A generalization of Larman-Rogers-Seidel's theorem

TL;DR

This paper extends the Larman-Rogers-Seidel theorem to s-distance sets in Euclidean space, establishing conditions under which certain distance functions are integers and proving finiteness of such sets beyond a size threshold.

Contribution

It generalizes a classical theorem from two-distance sets to s-distance sets, providing new bounds and finiteness results.

Findings

Certain functions of s distances are integers for large enough sets.

The number of s-distance sets is finite beyond a specific size threshold.

The extension applies to any s, not just two-distance sets.

Abstract

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.