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$B_h[g]$-vectors and finite families of linear forms

Part of: Sequences and sets Spaces with richer structures

Published online by Cambridge University Press:  13 July 2026

Diego Marques Open the ORCID record for Diego Marques [Opens in a new window]
Diego Marques*
Affiliation:
Universidade de Brasília , Brazil
*
e-mail: diego@mat.unb.br
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Abstract

In a recent work, Nathanson proved that, for $K=\mathbb {R}$ or $\mathbb {C}$, the set of $B_h$-vectors in $K^n$ is open and dense. Leonetti later extended this to $B_h[g]$-vectors. In this note, we isolate a finite linear functional principle behind these results. Namely, if $\mathcal F$ is a finite family of pairwise distinct linear functionals on $K^n$, then the set of points at which no value is assumed by more than g members of $\mathcal F$ is open and dense in $K^n$. As consequences, we recover the theorems of Nathanson and Leonetti for $B_h[g]$-vectors and obtain the following variant: for every linear form $L(x_1,\ldots ,x_h)=c_1x_1+\cdots +c_hx_h$, with nonzero integer coefficients, the corresponding set of $L[g]$-vectors, evaluated on distinct variables, is open and dense in $K^n$.

Keywords

B h-setB h[g]-setSidon setlinear formsopen and dense

MSC classification

Primary: 11B05: Density, gaps, topology 11B13: Additive bases, including sumsets
Secondary: 11B75: Other combinatorial number theory 54E52: Baire category, Baire spaces

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Article
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Canadian Mathematical Bulletin , First View , pp. 1 - 6
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© The Author(s), 2026. Published by Cambridge University Press on behalf of Canadian Mathematical Society

1 Introduction

The study of $B_h$ upper B Subscript h -sets is a classical topic in additive combinatorics. A subset A of an additive abelian semigroup G is called a $B_h$ upper B Subscript h -set if every element of G has at most one representation as the sum of h not necessarily distinct elements of A. The case $h=2$ h equals 2 corresponds to Sidon sets.

In the setting of real and complex numbers, Nathanson [Reference Nathanson1] introduced a topological viewpoint on finite $B_h$ upper B Subscript h -sets. Let $K=\mathbb {R}$ upper K equals double struck upper R or $\mathbb {C}$ double struck upper C . To a vector

$$\begin{align*}\mathbf a=(a_1,\ldots,a_n)\in K^n \end{align*}$$ bold a equals left parenthesis a 1 comma ellipsis comma a Subscript n Baseline right parenthesis element of upper K Superscript n

with pairwise distinct coordinates, one associates the set

$$\begin{align*}A=\{a_1,\ldots,a_n\}\subseteq K. \end{align*}$$ upper A equals StartSet a 1 comma ellipsis comma a Subscript n Baseline EndSet subset of or equal to upper K period

The vector $\mathbf a$ bold a is called a $B_h$ upper B Subscript h -vector if A is a $B_h$ upper B Subscript h -set. Nathanson proved that the set of $B_h$ upper B Subscript h -vectors is an open and dense subset of $K^n$ upper K Superscript n .

A natural generalization is the notion of a $B_h[g]$ upper B Subscript h Baseline left bracket g right bracket -set. For positive integers g and h, a subset A of G is called a $B_h[g]$ upper B Subscript h Baseline left bracket g right bracket -set if every element of G has at most g representations as a sum of h not necessarily distinct elements of A, where representations are counted up to order. Thus, the case $g=1$ g equals 1 recovers the notion of a $B_h$ upper B Subscript h -set. Nathanson asked whether the set of $B_h[g]$ upper B Subscript h Baseline left bracket g right bracket -vectors is also open and dense in $K^n$ upper K Superscript n , and this was answered affirmatively by Leonetti [Reference Leonetti2].

The common mechanism behind these results is that coincidences among finitely many linear expressions define a finite union of proper linear subspaces. The purpose of this note is to make this principle explicit in a form that applies both to the repeated-summand setting of $B_h[g]$ upper B Subscript h Baseline left bracket g right bracket -sets and to fixed linear forms.

We first prove an elementary finite-linear-functional theorem. Let $\mathcal F$ script upper F be a finite family of pairwise distinct linear functionals $\ell :K^n\to K$ script l colon upper K Superscript n Baseline right arrow upper K . We say that $\mathbf a\in K^n$ bold a element of upper K Superscript n is an $\mathcal F[g]$ script upper F left bracket g right bracket -vector if, for every $w\in K$ w element of upper K , the equation

$$\begin{align*}\ell(\mathbf a)=w, \qquad \ell\in\mathcal F, \end{align*}$$ StartLayout 1st Row script l left parenthesis bold a right parenthesis equals w comma script l element of script upper F comma EndLayout

has at most g solutions $\ell \in \mathcal F$ script l element of script upper F . We prove that the set of $\mathcal F[g]$ script upper F left bracket g right bracket -vectors is open and dense in $K^n$ upper K Superscript n .

This immediately recovers the Nathanson and Leonetti results. Indeed, repetitions in an h-fold sum are encoded by multiplicity vectors

$$\begin{align*}(m_1,\ldots,m_n)\in\mathbb{Z}_{\ge0}^n, \qquad m_1+\cdots+m_n=h, \end{align*}$$ left parenthesis m 1 comma ellipsis comma m Subscript n Baseline right parenthesis element of double struck upper Z Subscript greater than or equals 0 Superscript n Baseline comma m 1 plus midline horizontal ellipsis plus m Subscript n Baseline equals h comma

through the linear functionals

$$\begin{align*}\ell_m(\mathbf a)=m_1a_1+\cdots+m_na_n. \end{align*}$$ script l Subscript m Baseline left parenthesis bold a right parenthesis equals m 1 a 1 plus midline horizontal ellipsis plus m Subscript n Baseline a Subscript n Baseline period

Different multiplicity vectors give different linear functionals.

We also obtain a fixed-linear-form version. Let

$$\begin{align*}L(x_1,\ldots,x_h)=c_1x_1+\cdots+c_hx_h \end{align*}$$ upper L left parenthesis x 1 comma ellipsis comma x Subscript h Baseline right parenthesis equals c 1 x 1 plus midline horizontal ellipsis plus c Subscript h Baseline x Subscript h

be a linear form with $c_i\in \mathbb {Z}\setminus \{0\}$ c Subscript i Baseline element of double struck upper Z minus StartSet 0 EndSet . Negative coefficients are allowed because we are considering linear-form equations $L(x_1,\ldots ,x_h)=w$ upper L left parenthesis x 1 comma ellipsis comma x Subscript h Baseline right parenthesis equals w ; equivalently, terms with negative coefficients may be moved to the other side of the equation. For a vector $\mathbf a=(a_1,\ldots ,a_n)\in K^n$ bold a equals left parenthesis a 1 comma ellipsis comma a Subscript n Baseline right parenthesis element of upper K Superscript n with pairwise distinct coordinates, we evaluate L on distinct elements of the associated set $A=\{a_1,\ldots ,a_n\}$ upper A equals StartSet a 1 comma ellipsis comma a Subscript n Baseline EndSet . Thus, we consider values of the form

$$\begin{align*}L(a_{i_1},\ldots,a_{i_h}), \qquad 1\le i_1<\cdots<i_h\le n. \end{align*}$$ upper L left parenthesis a Subscript i 1 Baseline comma ellipsis comma a Subscript i Sub Subscript h Subscript Baseline right parenthesis comma 1 less than or equals i 1 less than midline horizontal ellipsis less than i Subscript h Baseline less than or equals n period

The resulting set of $L[g]$ upper L left bracket g right bracket -vectors is again open and dense in $K^n$ upper K Superscript n .

Remark 1.1 The restriction to distinct variables is essential for general signed linear forms. Unlike the classical $B_h[g]$ upper B Subscript h Baseline left bracket g right bracket case, where repetitions are encoded by multiplicity vectors, repetitions may force identities. For example, if $L(x_1,x_2)=x_1-x_2$ upper L left parenthesis x 1 comma x 2 right parenthesis equals x 1 minus x 2 , then $L(a_i,a_i)=0$ upper L left parenthesis a Subscript i Baseline comma a Subscript i Baseline right parenthesis equals 0 for every i, independently of $\mathbf a$ bold a . Thus, repeated variables may create unavoidable coincidences, and the fixed-linear-form result is therefore stated for distinct variables.

2 A finite-linear-functional principle

Let $K=\mathbb {R}$ upper K equals double struck upper R or $\mathbb {C}$ double struck upper C , and let $\mathcal F$ script upper F be a finite family of pairwise distinct linear functionals

$$\begin{align*}\ell:K^n\longrightarrow K. \end{align*}$$ script l colon upper K Superscript n Baseline long right arrow upper K period

Definition 2.1 For an integer $g\ge 1$ g greater than or equals 1 , a vector $\mathbf a\in K^n$ bold a element of upper K Superscript n is called an $\mathcal F[g]$ script upper F left bracket g right bracket -vector if, for every $w\in K$ w element of upper K , the equation

$$\begin{align*}\ell(\mathbf a)=w,\qquad \ell\in\mathcal F, \end{align*}$$ StartLayout 1st Row script l left parenthesis bold a right parenthesis equals w comma script l element of script upper F comma EndLayout

has at most g solutions $\ell \in \mathcal F$ script l element of script upper F . We denote the set of all $\mathcal F[g]$ script upper F left bracket g right bracket -vectors by $\mathcal B_{\mathcal F}[g]$ script upper B Subscript script upper F Baseline left bracket g right bracket .

We now state the main structural result of the article.

Theorem 2.1 Let $K=\mathbb {R}$ upper K equals double struck upper R or $\mathbb {C}$ double struck upper C , and let $\mathcal F$ script upper F be a finite family of pairwise distinct linear functionals on $K^n$ upper K Superscript n . Then, for every integer $g\ge 1$ g greater than or equals 1 , the set $\mathcal B_{\mathcal F}[g]$ script upper B Subscript script upper F Baseline left bracket g right bracket is open and dense in $K^n$ upper K Superscript n .

Proof If $|\mathcal F|\le g$ StartAbsoluteValue script upper F EndAbsoluteValue less than or equals g , then every vector in $K^n$ upper K Superscript n is an $\mathcal F[g]$ script upper F left bracket g right bracket -vector, and there is nothing to prove. Thus assume $|\mathcal F|\ge g+1$ StartAbsoluteValue script upper F EndAbsoluteValue greater than or equals g plus 1 .

For each ordered $(g+1)$ left parenthesis g plus 1 right parenthesis -tuple

$$\begin{align*}\boldsymbol\ell=(\ell_0,\ell_1,\ldots,\ell_g) \end{align*}$$ bold italic script l equals left parenthesis script l 0 comma script l 1 comma ellipsis comma script l Subscript g Baseline right parenthesis

of pairwise distinct elements of $\mathcal F$ script upper F , define

$$\begin{align*}E(\boldsymbol\ell) = \{\mathbf a\in K^n: \ell_0(\mathbf a)=\ell_1(\mathbf a)=\cdots=\ell_g(\mathbf a)\}. \end{align*}$$ StartLayout 1st Row upper E left parenthesis bold italic script l right parenthesis equals StartSet bold a element of upper K Superscript n Baseline colon script l 0 left parenthesis bold a right parenthesis equals script l 1 left parenthesis bold a right parenthesis equals midline horizontal ellipsis equals script l Subscript g Baseline left parenthesis bold a right parenthesis EndSet period EndLayout

Equivalently,

$$\begin{align*}E(\boldsymbol\ell) = \bigcap_{j=1}^g \ker(\ell_0-\ell_j). \end{align*}$$ upper E left parenthesis bold italic script l right parenthesis equals intersection Underscript j equals 1 Overscript g Endscripts kernel left parenthesis script l 0 minus script l Subscript j Baseline right parenthesis period

Since the functionals in $\mathcal F$ script upper F are pairwise distinct, each difference $\ell _0-\ell _j$ script l 0 minus script l Subscript j is a nonzero linear functional. Hence, each $\ker (\ell _0-\ell _j)$ kernel left parenthesis script l 0 minus script l Subscript j Baseline right parenthesis is a proper closed linear hyperplane in $K^n$ upper K Superscript n , and $E(\boldsymbol \ell )$ upper E left parenthesis bold italic script l right parenthesis is a proper closed linear subspace of $K^n$ upper K Superscript n .

Define the exceptional set

$$\begin{align*}\mathcal E = \bigcup_{\substack{ \boldsymbol\ell=(\ell_0,\ldots,\ell_g)\in\mathcal F^{g+1}\\ \ell_0,\ldots,\ell_g\ \text{pairwise distinct} }} E(\boldsymbol\ell). \end{align*}$$ script upper E equals union Underscript StartLayout 1st Row bold italic script l equals left parenthesis script l 0 comma ellipsis comma script l Subscript g Baseline right parenthesis element of script upper F Superscript g plus 1 Baseline 2nd Row script l 0 comma ellipsis comma script l Subscript g Baseline pairwise distinct EndLayout Endscripts upper E left parenthesis bold italic script l right parenthesis period

This is a finite union of proper closed linear subspaces of $K^n$ upper K Superscript n . Therefore, $\mathcal E$ script upper E is closed and has empty interior. Consequently, $K^n\setminus \mathcal E$ upper K Superscript n Baseline minus script upper E is open and dense in $K^n$ upper K Superscript n .

By construction, a vector $\mathbf a$ bold a belongs to $\mathcal E$ script upper E if and only if there exist $g+1$ g plus 1 pairwise distinct functionals $\ell _0,\ldots ,\ell _g\in \mathcal F$ script l 0 comma ellipsis comma script l Subscript g Baseline element of script upper F such that

$$\begin{align*}\ell_0(\mathbf a)=\ell_1(\mathbf a)=\cdots=\ell_g(\mathbf a). \end{align*}$$ StartLayout 1st Row script l 0 left parenthesis bold a right parenthesis equals script l 1 left parenthesis bold a right parenthesis equals midline horizontal ellipsis equals script l Subscript g Baseline left parenthesis bold a right parenthesis period EndLayout

Equivalently, $\mathbf a\notin \mathcal E$ bold a not an element of script upper E if and only if no value $w\in K$ w element of upper K is assumed by more than g members of $\mathcal F$ script upper F at $\mathbf a$ bold a . Thus,

$$\begin{align*}\mathcal B_{\mathcal F}[g]=K^n\setminus\mathcal E. \end{align*}$$ script upper B Subscript script upper F Baseline left bracket g right bracket equals upper K Superscript n Baseline minus script upper E period

Hence, $\mathcal B_{\mathcal F}[g]$ script upper B Subscript script upper F Baseline left bracket g right bracket is open and dense in $K^n$ upper K Superscript n .

We shall also use the following immediate variant, which incorporates the requirement that the coordinates of the vector be distinct.

Corollary 2.2 Let $K=\mathbb {R}$ upper K equals double struck upper R or $\mathbb {C}$ double struck upper C , and let $\mathcal F$ script upper F be a finite family of pairwise distinct linear functionals on $K^n$ upper K Superscript n . Let

$$\begin{align*}D = \{\mathbf a=(a_1,\ldots,a_n)\in K^n: a_i\ne a_j\ \text{whenever}\ i\ne j\}. \end{align*}$$ upper D equals StartSet bold a equals left parenthesis a 1 comma ellipsis comma a Subscript n Baseline right parenthesis element of upper K Superscript n Baseline colon a Subscript i Baseline not equals a Subscript j Baseline whenever i not equals j EndSet period

Then

$$\begin{align*}\mathcal B_{\mathcal F}^{\,*}[g] := \mathcal B_{\mathcal F}[g]\cap D \end{align*}$$ StartLayout 1st Row script upper B Subscript script upper F Superscript asterisk Baseline left bracket g right bracket colon equals script upper B Subscript script upper F Baseline left bracket g right bracket intersection upper D EndLayout

is open and dense in $K^n$ upper K Superscript n .

Proof We have

$$\begin{align*}D = K^n \setminus \bigcup_{1\le i<j\le n} \{\mathbf a\in K^n:a_i-a_j=0\}. \end{align*}$$ upper D equals upper K Superscript n Baseline minus union Underscript 1 less than or equals i less than j less than or equals n Endscripts StartSet bold a element of upper K Superscript n Baseline colon a Subscript i Baseline minus a Subscript j Baseline equals 0 EndSet period

Thus, D is the complement of a finite union of proper closed hyperplanes, and so D is open and dense in $K^n$ upper K Superscript n . By Theorem 2.1, $\mathcal B_{\mathcal F}[g]$ script upper B Subscript script upper F Baseline left bracket g right bracket is open and dense. Therefore, $\mathcal B_{\mathcal F}[g]\cap D$ script upper B Subscript script upper F Baseline left bracket g right bracket intersection upper D is open and dense in $K^n$ upper K Superscript n .

3 Applications

We now recover the classical $B_h[g]$ upper B Subscript h Baseline left bracket g right bracket result and then deduce the fixed-linear-form variant.

3.1 The repeated-summand case

Let $h,n\ge 1$ h comma n greater than or equals 1 , and define

$$\begin{align*}\mathcal M_{h,n} = \left\{ (m_1,\ldots,m_n)\in\mathbb{Z}_{\ge0}^n: m_1+\cdots+m_n=h \right\}. \end{align*}$$ script upper M Subscript h comma n Baseline equals StartSet left parenthesis m 1 comma ellipsis comma m Subscript n Baseline right parenthesis element of double struck upper Z Subscript greater than or equals 0 Superscript n Baseline colon m 1 plus midline horizontal ellipsis plus m Subscript n Baseline equals h EndSet period

For each $\textbf {m}=(m_1,\ldots ,m_n)\in \mathcal M_{h,n}$ bold m equals left parenthesis m 1 comma ellipsis comma m Subscript n Baseline right parenthesis element of script upper M Subscript h comma n , define a linear functional $\ell _{\textbf {m}}:K^n\to K$ script l Subscript bold m Baseline colon upper K Superscript n Baseline right arrow upper K by

$$\begin{align*}\ell_{\textbf{m}}(\mathbf a) = m_1a_1+\cdots+m_na_n. \end{align*}$$ script l Subscript bold m Baseline left parenthesis bold a right parenthesis equals m 1 a 1 plus midline horizontal ellipsis plus m Subscript n Baseline a Subscript n Baseline period

Let

$$\begin{align*}\mathcal F_{h,n} = \{\ell_{\textbf{m}}:\textbf{m}\in\mathcal M_{h,n}\}. \end{align*}$$ script upper F Subscript h comma n Baseline equals StartSet script l Subscript bold m Baseline colon bold m element of script upper M Subscript h comma n Baseline EndSet period

If $\textbf {m}\ne \textbf {m}'$ bold m not equals bold m prime , then $\ell _{\textbf {m}}\ne \ell _{\textbf {m}'}$ script l Subscript bold m Baseline not equals script l Subscript bold m prime . Hence, $\mathcal F_{h,n}$ script upper F Subscript h comma n is a finite family of pairwise distinct linear functionals.

Corollary 3.1 (Nathanson–Leonetti)

Let $K=\mathbb {R}$ upper K equals double struck upper R or $\mathbb {C}$ double struck upper C , and let $g,h,n$ g comma h comma n be positive integers. The set of $B_h[g]$ upper B Subscript h Baseline left bracket g right bracket -vectors in $K^n$ upper K Superscript n is open and dense in $K^n$ upper K Superscript n .

Proof Let $\mathbf a=(a_1,\ldots ,a_n)\in K^n$ bold a equals left parenthesis a 1 comma ellipsis comma a Subscript n Baseline right parenthesis element of upper K Superscript n have pairwise distinct coordinates, and let $A=\{a_1,\ldots ,a_n\}$ upper A equals StartSet a 1 comma ellipsis comma a Subscript n Baseline EndSet . Since representations are counted up to order, every representation of an element of K as a sum of h not necessarily distinct elements of A corresponds uniquely to a multiplicity vector

$$\begin{align*}\textbf{m}=(m_1,\ldots,m_n)\in\mathcal M_{h,n}. \end{align*}$$ bold m equals left parenthesis m 1 comma ellipsis comma m Subscript n Baseline right parenthesis element of script upper M Subscript h comma n Baseline period

The corresponding sum is precisely

$$\begin{align*}m_1a_1+\cdots+m_na_n=\ell_{\textbf{m}}(\mathbf a). \end{align*}$$ m 1 a 1 plus midline horizontal ellipsis plus m Subscript n Baseline a Subscript n Baseline equals script l Subscript bold m Baseline left parenthesis bold a right parenthesis period

Thus, $\mathbf a$ bold a is a $B_h[g]$ upper B Subscript h Baseline left bracket g right bracket -vector if and only if $\mathbf a\in \mathcal B_{\mathcal F_{h,n}}[g]\cap D$ bold a element of script upper B Subscript script upper F Sub Subscript h comma n Baseline left bracket g right bracket intersection upper D , where D is the set of vectors with pairwise distinct coordinates. The conclusion now follows from Corollary 2.2.

For $g=1$ g equals 1 , Corollary 3.1 gives Nathanson’s theorem on $B_h$ upper B Subscript h -vectors. For arbitrary $g\ge 1$ g greater than or equals 1 , it gives Leonetti’s theorem on $B_h[g]$ upper B Subscript h Baseline left bracket g right bracket -vectors.

3.2 A fixed-linear-form variant

Let

$$\begin{align*}L(x_1,\ldots,x_h)=c_1x_1+\cdots+c_hx_h \end{align*}$$ upper L left parenthesis x 1 comma ellipsis comma x Subscript h Baseline right parenthesis equals c 1 x 1 plus midline horizontal ellipsis plus c Subscript h Baseline x Subscript h

be a linear form with $c_1,\ldots ,c_h\in \mathbb {Z}\setminus \{0\}$ c 1 comma ellipsis comma c Subscript h Baseline element of double struck upper Z minus StartSet 0 EndSet . Let $n\ge h\ge 2$ n greater than or equals h greater than or equals 2 . Denote by

$$\begin{align*}\mathcal X_{h,n} = \{I=\{i_1<\cdots<i_h\}:1\le i_1<\cdots<i_h\le n\} \end{align*}$$ script upper X Subscript h comma n Baseline equals StartSet upper I equals left brace i 1 less than midline horizontal ellipsis less than i Subscript h Baseline right brace colon 1 less than or equals i 1 less than midline horizontal ellipsis less than i Subscript h Baseline less than or equals n EndSet

the family of h-element subsets of $\{1,\ldots ,n\}$ StartSet 1 comma ellipsis comma n EndSet . For $I=\{i_1<\cdots <i_h\}\in \mathcal X_{h,n}$ upper I equals left brace i 1 less than midline horizontal ellipsis less than i Subscript h Baseline right brace element of script upper X Subscript h comma n , define the linear functional $\ell _I:K^n\to K$ script l Subscript upper I Baseline colon upper K Superscript n Baseline right arrow upper K by

$$\begin{align*}\ell_I(\mathbf a) = c_1a_{i_1}+\cdots+c_ha_{i_h}. \end{align*}$$ script l Subscript upper I Baseline left parenthesis bold a right parenthesis equals c 1 a Subscript i 1 Baseline plus midline horizontal ellipsis plus c Subscript h Baseline a Subscript i Sub Subscript h Subscript Baseline period

Let

$$\begin{align*}\mathcal F_{L,n} = \{\ell_I:I\in\mathcal X_{h,n}\}. \end{align*}$$ script upper F Subscript upper L comma n Baseline equals StartSet script l Subscript upper I Baseline colon upper I element of script upper X Subscript h comma n Baseline EndSet period

Definition 3.1 A vector $\mathbf a=(a_1,\ldots ,a_n)\in K^n$ bold a equals left parenthesis a 1 comma ellipsis comma a Subscript n Baseline right parenthesis element of upper K Superscript n with pairwise distinct coordinates is called an $L[g]$ upper L left bracket g right bracket -vector if, for every $w\in K$ w element of upper K , the equation

$$\begin{align*}L(a_{i_1},\ldots,a_{i_h})=w, \qquad 1\le i_1<\cdots<i_h\le n, \end{align*}$$ upper L left parenthesis a Subscript i 1 Baseline comma ellipsis comma a Subscript i Sub Subscript h Subscript Baseline right parenthesis equals w comma 1 less than or equals i 1 less than midline horizontal ellipsis less than i Subscript h Baseline less than or equals n comma

has at most g solutions $I=\{i_1<\cdots <i_h\}\in \mathcal X_{h,n}$ upper I equals left brace i 1 less than midline horizontal ellipsis less than i Subscript h Baseline right brace element of script upper X Subscript h comma n . We denote the set of all $L[g]$ upper L left bracket g right bracket -vectors by $\mathcal B_L[g]$ script upper B Subscript upper L Baseline left bracket g right bracket .

Theorem 3.2 Let $K=\mathbb {R}$ upper K equals double struck upper R or $\mathbb {C}$ double struck upper C , and let

$$\begin{align*}L(x_1,\ldots,x_h)=c_1x_1+\cdots+c_hx_h \end{align*}$$ upper L left parenthesis x 1 comma ellipsis comma x Subscript h Baseline right parenthesis equals c 1 x 1 plus midline horizontal ellipsis plus c Subscript h Baseline x Subscript h

be a linear form with $c_1,\ldots ,c_h\in \mathbb {Z}\setminus \{0\}$ c 1 comma ellipsis comma c Subscript h Baseline element of double struck upper Z minus StartSet 0 EndSet . If $g\ge 1$ g greater than or equals 1 and $n\ge h\ge 2$ n greater than or equals h greater than or equals 2 , then $\mathcal B_L[g]$ script upper B Subscript upper L Baseline left bracket g right bracket is open and dense in $K^n$ upper K Superscript n .

Proof It is enough to show that the elements of $\mathcal F_{L,n}$ script upper F Subscript upper L comma n are pairwise distinct and then apply Corollary 2.2.

Let $I,J\in \mathcal X_{h,n}$ upper I comma upper J element of script upper X Subscript h comma n Baseline with $I\ne J$ upper I not equals upper J . Write

$$\begin{align*}I=\{i_1<\cdots<i_h\}, \qquad J=\{j_1<\cdots<j_h\}. \end{align*}$$ upper I equals left brace i 1 less than midline horizontal ellipsis less than i Subscript h Baseline right brace comma upper J equals left brace j 1 less than midline horizontal ellipsis less than j Subscript h Baseline right brace period

Since $I\ne J$ upper I not equals upper J , there exists an index t that belongs to exactly one of the two sets. Suppose, for instance, that $t\in I\setminus J$ t element of upper I minus upper J . Then $t=i_s$ t equals i Subscript s for some $s\in \{1,\ldots ,h\}$ s element of StartSet 1 comma ellipsis comma h EndSet . In the linear functional $\ell _I-\ell _J$ script l Subscript upper I Baseline minus script l Subscript upper J , the coefficient of the t-th coordinate is $c_s$ c Subscript s , because t does not occur among $j_1,\ldots ,j_h$ j 1 comma ellipsis comma j Subscript h Baseline . Since $c_s\ne 0$ c Subscript s Baseline not equals 0 , we have $\ell _I-\ell _J\ne 0$ script l Subscript upper I Baseline minus script l Subscript upper J Baseline not equals 0 . Hence, $\ell _I\ne \ell _J$ script l Subscript upper I Baseline not equals script l Subscript upper J .

Therefore, $\mathcal F_{L,n}$ script upper F Subscript upper L comma n is a finite family of pairwise distinct linear functionals. By Corollary 2.2,

$$\begin{align*}\mathcal B_{\mathcal F_{L,n}}[g]\cap D \end{align*}$$ script upper B Subscript script upper F Sub Subscript upper L comma n Baseline left bracket g right bracket intersection upper D

is open and dense in $K^n$ upper K Superscript n . By the definition above, this set is exactly $\mathcal B_L[g]$ script upper B Subscript upper L Baseline left bracket g right bracket . Hence, $\mathcal B_L[g]$ script upper B Subscript upper L Baseline left bracket g right bracket is open and dense in $K^n$ upper K Superscript n .

Acknowledgements

The author thanks the anonymous referees for their constructive comments, which led to a clearer formulation of the results and a better presentation of the manuscript.

Footnotes

The author gratefully acknowledges the financial support provided by the National Council for Scientific and Technological Development (CNPq).

References

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