More unit distances in arbitrary norms

Abstract

For $d\ge 2$ and any norm on ${\mathbb{R}}^d$, we prove that there exists a set of $n$ points that spans at least $(\frac{d}{2}-o(1))n{\log }_{2}n$ unit distances under this norm for every $n$. This matches the upper bound recently proved by Alon, Bucić, and Sauermann for typical norms (i.e., norms lying in a comeagre set). We also show that for $d\ge 3$ and a typical norm on ${\mathbb{R}}^d$, the unit distance graph of this norm contains a copy of $K_{d,m}$ for all $m$.

14 pages