Upper bounds for $s$-distance sets and equiangular lines

Abstract

The set of points in a metric space is called an $s$-distance set if pairwise distances between these points admit only $s$ distinct values. Two-distance spherical sets with the set of scalar products $\{\alpha ,-\alpha \}$, $\alpha \in [0,1)$, are called equiangular. The problem of determining the maximum size of $s$-distance sets in various spaces has a long history in mathematics. We suggest a new method of bounding the size of an $s$-distance set in compact two-point homogeneous spaces via zonal spherical functions. This method allows us to prove that the maximum size of a spherical two-distance set in ${\mathbb{R}}^n$, $n\ge 7$, is $\frac{n(n+1)}{2}$ with possible exceptions for some $n=(2k+1{)}^2-3$, $k\in \mathbb{N}$. We also prove the universal upper bound $\sim \frac{2}{3}n{a}^2$ for equiangular sets with $\alpha =\frac{1}{a}$ and, employing this bound, prove a new upper bound on the size of equiangular sets in all dimensions. Finally, we classify all equiangular sets reaching this new bound.

References in corpus (2)

• New upper bounds for kissing numbers from semidefinite programming
• New bounds for equiangular lines and spherical two-distance sets