## 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 \({\mathcal {G}}\):

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 \({\mathcal {G}}\):

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

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

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

Then

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

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

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

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

## 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

Define the set

### Lemma 2.2

Let \(n,s\ge 1\) be integers. Then

### 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

### 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

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

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

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

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

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

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

for each *i*, hence

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

Then

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## Acknowledgements

I am indebted to Lajos Rónyai for his useful remarks.

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### Cite this article

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

### Mathematics Subject Classification

- 52C45
- 15A03
- 12D99