
Mathematics 2009
A generalization of LarmanRogersSeidel's theoremAbstract: A finite set X in the ddimensional Euclidean space is called an sdistance set if the set of Euclidean distances between any two distinct points of X has size s. LarmanRogersSeidel proved that if the cardinality of a twodistance set is greater than 2d+3, then there exists an integer k such that a^2/b^2=(k1)/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 sdistance 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 sdistance sets is finite.
