Regular two distance sets

TL;DR

This paper thoroughly investigates regular two-distance sets in Euclidean spaces, exploring their properties, constructions, and connections to other mathematical structures like equiangular lines and designs.

Contribution

It introduces new properties, provides constructions of two-distance sets and frames, and explores their relationships with equiangular lines and quasi-symmetric designs.

Findings

Characterization of regular two-distance sets

Construction methods for two-distance frames

Connections to equiangular lines and designs

Abstract

This paper makes a deep study of regular two-distance sets. A set of unit vectors $X$ in Euclidean space $\RR^n$ is said to be regular two-distance set if the inner product of any pair of its vectors is either $\alpha$ or $\beta$, and the number of $\alpha$ (and hence $\beta$) on each row of the Gram matrix of $X$ are the same. We present various properties of these sets as well as focus on the case where they form tight frames for the underling space. We then give some constructions of regular two-distance sets, in particular, two-distance frames, both tight and non-tight cases. Connections among two-distance sets, equiangular lines and quasi-symmetric designs are also given.