Bulletin of the Iranian Mathematical Society ( IF 0.7 ) Pub Date : 2024-01-04 , DOI: 10.1007/s41980-023-00842-z Hiroshi Nozaki , Masashi Shinohara , Sho Suda
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A finite set of the Euclidean space is called an s-distance set provided that the number of Euclidean distances in the set is s. Determining the largest possible s-distance set for the Euclidean space of a given dimension is challenging. This problem was solved only when dealing with small values of s and dimensions. Lisoněk (J Combin Theory Ser A 77(2):318–338, 1997) achieved the classification of the largest 2-distance sets for dimensions up to 7, using computer assistance and graph representation theory. In this study, we consider a theory analogous to these results of Lisoněk for the pseudo-Euclidean space \(\mathbb {R}^{p,q}\). We consider an s-indefinite-distance set in a pseudo-Euclidean space that uses the value
$$\begin{aligned} || \varvec{x}-\varvec{y}||&=(x_1-y_1)^2 +\cdots +(x_p -y_p)^2 \\&\quad -(x_{p+1}-y_{p+1})^2-\cdots -(x_{p+q}-y_{p+q})^2 \end{aligned}$$instead of the Euclidean distance. We develop a representation theory for symmetric matrices in the context of s-indefinite-distance sets, which includes or improves the results of Euclidean s-distance sets with large s values. Moreover, we classify the largest possible 2-indefinite-distance sets for small dimensions.
中文翻译：
伪欧几里德空间中条目为不定距离的矩阵的嵌入维数
欧几里德空间的有限集称为s距离集，前提是该集合中欧几里德距离的数量为s。确定给定维度的欧几里得空间的最大可能s距离集具有挑战性。这个问题只有在处理较小的s值和维度时才能解决。Lisoněk（J Combin Theory Ser A 77(2):318–338, 1997）利用计算机辅助和图表示理论，实现了维度高达 7 的最大 2 距离集的分类。在本研究中，我们考虑了一种类似于 Lisoněk 对于伪欧几里得空间\(\mathbb {R}^{p,q}\)的结果的理论。我们考虑伪欧几里得空间中使用该值的s不定距离集
$$\begin{对齐} || \varvec{x}-\varvec{y}||&=(x_1-y_1)^2 +\cdots +(x_p -y_p)^2 \\&\quad -(x_{p+1}-y_{p) +1})^2-\cdots -(x_{p+q}-y_{p+q})^2 \end{对齐}$$而不是欧几里德距离。我们在s不定距离集的背景下开发了对称矩阵的表示理论，其中包括或改进了具有大s值的欧几里得s距离集的结果。此外，我们对小维度的最大可能 2-不定距离集进行分类。
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