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Obtuse almost-equiangular sets

Christine Bachoc, Bram Bekker, Philippe Moustrou, Fernando Mário de Oliveira Filho

TL;DR

This paper studies the maximal size $\alpha(n,t)$ of $t$-almost-equiangular sets on the $(n-1)$-sphere and develops a semidefinite programming bound derived from an extension of Lovász theta to $3$-uniform hypergraphs. The bound improves previous estimates and, together with refined spectral methods, shows $\alpha(n,t)\le 2(n+1)$ for all $n$ and $t\le 0$, with equality only at $t=-1/n$. The authors classify maximum obtuse almost-equiangular sets in dimensions $n\le 5$ (and enumerate all optimal constructions for $n\le 3$ and $t\le 0$), connecting these sets to spherical $2$-designs and to symmetric orthogonal matrices via a bijection. They also develop realizability criteria for anti-triangle-free graphs, yielding structural constraints and exact realizability results in low dimensions.

Abstract

For $t \in [-1, 1)$, a set of points on the $(n-1)$-dimensional unit sphere is called $t$-almost equiangular if among any three distinct points there is a pair with inner product $t$. We propose a semidefinite programming upper bound for the maximum cardinality $α(n, t)$ of such a set based on an extension of the Lovász theta number to hypergraphs. This bound is at least as good as previously known bounds and for many values of $n$ and $t$ it is better. We also refine existing spectral methods to show that $α(n, t) \leq 2(n+1)$ for all $n$ and $t \leq 0$, with equality only at $t = -1/n$. This allows us to show the uniqueness of the optimal construction at $t = -1/n$ for $n \leq 5$ and to enumerate all possible constructions for $n \leq 3$ and $t \leq 0$.

Obtuse almost-equiangular sets

TL;DR

This paper studies the maximal size of -almost-equiangular sets on the -sphere and develops a semidefinite programming bound derived from an extension of Lovász theta to -uniform hypergraphs. The bound improves previous estimates and, together with refined spectral methods, shows for all and , with equality only at . The authors classify maximum obtuse almost-equiangular sets in dimensions (and enumerate all optimal constructions for and ), connecting these sets to spherical -designs and to symmetric orthogonal matrices via a bijection. They also develop realizability criteria for anti-triangle-free graphs, yielding structural constraints and exact realizability results in low dimensions.

Abstract

For , a set of points on the -dimensional unit sphere is called -almost equiangular if among any three distinct points there is a pair with inner product . We propose a semidefinite programming upper bound for the maximum cardinality of such a set based on an extension of the Lovász theta number to hypergraphs. This bound is at least as good as previously known bounds and for many values of and it is better. We also refine existing spectral methods to show that for all and , with equality only at . This allows us to show the uniqueness of the optimal construction at for and to enumerate all possible constructions for and .

Paper Structure

This paper contains 25 sections, 11 theorems, 66 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Upper bounds through semidefinite programming
  3. Lower bounds through constructions
  4. Maximum obtuse almost-equiangular sets
  5. Preliminaries
  6. Hypergraphs
  7. Spaces of subsets
  8. Geometry
  9. Optimization bounds for the independence number of hypergraphs
  10. Hypergraphs on the sphere
  11. Solving the optimization problem
  12. Upper bounds from the 3-point bound
  13. Realizability of anti-triangle-free graphs
  14. The simplex
  15. The rhombus
  16. ...and 10 more sections

Key Result

Theorem 1.1

If $t \in [-1, 0]$ and $n \geq 3$, and if $f(n,t) \coloneqq p^2n(1-t)^2/(2(nt^2+1))$, where then $\alpha(n,t) \leq \left\lfloor f(n,t)\right\rfloor \leq \left\lfloor (16t-9)^2/(128t^2)\right\rfloor$.

Figures (9)

  • Figure 1: The classification of maximum-cardinality $t$-almost-equiangular sets in $S^1$ and $S^2$ for $t \in [-1, 0]$. The vertical axis is the cardinality of the set, the horizontal axis is the inner product $t$. Open bullets indicate that a point is excluded from the interval while closed bullets indicate that the point is included. Green and purple stripes indicate that in both dimensions the same maximum cardinality is attained. The numbers $t_1$ and $t_2$ are the first two roots of \ref{['eq:MoserSpindlePolynomial']} with $k=2$.
  • Figure 2: The numeric solutions of degree $4$ and $16$, and the analytic (Theorem \ref{['thm:interpolation']}) and asymptotic solutions to the bound \ref{['opt:sphere-kpb-dual']}. The asymptotic bound is the limit as $n \to \infty$ of the analytic bound.
  • Figure 3: From left to right: ${\rm S}(1, 2)$, ${\rm MS}(2)$, and ${\rm S}(2, 3)$. The $R_i$ are indicated with their respective colors.
  • Figure 4: A plot of $f$ and $g$ for $k = 2$. The horizontal axis is the inner product $t$. Clearly $-1/2 < g_2 \leq f_1 < 0 < f_2 \leq g_3$. If $n > 2$, then ${\rm S}(1, 2)$ is $(n, t)$-realizable if and only if $t > -1/2$, and $f \geq 0$ or $g \leq 0$. This plot shows that it is $(n, t)$-realizable if and only if $t > -1/2$.
  • Figure 5: The extended 2-rhombus on the left and the extended 3-rhombus on the right. The dotted lines are edges that follow from Lemma \ref{['lem:kRhombusRealizable']}, forcing $f$ to coincide with $e$.
  • ...and 4 more figures

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • proof
  • proof : Proof of Theorem \ref{['thm:interpolation']}
  • Lemma 5.1
  • ...and 12 more