Spherical sets avoiding a prescribed set of angles

TL;DR

This paper investigates the maximum surface measure of sets on the sphere avoiding certain angles, improving bounds in three dimensions and exploring existence of maximizers using harmonic analysis.

Contribution

It improves the upper bound for the measure of 0-avoiding sets in 3D from 1/3 to 0.313 and demonstrates the existence of maximizers in higher dimensions.

Findings

Upper bound for 3D case improved to 0.313

Existence of maximum measure X-avoiding sets shown for n ≥ 3

Example provided where maximizer does not exist in 2D

Abstract

Let $X$ be any subset of the interval $[-1,1]$. A subset $I$ of the unit sphere in $R^n$ will be called \emph{$X$-avoiding} if $<u,v >\notin X$ for any $u,v \in I$. The problem of determining the maximum surface measure of a $\{ 0 \}$-avoiding set was first stated in a 1974 note by Witsenhausen; there the upper bound of $1/n$ times the surface measure of the sphere is derived from a simple averaging argument. A consequence of the Frankl-Wilson theorem is that this fraction decreases exponentially, but until now the $1/3$ upper bound for the case $n=3$ has not moved. We improve this bound to $0.313$ using an approach inspired by Delsarte's linear programming bounds for codes, combined with some combinatorial reasoning. In the second part of the paper, we use harmonic analysis to show that for $n\geq 3$ there always exists an $X$-avoiding set of maximum measure. We also show with an example that a maximiser need not exist when $n=2$.