A generalization of Larman–Rogers–Seidel’s theorem

A 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 2 d + 3 , then there exists an integer k such that a 2 / b 2 = ( 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.