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 $|X|\geqslant 2N$ with distances $\alpha_1,\alpha_2,\cdots,\alpha_s$, where $N=\binom{d+s-1}{s-1}+\binom{d+s-2}{s-2}$, then the numbers $k_i=\prod_{j=1,2,\cdots,s,\text{ }j\neq i}\frac{\alpha_{j}^{2}}{\alpha_{j}^{2}-\alpha_{i}^{2}}$ 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 $|X|\geqslant 2\binom{d+s-1}{s-1}.$