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.
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1 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
2 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. \(\square \)
3 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. \(\square \)
4 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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Hegedüs, G. A new proof of a generalization of Gerzon’s bound. Period Math Hung 83, 49–53 (2021). https://doi.org/10.1007/s10998-020-00372-9
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DOI: https://doi.org/10.1007/s10998-020-00372-9