More unit distances in arbitrary norms

TL;DR

This paper demonstrates that for any norm in d, one can construct large point sets with many unit distances, matching known upper bounds, and shows the presence of complete bipartite subgraphs in the unit distance graph for typical norms.

Contribution

It establishes the existence of large point sets with many unit distances in arbitrary norms and characterizes the structure of their unit distance graphs for typical norms.

Findings

Existence of point sets with d/2-o(1) n log n unit distances in any norm.

Construction of unit distance graphs containing K_{d,m} for all m in typical norms.

Matches the upper bounds for the number of 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 $m$.