# Northcott numbers for generalized weighted Weil heights

Masao Okazaki, Kaoru Sano  National Institute of Technology (KOSEN), Oyama
College, Tochigi, 323-0806, Japan<EMAIL_ADDRESS>NTT Institute for
Fundamental Mathematics, NTT Communication Science Laboratories, NTT
Corporation, 2-4, Hikaridai, Seika-cho, Soraku-gun, Kyoto 619-0237, Japan
<EMAIL_ADDRESS>
###### Abstract.

We give a generalization of weighted Weil heights. These heights generalize
both Weil’s heights and Dobrowolski’s height. We study Northcott numbers for
our heights. Our results generalize the authors’ former work on Vidaux and
Videla’s question about the Northcott number. As an application, we evaluate
Northcott numbers for Talamanca’s spectral height on matrices.

###### Key words and phrases:

weighted height, Northcott number, finite extensions, operator height,
spectral height

###### 2020 Mathematics Subject Classification:

Primary 11G50, Secondary 11C20, 15B33

##  1\. Introduction

Let $S$ be a set and $\mathfrak{h}\colon S\longrightarrow\mathbb{R}_{\geq 0}$
be a function. We set

$B(S,\mathfrak{h},C)=\left\\{s\in S\mid\mathfrak{h}(s)<C\right\\}$

for each $C\in\mathbb{R}_{>0}$. We also set

$\operatorname{Nor}(S,\mathfrak{h})=\inf\left\\{C\in\mathbb{R}_{>0}\mid\\#B(S,\mathfrak{h},C)=\infty\right\\}.$

The non-negative real number $\operatorname{Nor}(S,\mathfrak{h})$ is called
the $\mathfrak{h}$-Northcott number of $S$, introduced in [PTW22] and [VV16].
Here note that we regard the value $\inf\emptyset$ to be $\infty$. It is said
that $S$ has the $\mathfrak{h}$-Northcott property or $\mathfrak{h}$-(N) for
short (resp. $\mathfrak{h}$-Bogomolov property or $\mathfrak{h}$-(B) for
short) if the equality $\operatorname{Nor}(S,\mathfrak{h})=\infty$ (resp. the
inequality $\operatorname{Nor}(S\setminus Z(S,\mathfrak{h}),\mathfrak{h})>0$)
holds, where

$Z(S,\mathfrak{h})=\\{a\in S\mid\mathfrak{h}(a)=0\\}$

(see [BZ01] and [PTW22]). Note that $\mathfrak{h}$-(B) follows from
$\mathfrak{h}$-(N).

In this paper, we fix an algebraic closure $\overline{\mathbb{Q}}$ of
$\mathbb{Q}$ and consider any algebraic extension of $\mathbb{Q}$ to be a
subset of $\overline{\mathbb{Q}}$. Let
$h\colon\overline{\mathbb{Q}}\longrightarrow\mathbb{R}_{\geq 0}$ be the
absolute (logarithmic) Weil height. We will recall the definition of $h$ in
Section 2. In number theory, it is a basic problem to calculate the
$h$-Northcott numbers. Vidaux and Videla gave the following.

######  Question 1.1 ([VV16, Question 6]).

Which real numbers can be realized as $\operatorname{Nor}(L,h)$ for some field
$L\subset\overline{\mathbb{Q}}$?

On the basis of the former works [Wid11] and [PTW22], the authors generalized
Question 1.1 and answered to it in [OS22].

######  Theorem 1.2 ([OS22, Theorem 1.4]).

Let $c$ be a positive real number and $\gamma$ a real number with $\gamma<1$.
Then we can construct a field $L\subset\overline{\mathbb{Q}}$ satisfying the
equality

$\operatorname{Nor}(L,h_{\gamma})=c,$

where $h_{\gamma}(a)=\deg(a)^{\gamma}h(a)$ ($a\in\overline{\mathbb{Q}}$,
$\deg(a)=[\mathbb{Q}(a):\mathbb{Q}]$) is the $\gamma$-weighted Weil height,
introduced in [PTW22].

######  Remark 1.3.

The case $\gamma=0$ of Theorem 1.2 answers Question 1.1. In [Hul23], Hultberg
applied this result to the finiteness of cycles of small height on projective
varieties.

In this paper, we introduce $w$-weighted heights $h^{w}$ that further
generalize the $\gamma$-weighted Weil heights. The purpose of this paper is to
generalize Theorem 1.2 to our heights $h^{w}$. We say that a function
$w\colon\mathbb{Z}_{>0}\longrightarrow\mathbb{R}_{>0}$ ia a weight if it is
eventually non-increasing or non-decreasing. For a weight $w$ and
$a\in\overline{\mathbb{Q}}$, we set

$h^{w}(a)=w(\deg(a))h(a).$

We call $h^{w}$ the $w$-weighted Weil height. We remark that the
$\gamma$-weighted Weil height $h_{\gamma}$ coincides with $h^{w_{\gamma}}$,
where $w_{\gamma}(d)=d^{\gamma}$. Another remarkable example is Dobrowolski’s
height. For the weight

$w_{\text{D}}(d)=\left(\frac{\log^{+}(d)}{\log^{+}(\log(d))}\right)^{3}d,$

Dobrowolski in [Dob79] proved that $\overline{\mathbb{Q}}$ has
$h^{w_{\text{D}}}$-(B), where $\log^{+}(\cdot)=\max\\{1,\log(\cdot)\\}$. A
merit to introduce our generalized heights is the fact that for any infinite
algebraic extension of $\mathbb{Q}$, there exists a weight $w$ such that
$\operatorname{Nor}(L\setminus\mu,h^{w})$ is a positive real number, where
$\mu$ is the set of roots of unity in $\overline{\mathbb{Q}}$ (see Corollary
3.5).

In addition, we address not only fields but also sets of algebraic numbers of
bounded degree: For a field $L\subset\overline{\mathbb{Q}}$ and
$N\in\mathbb{Z}_{>0}$, we set

$L^{(N)}=\left\\{a\in\overline{\mathbb{Q}}\mid[L(a):L]\leq N\right\\}.$

Our motivations to focus on the sets $L^{(N)}$ are a number of topics
regarding the Northcott property and the Bogomolov property (e.g., [ADZ14],
[AZ00], [Dob79], [DZ08, Theorem 2.1], [Sil07, Conjecture 3.24]).

Our main result is the following.

######  Theorem 1.4.

Let $c$ be a positive real number, $N$ a positive integer, and $w$ a weight
with the following two conditions.

  * (1)

limd→∞w(d)log(d)/d=0.

  * (2)

w(d)/d is eventually non-increasing.

Then we can construct a field $L\subset\overline{\mathbb{Q}}$ satisfying the
equality

$\operatorname{Nor}(L^{(N)},h^{w})=c.$

######  Remark 1.5.

Both conditions (1) and (2) in Theorem 1.4 demand that a weight $w$ is smaller
enough than the weight $w_{1}(d)=d$. Here, we refer to Lehmer’s conjecture,
which asserts that $\overline{\mathbb{Q}}$ has $h^{w_{1}}$-(B).

######  Remark 1.6.

If $N=1$ or the limit $\lim_{d\rightarrow\infty}w(d)$ converges to a positive
real number, we can remove the condition (2) as shown in the proof of Theorem
4.5.

As an application of Theorem 1.4, we study Northcott numbers of matrices. Let
$h_{\rm s}$ be the spectral height on
$\operatorname{M}_{n}(\overline{\mathbb{Q}})$, introduced in [Tal00]. The
height $h_{\rm s}$ is obtained from Tate’s limiting process starting with the
operator height $h_{\rm op}$ (see Section 5 for the definitions of $h_{\rm s}$
and $h_{\rm op}$). Hence, $h_{\rm s}$ is an analogue of the canonical heights
on abelian varieties. Now, we should refer to the following Talamanca’s work.

######  Theorem 1.7 ([Tal00, Theorem 4.2]).

For each $A\in{\rm M}_{n}(\overline{\mathbb{Q}})$, we denote by
$(\lambda_{1}^{(A)},\ldots,\lambda_{n}^{(A)})$ the $n$-tuple formed by the all
eigenvalues of $A$. Then, we have the equality

$h_{\rm s}(A)=h(\lambda_{1}^{(A)},\ldots,\lambda_{n}^{(A)}),$

where $h$ is the Weil height on
$\mathbb{P}^{n-1}(\overline{\mathbb{Q}})\cup\\{\bm{0}\\}$ (for the definition,
see Section 2).

Combining the product formula in Section 2, to study $h_{\rm s}$-Northcott
numbers, we should consider $h_{\rm s}$ to be a function on the set
$\operatorname{Conj}(\operatorname{M}_{n}(\overline{\mathbb{Q}}))$. Here, we
set

$\operatorname{Conj}(\operatorname{M}_{n}(L))=\operatorname{M}_{n}(L)/\mathord{\sim}_{\rm
conj}$

for each field $L\subset\overline{\mathbb{Q}}$, where $A\mathbin{\sim}_{\rm
conj}B$ ($A,B\in\operatorname{M}_{n}(L)$) means that there exist
$P\in\operatorname{GL}_{n}(\overline{\mathbb{Q}})$ and
$c\in\overline{\mathbb{Q}}^{\times}$ such that $B=c(P^{-1}AP)$. We denote by
$[A]_{\rm conj}$ the class of $A\in\operatorname{M}_{n}(L)$ in
$\operatorname{Conj}(\operatorname{M}_{n}(L))$. We also set

$\operatorname{Conj}(\operatorname{GL}_{n}(L))=\\{[A]_{\rm
conj}\in\operatorname{Conj}(\operatorname{M}_{n}(L))\mid
A\in\operatorname{GL}_{n}(L)\\}.$

We address $\operatorname{Conj}(\operatorname{M}_{n}(L))$ motivated by
thinking about the $\operatorname{PGL}_{n}(\overline{\mathbb{Q}})$-conjugacy
class of rational self-maps on the projective space
$\mathbb{P}^{n-1}(\overline{\mathbb{Q}})$. It is natural to ask about the
existence of inequality relations among $\operatorname{Nor}(L,h)$,
$\operatorname{Nor}(\operatorname{Conj}(\operatorname{M}_{n}(L)),h_{\rm s})$,
and $\operatorname{Nor}(\operatorname{Conj}(\operatorname{GL}_{n}(L)),h_{\rm
s})$. More precisely, we want to find inequalities

$\displaystyle\hphantom{=}c_{1}\operatorname{Nor}(L,h)\leq\operatorname{Nor}(\operatorname{Conj}(\operatorname{M}_{n}(L),h_{\rm
s})$ (1.1)
$\displaystyle\leq\operatorname{Nor}(\operatorname{Conj}(\operatorname{GL}_{n}(L),h_{\rm
s})\leq c_{2}\operatorname{Nor}(L,h)$ (1.2)

for some $c_{1},c_{2}\in\mathbb{R}_{>0}$. However, the first author proved the
following proposition.

######  Proposition 1.8 ([Oka19-1, Theorem 1]).

We have the inequalities

$\operatorname{Nor}(\mathbb{Q}^{\rm
tr},h)>\operatorname{Nor}(\operatorname{Conj}(\operatorname{M}_{n}(\mathbb{Q}^{\rm
tr})),h_{\rm s})=0,$

where $\mathbb{Q}^{\rm tr}$ is the field of totally real numbers.

######  Remark 1.9.

The fact that $\mathbb{Q}^{\rm tr}$ has $h$-(B) was first proven in [Sch73].

Proposition 1.8 reveals that it is impossible to seek for inequalities (1.2)
for a general field $L\subset\overline{\mathbb{Q}}$. Hence we will attempt to
give an example of a field $L\subset\overline{\mathbb{Q}}$ satisfying
inequalities (1.2). As we did for $h$, we address this problem not only for
$h_{\rm s}$ but also the $w$-weighted spectral height $h_{\rm s}^{w}$ (for the
definition, see Section 5). In Section 5, for a weight $w$ with several
conditions, we will give a field $L\subset\overline{\mathbb{Q}}$ and
$c_{1},c_{2}\in\mathbb{R}_{>0}$ satisfying the inequalities

$\displaystyle 0$
$\displaystyle<c_{1}\operatorname{Nor}(L,h^{w})\leq\operatorname{Nor}(\operatorname{Conj}(\operatorname{M}_{n}(L),h_{\rm
s}^{w})$ (1.3)
$\displaystyle\leq\operatorname{Nor}(\operatorname{Conj}(\operatorname{GL}_{n}(L),h_{\rm
s}^{w})\leq c_{2}\operatorname{Nor}(L,h^{w})<\infty$ (1.4)

by using results on Theorem 1.4.

### Notation

  * $\bullet$

For a number field $K$, we denote by $\mathcal{M}_{K}$ the set of all places
of $K$.

  * $\bullet$

For a field $L\subset\overline{\mathbb{Q}}$, we denote by $\mathbb{P}^{n}(L)$
the $n$-dimensional projective space over $L$. We denote by $[\bm{a}]$ (resp.
$[a_{0}:a_{1}:\cdots:a_{n}]$) the class of $\bm{a}\in
L^{n+1}\setminus\\{\bm{0}\\}$ (resp. $(a_{0},a_{1},\ldots,a_{n})\in
L^{n+1}\setminus\\{\bm{0}$}) in $\mathbb{P}^{n}(L)$.

  * $\bullet$

For a field $L\subset\overline{\mathbb{Q}}$ and
$\bm{a}=(a_{0},\ldots,a_{n})\in\overline{\mathbb{Q}}^{n+1}\setminus\\{\bm{0}\\}$,
we fix a non-zero entry $a_{i}$ and set

$\deg_{L}(\bm{a})=[L(a_{j}/a_{i}\mid 0\leq j\leq n):L],$

while we set $\deg_{L}(\bm{0})=1$. We may consider $\deg_{L}$ to be a function
on $\mathbb{P}^{n}(\overline{\mathbb{Q}})$. We simply denote
$\deg_{\mathbb{Q}}$ by $\deg$.

  * $\bullet$

We denote by $\mu$ the set of roots of unity in $\overline{\mathbb{Q}}$. We
also set $\mu_{0}=\\{0\\}\cup\mu$.

### Outline of this paper

In Section 2, we quickly summarize definitions and basic properties of the
Weil heights. In Section 3, we give two fundamental properties of the
$w$-weighted Weil heights. In Section 4, we prove Theorem 1.4. In Section 5,
we introduce the $w$-weighted spectral heights $h_{\rm s}^{w}$ on matrices and
study inequalities (1.4). Section 6 is an appendix. We address the $h_{\rm
op}$-version of inequalities (1.4).

##  2\. A quick review on Weil heights

In this section, we summarize the definitions and basic properties of the Weil
heights (see [HS00] for details). We will frequently use them in the latter
sections without mentioning them.

Let $K$ be a number field. For each $v\in\mathcal{M}_{K}$, we fix the absolute
value $|\cdot|_{v}\colon K\longrightarrow\mathbb{R}_{\geq 0}$ such that the
product formula

$\prod_{v\in\mathcal{M}_{K}}|a|_{v}^{[K_{v}:\mathbb{Q}_{v}]}=1$

holds for all $a\in K$, where $K_{v}$ (resp. $\mathbb{Q}_{v}$) is the
completion of $K$ (resp. $\mathbb{Q}$) by $|\cdot|_{v}$. For each
$\bm{a}=(a_{0},\ldots,a_{n})\in K^{n+1}\setminus\\{\bm{0}\\}$, we set

$h(\bm{a})=\sum_{v\in\mathcal{M}_{K}}\frac{[K_{v}:\mathbb{Q}_{v}]}{[K:\mathbb{Q}]}\log\left(\max_{0\leq
i\leq n}\\{|a_{i}|_{v}\\}\right)$

and $h(\bm{0})=0$. It is well-known that the value $h(\bm{a})$ is independent
of the choice of a number field $K$. Combining the product formula, we may
consider $h$ to be a function on $\mathbb{P}^{n}(\overline{\mathbb{Q}})$. The
function $h$ is called the Weil height on
$\mathbb{P}^{n}(\overline{\mathbb{Q}})$.

By abuse of notation, for each $a\in\overline{\mathbb{Q}}$, we set

$h(a)=h(1,a).$

The function $h$ is called the Weil height on $\overline{\mathbb{Q}}$. The
following are basic formulae of the Weil height on $\overline{\mathbb{Q}}$.

######  Lemma 2.1.

  * (1)

For each $a\in\overline{\mathbb{Q}}$ and $d\in\mathbb{Z}_{>0}$, we have

$h(a^{1/d})=\frac{1}{d}h(a).$

  * (2)

For each $p,q\in\mathbb{Z}\setminus\\{0\\}$ with $\gcd(p,q)=1$, we have

$h(p/q)=\log\left(\max\\{|p|,|q|\\}\right).$

  * (3)

For each $a,b\in\overline{\mathbb{Q}}$, we have

$h(ab)\leq h(a)+h(b).$

###### Proof.

The assertions follow from the definition of $h$. For proofs, see, e.g.,
[BG06, Section 1.5]. ∎

The following fundamental theorem for the Weil heights is known as Northcott’s
property.

######  Theorem 2.2.

Let $L\subset\overline{\mathbb{Q}}$ be a field with $h$-(N). Then, for any
positive integer $N$, the set

$\\{[\bm{a}]\in\mathbb{P}^{n}(\overline{\mathbb{Q}})\mid\deg_{L}(\bm{a})\leq
N\\}$

has $h$-(N). In particular, the set $L^{(N)}$ has $h$-(N).

###### Proof.

See, e.g., [DZ08, Theorem 2.1]. ∎

######  Remark 2.3.

Lemma 2.1 (2) and Theorem 2.2 imply that any number field has $h^{w}$-(N) for
any weight $w$.

We also give the following Kronecker’s theorem.

######  Theorem 2.4.

The equality $Z(\overline{\mathbb{Q}},h)=\mu_{0}$ holds.

###### Proof.

See, e.g., [BG06, Theorem 1.5.9]. ∎

##  3\. Weights

In this section, we give two fundamental properties of the $w$-weighted Weil
heights. However, we will not use these properties in the latter sections.

######  Proposition 3.1.

Let $w$ and $u$ be weights with $\lim_{d\rightarrow\infty}u(d)/w(d)=\infty$.
If a set $S\subset\overline{\mathbb{Q}}$ has $h^{w}$-(B), then the set
$S\setminus\mu$ has $h^{u}$-(N).

###### Proof.

Take any $C\in\mathbb{R}_{>0}$. We claim that there are only finitely many
$a\in S\setminus\mu_{0}$ satisfying the inequality $h^{u}(a)<C$. Since $S$ has
$h^{w}$-(B), there exists $D\in\mathbb{R}_{>0}$ independent of $a$ such that
$h^{w}(a)\geq D$. Hence, we have the inequality

$\frac{u(\deg(a))}{w(\deg(a))}=\frac{h^{u}(a)}{h^{w}(a)}<\frac{C}{D}.$

Combining the assumption that $\lim_{d\rightarrow\infty}u(d)/w(d)=\infty$, we
notice that there exists $M\in\mathbb{Z}_{>0}$ independent of $a$ such that

$\displaystyle\deg(a)\leq M.$ (3.1)

Thus we have the inequality

$\displaystyle h(a)=\frac{h^{u}(a)}{u(\deg(a))}<\frac{C}{\min\left\\{u(i)\mid
1\leq i\leq M\right\\}}.$ (3.2)

By Theorem 2.2, the inequalities (3.1) and (3.2) imply that there are only
finitely many choices for $a\in S\setminus\mu_{0}$. ∎

######  Remark 3.2.

Proposition 3.1 generalizes [OS22, Proposition 2.3].

######  Remark 3.3.

We did not use the fact that $w$ and $u$ are eventually non-decreasing or
eventually non-increasing in the above proof of Proposition 3.1.

######  Theorem 3.4.

Let $S\subset\overline{\mathbb{Q}}$ be a set with $\\#(S\cap\mu)<\infty$. Then
there exists a weight $w$ such that $\operatorname{Nor}(S,h^{w})$ is a
positive finite real number if and only if $\\{\deg(a)\mid a\in S\\}$ is an
infinite set.

###### Proof.

We may assume that $S\cap\mu_{0}=\emptyset$.

First, assume that there exists a weight $w$ such that
$\operatorname{Nor}(S,h^{w})$ is a positive real number. This implies that the
set $S$ does not have $h^{w}$-(N). Hence, there exists an infinite subset
$S^{\prime}\subset S$ such that $\sup\\{h^{w}(a)\mid a\in
S^{\prime}\\}<\infty$. If $\\{\deg(a)\mid a\in S^{\prime}\\}$ is a finite set,
we also have $\sup\\{h(a)\mid a\in S^{\prime}\\}<\infty$. This contradicts
Theorem 2.2.

Next, assume that $\\{\deg(a)\mid a\in S\\}$ is an infinite set. We divide the
proof into the following three cases.

  * (1)

$\operatorname{Nor}(S,h)=\infty$.

Since $S$ has $h$-(N), for each $d\in\mathbb{Z}_{>0}$, we can take an element
$a_{d}\in S$ satisfying that $\deg(a_{d})\geq d$ and

$h(a_{d})=\min\\{h(a)\mid a\in S,\,\deg(a)\geq d\\}.$

Note that $\\{a\in S\mid\deg(a)\geq d\\}\neq\emptyset$ for all
$d\in\mathbb{Z}_{>0}$ and in fact

$\displaystyle\\#\\{a_{d}\mid d\in\mathbb{Z}_{>0}\\}=\infty$ (3.3)

by the assumption that $\\{\deg(a)\mid a\in S\\}$ is an infinite set. We also
note that the sequence $(h(a_{d}))_{d\in\mathbb{Z}_{>0}}$ is non-decreasing.
Hence,

$w(d)=\frac{1}{h(a_{d})}\quad(d\in\mathbb{Z}_{>0})$

is a non-increasing weight. We remark that

$\displaystyle w(\deg(a_{d}))=\frac{1}{h(a_{d})}$ (3.4)

holds for all $d\in\mathbb{Z}_{>0}$. This is because
$h(a_{\deg(a_{d})})=h(a_{d})$ holds:

    * $\bullet$

By $\deg(a_{d})\geq d$, we have $h(a_{\deg(a_{d})})\geq h(a_{d})$;

    * $\bullet$

Since $a_{d}$ is an element of $\\{a\in S\mid\deg(a)\geq\deg(a_{d})\\}$, we
have $h(a_{\deg(a_{d})})\leq h(a_{d})$.

Now, we claim that $\operatorname{Nor}(S,h^{w})=1$.

    * $\bullet$

By (3.4), for all $d\in\mathbb{Z}_{>0}$, we have

$h^{w}(a_{d})=w(\deg(a_{d}))h(a_{d})=\frac{h(a_{d})}{h(a_{d})}=1.$

By (3.3), the above equalities imply that $\operatorname{Nor}(S,h^{w})\leq 1$
holds.

    * $\bullet$

For each $a\in S$, we set $d_{1}=\deg(a)$ and then observe that

$h^{w}(a)=\frac{h(a)}{h(a_{d_{1}})}\geq 1\quad\text{by the definition of
}a_{d_{1}}.$

The above inequality implies that $\operatorname{Nor}(S,h^{w})\geq 1$ also
holds.

Thus, we have $\operatorname{Nor}(S,h^{w})=1$. This means that $w$ is a weight
function that we want.

  * (2)

$\operatorname{Nor}(S,h)\in\mathbb{R}_{>0}$.

As this case is trivial, set $w(d)=1$.

  * (3)

$\operatorname{Nor}(S,h)=0$.

Let $d_{0}=\min\\{\deg(a)\mid a\in S\\}$. By Theorem 2.2, for each
$d\in\mathbb{Z}_{\geq d_{0}}$, we can take an element $a_{d}\in S$ satisfying
that $\deg(a_{d})\leq d$ and

$h(a_{d})=\min\\{h(a)\mid a\in S,\,\deg(a)\leq d\\}.$

Since $S$ does not have $h$-(B), it is implied that
$\lim_{d\rightarrow\infty}h(a_{d})=0$ (this is a special case of Lemma 4.9).
Hence we have

$\\#\left\\{a_{d}\mid d\in\mathbb{Z}_{>0}\right\\}=\infty.$ (3.5)

We also note that the sequence $(h(a_{d}))_{d\geq d_{0}}$ is non-increasing.
Hence,

$w(d)=\begin{cases}1&(d\in\mathbb{Z}_{>0},\,d<d_{0})\\\
1/h(a_{d})&(d\in\mathbb{Z}_{\geq d_{0}})\end{cases}$ (3.6)

is an eventually non-decreasing weight. We remark that

$\displaystyle w(\deg(a_{d}))=\frac{1}{h(a_{d})}$ (3.7)

holds for all $d\in\mathbb{Z}_{\geq d_{0}}$. This is because
$h(a_{\deg(a_{d})})=h(a_{d})$ holds:

    * $\bullet$

By $\deg(a_{d})\leq d$, we have $h(a_{\deg(a_{d})})\geq h(a_{d})$;

    * $\bullet$

Since $a_{d}$ is an element of $\\{a\in S\mid\deg(a)\leq\deg(a_{d})\\}$, we
have $h(a_{\deg(a_{d})})\leq h(a_{d})$.

Now, we claim that $\operatorname{Nor}(S,h^{w})=1$.

    * $\bullet$

By (3.7), for all $d\in\mathbb{Z}_{\geq d_{0}}$, we have

$h^{w}(a_{d})=w(\deg(a_{d}))h(a_{d})=\frac{h(a_{d})}{h(a_{d})}=1.$

By (3.5), the above equalities imply that $\operatorname{Nor}(S,h^{w})\leq 1$
holds.

    * $\bullet$

For each $a\in S$, we set $d_{2}=\deg(a)$ and then observe that

$h^{w}(a)=\frac{h(a)}{h(a_{d_{2}})}\geq 1\quad\text{by the definition of
}a_{d_{2}}.$

The above inequality implies that $\operatorname{Nor}(S,h^{w})\geq 1$ also
holds.

Thus, we have the equality $\operatorname{Nor}(S,h^{w})=1$. This means that
$w$ is a weight function that we want.

∎

######  Corollary 3.5.

Let $L\subset\overline{\mathbb{Q}}$ be an infinite extension of $\mathbb{Q}$.
Then, there exists a weight $w$ such that
$\operatorname{Nor}(L\setminus\mu,h^{w})$ is a positive real number.

##  4\. Northcott numbers for the generalized weighted heights

We will prove Theorem 1.4 in Subsection 4.3 of this section.

###  4.1. Preparations from group theory

All statements in this section will be used to prove several properties of the
tower of field extensions in Subsection 4.2 through the Galois theory. Let $d$
be a prime number. By abuse of notation, for each $a\in\mathbb{Z}$, we denote
$a+d\mathbb{Z}\in\mathbb{Z}/d\mathbb{Z}$ by $a$. Let $G$ be a semi-direct
product of $\mathbb{Z}/d\mathbb{Z}$ and $(\mathbb{Z}/d\mathbb{Z})^{\times}$
defined by the following binary operation:

$(a_{1},b_{1})\cdot(a_{2},b_{2})=(a_{1}+a_{2}b_{1},b_{1}b_{2})$

for
$(a_{1},b_{1}),(a_{2},b_{2})\in\mathbb{Z}/d\mathbb{Z}\times(\mathbb{Z}/d\mathbb{Z})^{\times}$.
We also let $\pi\colon G\longrightarrow(\mathbb{Z}/d\mathbb{Z})^{\times}$ be
the canonical projection.

######  Lemma 4.1.

  * (1)

The group $G$ has only one subgroup of order $d$.

  * (2)

The group $G$ has exactly $d$ subgroups of order $d-1$.

  * (3)

Any subgroup $H\subset G$ with $d\mid|H|$ contains the subgroup of $G$ of
order $d$.

  * (4)

Any subgroup $H\subset G$ with $d\nmid|H|$ is contained in a subgroup of $G$
of order $d-1$.

###### Proof.

  * (1)

We prove that

$G_{1}=\left\\{(a,1)\in G\mid a\in\mathbb{Z}/d\mathbb{Z}\right\\}\subset G$

is the only subgroup of order $d$. Note that any Sylow $d$-subgroup of $G$ is
a conjugate of $G_{1}$. On the other hand, the group $G_{1}$ is a normal
subgroup of $G$ by definition. Thus, $G_{1}$ is the only subgroup of $G$ of
order $d$.

  * (2)

Fix a generator $b_{0}\in(\mathbb{Z}/d\mathbb{Z})^{\times}$. Let $H\subset G$
be a subgroup with $|H|=d-1$. Since $\operatorname{Ker}(\pi|_{H})$ is a
subgroup of both $H$ and $\operatorname{Ker}(\pi)$, the value
$|\operatorname{Ker}(\pi|_{H})|$ divides both $|H|=d-1$ and
$|\operatorname{Ker}(\pi)|=d$. Thus, the equality
$|\operatorname{Ker}(\pi|_{H})|=1$ holds and $\pi|_{H}$ is an isomorphism.
Hence, the group $H$ is generated by an element of $\pi^{-1}(b_{0})$.
Therefore, $H$ is of the form

$\displaystyle H_{a}=\langle(a,b_{0})\rangle\subset G$ (4.1)

for some $a\in\mathbb{Z}/d\mathbb{Z}$. Thus, it is sufficient to prove that
the equality $|H_{a}|=d-1$ holds for all $a\in\mathbb{Z}/d\mathbb{Z}$ and that
$H_{a_{1}}\neq H_{a_{2}}$ holds for distinct two elements
$a_{1},a_{2}\in\mathbb{Z}/d\mathbb{Z}$.

First, let $a\in\mathbb{Z}/d\mathbb{Z}$ and $s\in\mathbb{Z}\cap[1,d-1]$.
Noting that $b_{0}-1\neq 0$, we have the equalities

$(a,b_{0})^{s}=\left(a\sum_{i=1}^{s}b_{0}^{i-1},b_{0}^{s}\right)=\left(a\frac{b_{0}^{s}-1}{b_{0}-1},b_{0}^{s}\right).$
(4.2)

Since $|(\mathbb{Z}/d\mathbb{Z})^{\times}|=d-1$ holds, the equalities (4.2)
yield that

$(a,b_{0})^{d-1}=\left(a\frac{b_{0}^{d-1}-1}{b_{0}-1},b_{0}^{d-1}\right)=(0,1).$

Thus, $|H_{a}|$ divides $d-1$. On the other hand, we have
$(a,b_{0})^{s}\neq(a,b_{0})^{t}$ for distinct $s,t\in\mathbb{Z}_{\leq d-1}$
since $b_{0}^{s}\neq b_{0}^{t}$ holds. Therefore, the equality $|H_{a}|=d-1$
holds.

Next, assume that the equality $H_{a_{1}}=H_{a_{2}}$ holds. Then, there exists
an integer $s\in[1,d-1]$ such that $(a_{1},b_{0})=(a_{2},b_{0})^{s}$. By
(4.2), we have the equalities

$(a_{1},b_{0})=(a_{2},b_{0})^{s}=\left(a_{2}\frac{b_{0}^{s}-1}{b_{0}-1},b_{0}^{s}\right).$

Thus, we get $s=1$. Therefore, we conclude that $a_{1}=a_{2}$.

  * (3)

Let $H\subset G$ be a subgroup with $d\mid|H|$. Then, $H$ has a Sylow
$d$-subgroup $H^{\prime}$. Since $H^{\prime}$ is also a Sylow subgroup of $G$,
the equality $H^{\prime}=G_{1}$ holds by (1), where $G_{1}$ is as in (1).

  * (4)

Let $H\subset G$ be a subgroup with $d\nmid|H|$. From the equality
$|G|=d(d-1)$, we know that $|H|$ divides $d-1$. We set
$m=(d-1)/|H|\in\mathbb{Z}_{>0}$. By the similar discussion of (2), we get the
injectivity of $\pi|_{H}$. Hence, $H$ is cyclic and generated by an element
$g_{0}\in\pi^{-1}(b_{0}^{m})$, where $b_{0}$ is as in (2). We write
$g_{0}=(a_{0},b_{0}^{m})$ ($a_{0}\in\mathbb{Z}/d\mathbb{Z}$). If $|H|=1$, then
the assertion immediately follows. Thus we assume that $|H|\geq 2$, i.e.,
$m\leq d-2$. We set $g=\left(a_{0}(b_{0}-1)(b_{0}^{m}-1)^{-1},b_{0}\right)$.
By (4.2), we have $g_{0}=g^{m}$. Hence, $H$ is contained in the subgroup
$H_{a_{0}(b_{0}-1)(b_{0}^{m}-1)^{-1}}$ of order $d-1$ defined in (4.1).

∎

###  4.2. Preparations from field theory

The following Proposition 4.2 plays a crucial role in the proof of Theorem
1.4. The purpose of this section is to prove it.

######  Proposition 4.2.

Let $N$ be a positive integer and $d$ be a prime number with $d>N$. Assume
that a primitive $d$-th root of unity $\zeta_{d}\in\overline{\mathbb{Q}}$
satisfies that $[K(\zeta_{d}):K]=d-1$. Let $\alpha$ be an algebraic number
whose minimal polynomial over $K$ is $x^{d}-\alpha^{d}\in K[x]$. We set
$M=K(\alpha)$. For each $a\in M^{(N)}$, we also set $s=[M(a):M]$. Then, if
$s<[K(a):K]$, there exists a non-negative integer $k\leq d-1$ such that
$K(\zeta_{d}^{k}\alpha)\subset K(a)$.

Let the notation be as in Proposition 4.2 and $\widetilde{M}$ the Galois
closure of $M/K$. We also let
$\sigma_{(a,b)}\in\operatorname{Gal}(\widetilde{M}/K)$ be an element defined
by

$\begin{cases}\sigma_{(a,b)}(\alpha)=\zeta_{d}^{a}\alpha,\\\
\sigma_{(a,b)}(\zeta_{d})=\zeta_{d}^{b}.\end{cases}$

Now, since $d=[M:K]$ divides
$[\widetilde{M}:K]=[\widetilde{M}:K(\zeta_{d})][K(\zeta_{d}):K]=[\widetilde{M}:K(\zeta_{d})](d-1)$
and $d$ is a prime, $d$ must divide $[\widetilde{M}:K(\zeta_{d})]$. Since
$\widetilde{M}$ is generated by $\alpha$ over $K(\zeta_{d})$, the inequality
$[\widetilde{M}:K(\zeta_{d})]\leq d$ holds. Thus, we have

$\displaystyle d=[\widetilde{M}:K(\zeta_{d})].$ (4.3)

Therefore, the map

$G\longrightarrow\operatorname{Gal}(\widetilde{M}/K);\,(a,b)\longmapsto\sigma_{(a,b)}$

is a group isomorphism, where $G$ is the group defined in Section 4.1. Thus,
we get the following assertion by Lemma 4.1 and Galois theory.

######  Lemma 4.3.

Let $\widetilde{M}/K$ be the field extension in Proposition 4.2.

  * (1)

The field $K(\zeta_{d})$ is the only intermediate field of $\widetilde{M}/K$
of degree $d-1$ over $K$.

  * (2)

The fields $K(\alpha),K(\zeta_{d}\alpha),\ldots,K(\zeta_{d}^{d-1}\alpha)$ are
all intermediate fields of $\widetilde{M}/K$ of degree $d$ over $K$.

  * (3)

Any intermediate field $M^{\prime}$ of $\widetilde{M}/K$ with
$d\nmid[M^{\prime}:K]$ is contained in $K(\zeta_{d})$.

  * (4)

Any intermediate field $M^{\prime}$ of $\widetilde{M}/K$ with
$d\mid[M^{\prime}:K]$ contains $K(\zeta_{d}^{k}\alpha)$ for some integer
$k\in[0,d-1]$.

To prove Proposition 4.2, we also employ the following lemma.

######  Lemma 4.4.

Let the notation be as in Proposition 4.2 and $\widetilde{M}$ the Galois
closure of $M/K$. Then, we have the following assertions.

  * (1)

d∣[K(a):K].

  * (2)

d∣[~M∩K(a):K].

$\textstyle{\widetilde{M}}$$\textstyle{M(a)}$$\textstyle{K(\zeta_{d})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\textstyle{K(a)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\widetilde{M}\cap
K(a)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d-1}$$\scriptstyle{d}$$\scriptstyle{(2)}$$\scriptstyle{\begin{matrix}\text{\tiny{\rm(1)}}\\\
\text{\tiny{$>s$}}\end{matrix}}$

###### Proof.

  * (1)

Note that $[K(a):K]$ divides $[M(a):K]=[M(a):M][M:K]=sd$. By the assumption
that $[K(a):K]>s$, the prime number $d$ must divide $[K(a):K]$.

  * (2)

We prove the assertion by contradiction. Suppose that
$d\nmid[\widetilde{M}\cap K(a):K]$ holds. Under the assumption, we claim that
$[K(a)(\alpha):K(a)]=d$ holds. It is sufficient to show that
$x^{d}-\alpha^{d}$ is irreducible over $K(a)$. Consider the irreducible
decomposition of $x^{d}-\alpha^{d}$ over $K(a)$. Clearly all coefficients of
the irreducible factors of $x^{d}-\alpha^{d}$ are in $\widetilde{M}$, so are
in $\widetilde{M}\cap K(a)$. By the assumption that $d\nmid[\widetilde{M}\cap
K(a):K]$ and Lemma 4.3 (3), the field $\widetilde{M}\cap K(a)$ is contained in
$K(\zeta_{d})$. Thus, the irreducible decomposition of $x^{d}-\alpha^{d}$ over
$K(a)$ is also a decomposition of it over $K(\zeta_{d})$. Now, note that
$[K(\zeta_{d})(\alpha):K(\zeta_{d})]=d$ holds by (4.3). Hence, the irreducible
decomposition of $x^{d}-\alpha^{d}$ over $K(a)$ is trivial. Thus, we have
$[K(a)(\alpha):K(a)]=d$.

$\textstyle{M(a)=K(a)(\alpha)}$$\textstyle{M\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s}$$\textstyle{K(a)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$$\textstyle{K\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{d}$$\scriptstyle{dt}$

By (1), we have $[K(a):K]=dt$ for some $t\in\mathbb{Z}_{>0}$. Hence we have
the equalities

$sd=[M(a):M][M:K]=[M(a):K(a)][K(a):K]=d^{2}t.$

This contradicts the assumption that $s\leq N<d$.

∎

###### Proof of Proposition 4.2.

By Lemma 4.4, the prime number $d$ divides $[\widetilde{M}\cap K(a):K]$.
Hence, $\widetilde{M}\cap K(a)$ contains $K(\zeta_{d}^{k}\alpha)$ for some
integer $k\in[0,d-1]$ by Lemma 4.3 (4). By the inclusion $\widetilde{M}\cap
K(a)\subset K(a)$, the assertion follows. ∎

###  4.3. Proof of Theorem 1.4

In this subsection, we prove Theorem 1.4. More precisely, we prove the
following explicit one.

######  Theorem 4.5.

Let $c$ be a positive real number, $N$ be a positive integer, and $w$ be a
weight with the following two conditions.

  * (1)

limd→∞w(d)log(d)/d=0.

  * (2)

w(d)/d is eventually non-increasing.

Fix $d_{0}\in\mathbb{Z}_{>0}$ such that the following two conditions hold.

  * ($d_{0}$-1)

w(d) is non-increasing or non-decreasing on $\mathbb{Z}_{\geq d_{0}}$.

  * ($d_{0}$-2)

w(d)/d is non-increasing on $\mathbb{Z}_{\geq d_{0}}$.

We also let $(d_{i})_{i\in\mathbb{Z}_{>0}}$, $(p_{i})_{i\in\mathbb{Z}_{>0}}$,
and $(q_{i})_{i\in\mathbb{Z}_{>0}}$ be strictly increasing sequences of prime
numbers satisfying the inequalities

$\displaystyle d_{1}\geq d_{0},$ (4.4)
$\displaystyle\exp\left(\frac{Nd_{1}}{w(ND_{1})}c\right)\geq 2,$ (4.5)
$\displaystyle
4\exp\left(\frac{Nd_{i}}{w(ND_{i})}c\right)\leq\exp\left(\frac{Nd_{i+1}}{w(ND_{i+1})}c\right),$
(4.6) $\displaystyle\exp\left(\frac{Nd_{i}}{w(ND_{i})}c\right)\leq p_{i}\leq
2\exp\left(\frac{Nd_{i}}{w(ND_{i})}c\right),$ (4.7) and $\displaystyle
p_{i}<q_{i}<2p_{i}$ (4.8)

for all $i\in\mathbb{Z}_{>0}$, where

$D_{i}=\begin{cases}d_{i}&(\text{if }\lim_{d\rightarrow\infty}w(d)>0),\\\
\prod_{j=1}^{i}d_{j}&(\text{if }\lim_{d\rightarrow\infty}w(d)=0).\end{cases}$

If $\lim_{d\rightarrow\infty}w(d)=0$, we furthermore assume that

$\displaystyle\lim_{i\rightarrow\infty}\frac{iw(ND_{i})}{w(ND_{i-1})}=0.$
(4.9)

We set $L=\mathbb{Q}((p_{i}/q_{i})^{1/d_{i}}\mid i\in\mathbb{Z}_{>0})$. Then,
the equality $\operatorname{Nor}(L^{(N)},h^{w})=c$ holds.

######  Remark 4.6.

We can take a sequences $(d_{i})_{i\in\mathbb{Z}_{>0}}$ satisfying (4.5) and
(4.8) by the condition (1), that implies that
$\lim_{d\rightarrow\infty}d/w(d)=\infty$. The inequalities (4.6), (4.7), and
(4.8) imply that $\\{p_{i}\ |\ I\in\mathbb{Z}_{\geq 0}\\}\cap\\{q_{i}\ |\
\mathbb{Z}_{\geq 0}\\}=\emptyset$. We can also take sequences
$(p_{i})_{i\in\mathbb{Z}_{>0}}$ and $(q_{i})_{i\in\mathbb{Z}_{>0}}$ satisfying
(4.6) and (4.7) by (4.5) and Bertrand-Chebyshev’s theorem. If
$\lim_{d\rightarrow\infty}w(d)=0$, then we can always take a sequence
$(d_{i})_{i\in\mathbb{Z}_{>0}}$ satisfying (4.9) by letting, for example, a
prime $d_{i}$ be so large that

$w(ND_{i})<\frac{w(ND_{i-1})}{i^{2}}.$

The proof of Theorem 4.5 is further based on the following three lemmata.

######  Lemma 4.7.

Let $K^{\prime}/M^{\prime}/K$ be extensions of number fields. Then

$D_{K^{\prime}/K}=N_{M^{\prime}/K}(D_{K^{\prime}/M^{\prime}})D_{M^{\prime}/K}^{[K^{\prime}:M^{\prime}]}.$

###### Proof.

See, e.g., [Neu99, p.202, (2.10)]. ∎

######  Lemma 4.8.

Let $K$ be a number field. Assume that $a\in\overline{\mathbb{Q}}$ satisfies
that $[K(a):K]>1$. We set $M=K(a)$ and $m=[M:K]$. Then

$h(a)\geq\frac{1}{2(m-1)}\left(\frac{\log(N_{K/\mathbb{Q}}(D_{M/K}))}{m[K:\mathbb{Q}]}-\log(m)\right)$

holds, where $N_{K/\mathbb{Q}}$ is the usual norm and $D_{M/K}$ is the
relative discriminant ideal of the extension $M/K$.

###### Proof.

See [Sil84, Theorem 2]. ∎

######  Lemma 4.9.

Let $S\subset\overline{\mathbb{Q}}$ be a subset and $w$ be a weight. Assume
that $S_{0}\subsetneq S_{1}\subsetneq S_{2}\subsetneq\cdots$ is an ascending
chain of non-empty subsets of $S$ satisfying the following two conditions.

  * (1)

$S_{i}$ has $h^{w}$-(N) for all $i\in\mathbb{Z}_{\geq 0}$.

  * (2)

$S=\bigcup_{i\in\mathbb{Z}_{\geq 0}}S_{i}$.

Then we have

$\operatorname{Nor}(A,h^{w})=\liminf_{i\rightarrow\infty}\inf\left(S_{i}\setminus
S_{i-1}\right).$

###### Proof.

See [PTW22, Lemma 6]. ∎

###### Proof of Theorem 4.5.

We set $\alpha_{i}=(p_{i}/q_{i})^{1/d_{i}}$, $K_{0}=\mathbb{Q}$, and
$K_{i}=K_{i-1}(\alpha_{i})$ for each $i\in\mathbb{Z}_{>0}$. First, we apply
Lemma 4.9 to $L^{(N)}=\bigcup_{i\in\mathbb{Z}_{\geq 0}}K_{i}^{(N)}$. Take any
$i\in\mathbb{Z}_{>0}$ with $d_{i}>N$ and $a\in K_{i}^{(N)}\setminus
K_{i-1}^{(N)}$. By Proposition 4.2, there exists an integer $k\in[0,d_{i}-1]$
such that $K_{i-1}(\zeta_{d_{i}}^{k}\alpha_{i})\subset K_{i-1}(a)$. We set
$s=[K_{i-1}(a):K_{i-1}(\zeta_{d_{i}}^{k}\alpha_{i})]$. Note that the
inequalities

$\displaystyle sd_{i}\leq\deg(a)\leq s\prod_{j=1}^{i}d_{j}$ (4.10)

hold by the equalities

$sd_{i}=[K_{i-1}(a):K_{i-1}(\zeta_{d_{i}}^{k}\alpha_{i})][K_{i-1}(\zeta_{d_{i}}^{k}\alpha_{i}):K_{i-1}]=[K_{i-1}(a):K_{i-1}]$

and

$\displaystyle s\prod_{j=1}^{i}d_{j}$
$\displaystyle=[K_{i-1}(a):K_{i-1}(\zeta_{d_{i}}^{k}\alpha_{i})][K_{i-1}(\zeta_{d_{i}}^{k}\alpha_{i}):K_{i-1}][K_{i-1}:\mathbb{Q}]$
$\displaystyle=[K_{i-1}(a):\mathbb{Q}].$

We also have

$\displaystyle s$
$\displaystyle=\frac{[K_{i-1}(a):K_{i-1}(\zeta_{d_{i}}^{k}\alpha_{i})][K_{i-1}(\zeta_{d_{i}}^{k}\alpha_{i}):K_{i-1}]}{[K_{i-1}(\zeta_{d_{i}}^{k}\alpha_{i}):K_{i-1}]}=\frac{[K_{i-1}(a):K_{i-1}]}{d_{i}}$
(4.11)
$\displaystyle\leq\frac{[K_{i}(a):K_{i-1}]}{d_{i}}=\frac{[K_{i}(a):K_{i}][K_{i}:K_{i-1}]}{d_{i}}=[K_{i}(a):K_{i}]$
(4.12) $\displaystyle\leq N.$ (4.13)

Applying Lemma 4.7 to the extensions

$\begin{matrix}K_{i-1}(a)&&/K_{i-1}&/\mathbb{Q},\\\
K_{i-1}(a)&/K_{i-1}(\zeta_{d_{i}}^{k}\alpha_{i})&&/\mathbb{Q},&\text{and}\\\
&\ \,K_{i-1}(\zeta_{d_{i}}^{k}\alpha_{i})&/K_{i-1}&/\mathbb{Q},\end{matrix}$

we have

$\displaystyle D_{K_{i-1}(a)/\mathbb{Q}}$
$\displaystyle=N_{K_{i-1}/\mathbb{Q}}(D_{K_{i-1}(a)/K_{i-1}})D_{K_{i-1}/\mathbb{Q}}^{[K_{i-1}(a):K_{i-1}]},$
$\displaystyle D_{K_{i-1}(a)/\mathbb{Q}}$
$\displaystyle=N_{K_{i-1}(\zeta_{d_{i}}^{k}\alpha_{i})/\mathbb{Q}}(D_{K_{i-1}(a)/K_{i-1}(\zeta_{d_{i}}^{k}\alpha_{i})})D_{K_{i-1}(\zeta_{d_{i}}^{k}\alpha_{i})/\mathbb{Q}}^{s},\text{
and}$ $\displaystyle D_{K_{i-1}(\zeta_{d_{i}}^{k}\alpha_{i})/\mathbb{Q}}$
$\displaystyle=N_{K_{i-1}/\mathbb{Q}}(D_{K_{i-1}(\zeta_{d_{i}}^{k}\alpha_{i})/K_{i-1}})D_{K_{i-1}/\mathbb{Q}}^{[K_{i-1}(\zeta_{d_{i}}^{k}\alpha_{i}):K_{i-1}]}$

Since $p_{i}$ and $q_{i}$ does not ramify in $K_{i-1}$ and ramify totally in
$\mathbb{Q}(\zeta_{d_{i}}^{k}\alpha_{i})$, we have

$p_{i},q_{i}\nmid D_{K_{i-1}/\mathbb{Q}}\ \text{ and }\
(p_{i}q_{i})^{[K_{i-1}:\mathbb{Q}](d_{i}-1)}\mid
N_{K_{i-1}/\mathbb{Q}}(D_{K_{i-1}(\zeta_{d_{i}}^{k}\alpha_{i})/K_{i-1}})$

by [Oka22, Lemma 3.2]. Thus, we get

$(p_{i}q_{i})^{[K_{i-1}:\mathbb{Q}](d_{i}-1)s}\mid
N_{K_{i-1}/\mathbb{Q}}(D_{K_{i-1}(a)/K_{i-1}}).$

By (4.7), we conclude that

$\displaystyle
2[K_{i-1}:\mathbb{Q}](d_{i}-1)s\log(p_{i})<\log(N_{K_{i-1}/\mathbb{Q}}(D_{K_{i-1}(a)/K_{i-1}})).$
(4.14)

Applying Lemma 4.8 to the extension $K_{i-1}(a)/K_{i-1}$, we get the following
inequalities

$\displaystyle h(a)\geq$
$\displaystyle~{}\frac{1}{2(sd_{i}-1)}\left(\frac{\log(N_{K_{i-1}/\mathbb{Q}}(D_{K_{i-1}(a)/K_{i-1}}))}{sd_{i}[K_{i-1}:\mathbb{Q}]}-\log(sd_{i})\right)$
(4.15) $\displaystyle>$
$\displaystyle~{}\frac{1}{2(sd_{i}-1)}\left(\frac{2(d_{i}-1)\log(p_{i})}{d_{i}}-\log(sd_{i})\right)$
(4.16) $\displaystyle\geq$
$\displaystyle~{}\frac{1}{2(sd_{i}-1)}\left(\frac{2N(d_{i}-1)}{w(ND_{i})}c-\log(sd_{i})\right),$
(4.17)

where the second and the third inequalities follow from (4.14) and (4.6),
respectively. If $\lim_{d\rightarrow\infty}w(d)=\infty\text{ or }0$, then we
have

$\displaystyle~{}h^{w}(a)$ $\displaystyle>$
$\displaystyle~{}\frac{w(sD_{i})}{2(sd_{i}-1)}\left(\frac{2N(d_{i}-1)}{w(ND_{i})}c-\log(sd_{i})\right)$
by ($d_{0}$-1), (4.4), (4.10), and (4.17) $\displaystyle=$
$\displaystyle~{}\frac{N(d_{i}-1)}{sd_{i}-1}\frac{s}{N}\frac{w(sD_{i})}{sD_{i}}\frac{ND_{i}}{w(ND_{i})}c-\frac{w(sD_{i})\log(sd_{i})}{2(sd_{i}-1)}$
$\displaystyle\geq$
$\displaystyle~{}\frac{N(d_{i}-1)}{sd_{i}-1}\frac{s}{N}c-\frac{w(sD_{i})\log(sd_{i})}{2(sd_{i}-1)}$
by ($d_{0}$-2), (4.4), and (4.13) $\displaystyle\rightarrow$
$\displaystyle~{}c\quad(\text{as $i\rightarrow\infty$})$ by (1).

On the other hand, if $\lim_{d\rightarrow\infty}w(d)$ is positive and finite,
then we have

$\displaystyle h^{w}(a)>$
$\displaystyle~{}\frac{w(\deg(a))}{w(ND_{i})}\frac{N(d_{i}-1)}{sd_{i}-1}c-\frac{w(\deg(a))\log(sd_{i})}{2(sd_{i}-1)}$
by (4.17) $\displaystyle\rightarrow$
$\displaystyle~{}\frac{N}{s}c\quad(\text{as $i\rightarrow\infty$})$ by (1) and
(4.10) $\displaystyle\geq$ $\displaystyle~{}c$ by (4.13).

Therefore, by Lemma 4.9, we have the inequality

$\displaystyle\operatorname{Nor}(L^{(N)},h^{w})\geq c$ (4.18)

in any cases.

Next, if $\lim_{d\rightarrow\infty}w(d)>0$, then the inequalities

$\displaystyle h^{w}(\alpha_{i}^{1/N})=$
$\displaystyle~{}\frac{w(Nd_{i})}{Nd_{i}}\log(q_{i})$ by (4.7)
$\displaystyle<$
$\displaystyle~{}\frac{w(Nd_{i})}{Nd_{i}}\left(\frac{Nd_{i}}{w(Nd_{i})}c+\log(4)\right)$
by (4.6) and (4.7) $\displaystyle=$
$\displaystyle~{}c+\frac{w(Nd_{i})}{Nd_{i}}\log(4)$ $\displaystyle\rightarrow$
$\displaystyle~{}c\quad(\text{as $i\rightarrow\infty$})$ by (1)

hold. On the other hand, if $\lim_{d\rightarrow\infty}w(d)=0$, then the
inequalities

$\displaystyle~{}h^{w}\left(\left(\prod_{j=1}^{i}\alpha_{j}\right)^{1/N}\right)$
$\displaystyle=$
$\displaystyle~{}\frac{w(ND_{i})}{N}h\left(\prod_{j=1}^{i}\alpha_{j}\right)$
$\displaystyle\leq$
$\displaystyle~{}\frac{w(ND_{i})}{N}\sum_{j=1}^{i}h(\alpha_{j})$
$\displaystyle=$
$\displaystyle~{}\frac{w(ND_{i})}{N}\sum_{j=1}^{i}\frac{\log(q_{j})}{d_{j}}$
by (4.7) $\displaystyle<$
$\displaystyle~{}\frac{w(ND_{i})}{N}\left(\sum_{j=1}^{i}\frac{Nc}{w(ND_{j})}+i\log(4)\right)$
by (4.6) and (4.7) $\displaystyle\leq$
$\displaystyle~{}c+\frac{iw(ND_{i})}{w(ND_{i-1})}c+iw(ND_{i})\frac{\log(4)}{N}$
by ($d_{0}$-1) and (4.4) $\displaystyle\rightarrow$
$\displaystyle~{}c\quad(\text{as $i\rightarrow\infty$})$ by (4.9)

hold. Therefore, we have the inequality

$\displaystyle\operatorname{Nor}(L^{(N)},h^{w})\leq c$ (4.19)

in any cases.

The inequalities (4.18) and (4.19) complete the proof. ∎

Then, we get the following stronger result, which will be used in the proof of
Proposition 5.2.

######  Theorem 4.10.

Let $w$, $c$, and $L$ be as in Theorem 4.5 of the case $N=1$. We further
assume that $w$ satisfies the condition

  * (3)

For all $M\in\mathbb{Z}_{>0}$, the limit

$l_{w}(M)=\lim_{d\rightarrow\infty}\frac{w(Md)}{w(d)}$

exists and is positive.

Then

$\operatorname{Nor}(L^{(M)},h^{w})=\frac{l_{w}(M)}{M}c$

holds for all $M\in\mathbb{Z}_{>0}$.

###### Proof.

First, we see that $\operatorname{Nor}(L^{(M)},h^{w})\geq(l_{w}(M)/M)c$. Let
$K_{i}$ be as in the proof of Theorem 4.5. Take any $i\in\mathbb{Z}_{>0}$ with
$d_{i}>M$ and $a\in K_{i}^{(M)}\setminus K_{i-1}^{(M)}$. We also let $s$ be as
that in the proof of Theorem 4.5. Then by the same discussion in the proof of
Theorem 4.5, if $\lim_{d\rightarrow\infty}w(d)=\infty\text{ or }0$, we have

$\displaystyle h^{w}(a)>$
$\displaystyle~{}\frac{w(sD_{i})}{2(sd_{i}-1)}\left(\frac{2(d_{i}-1)}{w(D_{i})}c-\log(sd_{i})\right)$
$\displaystyle=$
$\displaystyle~{}\frac{d_{i}-1}{sd_{i}-1}\frac{w(sD_{i})}{w(D_{i})}c-\frac{w(sD_{i})\log(sd_{i})}{2(sd_{i}-1)}$
$\displaystyle\rightarrow$ $\displaystyle~{}\frac{l_{w}(s)}{s}c\quad(\text{as
}i\rightarrow\infty)$ $\displaystyle\geq$
$\displaystyle~{}\frac{l_{w}(M)}{M}c.$

If $\lim_{d\rightarrow\infty}w(d)$ is positive and finite, we have

$\displaystyle h^{w}(a)>$
$\displaystyle~{}\frac{w(\deg(a))}{w(D_{i})}\frac{d_{i}-1}{sd_{i}-1}c-\frac{w(\deg(a))\log(sd_{i})}{2(sd_{i}-1)}$
$\displaystyle\rightarrow$ $\displaystyle~{}\frac{1}{s}c\quad(\text{as
}i\rightarrow\infty)$ $\displaystyle\geq$ $\displaystyle~{}\frac{1}{M}c$
$\displaystyle=$ $\displaystyle~{}\frac{l_{w}(M)}{M}c.$

Next, we see that $\operatorname{Nor}(L^{(M)},h^{w})\leq(l_{w}(M)/M)c$. If
$\lim_{d\rightarrow\infty}w(d)>0$, we have

$\displaystyle h^{w}(\alpha_{i}^{1/M})<$
$\displaystyle~{}\frac{w(Md_{i})}{Md_{i}}\left(\frac{d_{i}}{w(d_{i})}c+\log(4)\right)$
$\displaystyle=$
$\displaystyle~{}\frac{1}{M}\frac{w(Md_{i})}{w(d_{i})}c+\frac{w(Md_{i})}{Md_{i}}\log(4)$
$\displaystyle\rightarrow$ $\displaystyle~{}\frac{l(M)}{M}c.\quad(\text{as
}i\rightarrow\infty)$

If $\lim_{d\rightarrow\infty}w(d)=0$, we have

$\displaystyle~{}h^{w}\left(\left(\prod_{j=1}^{i}\alpha_{j}\right)^{1/M}\right)$
$\displaystyle<$
$\displaystyle~{}\frac{w(MD_{i})}{M}\left(\sum_{j=1}^{i}\frac{c}{w(D_{j})}+i\log(4)\right)$
$\displaystyle\leq$
$\displaystyle~{}\frac{1}{M}\frac{w(MD_{i})}{w(D_{i})}c+\frac{iw(MD_{i})}{w(D_{i-1})}c+iw(MD_{i})\frac{\log(4)}{M}$
$\displaystyle=$
$\displaystyle~{}\frac{1}{M}\frac{w(MD_{i})}{w(D_{i})}c+\frac{iw(D_{i})}{w(D_{i-1})}\frac{w(MD_{i})}{w(D_{i})}c+iw(MD_{i})\frac{\log(4)}{M}$
$\displaystyle\rightarrow$ $\displaystyle~{}\frac{l(M)}{M}c.\quad(\text{as
}i\rightarrow\infty)$

∎

######  Example 4.11.

For $\gamma\in\mathbb{R}_{<1}$ and $w(d)=d^{\gamma}/\log(d)$, $d^{\gamma}$, or
$d^{\gamma}\log(d)$, the field $L$ in Theorem 4.10 coincides with that dealt
in [OS22, Theorem 4.1]. For all $M\in\mathbb{Z}_{>0}$ and each of three
weights $w$, we observe that $l_{w}(M)=M^{\gamma}$. Therefore we have

$\operatorname{Nor}(L^{(M)},h^{w})=\frac{c}{M^{1-\gamma}}.$

######  Example 4.12.

For a weight $w(d)=d^{-d^{2}}$, the field $L$ in Theorem 4.5 of the case $N=1$
is an example of those in [OS22, Proposition 5.5]. However, we observe that
$l_{w}(M)=0$ for all $M\in\mathbb{Z}_{\geq 2}$.

##  5\. Northcott numbers for the weighted spectral heights

In this section, we apply Theorem 4.10 to study the inequalities (1.2). First,
we quickly recall the spectral height $h_{\rm s}$. For a number field $K$ and
each $\bm{a}=(a_{0},\ldots,a_{n})\in K^{n+1}\setminus\\{\bm{0}\\}$, we set

$\displaystyle h_{2}(\bm{a})=$
$\displaystyle~{}\sum_{\begin{subarray}{c}v\in\operatorname{M}_{K}\\\
v:\text{archimedean}\end{subarray}}\frac{[K_{v}:\mathbb{Q}_{v}]}{[K:\mathbb{Q}]}\log\left(\max_{0\leq
i\leq n}\\{|a_{i}|_{v}\\}\right)$
$\displaystyle~{}+\sum_{\begin{subarray}{c}v\in\operatorname{M}_{K}\\\
v:\text{non-
archimedean}\end{subarray}}\frac{[K_{v}:\mathbb{Q}_{v}]}{[K:\mathbb{Q}]}\log\left(\sqrt{\sum_{i=0}^{n}|a_{i}|_{v}^{2}}\right)$

and $h_{2}(\bm{0})=0$. As the Weil height $h$, we may regard $h_{2}$ as a
function on $\mathbb{P}^{n}(\overline{\mathbb{Q}})$. For each
$A\in\operatorname{M}_{n}(\overline{\mathbb{Q}})$, we set

$\displaystyle h_{\rm op}(A)$
$\displaystyle=\sup_{\bm{x}\in\overline{\mathbb{Q}}^{n}}\left\\{h_{2}(A\bm{x})-h_{2}(\bm{x})\right\\}\
\text{ and}$ $\displaystyle h_{\rm s}(A)$
$\displaystyle=\lim_{k\rightarrow\infty}\frac{h_{\rm op}(A^{k})}{k}.$

For each
$A=(a_{ij})\in\operatorname{M}_{n}(\overline{\mathbb{Q}})\setminus\\{O\\}$, we
fix a non-zero entry $a_{ij}$ and set

$\deg(A)=[\mathbb{Q}(a_{kl}/a_{ij}\mid 1\leq k,l\leq n):\mathbb{Q}],$

while we set $\deg(O)=1$. Here, we remark that, for example,

$A=\begin{pmatrix}1&1\\\ -1&1\end{pmatrix}\text{ and
}B=\begin{pmatrix}1+\sqrt{-1}&0\\\ 0&1-\sqrt{-1}\end{pmatrix}$

satisfy that $A\sim_{\rm conj}B$ and $\deg(A)=1<2=\deg(B)$. Hence we also set

$\deg([A]_{\rm conj})=\min_{B\in[A]_{\rm conj}}\\{\deg(B)\\}$

for $A\in\operatorname{M}_{n}(\overline{\mathbb{Q}})$.

######  Lemma 5.1.

For all $A\in{\rm M}_{n}(\overline{\mathbb{Q}})$, it holds that

$n!\deg([A]_{\rm conj})\geq\deg(\lambda_{1}^{(A)},\ldots,\lambda_{n}^{(A)}).$

###### Proof.

Take any
$A=(a_{ij})\in\operatorname{M}_{n}(\overline{\mathbb{Q}})\setminus\\{O\\}$. We
may assume that $a_{ij}=1$ for some integers $i,j\in[1,n]$. Thus we have

$\displaystyle\deg(A)=[\mathbb{Q}(a_{kl}\mid 1\leq k,l\leq n):\mathbb{Q}].$
(5.1)

Let $X^{n}+b_{1}X^{n-1}+\cdots+b_{n-1}X+b_{n}\in\overline{\mathbb{Q}}[X]$ be
the characteristic polynomial of $A$. We know that
$b_{m}\in\mathbb{Q}(a_{kl}\mid 1\leq k,l\leq n)$ for all $m\in[1,n]$. Then we
have

$\displaystyle[\mathbb{Q}(b_{m}\mid 1\leq m\leq
n]):\mathbb{Q}]\leq[\mathbb{Q}(a_{kl}\mid 1\leq k,l\leq n):\mathbb{Q}].$ (5.2)

Now, note that
$b_{k}\in\mathbb{Q}(\lambda_{1}^{(A)},\ldots,\lambda_{n}^{(A)})$ holds for all
$0\leq k\leq n-1$. Thus, we conclude that

$\displaystyle~{}\deg(\lambda_{1}^{(A)},\ldots,\lambda_{n}^{(A)})$
$\displaystyle\leq$ $\displaystyle~{}[\mathbb{Q}(\lambda_{m}^{(A)}\mid 1\leq
m\leq n):\mathbb{Q}]$ $\displaystyle=$
$\displaystyle~{}[\mathbb{Q}(b_{m},\lambda_{m}^{(A)}\mid 1\leq m\leq
n):\mathbb{Q}]$ $\displaystyle=$
$\displaystyle~{}[\mathbb{Q}(b_{m},\lambda_{m}^{(A)}\mid 1\leq m\leq
n):\mathbb{Q}(b_{m}\mid 1\leq m\leq n)]$
$\displaystyle~{}\times[\mathbb{Q}(b_{m}\mid 1\leq m\leq n):\mathbb{Q}]$
$\displaystyle\leq$ $\displaystyle~{}[\mathbb{Q}(\lambda_{m}^{(A)}\mid 1\leq
m\leq n):\mathbb{Q}][\mathbb{Q}(b_{m}\mid 1\leq m\leq n):\mathbb{Q}]$
$\displaystyle\leq$ $\displaystyle~{}n!\deg(A)\text{\hskip 120.00018pt by
\eqref{7.4} and \eqref{7.5}}.$

These complete the proof. ∎

Now, we define our weighted spectral heights $h_{\rm s}^{w}$. For a weight $w$
and $[A]_{\rm
conj}\in\operatorname{Conj}(\operatorname{M}_{n}(\overline{\mathbb{Q}}))$, we
set

$h_{\rm s}^{w}([A]_{\rm conj})=w(\deg([A]_{\rm conj}))h_{\rm s}(A).$

The purpose of this section is to prove the following.

######  Proposition 5.2.

Let $w$ and $L$ be as in Theorem 4.10. We furthermore assume that $w$ is non-
decreasing. Then they hold that

$\displaystyle\hphantom{=}\frac{l_{w}(n^{2})}{n^{2}n!}\operatorname{Nor}(L,h^{w})\leq\operatorname{Nor}(\operatorname{Conj}({\rm
M}_{n}(L),h_{\rm s}^{w})$ (5.3)
$\displaystyle\leq\operatorname{Nor}(\operatorname{Conj}(\operatorname{GL}_{n}(L)),h_{\rm
s}^{w})\leq\frac{1}{n-1}\operatorname{Nor}(L,h^{w}).$ (5.4)

###### Proof.

First, we prove the left inequality. Take any $C\in\mathbb{R}_{>0}$ with

$C<\frac{l_{w}(n^{2})}{n^{2}n!}\operatorname{Nor}(L,h^{w}).$

It is sufficient to prove that the set
$B(\operatorname{Conj}(\operatorname{M}_{n}(L)),h_{\rm s}^{w},C)$ is finite.
Take any element $[A]_{\rm conj}\in
B(\operatorname{Conj}(\operatorname{M}_{n}(L)),h_{\rm s}^{w},C)$. Considering
the Jordan form, there are only finitely many $[B]_{\rm
conj}\in\operatorname{Conj}(\operatorname{M}_{n}(\overline{\mathbb{Q}}))$ such
that $(\lambda_{1}^{(B)},\ldots,\lambda_{n}^{(B)})=\bm{0}$. Therefore, we may
assume $\lambda^{(A)}_{1}\neq 0$. Fix $M\in\mathbb{Z}_{>0}$ such that $w(d)/d$
is non-increasing on $\mathbb{Z}_{>M}$. By Lemma 5.1 and Theorem 2.2, there
are only finitely many $[B]_{\rm conj}\in
B(\operatorname{Conj}(\operatorname{M}_{n}(L)),h_{\rm s}^{w},C)$ such that
$\deg([B]_{\rm conj})\leq M$. Therefore, we may assume that $\deg([A]_{\rm
conj})>M$. Thus, we have

$\frac{w(\deg([A]_{\rm conj}))}{\deg([A]_{\rm
conj})}\geq\frac{w(n!\deg([A]_{\rm conj}))}{n!\deg([A]_{\rm conj})}.$

Hence, the inequality

$w(\deg([A]_{\rm conj}))\geq\frac{1}{n!}w(n!\deg([A]_{\rm conj}))$

holds. Therefore, we have

$\displaystyle~{}\frac{l_{w}(n^{2})}{n^{2}}\operatorname{Nor}(L,h^{w})$
$\displaystyle>$ $\displaystyle~{}n!C$ $\displaystyle>$
$\displaystyle~{}n!h^{w}([A]_{\rm conj})$ $\displaystyle=$
$\displaystyle~{}n!w(\deg([A]_{\rm
conj}))h(\lambda_{1}^{(A)},\ldots,\lambda_{n}^{(A)})$ by Theorem 1.7
$\displaystyle\geq$ $\displaystyle~{}w(n!\deg([A]_{\rm
conj}))h(\lambda_{1}^{(A)},\ldots,\lambda_{n}^{(A)})$ $\displaystyle\geq$
$\displaystyle~{}w(\deg(\lambda_{1}^{(A)},\ldots,\lambda_{n}^{(A)}))h(\lambda_{1}^{(A)},\ldots,\lambda_{n}^{(A)})$
by Lemma 5.1 $\displaystyle\geq$
$\displaystyle~{}w(\deg(\lambda_{i}^{(A)}/\lambda_{1}^{(A)}))h(\lambda_{i}^{(A)}/\lambda_{1}^{(A)})$
$\displaystyle=$ $\displaystyle~{}h^{w}(\lambda_{i}^{(A)}/\lambda_{1}^{(A)})$

for all integers $i$ in $[2,n]$. Since $\lambda_{i}^{(A)}/\lambda_{1}^{(A)}\in
L^{(n^{2})}$ holds, Theorem 4.5 implies that there are only finitely many
choices for the value of $\lambda_{i}^{(A)}/\lambda_{1}^{(A)}$. Thus, we
conclude that there are only finitely many possibilities for $[A]_{\rm conj}$.

Next, we prove the right inequality. For each $i\in\mathbb{Z}_{>0}$, let
$d_{i}$ and $\alpha_{i}$ be as in Theorem 4.5 and Theorem 4.10. We set

$A_{i}=\begin{cases}\begin{pmatrix}1&0\\\ 0&\alpha_{i}\\\
\end{pmatrix}\in\operatorname{M}_{2}(L)&(n=2),\\\
\left(\begin{array}[]{c:c}1&\huge{O}\\\
\hdashline\huge{O}&{\begin{matrix}0&0&0&\cdots&0&\alpha_{i}\\\
1&0&0&\cdots&0&0\\\ 0&1&0&\cdots&0&0\\\ 0&0&1&\cdots&0&0\\\
\vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\ 0&0&0&\cdots&1&0\end{matrix}}\\\
\end{array}\right)\in\operatorname{M}_{n}(L)&(n>2).\end{cases}$

Whether $n=2$ or not, they hold that $\deg(A_{i})=d_{i}$ and

$\displaystyle\left(\lambda_{1}^{(A_{i})},\ldots,\lambda_{n}^{(A_{i})}\right)=\left(1,\alpha_{i}^{\frac{1}{n-1}},\alpha_{i}^{\frac{1}{n-1}}\zeta_{n-1},\ldots,\alpha_{i}^{\frac{1}{n-1}}\zeta_{n-1}^{n-1}\right),$
(5.5)

where $\zeta_{n-1}$ is a primitive $(n-1)$-th root of unity. Here, note that

$\displaystyle
h\left(1,\alpha_{i}^{\frac{1}{n-1}},\alpha_{i}^{\frac{1}{n-1}}\zeta_{n-1},\ldots,\alpha_{i}^{\frac{1}{n-1}}\zeta_{n-1}^{n-1}\right)=h\left(\alpha_{i}^{\frac{1}{n-1}}\right)$
(5.6)

holds since we have $|a\zeta|_{v}=|a|_{v}$ for all
$a\in\overline{\mathbb{Q}}$, $\zeta\in\mu$, and
$v\in\mathcal{M}_{\mathbb{Q}(\zeta,a)}$. Therefore, we observe that

$\displaystyle~{}h_{\rm s}^{w}([A_{i}]_{\rm conj})$ (5.7) $\displaystyle\leq$
$\displaystyle~{}w(d_{i})h_{\rm s}([A_{i}]_{\rm conj})$ (5.8) $\displaystyle=$
$\displaystyle~{}w(d_{i})h\left(\lambda_{1}^{(A_{i})},\ldots,\lambda_{n}^{(A_{i})}\right)$
by Theorem 1.7 (5.9) $\displaystyle=$
$\displaystyle~{}w(d_{i})h\left(\alpha_{i}^{\frac{1}{n-1}}\right)$ by (5.5)
and (5.6) (5.10) $\displaystyle=$
$\displaystyle~{}\frac{1}{n-1}h^{w}(\alpha_{i})$ (5.11)
$\displaystyle\rightarrow$
$\displaystyle~{}\frac{1}{n-1}\operatorname{Nor}(L,h^{w})\quad(\text{as
}i\rightarrow\infty)$ by the proof of Theorem 4.10. (5.12)

The inequalities above imply the right inequality. ∎

##  6\. Appendix: Northcott numbers for weighted operator heights

In Section 5, we studied relations between the $h^{w}$-Northcott number of a
field $L$ and that of the matrices with the entries in $L$. In this additional
section, we address the same topic for $h_{\rm op}$. For each weight $w$ and
$A\in\operatorname{M}_{n}(\overline{\mathbb{Q}})$, we set

$h_{\rm op}^{w}(A)=w(\deg(A))h_{\rm op}(A).$

We may consider $h_{\rm op}^{w}(A)$ to be a function on
$\mathbb{P}(\operatorname{M}_{n}(\overline{\mathbb{Q}}))$. Here for each field
$L\subset\overline{\mathbb{Q}}$, we set

$\mathbb{P}(\operatorname{M}_{n}(L))=\operatorname{M}_{n}(L)/\mathord{\sim},$

where $A\mathbin{\sim}B$ ($A,B\in\operatorname{M}_{n}(L)$) means that there
exists $c\in\overline{\mathbb{Q}}^{\times}$ such that $B=cA$. We denote by
$[A]$ the class of $A\in\operatorname{M}_{n}(L)$ in
$\mathbb{P}(\operatorname{M}_{n}(L))$. We also set

$\displaystyle\mathbb{P}(\operatorname{M}_{n}(L))_{\geq 2}$
$\displaystyle=\\{[A]\in\mathbb{P}(\operatorname{M}_{n}(L))\mid\operatorname{rank}(A)\geq
2\\}\ \text{ and}$ (6.1) $\displaystyle\mathbb{P}(\operatorname{GL}_{n}(L))$
$\displaystyle=\\{[A]\in\mathbb{P}(\operatorname{M}_{n}(L))\mid
A\in\operatorname{GL}_{n}(L)\\}.$ (6.2)

######  Proposition 6.1.

For a non-decreasing weight $w$ and a field $L\subset\overline{\mathbb{Q}}$,
they hold that

$\displaystyle\frac{1}{2}\operatorname{Nor}(L,h^{w})\leq$
$\displaystyle\operatorname{Nor}(\mathbb{P}({\rm M}_{n}(L))_{\geq 2},h_{\rm
op}^{w})$ (6.3) $\displaystyle\leq$
$\displaystyle\operatorname{Nor}(\mathbb{P}(\operatorname{GL}_{n}(L)),h_{\rm
op}^{w})\leq\operatorname{Nor}(L,h^{w}).$ (6.4)

######  Remark 6.2.

We dealt with $\mathbb{P}({\rm M}_{n}(L))_{\geq 2}$ in Proposition 6.1 since
it is known that there are infinitely many
$[A]\in\mathbb{P}(\operatorname{M}_{n}(\mathbb{Q}))$ of rank $1$ such that
$h_{\rm op}(A)=0$ (see [Oka19-2, Theorem 1.1 (2)]) or $h_{\rm
op}(A)=\log(2)/2$ (see [Tal00, p.103]), respectively.

For each weight $w$ and $[\bm{a}]\in\mathbb{P}^{n}(\overline{\mathbb{Q}})$, we
set

$h^{w}(\bm{a})=w(\deg(\bm{a}))h(\bm{a})\\\ \ \text{ and }\
h_{2}^{w}(\bm{a})=w(\deg(\bm{a}))h_{2}(\bm{a}).$

Our proof of Proposition A.1 is based on the following Lemma 6.3, 6.4, and
6.5.

######  Lemma 6.3.

Let $w$ be a weight and $a_{1},a_{2},\ldots,a_{n}\in\overline{\mathbb{Q}}$.
Then it holds that

$h_{\rm op}^{w}\left(\left(\begin{array}[]{c c}{\begin{array}[]{cc}a_{1}&\\\
&a_{2}\end{array}}&\huge{O}\\\ \huge{O}&{\begin{array}[]{c c}\ddots&\\\
&a_{n}\end{array}}\end{array}\right)\right)=h^{w}(a_{1},a_{2},\ldots,a_{n}).$

###### Proof.

The assertion immediately follows from [Tal00, Theorem 3.3 (a)] and well-known
explicit formulae for operator norms (See, e.g., [Oka19-2, (3.1) and (3.2)]).
∎

######  Lemma 6.4.

Let $w$ be a non-decreasing weight. For each matrix $A\in{\rm
M}_{n}(\overline{\mathbb{Q}})$ and integer $j\in[1,n]$, we denote by
$\bm{a}_{j}$ the $j$-th column vector of $A$. Then

$h_{2}^{w}(\bm{a}_{j})\leq h_{\rm op}^{w}(A)$

holds for all integers $j$ in $[1,n]$.

###### Proof.

Consider the vector

$\bm{e}_{j}={}^{t}(0,\ldots,0,\underset{\underset{\tiny{j\mbox{-th}}}{\wedge}}{1},0,\ldots,0)\in\overline{\mathbb{Q}}^{n}$

for each integer $j\in[1,n]$. Then we see that

$\displaystyle h_{\rm op}^{w}(A)$ $\displaystyle\geq
w(\deg(A))\left(h_{2}(A\bm{e}_{j})-h_{2}(\bm{e}_{j})\right)$
$\displaystyle=w(\deg(A))h_{2}(\bm{a}_{j})$ $\displaystyle\geq
w(\deg(\bm{a}_{j}))h_{2}(\bm{a}_{j})$ since $w$ is non-decreasing
$\displaystyle=h_{2}^{w}(\bm{a}_{j})$

for all integers $j$ in $[1,n]$. ∎

######  Lemma 6.5.

Let $w$ be an eventually non-decreasing weight and
$L\subset\overline{\mathbb{Q}}$ be a field. Then, it holds that

$\operatorname{Nor}(\mathbb{P}^{n}(L),h_{2}^{w})\geq\operatorname{Nor}(L,h^{w}).$

###### Proof.

The lemma immediately follows from the fact that

$h_{2}^{w}(\bm{a})\geq h^{w}(a_{j}/a_{i})$

holds for all
$[\bm{a}]=[a_{0}:\cdots:a_{n}]\in\mathbb{P}^{n}(\overline{\mathbb{Q}})$ and
integers $i,j$ in $[1,n]$. ∎

###### Proof of Proposition 6.1.

If $\operatorname{Nor}(L,h_{\rm op}^{w})=0$, then the assertion immediately
follows from Lemma 6.3. Assume that we have the inequality
$\operatorname{Nor}(L,h^{w})>0$. The inequality
$\operatorname{Nor}(\mathbb{P}(\operatorname{M}_{n}(L))_{\geq 2},h_{\rm
op}^{w})\leq\operatorname{Nor}(L,h^{w})$ immediately follows from Lemma 6.3.
We prove the inequality
$\operatorname{Nor}(L,h^{w})/2\leq\operatorname{Nor}(\mathbb{P}(\operatorname{M}_{n}(L))_{\geq
2},h_{\rm op}^{w})$. Take any constant $C\in\mathbb{R}_{>0}$ with
$C<\operatorname{Nor}(L,h^{w})/2$. It is sufficient to show that the set
$B(\mathbb{P}(\operatorname{M}_{n}(L))_{\geq 2},h_{\rm op}^{w},C)$ is finite.
Take any $[A]=[(\bm{a}_{1},\ldots,\bm{a}_{n})]\in
B(\mathbb{P}(\operatorname{M}_{n}(L))_{\geq 2},h_{\rm op}^{w},C)$. By Lemma
6.4, each vector $\bm{a}_{j}$ is an element of $B(L^{n},h^{2}_{\gamma},C)$ for
all $j$. Since the inequalities

$\displaystyle
C<\operatorname{Nor}(L,h^{w})/2\leq\operatorname{Nor}(\mathbb{P}^{n-1}(L),h_{2}^{w})/2<\operatorname{Nor}(\mathbb{P}^{n-1}(L),h_{2}^{w})$
(6.5)

hold by Lemma 6.5, we can chose $\bm{x}_{1},\ldots,\bm{x}_{m}\in L^{n}$
satisfying the following condition.

  * $\bullet$

For any $\bm{y}\in B(L^{n},h_{2}^{w},C)$, there only exists an integer
$j(\bm{y})\in[1,m]$ such that $\bm{y}$ is a non-zero scalar multiple of
$\bm{x}_{j(\bm{y})}$.

For each $\bm{y}\in B(L^{n},h_{2}^{w},C)\setminus\\{\bm{0}\\}$, we let
$\lambda(\bm{y})\in L^{\times}$ satisfy that
$\bm{y}=\lambda(\bm{y})\bm{x}_{j(\bm{y})}$ (we consider $\lambda(\bm{0})=1$).
We may assume $m\geq 3$ since the inequality $\operatorname{rank}(A)\geq 2$
holds. We divide the cases into whether there exists a column vector
$\bm{a}_{p}$ that has at least two non-zero entries.

First, we assume that such a column vector
$\bm{a}_{p}={}^{t}(a_{1p},\ldots,a_{np})$ exists. Fix non-zero entries
$a_{qp}$ and $a_{rp}$ ($1\leq q<r\leq n$). Note that the equalities
$a_{qp}=\lambda(\bm{a}_{p})x_{qp}$ and $a_{rp}=\lambda(\bm{a}_{p})x_{rp}$
hold, where $x_{qp}$ (resp. $x_{rp}$) is the $q$-th (resp. $r$-th) entry of
$\bm{x}_{j(\bm{a}_{p})}$. We also fix any non-zero column vectors $\bm{a}_{k}$
and $\bm{a}_{l}$ satisfying that $1\leq k<l\leq n$ and
$\bm{x}_{j(\bm{a}_{k})}\neq\bm{x}_{j(\bm{a}_{l})}$. We note that such $k$ and
$l$ exist since $\operatorname{rank}(A)\geq 2$. We remark that the inequality

$\displaystyle\deg(a_{qp}\bm{a}_{k}+a_{rp}\bm{a}_{l})\leq\deg(A)$ (6.6)

holds. Now, we set

$\bm{a}_{p}^{\prime}={}^{t}(0,\ldots,0,\underset{\underset{\tiny{k\mbox{-th}}}{\wedge}}{a_{qp}},0,\ldots,0,\underset{\underset{\tiny{l\mbox{-th}}}{\wedge}}{a_{rp}},0,\ldots,0)\in
L^{n}.$

Then, we have the inequalities

$\displaystyle~{}h_{2}^{w}\left(\bm{x}_{j(\bm{a}_{k})}+\frac{x_{rp}}{x_{qp}}\frac{\lambda(\bm{a}_{l})}{\lambda(\bm{a}_{k})}\bm{x}_{j(\bm{a}_{l})}\right)$
$\displaystyle=$
$\displaystyle~{}h_{2}^{w}(a_{qp}\bm{a}_{k}+a_{rp}\bm{a}_{l})$
$\displaystyle=$
$\displaystyle~{}w(\deg(a_{qp}\bm{a}_{k}+a_{rp}\bm{a}_{l}))\left(h_{2}(A\bm{a}_{p}^{\prime})-h_{2}(\bm{a}_{p}^{\prime})+h_{2}(\bm{a}_{p}^{\prime})\right)$
$\displaystyle\leq$
$\displaystyle~{}w(\deg(A))\left(h_{2}(A\bm{a}_{p}^{\prime})-h_{2}(\bm{a}_{p}^{\prime})\right)+w(\deg(A))h_{2}(\bm{a}_{p})$
by (6.6) $\displaystyle\leq$ $\displaystyle~{}w(\deg(A))h_{\rm
op}(A)+w(\deg(A))h_{\rm op}(A)$ by Lemma 6.4 $\displaystyle<$
$\displaystyle~{}2C$ $\displaystyle<$
$\displaystyle~{}\operatorname{Nor}(\mathbb{P}^{n-1}(L),h_{2}^{w})$
$\displaystyle\text{by \eqref{8.1}}.$

Since $\bm{x}_{j(\bm{a}_{k})}$ and $\bm{x}_{j(\bm{a}_{l})}$ are linearly
independent, there are only finitely many possibilities for
$\bm{x}_{j(\bm{a}_{k})}+(x_{rp}/x_{qp})(\lambda(\bm{a}_{l})/\lambda(\bm{a}_{k}))\bm{x}_{j(\bm{a}_{l})}$
by Lemma 6.5. Here, note that there are only finitely many choices for the
value of $x_{rp}/x_{qp}$ ($\neq 0$). Thus, we know that there are also only
finitely many choices for the value of
$\lambda(\bm{a}_{l})/\lambda(\bm{a}_{k})$. Thus, all $[A]\in
B(\mathbb{P}(\operatorname{M}_{n}^{2}(L),h_{\rm op}^{w},C)$ of the case is a
class of a matrix $A$ constructed by the following process.

  * (1)

Choose any integers $k\in[1,n]$ and $j_{k}\in[1,m]$. We set
$\bm{a}_{k}=\bm{x}_{j_{k}}$.

  * (2)

Choose any integers $l\in[1,n]\setminus\\{k\\}$ and $j_{l}\in[1,m]$ such that
$\bm{x}_{j_{k}}$ is linearly independent of $\bm{x}_{j_{k}}$. We set
$\bm{a}_{l}=\lambda_{l}\bm{x}_{j_{l}}$ for some $\lambda_{l}\in L^{\times}$ of
finitely many choices.

  * (3)

For each integer $k^{\prime}\in[1,n]\setminus\\{k,l\\}$, choose any integer
$j_{k^{\prime}}\in[1,m]$. Set
$\bm{a}_{k^{\prime}}=\lambda_{k^{\prime}}\bm{x}_{j_{k^{\prime}}}$ for some
$\lambda_{k^{\prime}}\in L^{\times}$ of finitely many choices:

indep.$\textstyle{A=(\
\cdots}$$\textstyle{\bm{a}_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$finitely
many
choices$\textstyle{\cdots}$$\textstyle{\bm{a}_{l}}$$\textstyle{\cdots}$$\textstyle{\bm{a}_{k^{\prime}}}$$\textstyle{\cdots\
)}$ dep.indep.indep.$\textstyle{A=(\
\cdots}$$\textstyle{\bm{a}_{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\begin{matrix}\text{\tiny{finitely
many}}\\\
\text{\tiny{choices}}\end{matrix}}$$\textstyle{\cdots}$$\textstyle{\bm{a}_{l}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\begin{matrix}\text{\tiny{finitely
many}}\\\
\text{\tiny{choices}}\end{matrix}}$$\textstyle{\cdots}$$\textstyle{\bm{a}_{k^{\prime}}}$$\textstyle{\cdots\
)}$

Therefore, we conclude that there are only finitely many choices for $[A]$ of
the case.

On the other hand, assume that any column vector of $A$ has at most one non-
zero entry. Fix any non-zero entries $a_{ij},a_{kl}\neq 0$ with $1\leq i\neq
k\leq n$ and $1\leq j\neq l\leq n$. We note that such non-zero entries exist
since $\operatorname{rank}(A)\geq 2$. We set

$\overline{\mathbb{Q}}^{n}_{jl}=\left\\{{}^{t}(x_{1},\ldots,x_{n})\in\overline{\mathbb{Q}}^{n}\mid
x_{m}=0\text{ for all }m\neq j,l\right\\}.$

Then we have the inequalities

$\displaystyle\operatorname{Nor}(L,h^{w})$
$\displaystyle>\frac{\operatorname{Nor}(L,h^{w})}{2}$ $\displaystyle>C$
$\displaystyle>h_{\rm op}^{w}(A)$ $\displaystyle\geq
w(\deg(A))\sup_{\bm{x}\in\overline{\mathbb{Q}}^{n}_{jl}}\left\\{h_{2}(A\bm{x})-h_{2}(\bm{x})\right\\}$
$\displaystyle=w(\deg(A))h_{\rm op}\left(\begin{pmatrix}a_{ij}&0\\\
0&a_{kl}\end{pmatrix}\right)$ $\displaystyle\geq
w\left(\deg\left(\begin{pmatrix}a_{ij}&0\\\
0&a_{kl}\end{pmatrix}\right)\right)h_{\rm op}\left(\begin{pmatrix}a_{ij}&0\\\
0&a_{kl}\end{pmatrix}\right)$ $\displaystyle=h_{\rm
op}^{w}\left(\begin{pmatrix}a_{ij}&0\\\ 0&a_{kl}\end{pmatrix}\right)$
$\displaystyle=h^{w}(a_{ij},a_{kl})\text{\hskip 80.00012pt by Lemma
}\ref{diagonalop}.$

Thus, there are only finitely many choices for the value of $a_{ij}/a_{kl}\in
L$. Therefore, we also conclude that there are only finitely many choices for
$[A]$ of the case.

These complete the proof. ∎

######  Remark 6.6.

The proof above is greatly based on the proof of [Tal00, Theorem 4.3 (a)].

######  Acknowledgements.

The authors would like to express their gratitude to Professor Fabien Pazuki
for detecting certain notational inaccuracies and bringing reference [Hul23]
to their attention. The first author was supported until March 2022 by JST
SPRING, Grant Number JPMJSP2136. The second author is supported by JSPS
KAKENHI Grant Number JP20K14300.

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