p-adic Equiangular Lines and p-adic van Lint-Seidel Relative Bound

1. Introduction

Let $d\in \mathbb{N}$ and $\gamma \in [0,1]$. Recall that a collection ${\left\{{\tau }_j\right\}}_{j=1}^n$ of unit vectors in ${\mathbb{R}}^d$ is said to be $\gamma$-equiangular lines [1,2] if

$$\begin{array}{c}\hfill |\langle {\tau }_j,{\tau }_k\rangle |=\gamma ,\forall 1\le j,k\le n,j\ne k.\end{array}$$

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 ${\left\{{\tau }_j\right\}}_{j=1}^n$ of γ-equiangular lines in ${\mathbb{R}}^d$?

An answer to Problem 1.1 which is fundamental driving force in the study of equiangular lines is the following result of van Lint and Seidel [2,3].

Theorem 1.2.

[2,3] (van Lint-Seidel Relative Bound) Let ${\left\{{\tau }_j\right\}}_{j=1}^n$ be γ-equiangular lines in ${\mathbb{R}}^d$. Then

$$\begin{array}{c}\hfill n(1-d{\gamma }^2)\le d(1-{\gamma }^2).\end{array}$$

In particular, if

$$\begin{array}{c}\hfill \gamma <{\displaystyle \frac{1}{\sqrt{d}}},\end{array}$$

then

$$\begin{array}{c}\hfill n\le {\displaystyle \frac{d(1-{\gamma }^2)}{1-d{\gamma }^2}}.\end{array}$$

While deriving p-adic Welch bounds, the notion of p-adic equiangular lines is hinted in [4]. In this paper, we make it more rigorous and derive a fundamental relation which complements Theorem 1.2.

2. p-adic Equiangular Lines

We begin by recalling the notion of p-adic Hilbert space. We refer [5,6,7,8,9] for more on p-adic Hilbert spaces.

Definition 2.1.

[6,7] Let $\mathbb{K}$ be a non-Archimedean valued field (with valuation $|·|$) and $\mathcal{X}$ be a non-Archimedean Banach space (with norm $\parallel ·\parallel$) over $\mathbb{K}$. We say that $\mathcal{X}$ is a p-adic Hilbert space if there is a map (called as p-adic inner product) $\langle ·,·\rangle :\mathcal{X}\times \mathcal{X}\to \mathbb{K}$ satisfying following. (i)(i)

• If $x\in \mathcal{X}$ is such that $\langle x,y\rangle =0$ for all $y\in \mathcal{X}$, then $x=0$.
• $\langle x,y\rangle =\langle y,x\rangle$ for all $x,y\in \mathcal{X}$.
• $\langle \alpha x,y+z\rangle =\alpha \langle x,y\rangle +\langle x,z\rangle$ for all $\alpha \in \mathbb{K}$, for all $x,y,z\in \mathcal{X}$.
• $|\langle x,y\rangle |\le \parallel x\parallel \parallel y\parallel$ for all $x,y\in \mathcal{X}$.

Following is the standard example which we consider in the paper.

Example 2.2.

[5] Let p be a prime. For $d\in \mathbb{N}$, let ${\mathbb{Q}}_p^d$ be the standard p-adic Hilbert space equipped with the inner product

$$\begin{array}{c}\hfill \langle {\left(a_j\right)}_{j=1}^d,{\left(b_j\right)}_{j=1}^d\rangle :=\sum _{j=1}^d{a}_j{b}_j,\forall {\left(a_j\right)}_{j=1}^d,{\left(b_j\right)}_{j=1}^d\in {\mathbb{Q}}_p^d\end{array}$$

and norm

$$\begin{array}{c}\hfill \parallel {\left(x_j\right)}_{j=1}^d\parallel :=\underset{1\le j\le d}{max}\left|x_j\right|,\forall {\left(x_j\right)}_{j=1}^d\in {\mathbb{Q}}_p^d.\end{array}$$

Through various trials, we believe following is the correct definition of equiangular lines in the p-adic setting.

Definition 2.3.

Let p be a prime, $d\in \mathbb{N}$ and $\gamma \ge 0$. A collection ${\left\{{\tau }_j\right\}}_{j=1}^n$ in ${\mathbb{Q}}_p^d$ is said to be p-adic $\gamma$-equiangular lines if the following conditions hold: (i)(i)

• $\langle {\tau }_j,{\tau }_j\rangle =1,\forall 1\le j\le n$.
• $|\langle {\tau }_j,{\tau }_k\rangle |=\gamma ,\forall 1\le j,k\le n,j\ne k$.
• The operator

  $$\begin{array}{c}\hfill S_{\tau }:{\mathbb{Q}}_p^d\ni x\mapsto \sum _{j=1}^n\langle x,{\tau }_j\rangle {\tau }_j\in {\mathbb{Q}}_p^d\end{array}$$
  is similar (through invertible operator) to a diagonal operator over ${\mathbb{Q}}_p$ with eigenvalues ${\lambda }_1,\cdots ,{\lambda }_d\in {\mathbb{Q}}_p$ satisfying

  $$\begin{array}{c}\hfill {\left|\sum _{j=1}^d{\lambda }_j\right|}^2\le \left|d\right|\left|\sum _{j=1}^d{\lambda }_j^2\right|.\end{array}$$

The result of this paper is the following p-adic version of Theorem 1.2.

Theorem 2.4.

(p-adic van Lint-Seidel Relative Bound) Let p be a prime, $d\in \mathbb{N}$ and $\gamma \ge 0$. If ${\left\{{\tau }_j\right\}}_{j=1}^n$ is p-adic γ-equiangular lines in ${\mathbb{Q}}_p^d$, then

$$\begin{array}{c}\hfill {\left|n\right|}^2\le \left|d\right|max\left\{\right|n|,{\gamma }^2\}.\end{array}$$

In particular, we have following. (i)(i)

• If $\left|n\right|\le {\gamma }^2$, then

  $$\begin{array}{c}\hfill {\left|n\right|}^2\le \left|d\right|{\gamma }^2.\end{array}$$
• If $\left|n\right|\ge {\gamma }^2$, then

  $$\begin{array}{c}\hfill \left|n\right|\le \left|d\right|.\end{array}$$

Proof. 

We see that

$$\begin{array}{cc}\hfill & Tra\left(S_{\tau }\right)=\sum _{j=1}^n\langle {\tau }_j,{\tau }_j\rangle ,\hfill \\ \hfill & Tra\left(S_{\tau }^2\right)=\sum _{j=1}^n\sum _{k=1}^n\langle {\tau }_j,{\tau }_k\rangle \langle {\tau }_k,{\tau }_j\rangle =\sum _{j=1}^n\sum _{k=1}^n{\langle {\tau }_j,{\tau }_k\rangle }^2.\hfill \end{array}$$

Using Definition 2.3,

$$\begin{array}{cc}\hfill {\left|n\right|}^2& ={\left|\sum _{j=1}^n\langle {\tau }_j,{\tau }_j\rangle \right|}^2=|Tra\left(S_{\tau }\right){|}^2={\left|\sum _{j=1}^d{\lambda }_j\right|}^2\le \left|d\right|\left|\sum _{j=1}^d{\lambda }_j^2\right|\hfill \\ \hfill & =\left|d\right|\left|\sum _{j=1}^n\sum _{k=1}^n{\langle {\tau }_j,{\tau }_k\rangle }^2\right|=\left|d\right|\left|\sum _{j=1}^n{\langle {\tau }_j,{\tau }_j\rangle }^2+\sum _{1\le j,k\le n,j\ne k}{\langle {\tau }_j,{\tau }_k\rangle }^2\right|\hfill \\ \hfill & =\left|d\right|\left|\sum _{j=1}^{n}1+\sum _{1\le j,k\le n,j\ne k}{\langle {\tau }_j,{\tau }_k\rangle }^2\right|=\left|d\right|\left|n+\sum _{1\le j,k\le n,j\ne k}{\langle {\tau }_j,{\tau }_k\rangle }^2\right|\hfill \\ \hfill & \le \left|d\right|max\left\{\left|n\right|,\left|\sum _{1\le j,k\le n,j\ne k}{\langle {\tau }_j,{\tau }_k\rangle }^2\right|\right\}\hfill \\ \hfill & \le \left|d\right|max\left\{\left|n\right|,\underset{1\le j,k\le n,j\ne k}{max}{\left|\langle {\tau }_j,{\tau }_k\rangle \right|}^2\right\}=\left|d\right|max\left\{\right|n|,{\gamma }^2\}.\hfill \end{array}$$

□

Corollary 2.5.

Let ${\left\{{\tau }_j\right\}}_{j=1}^n$ be a collection in ${\mathbb{Q}}_p^d$ satisfying following. (i)(i)

• $\langle {\tau }_j,{\tau }_j\rangle =1,\forall 1\le j\le n$.
• There exists a nonzero element $b\in {\mathbb{Q}}_p$ such that

  $$\begin{array}{c}\hfill bx=\sum _{j=1}^n\langle x,{\tau }_j\rangle {\tau }_j,\forall x\in {\mathbb{Q}}_p^d.\end{array}$$

  Then

  $$\begin{array}{c}\hfill {\left|n\right|}^2\le \left|d\right|max\left\{\right|n|,{\gamma }^2\}.\end{array}$$

A careful observation of proof of Theorem 2.4 gives following general p-adic Welch bound.

Theorem 2.6.

(General p-adic Welch Bound) Let ${\left\{{\tau }_j\right\}}_{j=1}^n$ be a collection in ${\mathbb{Q}}_p^d$ satisfying following. (i)(i)

• $\langle {\tau }_j,{\tau }_j\rangle =1,\forall 1\le j\le n$.
• The operator

  $$\begin{array}{c}\hfill S_{\tau }:{\mathbb{Q}}_p^d\ni x\mapsto \sum _{j=1}^n\langle x,{\tau }_j\rangle {\tau }_j\in {\mathbb{Q}}_p^d\end{array}$$

  is similar to a diagonal operator over ${\mathbb{Q}}_p$ with eigenvalues ${\lambda }_1,\cdots ,{\lambda }_d\in {\mathbb{Q}}_p$ satisfying

  $$\begin{array}{c}\hfill {\left|\sum _{j=1}^d{\lambda }_j\right|}^2\le \left|d\right|\left|\sum _{j=1}^d{\lambda }_j^2\right|.\end{array}$$

Then

$$\begin{array}{c}\hfill {\left|n\right|}^2\le \left|d\right|\underset{1\le j,k\le n,j\ne k}{max}\left\{\left|n\right|,|\langle {\tau }_j,{\tau }_k\rangle {|}^2\right\}.\end{array}$$

We can generalize Definition 2.3 in the following way.

Definition 2.7.

Let p be a prime, $d\in \mathbb{N}$, $\gamma \ge 0$ and $a\in {\mathbb{Q}}_p$ be nonzero. A collection ${\left\{{\tau }_j\right\}}_{j=1}^n$ in ${\mathbb{Q}}_p^d$ is said to be p-adic $(\gamma ,a)$-equiangular lines if the following conditions hold: (i)(i)

• $\langle {\tau }_j,{\tau }_j\rangle =a,\forall 1\le j\le n$.
• $|\langle {\tau }_j,{\tau }_k\rangle |=\gamma ,\forall 1\le j,k\le n,j\ne k$.
• The operator

  $$\begin{array}{c}\hfill S_{\tau }:{\mathbb{Q}}_p^d\ni x\mapsto \sum _{j=1}^n\langle x,{\tau }_j\rangle {\tau }_j\in {\mathbb{Q}}_p^d\end{array}$$

  is similar to a diagonal operator over ${\mathbb{Q}}_p$ with eigenvalues ${\lambda }_1,\cdots ,{\lambda }_d\in {\mathbb{Q}}_p$ satisfying

  $$\begin{array}{c}\hfill {\left|\sum _{j=1}^d{\lambda }_j\right|}^2\le \left|d\right|\left|\sum _{j=1}^d{\lambda }_j^2\right|.\end{array}$$

Note that division by norm of an element is not allowed in a p-adic Hilbert space. Thus we can not reduce Definition 2.7 to Definition 2.3 (unlike the real case). By modifying earlier proofs, we easily get following theorems.

Theorem 2.8.

If ${\left\{{\tau }_j\right\}}_{j=1}^n$ is p-adic $(\gamma ,a)$-equiangular lines in ${\mathbb{Q}}_p^d$, then

$$\begin{array}{c}\hfill {\left|n\right|}^2\le \left|d\right|max\left\{\left|n\right|,{\displaystyle \frac{{\gamma }^2}{|a^2|}}\right\}.\end{array}$$

In particular, we have following. (i)(i)

• If $|a^{2}n|\le {\gamma }^2$, then

  $$\begin{array}{c}\hfill {\left|n\right|}^2\le \left|d\right|{\displaystyle \frac{{\gamma }^2}{|a^2|}}.\end{array}$$
• If $|a^{2}n|\ge {\gamma }^2$, then

  $$\begin{array}{c}\hfill \left|n\right|\le \left|d\right|.\end{array}$$

Corollary 2.9.

Let ${\left\{{\tau }_j\right\}}_{j=1}^n$ be a collection in ${\mathbb{Q}}_p^d$ satisfying following. (i)(i)

• There exists a nonzero element $a\in {\mathbb{Q}}_p$ such that $\langle {\tau }_j,{\tau }_j\rangle =a,\forall 1\le j\le n$.
• There exists a nonzero element $b\in {\mathbb{Q}}_p$ such that

  $$\begin{array}{c}\hfill bx=\sum _{j=1}^n\langle x,{\tau }_j\rangle {\tau }_j,\forall x\in {\mathbb{Q}}_p^d.\end{array}$$

  Then

  $$\begin{array}{c}\hfill {\left|n\right|}^2\le \left|d\right|max\left\{\left|n\right|,{\displaystyle \frac{{\gamma }^2}{|a^2|}}\right\}.\end{array}$$

Theorem 2.10.

Let ${\left\{{\tau }_j\right\}}_{j=1}^n$ be a collection in ${\mathbb{Q}}_p^d$ satisfying following. (i)(i)

• There exists a nonzero element $a\in {\mathbb{Q}}_p$ such that $\langle {\tau }_j,{\tau }_j\rangle =a,\forall 1\le j\le n$.
• The operator

  $$\begin{array}{c}\hfill S_{\tau }:{\mathbb{Q}}_p^d\ni x\mapsto \sum _{j=1}^n\langle x,{\tau }_j\rangle {\tau }_j\in {\mathbb{Q}}_p^d\end{array}$$

  is similar to a diagonal operator over ${\mathbb{Q}}_p$ with eigenvalues ${\lambda }_1,\cdots ,{\lambda }_d\in {\mathbb{Q}}_p$ satisfying

  $$\begin{array}{c}\hfill {\left|\sum _{j=1}^d{\lambda }_j\right|}^2\le \left|d\right|\left|\sum _{j=1}^d{\lambda }_j^2\right|.\end{array}$$

Then

$$\begin{array}{c}\hfill {\left|n\right|}^2\le \left|d\right|\underset{1\le j,k\le n,j\ne k}{max}\left\{\left|n\right|,{\displaystyle \frac{|\langle {\tau }_j,{\tau }_k\rangle {|}^2}{|a^2|}}\right\}.\end{array}$$

Note that there is a universal bound for equiangular lines known as Gerzon bound.

Theorem 2.11.

[10] (Gerzon Universal Bound) Let ${\left\{{\tau }_j\right\}}_{j=1}^n$ be γ-equiangular lines in ${\mathbb{R}}^d$. Then

$$\begin{array}{c}\hfill n\le {\displaystyle \frac{d(d+1)}{2}}.\end{array}$$

We are unable to derive p-adic version of Theorem 2.11. It is clear that, in the paper, we can replace ${\mathbb{Q}}_p$ by any non-Archimedean field.