2 research outputs found
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 2d+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.Comment: 12 pages, no figur