Distinct Distances with $\ell_p$ Spaces

TL;DR

This paper investigates the number of distinct distances determined by point sets under $ ext{L}_p$ metrics, improving bounds for the problem and characterizing minimal configurations for specific metrics.

Contribution

It advances the understanding of Erdős's distinct distances problem for $ ext{L}_p$ spaces, providing improved bounds and characterizations for minimal distance sets.

Findings

Improved lower bound from $ ext{Ω}(n^{4/5})$ to $ ext{Ω}(n^{6/7- ext{ε}})$ for $ ext{L}_p$ metrics.

Characterized sets with minimal distinct distances under $ ext{L}_1$ and $ ext{L}_ ext{∞}$ metrics.

Abstract

We study Erd\H os's distinct distances problem under $\ell_p$ metrics with integer $p$. We improve the current best bound for this problem from $\Omega(n^{4/5})$ to $\Omega(n^{6/7-\epsilon})$, for any $\epsilon>0$. We also characterize the sets that span an asymptotically minimal number of distinct distances under the $\ell_1$ and $\ell_\infty$ metrics.