More unit distances in arbitrary norms

Abstract

For $d\geq 2$ and any norm on $\mathbb R^d$, we prove that there exists a set of $n$ points that spans at least $(\tfrac d2-o(1))n\log_2n$ unit distances under this norm for every $n$. This matches the upper bound recently proved by Alon, Buci\'c, and Sauermann for typical norms (i.e., norms lying in a comeagre set). We also show that for $d\geq 3$ and a typical norm on $\mathbb R^d$, the unit distance graph of this norm contains a copy of $K_{d,m}$ for all $m$.