Upper bounds for $s$-distance sets and equiangular lines
By Alexey A. Glazyrin, Wei-Hsuan Yu. 2016
64 citations recorded by the source corpus.
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 $\mathbb{R}^n$, $n\geq 7$, 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.
Subfields: Computer Graphics and Computer-Aided Design, Computer Vision and Pattern Recognition, Numerical Analysis
View Paper: https://www.semanticscholar.org/paper/febf3f50156a9440ffab52245c351f8c6d477c47
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Numerical Analysis
Citations in the corpus
- Equiangular lines with a fixed angle (2019, background): “We focus on the problem for fixed α and large d and refer the readers to [7] for discussion on bounds for smaller values of d.”