A generalization of Larman–Rogers–Seidel’s theorem

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