Embedding dimensions of matrices whose entries are indefinite distances in the pseudo-Euclidean space

TL;DR

This paper extends the classification of distance sets from Euclidean to pseudo-Euclidean spaces, developing a representation theory for indefinite distances and classifying largest 2-distance sets in small dimensions.

Contribution

It introduces a theory for indefinite-distance sets in pseudo-Euclidean spaces and classifies the largest 2-distance sets for small dimensions, advancing understanding beyond Euclidean cases.

Findings

Developed a representation theory for symmetric matrices in pseudo-Euclidean spaces.

Classified the largest 2-indefinite-distance sets for small dimensions.

Extended results of Euclidean s-distance sets to indefinite distances.

Abstract

A finite set of the Euclidean space is called an $s$-distance set provided 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. Lison\v{e}k (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 Lison\v{e}k 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 \[ || x-y ||=(x_1-y_1)^2 +\cdots +(x_p -y_p)^2-(x_{p+1}-y_{p+1})^2-\cdots -(x_{p+q}-y_{p+q})^2 \] 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.