# A new proof of a generalization of Gerzon’s bound

## Abstract

In this paper we give a short, new proof of a natural generalization of Gerzon’s bound. This bound improves the Delsarte, Goethals and Seidel’s upper bound in a special case. Our proof is a simple application of the linear algebra bound method.

## Introduction

In this paper we give a linear algebraic proof of the known upper bound for the size of some special spherical s-distance sets. This result generalizes Gerzon’s general upper bound for the size of equiangular spherical set.

In the sequel, $${\mathbb {R}}[x_1, \ldots , x_n]={\mathbb {R}}[{\mathbf {x}}]$$ denotes the ring of polynomials in commuting variables $$x_1, \ldots , x_n$$ over $${\mathbb {R}}$$.

Let $${\mathcal {G}}\subseteq {{\mathbb {R}}}^n$$ be an arbitrary set. Denote by $$d({\mathcal {G}})$$ the set of (non-zero) distances among points of $${\mathcal {G}}$$:

\begin{aligned} d({\mathcal {G}}):=\{d({\mathbf {p}}_1,{\mathbf {p}}_2):~ {\mathbf {p}}_1,{\mathbf {p}}_2\in {\mathcal {G}},{\mathbf {p}}_1\ne {\mathbf {p}}_2\}. \end{aligned}

An s-distance set is any subset $$\mathcal {H} \subseteq {{\mathbb {R}}}^n$$ such that $$|d(\mathcal {H} )|\le s$$.

Let $$({\mathbf {x}},{\mathbf {y}})$$ stand for the standard scalar product. Let $$s({\mathcal {G}})$$ denote the set of scalar products between the distinct points of $${\mathcal {G}}$$:

\begin{aligned} s({\mathcal {G}}):=\{({\mathbf {p}}_1,{\mathbf {p}}_2):~ {\mathbf {p}}_1,{\mathbf {p}}_2\in {\mathcal {G}},{\mathbf {p}}_1\ne {\mathbf {p}}_2\}. \end{aligned}

A spherical s-distance set means any subset $$\mathcal G\subseteq {{\mathbb {S}}}^{n-1}$$ such that $${|s({\mathcal {G}})|\le s}$$.

Let $$n,s\ge 1$$ be integers. Define

\begin{aligned} M(n,s):={n+s-1\atopwithdelims ()s}+{n+s-2\atopwithdelims ()s-1}. \end{aligned}

Delsarte, Goethals and Seidel investigated the spherical s-distance sets. They proved a general upper bound for the size of a spherical s-distance set in [1].

### Theorem 1.1

(Delsarte et al. [1]) Suppose that $${\mathcal {F}}\subseteq {{\mathbb {S}}}^{n-1}$$ is a set satisfying $$|s({\mathcal {F}})|\le s$$. Then

\begin{aligned} |{\mathcal {F}}|\le M(n,s). \end{aligned}

Barg and Musin gave an improved upper bound for the size of a spherical s-distance set in a special case in [2]. Their proof builds upon Delsarte’s ideas (see [1]) and they used Gegenbauer polynomials in their argument.

### Theorem 1.2

(Barg and Musin [2]) Let $$n\ge 1$$ be a positive integer and let $$s>0$$ be an even integer. Let $${\mathcal {S}}\subseteq {{\mathbb {S}}}^{n-1}$$ denote a spherical s-distance set with inner products $$t_1,\ldots ,t_s$$ such that

\begin{aligned} t_1+\cdots +t_s\ge 0. \end{aligned}

Then

\begin{aligned} |{\mathcal {S}}|\le M(n,s-2)+\frac{n+2s-2}{s}{n+s-1\atopwithdelims ()s-1}. \end{aligned}

We point out the following special case of Theorem 1.1.

### Corollary 1.3

Suppose that $${\mathcal {F}}\subseteq {{\mathbb {S}}}^{n-1}$$ is a set satisfying $$|s({\mathcal {F}})|\le 2$$. Then

\begin{aligned} |{\mathcal {S}}|\le \frac{n(n+3)}{2} . \end{aligned}

An equiangular spherical set means a two-distance spherical set with scalar products $$\alpha$$ and $$-\alpha$$. Let M(n) denote the maximum cardinality of an equiangular spherical set. There is a very extensive literature devoted to the determination of the precise value of M(n) (see [3, 4]). Gerzon gave the first general upper bound for M(n)/

### Theorem 1.4

(Gerzon [3, Theorem 8]) Let $$n\ge 1$$ be a positive integer. Then

\begin{aligned} M(n)\le \frac{n(n+1)}{2}. \end{aligned}

Musin proved a stronger version of Gerzon’s Theorem in [5, Theorem 1]. He used the linear algebra bound method in his proof.

### Theorem 1.5

(Musin [5, Theorem 1]) Let $${\mathcal {S}}$$ be a spherical two-distance set with inner products a and b. Suppose that $$a + b\ge 0$$. Then

\begin{aligned} |{\mathcal {S}}|\le \frac{n(n+1)}{2}. \end{aligned}

De Caen gave a lower bound for the size of an equiangular spherical set.

### Theorem 1.6

(de Caen [6]) Let $$t>0$$ be a positive integer. For each $$n = 3 \cdot 2^{2t-1} - 1$$ there exists an equiangular spherical set of $$\frac{2}{9} (n + 1)^2$$ vectors.

Our main result is an alternative proof of a natural generalization of Gerzon’s bound, which improves the Delsarte, Goethals and Seidel’s upper bound in a special case. Our proof uses the linear algebra bound method. The following statement was proved in [7] Theorem 6.1. The original proof builds upon matrix techniques and the addition formula for Jacobi polynomials.

### Theorem 1.7

(Delsarte et al. [7, Theorem 6.1]) Let $$s=2t>0$$ be an even number and $$n>0$$ be a positive integer. Let $${\mathcal {S}}$$ denote a spherical s-distance set with inner products $$a_1,\ldots ,a_t,-a_1,\ldots ,-a_t$$. Suppose that $$0<a_i< 1$$ for each i. Then

\begin{aligned} |{\mathcal {S}}|\le {n+s-1\atopwithdelims ()s}. \end{aligned}

## Preliminaries

We prove our main result using the linear algebra bound method and the Determinant Criterion (see [8, Proposition 2.7]). We recall here this principle for the reader’s convenience.

### Proposition 2.1

(Determinant Criterion) Let $${\mathbb {F}}$$ be an arbitrary field. Let $$f_i:\Omega \rightarrow {\mathbb {F}}$$ be functions and $${\mathbf {v}}_j\in \Omega$$ elements for each $$1\le i,j\le m$$ such that the $$m \times m$$ matrix $$B=(f_i({\mathbf {v}}_j))_{i,j=1}^m$$ is non-singular. Then $$f_1,\ldots ,f_m$$ are linearly independent.

Consider the set of vectors

\begin{aligned} {\mathcal {M}}(n,s):=\left\{ \alpha =(\alpha _1, \ldots , \alpha _n)\in {{\mathbb {N}}}^n:~ \alpha _1\le 1, \sum _{i=1}^n \alpha _i \text{ is } \text{ even } ,\ \sum _{i=1}^n\alpha _i\le s\right\} . \end{aligned}

Define the set

\begin{aligned} {\mathcal {N}}(n,s)=\left\{ \alpha =(\alpha _1, \ldots , \alpha _n)\in {{\mathbb {N}}}^n:~ \sum _{i=1}^n \alpha _i\le s\right\} . \end{aligned}

### Lemma 2.2

Let $$n,s\ge 1$$ be integers. Then

\begin{aligned} |{\mathcal {N}}(n,s)|={n+s \atopwithdelims ()s}. \end{aligned}

### Proof

For a simple proof of this fact see [9, Section 9.2 Lemma 4]. $$\square$$

We need the following combinatorial statement.

### Lemma 2.3

Let $$n>0$$ be a positive integer and $$s>0$$ be an even integer. Then

\begin{aligned} |{\mathcal {M}}(n,s)|={n+s-1\atopwithdelims ()s}. \end{aligned}

### Proof

It is easy to check that there exists a bijection $$f:{\mathcal {M}}(n,s)\rightarrow {\mathcal {N}}(n-1,s)$$, since s is even. Namely, let $$\alpha =(\alpha _1, \ldots , \alpha _n)\in {\mathcal {M}}(n,s)$$ be an arbitrary element. Define $$f(\alpha ):=(\alpha _2, \ldots , \alpha _n)$$. It is easy to verify that $$f({\mathcal {M}}(n,s))\subseteq {\mathcal {N}}(n-1,s)$$ and f is a bijection.

Hence $$|{\mathcal {M}}(n,s)|=|{\mathcal {N}}(n-1,s)|$$ and Lemma 2.2 gives the result. $$\square$$

## Proof of Theorem 1.7

Consider the real polynomial

\begin{aligned} g(x_1,\ldots ,x_n)=\left( \sum _{m=1}^n x_m^2\right) -1\in {\mathbb {R}}[x_1, \ldots ,x_n]. \end{aligned}

Let $${\mathcal {S}}=\{{\mathbf {v}}_1, \ldots , {\mathbf {v}}_r\}$$ denote a spherical s-distance set with inner products $$a_1,\ldots ,a_t,-a_1,\ldots ,-a_t$$. Here $$r=|{\mathcal {S}}|$$. Define the polynomials

\begin{aligned} P_i(x_1, \ldots ,x_n):= \prod _{m=1}^t \Big ((\langle {\mathbf {x}}, {\mathbf {v}}_i\rangle )^2-(a_m)^2 \Big )\in {\mathbb {R}}[{\mathbf {x}}] \end{aligned}

for each $$1\le i\le r$$. Clearly $$\text{ deg }(P_i)\le s=2t$$ for each $$1\le i\le r$$.

Consider the set of vectors

\begin{aligned} {\mathcal {E}}(n,s):=\left\{ \alpha =(\alpha _1, \ldots , \alpha _n)\in {{\mathbb {N}}}^n:~ \sum _i \alpha _i \text{ is } \text{ even } ,\ \sum \alpha _i\le s\right\} \end{aligned}

It is easy to verify that if we can expand $$P_i$$ as a linear combination of monomials, then we get

\begin{aligned} P_i(x_1, \ldots ,x_n)=\sum _{\alpha \in {\mathcal {E}}(n,s)} c_{\alpha }x^{\alpha }, \end{aligned}
(3.1)

where $$c_{\alpha }\in {\mathbb {R}}$$ are real coefficients for each $${\alpha }\in {\mathcal {E}}(n,s)$$ and $$x^{\alpha }$$ denotes the monomial $$x_{1}^{\alpha _{1}}\cdot \ldots \cdot x_{n}^{\alpha _{n}}$$.

Since $${\mathbf {v}}_i\in {{\mathbb {S}}}^{n-1}$$, this means that the equation

\begin{aligned} x_1^2=1-\sum _{j=2}^n x_j^2 \end{aligned}
(3.2)

is true for each $${\mathbf {v}}_i$$, where $$1\le i\le r$$. Let $$Q_i$$ denote the polynomial obtained by writing $$P_i$$ as a linear combination of monomials and replacing, repeatedly, each occurrence of $$x_1^t$$, where $$t\ge 2$$, by a linear combination of other monomials, using the relations (3.2).

Since $$g({\mathbf {v}}_i)=0$$ for each i, hence $$Q_i({\mathbf {v}}_j)=P_i({\mathbf {v}}_j)$$ for each $$1\le i\ne j\le r$$.

We prove that the set of polynomials $$\{Q_i:~ 1\le i\le r\}$$ is linearly independent. This fact follows from the Determinant Criterion, when we define $${\mathbb {F}}:={\mathbb {R}}$$, $$\Omega ={{\mathbb {S}}}^{n-1}$$ and $$f_i:=Q_i$$ for each i. It is enough to prove that $$Q_i({\mathbf {v}}_i)=P_i({\mathbf {v}}_i)\ne 0$$ for each $$1\le i\le r$$ and $$Q_i({\mathbf {v}}_j)=P_i({\mathbf {v}}_j)=0$$ for each $$1\le i\ne j\le r$$, since then we can apply the Determinant Criterion.

But $$P_i({\mathbf {v}}_i)=\prod _{i=1}^m (1-a_m^2)$$ and $$P_i({\mathbf {v}}_j)=0$$, because $${\mathcal {S}}=\{{\mathbf {v}}_1, \ldots , {\mathbf {v}}_r\}$$ is a spherical s-distance set with inner products $$a_1,\ldots ,a_t,-a_1,\ldots ,-a_t$$.

It is easy to check that we can write $$Q_i$$ as a linear combination of monomials in the form

\begin{aligned} Q_i=\sum _{\alpha \in {\mathcal {M}}(n,s)} d_{\alpha }x^{\alpha }, \end{aligned}

where $$d_{\alpha }\in {\mathbb {R}}$$ are the real coefficients for each $${\alpha }\in {\mathcal {M}}(n,s)$$. This follows immediately from the expansion (3.1) and from the relation (3.2).

Since the polynomials $$\{Q_i:~ 1\le i\le r\}$$ are linearly independent and if we expand $$Q_i$$ as a linear combination of monomials, then all monomials appearing in this linear combination contained in the set of monomials

\begin{aligned} \{x^{\alpha }:~ \alpha \in {\mathcal {M}}(n,s)\} \end{aligned}

for each i, hence

\begin{aligned} r=|{\mathcal {S}}|\le |{\mathcal {M}}(n,s)|={n+s-1\atopwithdelims ()s}, \end{aligned}

by Lemma 2.3. $$\square$$

## Concluding remarks

The following Conjecture is a natural generalization of Theorem 1.5 and a strengthening of Theorem 1.2.

### Conjecture 4.1

Let $$n\ge 1$$ be a positive integer and let $$s>0$$ be an even integer. Let $${\mathcal {S}}\subseteq {{\mathbb {S}}}^{n-1}$$ denote a spherical s-distance set with inner products $$t_1,\ldots ,t_s$$ such that

\begin{aligned} t_1+\ldots +t_s\ge 0. \end{aligned}

Then

\begin{aligned} |{\mathcal {S}}|\le {n+s-1\atopwithdelims ()s}. \end{aligned}

## References

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## Acknowledgements

I am indebted to Lajos Rónyai for his useful remarks.

## Funding

Open access funding provided by Óbuda University

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Correspondence to Gábor Hegedüs.

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Hegedüs, G. A new proof of a generalization of Gerzon’s bound. Period Math Hung (2020). https://doi.org/10.1007/s10998-020-00372-9

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### Keywords

• Gerzon’s bound
• Distance problem
• Linear algebra bound method

• 52C45
• 15A03
• 12D99