More unit distances in arbitrary norms

For (Formula presented.) and any norm on (Formula presented.), we prove that there exists a set of (Formula presented.) points that spans at least (Formula presented.) unit distances under this norm for every (Formula presented.). 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 (Formula presented.) and a typical norm on (Formula presented.), the unit distance graph of this norm contains a copy of (Formula presented.) for all (Formula presented.).