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

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

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Acknowledgements

I am indebted to Lajos Rónyai for his useful remarks.

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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

Mathematics Subject Classification

  • 52C45
  • 15A03
  • 12D99