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Functional van Lint-Seidel Relative and Gerzon Universal Bounds

Published Oct 26, 2026Fresh — added to the Hub within the last 30 daysv12 min read

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Abstract

We introduce the notion of functional equiangular lines in finite rank modules over subrings of $\mathbb{R}$ . We show that van Lint-Seidel relative and Gerzon universal bounds hold for this most general lines.

Introduction

Let $d \in\mathbb{N}$ and $\gamma\in[0,1]$ . Recall that a collection $\{\tau_j\}_{j=1}^{n}$ of unit vectors in $\mathbb{R}^{d}$ is said to be $\gamma$ -equiangular lines [1, 3] if

\lvert\langle\tau_j,\tau_k\rangle\rvert= \gamma,\quad\forall1 \leq j,k \leq n,\ j \ne k.

A fundamental problem associated with equiangular lines is the following.

Problem 1.1. Given $d \in\mathbb{N}$ and $\gamma\in[0,1]$ , what is the upper bound on $n$ such that there exists a collection $\{\tau_j\}_{j=1}^{n}$ of $\gamma$ -equiangular lines in $\mathbb{R}^{d}$ ?

Two answers to Problem (1.1) which are driving forces in the study of equiangular lines is the following relative bound of van Lint and Seidel [3, 4] and universal bound of Gerzon [5].

Theorem 1.2. [3, 4] (van Lint-Seidel Relative Bound) Let $\{\tau_j\}_{j=1}^{n}$ be $\gamma$ -equiangular lines in $\mathbb{R}^{d}$ . Then

n(1-d\gamma^{2}) \leq d(1-\gamma^{2}).

In particular, if

\gamma< \frac{1}{\sqrt{d}},

then

n \leq\frac{d(1-\gamma^{2})}{1-d\gamma^{2}}.

Theorem 1.3. [5] (Gerzon Universal Bound) Let $\{\tau_j\}_{j=1}^{n}$ be $\gamma$ -equiangular lines in $\mathbb{R}^{d}$ . Then

n \leq\frac{d(d+1)}{2}.

The notion of functional equiangular lines is hinted in [2]. In this paper, we define it in most general form and derive functional forms of Theorems 1.2 and 1.3.

Functional Equiangular Lines

In the paper, $\mathcal{R}$ denotes a subring of $\mathbb{R}$ and $\mathcal{M}$ is a rank $d$ module over $\mathcal{R}$ . By $\mathcal{M}^{*}$ we mean the module of all homomorphisms from $\mathcal{M}$ to $\mathcal{R}$ . Module of all homomorphisms from $\mathcal{M}$ to $\mathcal{M}$ is denoted by $\operatorname{Mor}(\mathcal{M})$ .

Definition 2.1. Let $\{\tau_j\}_{j=1}^{n}$ be a collection in a module $\mathcal{M}$ of rank $d$ over $\mathcal{R}$ and $\{f_j\}_{j=1}^{n}$ be a collection in $\mathcal{M}^{*}$ . Let $\gamma\ge0$ . The pair $(\{f_j\}_{j=1}^{n},\{\tau_j\}_{j=1}^{n})$ is said to be functional $\gamma$ -equingular if following conditions hold.

(i) $f_j(\tau_j)=1$ for all $1 \le j \le n$ .

(ii) $\lvert f_j(\tau_k)f_k(\tau_j)\rvert=\gamma^2$ for all $1 \le j,k \le n$ , $j \ne k$ .

(iii) The operator

S_{f,\tau}:\mathcal{M}\ni x\mapsto\sum_{j=1}^{n}f_j(x)\tau_j\in\mathcal{M}.

is similar (through invertible operator) to a diagonal operator.

Theorem 2.2. (Functional van Lint-Seidel Relative Bound) If $(\{f_j\}_{j=1}^{n},\{\tau_j\}_{j=1}^{n})$ is functional $\gamma$ -equiangular for rank $d$ module $\mathcal{M}$ , then

n(1-d\gamma^2)\le d(1-\gamma^2).

In particular, if

\gamma<\frac{1}{\sqrt{d}},

then

n\le\frac{d(1-\gamma^2)}{1-d\gamma^2}.

Proof. Define

S_{f,\tau}:\mathcal{M}\ni x\mapsto S_{f,\tau}x:=\sum_{j=1}^{n}f_j(x)\tau_j\in\mathcal{M}.

Let $\lambda_1,\ldots,\lambda_d$ be eigenvalues of $S_{f,\tau}$ . Since $S_{f,\tau}$ is diagonalizable,

\begin{aligned} n^2=\left(\sum_{j=1}^{n}f_j(\tau_j)\right)^2=(\operatorname{Tra}(S_{f,\tau}))^2=\left(\sum_{k=1}^{d}\lambda_k\right)^2 \\ &\le d\sum_{k=1}^{d}\lambda_k^2=d\operatorname{Tra}(S_{f,\tau}^2)=d\sum_{j=1}^{n}\sum_{k=1}^{n}f_j(\tau_k)f_k(\tau_j) \\ &=d\sum_{j=1}^{n}f_j(\tau_j)^2+d\sum_{\substack{j,k=1\\j\ne k}}^{n}f_j(\tau_k)f_k(\tau_j)=dn+d\sum_{\substack{j,k=1\\j\ne k}}^{n}f_j(\tau_k)f_k(\tau_j) \\ &\le dn+d\sum_{\substack{j,k=1\\j\ne k}}^{n}\lvert f_j(\tau_k)f_k(\tau_j)\rvert=dn+d(n^2-n)\gamma^2. \end{aligned}

Therefore

n\le d+d(n-1)\gamma^2\Longrightarrow n(1-d\gamma^2)\le d(1-\gamma^2).

We are unable to derive Gerzon universal bound for functional equiangular lines. However, we derive following Gerzon bound for functionals.

Theorem 2.3. (Functional Gerzon Universal Bound) Let $\{\tau_j\}_{j=1}^{n}$ be a collection in a module $\mathcal{M}$ of rank $d$ over $\mathbb{R}$ and $\{f_j\}_{j=1}^{n}$ be a collection in $\mathcal{M}^*$ . Assume the following.

(i) $f_j(\tau_j)=1$ for all $1 \le j \le n$ .

(ii) There is a $\gamma\ne1$ such that $f_j(\tau_k)f_k(\tau_j)=\gamma^2$ for all $1 \le j,k \le n$ , $j \ne k$ .

Then

n \le d^2

Proof. For $1 \le j \le n$ , define

\tau_j \otimes f_j:\mathcal{M} \ni x \mapsto(\tau_j \otimes f_j)(x):=f_j(x)\tau_j \in\mathcal{M}

We show that the collection $\{\tau_j \otimes f_j\}_{j=1}^{n}$ in $\operatorname{Mor}(\mathcal{M})$ is linearly independent over $\mathbb{R}$ . Let $c_1,\ldots,c_n \in\mathbb{R}$ be such that

\sum_{j=1}^{n}c_j(\tau_j \otimes f_j)=0.

Let $1 \le k \le n$ be fixed. Then previous equation gives

\sum_{j=1}^{n}c_j(\tau_j \otimes f_j)(\tau_k \otimes f_k)=0.

By taking trace we get

\begin{aligned} 0=\sum_{j=1}^{n}c_j\operatorname{Tra}((\tau_j \otimes f_j)(\tau_k \otimes f_k))=\sum_{j=1}^{n}c_jf_j(\tau_k)f_k(\tau_j) \\ &=\sum_{\substack{j=1\\j\ne k}}^{n}c_jf_j(\tau_k)f_k(\tau_j)+c_kf_k(\tau_k)^2=\sum_{\substack{j=1\\j\ne k}}^{n}c_j\gamma^2+c_k \\ &=\gamma^2\left(\sum_{j=1}^{n}c_j-c_k\right)+c_k=\gamma^2\left(\sum_{j=1}^{n}c_j\right)+(1-\gamma^2)c_k. \end{aligned}

Therefore

c_k=\frac{\gamma^2}{\gamma^2-1}\sum_{j=1}^{n}c_j=:c,\qquad\forall1\le k\le n.

Now

\begin{aligned} 0=\operatorname{Tra}\left(\sum_{j=1}^{n}c_j(\tau_j\otimes f_j)\right)=\operatorname{Tra}\left(\sum_{j=1}^{n}c(\tau_j\otimes f_j)\right) \\ &=\sum_{j=1}^{n}c\operatorname{Tra}(\tau_j\otimes f_j)=\sum_{j=1}^{n}cf_j(\tau_j)=cn. \end{aligned}

Hence $c=0$ . Therefore $\{\tau_j\otimes f_j\}_{j=1}^{n}$ is linearly independent. Since the rank of $\operatorname{Mor}(\mathcal{M})$ is $d^2$ we must have $n\le d^2$ .

References

[1]J. Haantjes. Equilateral point-sets in elliptic two- and three-dimensional spaces. Nieuw Arch. Wiskunde (2), 22:355–362, 1948.
[2]K. Mahesh Krishna. Discrete and continuous Welch bounds for Banach spaces with applications. J. Class. Anal., 22(2):81–111, 2023.DOI
[3]P. W. H. Lemmens and J. J. Seidel. Equiangular lines. J. Algebra, 24:494–512, 1973.
[4]J. H. van Lint and J. J. Seidel. Equilateral point sets in elliptic geometry. Indag. Math., 28:335–348, 1966.
[5]Shayne F. D. Waldron. An introduction to finite tight frames. Applied and Numerical Harmonic Analysis. Birkhäuser/Springer, New York, 2018.DOI

Functional van Lint-Seidel Relative and Gerzon Universal Bounds

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Functional Equiangular Lines

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