Spherical two-distance sets and related topics in harmonic analysis

Abstract

This dissertation is devoted to the study of applications of

harmonic analysis. The maximum size of spherical few-distance sets

had been studied by Delsarte at al. in the 1970s. In particular,

the maximum size of spherical two-distance sets in $\mathbb{R}^n$

had been known for $n \leq 39$ except $n=23$ by linear programming

methods in 2008. Our contribution is to extend the known results

of the maximum size of spherical two-distance sets in

$\mathbb{R}^n$ when $n=23$, $40 \leq n \leq 93$ and $n \neq 46,

78$. The maximum size of equiangular lines in $\mathbb{R}^n$ had

been known for all $n \leq 23$ except $n=14, 16, 17, 18, 19$ and

$20$ since 1973. We use the semidefinite programming method to

find the maximum size for equiangular line sets in $\mathbb{R}^n$

when $24 \leq n \leq 41$ and $n=43$.

We suggest a method of constructing spherical two-distance sets

that also form tight frames. We derive new structural properties

of the Gram matrix of a two-distance set that also forms a tight

frame for $\mathbb{R}^n$. One of the main results in this part is

a new correspondence between two-distance tight frames and certain

strongly regular graphs. This allows us to use spectral properties

of strongly regular graphs to construct two-distance tight

frames. Several new examples are obtained using this

characterization.

Bannai, Okuda, and Tagami proved that a tight spherical designs of

harmonic index 4 exists if and only if there exists an equiangular

line set with the angle $\arccos (1/(2k-1))$ in the Euclidean

space of dimension $3(2k-1)^2-4$ for each integer $k \geq 2$. We

show nonexistence of tight spherical designs of harmonic index $4$

on $S^{n-1}$ with $n\geq 3$ by a modification of the semidefinite

programming method. We also derive new relative bounds for

equiangular line sets. These new relative bounds are usually

tighter than previous relative bounds by Lemmens and Seidel.