[Paper Review] New bounds for equiangular lines and spherical two-distance sets

Wei-Hsuan Yu|arXiv (Cornell University)|Sep 5, 2016

Mathematical Approximation and Integration11 references3 citations

TL;DR

This paper establishes new upper bounds for equiangular lines in ℝⁿ using symbolic semidefinite programming relaxations, proving that for any integer a ≥ 3 and dimensions n in the interval [a²−2, 3a²−16], the maximum number of equiangular lines with angle arccos(1/a) is at most ½(a²−2)(a²−1). These bounds apply to infinitely many dimensions and extend to spherical two-distance sets, improving known results up to n ≤ 417.

ABSTRACT

A set of lines in $\mathbb{R}^n$ is called equiangular if the angle between each pair of lines is the same. We derive new upper bounds on the cardinality of equiangular lines. Let us denote the maximum cardinality of equiangular lines in $\mathbb{R}^n$ with the common angle $\arccos α$ by $M_α(n)$. We prove that $M_{\frac 1 a} (n) \leq \frac 1 2 ( a^2-2) ( a^2-1)$ for any $n \in \mathbb{N}$ in the interval $a^2 -2 \leq n \leq 3 a^2-16$ and $a \geq 3$. Moreover, we discuss the relation between equiangular lines and spherical two-distance sets and we obtain the new results on the maximum spherical two-distance sets in $\mathbb{R}^n$ up to $n \leq 417$.

Motivation & Objective

• To derive new upper bounds on the maximum number of equiangular lines in ℝⁿ for infinitely many dimensions, not limited to finite cases.
• To establish a connection between equiangular lines and spherical two-distance sets to extend bounds on the latter up to n=417.
• To provide hand-computable bounds without relying on convex optimization software, using symbolic relaxation of semidefinite programs.
• To resolve open cases in equiangular line configurations by proving nonexistence of certain line sets in dimensions like n=14, 16, 19, 20, and 42.
• To support a conjecture that the maximum size of equiangular lines in ℝⁿ is determined by the nearest odd integer reciprocal angle for large n.

Proposed method

• Derives symbolic semidefinite programming relaxations to avoid reliance on numerical solvers like CVX.
• Applies the relaxation to matrix inequalities associated with equiangular line configurations to obtain bounds valid across intervals of dimensions.
• Uses the bound M₁/ₐ(n) ≤ ½(a²−2)(a²−1) for a ≥ 3 and n ∈ [a²−2, 3a²−16], derived from algebraic manipulation of SDP constraints.
• Leverages the duality between equiangular lines and spherical two-distance sets to transfer bounds from one problem to the other.
• Employs the relative bound formula fₐ(n) = n(1−a²)/(1−na²) and proves its monotonicity in n and a to justify interval dominance.
• Validates results via known bounds and numerical checks from prior works (e.g., [7], [14]) up to n=400.

Experimental results

Research questions

• RQ1 What are the tightest possible upper bounds on the number of equiangular lines in ℝⁿ for infinitely many dimensions?
• RQ2 Can symbolic semidefinite programming relaxations yield hand-computable bounds without numerical solvers?
• RQ3 How do bounds on equiangular lines relate to the maximum size of spherical two-distance sets in ℝⁿ?
• RQ4 For which dimensions n is the Gerzon bound M(n) = n(n+1)/2 not achievable, and can we prove nonexistence of configurations near it?
• RQ5 Is there a universal bound pattern for equiangular lines based on the reciprocal of odd integers, valid across large intervals of n?

Key findings

• For any integer a ≥ 3 and n ∈ [a²−2, 3a²−16], the maximum number of equiangular lines with angle arccos(1/a) is at most ½(a²−2)(a²−1), providing a new infinite family of bounds.
• The bound for a=3 (angle arccos(1/3)) gives M₁/₃(n) ≤ 28 for dimensions 7 ≤ n ≤ 11, improving prior knowledge.
• For a=5, the bound gives M₁/₅(n) ≤ 276 for 23 ≤ n ≤ 59, extending known results beyond previous limits.
• The paper proves that the maximum size of a spherical two-distance set in ℝⁿ is n(n+1)/2 for 7 ≤ n ≤ 417, except for n = (2k+1)²−3 with k=2 to 9, where tighter bounds apply.
• The results confirm that no 30 equiangular lines exist in ℝ¹⁴, no 42 in ℝ¹⁶, no 76 in ℝ¹⁹, and no 96 in ℝ²⁰, resolving open cases.
• If a tight spherical 5-design exists in ℝⁿ for n = (2k+1)²−2, then M(n) = ½((2k+1)²−2)((2k+1)²−1) holds for all n in [n, 3(2k+1)²−16], solving over 200 cases at once.