A generalization of Larman–Rogers–Seidel’s theorem

A finite set XX in the dd-dimensional Euclidean space is called an ss-distance set if the set of Euclidean distances between any two distinct points of XX has size ss. Larman–Rogers–Seidel proved that if the cardinality of a two-distance set is greater than 2d+32d+3, then there exists an integer kk such that a2/b2=(k−1)/ka2/b2=(k−1)/k, where aa and bb are the distances. In this paper, we give an extension of this theorem for any ss. Namely, if the size of an ss-distance set is greater than some value depending on dd and ss, then certain functions of ss distances become integers. Moreover, we prove that if the size of XX is greater than the value, then the number of ss-distance sets is finite.