Sets in $\mathbb{R}^d$ determining $k$ taxicab distances

TL;DR

This paper investigates the maximum size of point sets in Euclidean space that determine a limited number of taxicab distances, providing complete classifications in low dimensions and partial results in higher dimensions.

Contribution

It fully characterizes the maximum size of such sets in two dimensions and the case of one distance in three dimensions, offering partial solutions for the general case.

Findings

Complete classification in $oldsymbol{d=2}$

Solution for $oldsymbol{k=1}$ in $oldsymbol{d=3}$

Partial results for the general case with additional assumptions

Abstract

We address an analog of a problem introduced by Erd\H{o}s and Fishburn, itself an inverse formulation of the famous Erd\H{o}s distance problem, in which the usual Euclidean distance is replaced with the metric induced by the $\ell^1$-norm, commonly referred to as the $\textit{taxicab metric}$. Specifically, we investigate the following question: given $d,k\in \mathbb{N}$, what is the maximum size of a subset of $\mathbb{R}^d$ that determines at most $k$ distinct taxicab distances, and can all such optimal arrangements be classified? We completely resolve the question in dimension $d=2$, as well as the $k=1$ case in dimension $d=3$, and we also provide a full resolution in the general case under an additional hypothesis.