Embedding Dimensions of Matrices Whose Entries are Indefinite Distances in the Pseudo-Euclidean Space

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 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 + ⋯ + ( x p - y p ) 2 - ( x p + 1 - y p + 1 ) 2 - ⋯ - ( 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.