Distinct Distances with ${\ell }_p$ Spaces

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.

15 pages, 4 figures

References in corpus (2)

• Distinct distances in the complex plane
• Sets in ${\mathbb{R}}^d$ determining $k$ taxicab distances