More unit distances in arbitrary norms

IF 0.9 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society Pub Date : 2025-07-29 DOI:10.1112/blms.70133

Josef Greilhuber, Carl Schildkraut, Jonathan Tidor

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Abstract

For $d ⩾ 2$ and any norm on $R^{d}$ , 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 $d ⩾ 3$ and a typical norm on $R^{d}$ , the unit distance graph of this norm contains a copy of $K_{d, m}$ for all $m$ .

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任意范数中更多的单位距离

对于d小于2 $d\geqslant 2$和R d $\mathbb {R}^d$上的任何规范，我们证明了存在一个n个$n$点的集合，它张成至少（d 2−o (1)）) n log 2 n $(\tfrac{d}{2}-o(1))n\log _2n$在这个范数下的单位距离对于每一个n $n$。这与最近由Alon， buciki和Sauermann证明的典型范数（即，位于一个趋同集中的范数）的上界相匹配。我们还显示，对于d小于3 $d\geqslant 3$和R d $\mathbb {R}^d$上的典型范数，该范数的单位距离图包含K d的副本，M $K_{d,m}$代表所有M $m$。

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