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

Abstract

We address an analog of a problem introduced by Erdős and Fishburn, itself an inverse formulation of the famous Erdős distance problem, in which the usual Euclidean distance is replaced with the metric induced by the ${\ell }^1$-norm, commonly referred to as the taxicab metric. Specifically, we investigate the following question: given $d,k\in \mathbb{N}$, what is the maximum size of a subset of ${ℝ}^d$ 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.