More unit distances in arbitrary norms

2024/10/10 by Greilhuber, Josef, Schildkraut, Carl, Tidor, Jonathan
#05C62 (Secondary) #52C10 (Primary) 52A20 #Combinatorics (math.CO) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2410.07557

Abstract

For d≥ 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))nlog₂n 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≥ 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.

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