Larman-Rogers-Seidel定理的推广
A generalization of Larman-Rogers-Seidel's theorem
摘要：
d维欧几里得空间中的有限集X称为s-距离集合，如果X中任意两个不同点之间的欧氏距离的集合的大小为s。Larman-Rogers-Seidel证明了如果一个二距离集的基数大于2d+3，则存在一个整数k使得a^2/b^2=（k-1）/k，其中a和b是距离。本文给出了这一定理的一个推广。也就是说，如果s-距离集的大小大于某个取决于d和s的值，则s距离的某些函数变为整数。证明了如果X的大小大于该值，则s-距离集的个数是有限的.
英文摘要：
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.
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