Distinct Distances with ${\mathrm{\ell }}_p$ Spaces

We study Erd\H os’s distinct distances problem under ${\mathrm{\ell }}_p$ metrics with integer $p$. We improve the current best bound for this problem from $\mathrm{\Omega }(n^{4\mathrm{/}5})$ to $\mathrm{\Omega }(n^{6\mathrm{/}7-ϵ})$, for any $ϵ>0$. We also characterize the sets that span an asymptotically minimal number of distinct distances under the ${\mathrm{\ell }}_1$ and ${\mathrm{\ell }}_{\mathrm{\infty }}$ metrics.