Lattice Configurations Determining Few Distances

TL;DR

This paper revisits Erdős and Fishburn's question on the maximum number of points with limited distinct distances, verifying previous claims, analyzing lattice configurations, correcting errors, and proposing new conjectures.

Contribution

It rigorously verifies prior claims, analyzes lattice configurations, corrects errors, and offers new conjectures on optimal point arrangements with few distances.

Findings

Regular polygons with or without a center are not optimal for k ≥ 7.

Collected extensive data on lattice configurations.

Proposed new conjectures based on data analysis.

Abstract

We begin by revisiting a paper of Erd\H{o}s and Fishburn, which posed the following question: given $k\in \mathbb{N}$, what is the maximum number of points in a plane that determine at most $k$ distinct distances, and can such optimal configurations be classified? We rigorously verify claims made in remarks in that paper, including the fact that the vertices of a regular polygon, with or without an additional point at the center, cannot form an optimal configuration for any $k\geq 7$. Further, we investigate configurations in both triangular and rectangular lattices studied by Erd\H{o}s and Fishburn. We collect a large amount of data related to these and other configurations, some of which correct errors in the original paper, and we use that data and additional analysis to provide explanations and make conjectures.