Improvement of generalization of Larman-Rogers-Seidel's theorem

Abstract

A finite set $X$ in the $d$-dimensional Euclidean space is called an $s$-distance set if the set of distances between any two distinct points of $X$ has size $s$. In 1977, Larman-Rogers-Seidel proved that if the cardinality of an two-distance set is large enough, then there exists an integer $k$ such that the two distances $\alpha$, $\beta$ $(\alpha <\beta )$ having the integer condition, namely, $\frac{{\alpha }^2}{{\beta }^2}=\frac{k-1}{k}$. In 2011, Nozaki generalized Larman-Rogers-Seidel's theorem to the case of $s$-distance sets, i.e. if the cardinality of an $s$-distance set $|X|⩾2N$ with distances ${\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_s$, where $N=\left(\genfrac{}{}{0ex}{}{d+s-1}{s-1}\right)+\left(\genfrac{}{}{0ex}{}{d+s-2}{s-2}\right)$, then the numbers $k_i={\prod }_{j=1,2,\cdots ,s,j\ne i}\frac{{\alpha }_j^2}{{\alpha }_j^2-{\alpha }_i^2}$ are integers. In this note, we reduce the lower bound of the requirement of integer condition of $s$-distance sets in ${\mathbb{R}}^d$. Furthermore, we can show that there are only finitely many $s$-distance sets $X$ in ${\mathbb{R}}^d$ with $|X|⩾2\left(\genfrac{}{}{0ex}{}{d+s-1}{s-1}\right).$

6 pages, no figure