Distinct Distances with $ell_p$ Spaces

We study ErdH oss 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.