations in the Chebyshev norm. A line-like crescent configuration is a line-like configuration for which no three points lie on a common line and no four points lie on a common $\|\cdot \|$ circle. We prove that for $n \geq 7$, every line-like crescent configuration of size $n$ in the Chebyshev norm must have a rigid structure. Specifically, it must be a perpendicular perturbation of equally spaced points on a horizontal or vertical line.