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

K. MAHESH KRISHNA

doi:10.20944/preprints202407.1562.v1

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

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K. MAHESH KRISHNA^*

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Abstract

We introduce the notion of p-adic equiangular lines and derive the first fundamental relation between common angle, dimension of the space and the number of lines. More precisely, we show that if {τj} n j=1 is p-adic γ-equiangular lines in Qd p , then (1) |n| 2 ≤ |d| max{|n|, γ2 }. We call Inequality (1) as the p-adic van Lint-Seidel relative bound. We believe that this complements fundamental van Lint-Seidel [Indag. Math., 1966] relative bound for equiangular lines in the p-adic case. Keywords: Equiangular lines, p-adic Hilbert space.

Keywords:

Equiangular lines

;

p-adic Hilbert space

Subject:

Computer Science and Mathematics - Analysis

MSC: 12J25; 46S10; 47S10; 11D88

1. Introduction

Let

d \in N

and

γ \in [0, 1]

. Recall that a collection

{τ_{j}}_{j = 1}^{n}

of unit vectors in

R^{d}

is said to be $γ$ -equiangular lines [1,2] if

\begin{matrix} | 〈 τ_{j}, τ_{k} 〉 | = γ, \forall 1 \leq j, k \leq n, j \neq k . \end{matrix}

A fundamental problem associated with equiangular lines is the following.

Problem 1.1.

Given

d \in N

and

γ \in [0, 1]

, what is the upper bound on n such that there exists a collection

{τ_{j}}_{j = 1}^{n}

of γ-equiangular lines in

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

{τ_{j}}_{j = 1}^{n}

be γ-equiangular lines in

R^{d}

. Then

\begin{matrix} n (1 - d γ^{2}) \leq d (1 - γ^{2}) . \end{matrix}

In particular, if

\begin{matrix} γ < \frac{1}{\sqrt{d}}, \end{matrix}

then

\begin{matrix} n \leq \frac{d (1 - γ^{2})}{1 - d γ^{2}} . \end{matrix}

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

K

be a non-Archimedean valued field (with valuation

| \cdot |

) and

X

be a non-Archimedean Banach space (with norm

∥ \cdot ∥

) over

K

. We say that

X

is a p-adic Hilbert space if there is a map (called as p-adic inner product)

〈 \cdot, \cdot 〉 : X \times X \to K

satisfying following. (i)(i)

If $x \in X$ is such that $〈 x, y 〉 = 0$ for all $y \in X$ , then $x = 0$ .
$〈 x, y 〉 = 〈 y, x 〉$ for all $x, y \in X$ .
$〈 α x, y + z 〉 = α 〈 x, y 〉 + 〈 x, z 〉$ for all $α \in K$ , for all $x, y, z \in X$ .
$| 〈 x, y 〉 | \leq ∥ x ∥ ∥ y ∥$ for all $x, y \in X$ .

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

Example 2.2.

[5] Let p be a prime. For

d \in N

, let

Q_{p}^{d}

be the standard p-adic Hilbert space equipped with the inner product

\begin{matrix} 〈 {(a_{j})}_{j = 1}^{d}, {(b_{j})}_{j = 1}^{d} 〉 : = \sum_{j = 1}^{d} a_{j} b_{j}, \forall {(a_{j})}_{j = 1}^{d}, {(b_{j})}_{j = 1}^{d} \in Q_{p}^{d} \end{matrix}

and norm

\begin{matrix} ∥ {(x_{j})}_{j = 1}^{d} ∥ : = max_{1 \leq j \leq d} | x_{j} |, \forall {(x_{j})}_{j = 1}^{d} \in Q_{p}^{d} . \end{matrix}

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 N

and

γ \geq 0

. A collection

{τ_{j}}_{j = 1}^{n}

Q_{p}^{d}

is said to be p-adic $γ$ -equiangular lines if the following conditions hold: (i)(i)

$〈 τ_{j}, τ_{j} 〉 = 1, \forall 1 \leq j \leq n$ .
$| 〈 τ_{j}, τ_{k} 〉 | = γ, \forall 1 \leq j, k \leq n, j \neq k$ .
The operator

$\begin{matrix} S_{τ} : Q_{p}^{d} ∋ x \mapsto \sum_{j = 1}^{n} 〈 x, τ_{j} 〉 τ_{j} \in Q_{p}^{d} \end{matrix}$

is similar (through invertible operator) to a diagonal operator over $Q_{p}$ with eigenvalues $λ_{1}, \dots, λ_{d} \in Q_{p}$ satisfying

$\begin{matrix} {|\sum_{j = 1}^{d} λ_{j}|}^{2} \leq | d | |\sum_{j = 1}^{d} λ_{j}^{2}| . \end{matrix}$

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 N

and

γ \geq 0

. If

{τ_{j}}_{j = 1}^{n}

is p-adic γ-equiangular lines in

Q_{p}^{d}

, then

\begin{matrix} {| n |}^{2} \leq | d | max {| n |, γ^{2}} . \end{matrix}

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

If $| n | \leq γ^{2}$ , then

$\begin{matrix} {| n |}^{2} \leq | d | γ^{2} . \end{matrix}$
If $| n | \geq γ^{2}$ , then

$\begin{matrix} | n | \leq | d | . \end{matrix}$

Proof.

We see that

\begin{matrix} Tra (S_{τ}) = \sum_{j = 1}^{n} 〈 τ_{j}, τ_{j} 〉, \\ Tra (S_{τ}^{2}) = \sum_{j = 1}^{n} \sum_{k = 1}^{n} 〈 τ_{j}, τ_{k} 〉 〈 τ_{k}, τ_{j} 〉 = \sum_{j = 1}^{n} \sum_{k = 1}^{n} {〈 τ_{j}, τ_{k} 〉}^{2} . \end{matrix}

Using Definition 2.3,

\begin{matrix} {| n |}^{2} & = {|\sum_{j = 1}^{n} 〈 τ_{j}, τ_{j} 〉|}^{2} = | Tra (S_{τ}) |^{2} = {|\sum_{j = 1}^{d} λ_{j}|}^{2} \leq | d | |\sum_{j = 1}^{d} λ_{j}^{2}| \\ = | d | |\sum_{j = 1}^{n} \sum_{k = 1}^{n} {〈 τ_{j}, τ_{k} 〉}^{2}| = | d | |\sum_{j = 1}^{n} {〈 τ_{j}, τ_{j} 〉}^{2} + \sum_{1 \leq j, k \leq n, j \neq k} {〈 τ_{j}, τ_{k} 〉}^{2}| \\ = | d | |\sum_{j = 1}^{n} 1 + \sum_{1 \leq j, k \leq n, j \neq k} {〈 τ_{j}, τ_{k} 〉}^{2}| = | d | |n + \sum_{1 \leq j, k \leq n, j \neq k} {〈 τ_{j}, τ_{k} 〉}^{2}| \\ \leq | d | max \{| n |, |\sum_{1 \leq j, k \leq n, j \neq k} {〈 τ_{j}, τ_{k} 〉}^{2}|\} \\ \leq | d | max \{| n |, max_{1 \leq j, k \leq n, j \neq k} {| 〈 τ_{j}, τ_{k} 〉 |}^{2}\} = | d | max {| n |, γ^{2}} . \end{matrix}

□

Corollary 2.5.

Let

{τ_{j}}_{j = 1}^{n}

be a collection in

Q_{p}^{d}

satisfying following. (i)(i)

$〈 τ_{j}, τ_{j} 〉 = 1, \forall 1 \leq j \leq n$ .
There exists a nonzero element $b \in Q_{p}$ such that

$\begin{matrix} b x = \sum_{j = 1}^{n} 〈 x, τ_{j} 〉 τ_{j}, \forall x \in Q_{p}^{d} . \end{matrix}$

Then

$\begin{matrix} {| n |}^{2} \leq | d | max {| n |, γ^{2}} . \end{matrix}$

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

{τ_{j}}_{j = 1}^{n}

be a collection in

Q_{p}^{d}

satisfying following. (i)(i)

$〈 τ_{j}, τ_{j} 〉 = 1, \forall 1 \leq j \leq n$ .
The operator

$\begin{matrix} S_{τ} : Q_{p}^{d} ∋ x \mapsto \sum_{j = 1}^{n} 〈 x, τ_{j} 〉 τ_{j} \in Q_{p}^{d} \end{matrix}$

is similar to a diagonal operator over $Q_{p}$ with eigenvalues $λ_{1}, \dots, λ_{d} \in Q_{p}$ satisfying

$\begin{matrix} {|\sum_{j = 1}^{d} λ_{j}|}^{2} \leq | d | |\sum_{j = 1}^{d} λ_{j}^{2}| . \end{matrix}$

Then

\begin{matrix} {| n |}^{2} \leq | d | max_{1 \leq j, k \leq n, j \neq k} \{| n |, | 〈 τ_{j}, τ_{k} 〉 |^{2}\} . \end{matrix}

We can generalize Definition 2.3 in the following way.

Definition 2.7.

Let p be a prime,

d \in N

γ \geq 0

and

a \in Q_{p}

be nonzero. A collection

{τ_{j}}_{j = 1}^{n}

Q_{p}^{d}

is said to be p-adic $(γ, a)$ -equiangular lines if the following conditions hold: (i)(i)

$〈 τ_{j}, τ_{j} 〉 = a, \forall 1 \leq j \leq n$ .
$| 〈 τ_{j}, τ_{k} 〉 | = γ, \forall 1 \leq j, k \leq n, j \neq k$ .
The operator

$\begin{matrix} S_{τ} : Q_{p}^{d} ∋ x \mapsto \sum_{j = 1}^{n} 〈 x, τ_{j} 〉 τ_{j} \in Q_{p}^{d} \end{matrix}$

is similar to a diagonal operator over $Q_{p}$ with eigenvalues $λ_{1}, \dots, λ_{d} \in Q_{p}$ satisfying

$\begin{matrix} {|\sum_{j = 1}^{d} λ_{j}|}^{2} \leq | d | |\sum_{j = 1}^{d} λ_{j}^{2}| . \end{matrix}$

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.

{τ_{j}}_{j = 1}^{n}

is p-adic

(γ, a)

-equiangular lines in

Q_{p}^{d}

, then

\begin{matrix} {| n |}^{2} \leq | d | max \{| n |, \frac{γ^{2}}{| a^{2} |}\} . \end{matrix}

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

If $| a^{2} n | \leq γ^{2}$ , then

$\begin{matrix} {| n |}^{2} \leq | d | \frac{γ^{2}}{| a^{2} |} . \end{matrix}$
If $| a^{2} n | \geq γ^{2}$ , then

$\begin{matrix} | n | \leq | d | . \end{matrix}$

Corollary 2.9.

Let

{τ_{j}}_{j = 1}^{n}

be a collection in

Q_{p}^{d}

satisfying following. (i)(i)

There exists a nonzero element $a \in Q_{p}$ such that $〈 τ_{j}, τ_{j} 〉 = a, \forall 1 \leq j \leq n$ .
There exists a nonzero element $b \in Q_{p}$ such that

$\begin{matrix} b x = \sum_{j = 1}^{n} 〈 x, τ_{j} 〉 τ_{j}, \forall x \in Q_{p}^{d} . \end{matrix}$

Then

$\begin{matrix} {| n |}^{2} \leq | d | max \{| n |, \frac{γ^{2}}{| a^{2} |}\} . \end{matrix}$

Theorem 2.10.

Let

{τ_{j}}_{j = 1}^{n}

be a collection in

Q_{p}^{d}

satisfying following. (i)(i)

There exists a nonzero element $a \in Q_{p}$ such that $〈 τ_{j}, τ_{j} 〉 = a, \forall 1 \leq j \leq n$ .
The operator

$\begin{matrix} S_{τ} : Q_{p}^{d} ∋ x \mapsto \sum_{j = 1}^{n} 〈 x, τ_{j} 〉 τ_{j} \in Q_{p}^{d} \end{matrix}$

is similar to a diagonal operator over $Q_{p}$ with eigenvalues $λ_{1}, \dots, λ_{d} \in Q_{p}$ satisfying

$\begin{matrix} {|\sum_{j = 1}^{d} λ_{j}|}^{2} \leq | d | |\sum_{j = 1}^{d} λ_{j}^{2}| . \end{matrix}$

Then

\begin{matrix} {| n |}^{2} \leq | d | max_{1 \leq j, k \leq n, j \neq k} \{| n |, \frac{| 〈 τ_{j}, τ_{k} 〉 |^{2}}{| a^{2} |}\} . \end{matrix}

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

Theorem 2.11.

[10] (Gerzon Universal Bound) Let

{τ_{j}}_{j = 1}^{n}

be γ-equiangular lines in

R^{d}

. Then

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

We are unable to derive p-adic version of Theorem 2.11. It is clear that, in the paper, we can replace

Q_{p}

by any non-Archimedean field.

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

Abstract

Keywords:

Subject:

1. Introduction

2. p-adic Equiangular Lines

References

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