# On the generalisation of Roth’s theorem

Paolo Dolce  Francesco Zucconi

###### Abstract

We present two possible generalisations of Roth’s approximation theorem on
proper adelic curves, assuming some technical conditions on the behavior of
the logarithmic absolute values. We illustrate how tightening such assumptions
makes our inequalities stronger. As special cases we recover Corvaja’s results
[Cor97] for fields admitting a product formula, and Vojta’s ones [Voj21] for
arithmetic function fields.

toc

##  0 Introduction

###  0.1 History

The celebrated Roth’s theorem proved in [Rot55] asserts that the irrationality
measure of a real algebraic number is $2$. An equivalent, but more detailed
statement is the following:

######  Theorem 0.1 (Roth).

Let $\alpha$ be a real algebraic number and let $\varepsilon>0$ be a real
number. Then there exists a real constant $C(\alpha,\varepsilon)>0$ such that
for every pair of coprime integers $(p,q)$ with $q>C(\alpha,\varepsilon)$, it
holds that:

$\left|\alpha-\frac{p}{q}\right|>q^{-2-\varepsilon}$

The above statement can be naturally generalised in different directions by
considering a number field $K$ instead of $\mathbb{Q}$ and simultaneous
approximation of the elements $\alpha_{1},\ldots,\alpha_{n}$ algebraic over
$K$ by an element of $K$ with respect to different valuations (see [Lan83,
Chapter 7]). Actually, the statement proved in [Lan83, Chapter 7] holds for
fields that are more general than number fields. At the moment we don’t have
an _effective_ version of Roth’s theorem (i.e. a bound for the constant
$C(\alpha,\varepsilon)$, see [Cor95]), but we have a _quantitative_ version
(i.e. bounds, in terms of $\alpha$ and $\varepsilon$, on the number of good
approximants see [BvdP88], [BvdP90], [Eve97]).

Let $k$ be a field of characteristic $0$ and let $\mathcal{V}_{k}$ be a set in
bijection with a family of absolute values111We recall that as a
representative for the complex place it is customary to consider the square of
the euclidiean norm, even if this is not properly an absolute value. of $k$
which are not pairwise equivalent. The bijection is denoted by
$v\mapsto|\cdot|_{v}$, for any $v\in\mathcal{V}_{k}$. The couple
$(k,\mathcal{V}_{k})$ _satisfies the product formula_ if for any element $a\in
k^{\times}$ the series $\sum_{v\in\mathcal{V}_{k}}\log|a|_{v}$ converges
absolutely and moreover $\sum_{v\in\mathcal{V}_{k}}\log|a|_{v}=0$. In this
setting one also has a natural notion of _logarithmic height_ for $a\in
k^{\times}$:

$h(a):=\sum_{v\in\mathcal{V}_{K}}\log^{+}|a|_{v}\,,$

and we set $H(a):=e^{h(a)}$. Roth’s theorem was generalised by Corvaja in
[Cor97] for any couple $(k,\mathcal{V}_{k})$ satisfying the product formula:

######  Theorem 0.2.

[Cor97, Corollaire 1]. Assume that $(k,\mathcal{V}_{k})$ satisfies the product
formula. Let $\alpha_{1},\ldots,\alpha_{n}$ be distinct elements algebraic on
$k$, and let $|\cdot|_{v_{1}},\ldots,|\cdot|_{v_{n}}$ be distinct absolute
values of $k$, with $v_{1},\ldots,v_{n}\in\mathcal{V}_{k}$. For any
$i=1,\ldots,n$ let’s fix an appropriately normalised extension of
$|\cdot|_{v_{i}}$ to $k(\alpha_{i})$ (and by abuse of notation denote it with
the same symbol $|\cdot|_{v_{i}}$). Then for any $\varepsilon>0$ there exists
a constant
$C=C(\mathcal{V}_{k},\alpha_{1},\ldots,\alpha_{n},v_{1},\ldots,v_{n},\varepsilon)>0$
such that for all $\beta\in k$ with $H(\beta)>C$ it holds that:

$\sum^{n}_{i=1}\log|\alpha_{i}-\beta|_{v_{i}}>-(2+\varepsilon)h(\beta)\,.$

Notice that there are no further assumptions on the field $k$, which might be
for instance a function field; therefore, theorem 0.2 is a unifying result, as
well as a generalisation of the classical Roth’s theorem. Moreover, in the
same paper Corvaja obtained also a quantitative version of Theorem 0.2 (see
[Cor97, Corollaire 3.7]).

An _arithmetic function field_ , is a finitely generated field over
$\mathbb{Q}$. The “geometrisation” of these fields in terms of Arakelov
geometry is wonderfully explained in [Mor00]. Recently, Vojta in [Voj21]
proved a version of Roth’s theorem for arithmetic function fields with a big
polarisation. For obvious reasons this can be considered as a “higher
dimensional” generalisation of Roth’s theorem. We conclude this short historic
overview by explaining the statement of Vojta’s result: a big polarisation of
an arithmetic function field $K$ consists of a couple $(X,\mathcal{L})$ where:

  * $(i)$

$X\to\operatorname{Spec}\mathbb{Z}$ is a normal arithmetic variety whose
function field is isomorphic to $K$.

  * $(ii)$

If we denote with $d$ the relative dimension of $X$ over
$\operatorname{Spec}\mathbb{Z}$, then
$\mathcal{L}=\\{\mathscr{L}_{1},\ldots\mathscr{L}_{d}\\}$ is a collection of
hermitian, arithmetically big line bundles.

Now we fix an arithmetic function field $K$ with a big polarisation; then we
can define a geometric height function $h_{\mathcal{L}}$ for a prime divisor
$Y$ as the Arakelov intersection number of the hermitian line bundles
$\mathscr{L}_{1},\ldots,\mathscr{L}_{d}$ restricted to $Y$. We can now define
a non-archimedean absolute value on $K$, associated to $Y$:

$|a|_{Y}:=e^{-h_{\mathcal{L}}(Y)\mathrm{ord}\,_{Y}(a)}\quad\forall a\in K\,.$

Moreover, for any closed point $p\in
X_{\mathbb{C}}=X\times_{\operatorname{Spec}\mathbb{Z}}\operatorname{Spec}\mathbb{C}$
that doesn’t come from the base change of a divisor on $X$ we have the
following archimedean absolute value:

$|a|_{p}:=\sqrt{a(p)\overline{a(p)}}\quad\forall a\in K\,.$

By putting all together, we get a set of absolute values $M_{K}$ which turns
out to be a measure space with a measure that we denote with $\mu$. The notion
of product formula holds true for the couple $(K,M_{K})$ in the following
form:

$\int_{M_{K}}\log|a|_{\nu}d\mu(\nu)=0\,,\quad\forall a\in K^{\times}$

Moreover there is a notion of height for any element of $a\in K^{\times}$:

$h_{K}(a):=\int_{M_{K}}\log^{+}|a|_{v}\,d\mu(v)\,.$

We set $H_{K}(a):=e^{h_{K}(a)}$. One of the many equivalent versions of
Vojta’s generalisation of Roth’s theorem is given below.

######  Theorem 0.3.

[Voj21, Theorem 4.5] Consider a couple $(K,M_{K})$ where $K$ is an arithmetic
function field with a big polarisation and $M_{K}$ is a set of absolute values
as explained before. Moreover let $S$ be a subset of $M_{K}$ of finite measure
and let $\alpha_{1},\ldots,\alpha_{n}$ be distinct elements of $K$. Then for
any $\varepsilon>0$ there exists a real constant $C>0$ (depending on the fixed
data) such that for any $\beta\in\mathbb{K}$ with $H_{K}(\beta)>C$ it holds
that

$\int_{S}\,\max_{1\leq i\leq
n}\left(|\beta-\alpha_{i}|_{v}\right)d\mu(v)>-(2+\varepsilon)h_{K}(\beta)$

###  0.2 Results in this paper

The goal of this paper is to generalise Roth’s theorem in a wider framework
which includes Corvaja’s and Vojta’s settings.

The theory of adelic curves introduced by Chen and Moriwaki in [CM20] provides
such natural framework: an adelic curve $\mathbb{X}$ consists of a field
$\mathbb{K}$ of characteristic $0$ and a measure space
$(\Omega,\mathcal{A},\mu)$ endowed with a function
$\phi:\omega\mapsto|\cdot|_{\omega}$ that maps $\Omega$ into the set of places
of $\mathbb{K}$ and such that $\omega\mapsto\log|a|_{\omega}$ is in
$L^{1}(\Omega,\mu)$ for any $a\in\mathbb{K}^{\times}$. On $\mathbb{X}$ we have
a well defined notion of height $h_{\mathbb{X}}$, and moreover a “product
formula” which is expressed as an integral over $\Omega$. The adelic curves
satisfying this product formula are called _proper_. The fields with a product
formula of [Cor97] are trivially proper adelic curves when the set of places
is endowed with the counting measure. Moreover in [CM20, 3.2.6] it is shown
that arithmetic function fields can be endowed with a structure of proper
adelic curve.

In this paper we prove two generalisations of Roth’s theorem for proper adelic
curves. The core of the results will be some inequalities involving the
measurable functions $\omega\mapsto\log^{-}|\beta-\alpha_{i}|_{\omega}$ where
$\alpha_{1},\ldots,\alpha_{n}\in\mathbb{K}$ are the elements we want to
approximate by an approximant $\beta\in\mathbb{K}$. Unfortunately, the bare
definition of proper adelic curves is too general to give any meaningful
result in the direction we want, in other words the functions
$\log^{-}|\beta-\alpha_{i}|$ can be in principle very “wild”. Therefore, we
have to impose some analytic conditions on such functions in order to get
Roth’s theorem. We will see that these assumptions are not too artificial, in
fact we recover Theorems 0.2 and 0.3 as special cases.

The first condition we assume on our adelic curves is the _$\mu$
-equicontinuity_ (see definition 4.1): roughly speaking it means that for any
set of finite measure $S\subset\Omega$ one requires for the functions
$\omega\mapsto\log|\beta|_{\omega}$, for any $\beta\in\mathbb{K}^{\times}$, to
be “equicontinuous” on $S$ minus a set of negligible measure. Our first
version of Roth’s theorem is the following:

######  Theorem A.

Let $\mathbb{X}=(\mathbb{K},\Omega,\phi)$ be a proper adelic curve such that
for any finite Galois extension $\mathbb{K}^{\prime}|\mathbb{K}$ the adelic
curve $\mathbb{X}=(\mathbb{K}^{\prime},\Omega^{\prime},\phi^{\prime})$
satisfies the $\mu^{\prime}$-equicontinuity condition. Let $S=S_{1}\sqcup
S_{2}\sqcup\ldots\sqcup S_{n}$ be the disjoiunt union of measurable subsets of
$\Omega$ with finite measure. Fix some distinct elements
$\alpha_{1},\dots,\alpha_{n}\in\overline{\mathbb{K}}$, and choose some
$\mathbb{K}$-embeddings
$\iota_{\omega,i}\colon\mathbb{K}(\alpha_{i})\hookrightarrow\overline{\mathbb{K}}_{\omega}$
for all $i=1,\ldots,n$ and all $\omega\in S$ in such a way that the function
$\omega\mapsto\log^{-}|\iota_{\omega,i}(\beta-\alpha_{i})|_{\omega}$ is
measurable for all $i$ and any $\beta\in
K\setminus\\{\alpha_{1},\ldots,\alpha_{n}\\}$. Assume also that
$\iota_{\omega,i}(\alpha_{i})\neq\iota_{\omega,i^{\prime}}(\alpha_{i^{\prime}})$
for $i\neq i^{\prime}$. Then for any $\varepsilon>0$ there exists a real
constant $C>0$ (depending on the fixed data) such that for any
$\beta\in\mathbb{K}$ with $h_{\mathbb{X}}(\beta)>C$ it holds that:

$\sum^{n}_{i=1}\int_{S_{i}}\log^{-}|\iota_{\omega,i}(\beta-\alpha_{i})|_{\omega}d\mu(\omega)>-(2+\varepsilon)h_{\mathbb{X}}(\beta)\,.$
(1)

In the previous statement $\overline{\mathbb{K}}_{\omega}$ denotes the
algebraic closure of the completion $\mathbb{K}_{\omega}$ (with respect to the
absolute value $|\cdot|_{\omega}$). It is immediate to show that the
$\mu$-equicontinuity trivially holds for the fields considered in [Cor97].
Therefore Theorem A implies Theorem 0.2. On the other hand, despite [Voj21,
Proposition 8.9] shows that the $\mu$-equicontinuity holds in the case of
arithmetic functions fields, Theorem A is weaker than Theorem 0.3. Furthermore
we show that the $\mu$-equicontinuity condition doesn’t hold in certain cases,
such as for $\overline{\mathbb{Q}}$ (see Example 4.5).

Nevertheless, if in addition we assume that the functions
$\omega\to\log^{-}|\beta|_{\omega}$ are _uniformly integrable_ while $\beta$
varies (see definition 5.1) then we get the following:

######  Theorem B.

Let $\mathbb{X}=(\mathbb{K},\Omega,\phi)$ be a proper adelic curve satisfying:
the $\mu$-equicontinuity condition, and the uniform integrability condition.
Fix some distinct elements $\alpha_{1},\dots,\alpha_{n}\in\mathbb{K}$. Let $S$
be a measurable subset of $\Omega$ of finite measure. Then for any
$\varepsilon>0$ there exists a real constant $C>0$ (depending on the fixed
data) such that for any $\beta\in\mathbb{K}$ with $h_{\mathbb{X}}(\beta)>C$ it
holds that:

$\int_{S}\,\max_{1\leq i\leq
n}\left(\log^{-}|\beta-\alpha_{i}|_{\omega}\right)d\mu(\omega)>-(2+\varepsilon)h_{\mathbb{X}}(\beta)$
(2)

For arithmetic function fields Vojta proved the uniform integrality in [Voj21,
Proposition 8.8]. Hence, as a special case of Theorem B we recover Theorem
0.3.

######  Remark 0.4.

If one wants a version of Theorem B for algebraic extensions, then it is
enough to assume the uniform integrability conditions for all Galois
extensions, similarly to Theorem A. This time we avoided this further
technicality in the statement in order to help the quick comparison between B
and Theorem 0.3. For more details about possible equivalent statement we refer
to [Voj21, Section 4].

The advantage of our approach is that our proofs rather short and elementary,
and in the case of arithmetic function fields (after assuming the technical
hypotheses) we don’t need any heavy Arakelov geometry. On the other hand we
don’t have, at the moment, any examples of interesting proper adelic curve
different from the ones already known. It is a much harder problem to find new
concrete cases in which Roth’s theorem holds. Afterall, a highly nontrivial
achievement of Vojta’s work consists in showing that the $\mu$-equicontinuity
and uniform intrgrality conditions hold for arithmetic function fields. He
does this by using the geometry of complex fibres at infinity appearing in
Arakelov geometry.

Most of our the ideas are inspired by Corvaja’s paper [Cor97], and the very
coarse overview of the proofs is the following: Roth’s theorem is about the
simultaneous approximation of some distinct elements
$\alpha_{1},\ldots,\alpha_{n}$ that by simplicity (in fact it will be enough
to consider this case) can be fixed in $\mathbb{K}$, with an element
$\beta\in\mathbb{K}$. The existence of a very special “interpolating
polynomial” $\delta$ for such elements (section 2) allows us to write some
integral estimates for measurable functions on $\Omega$ satisfying some
technical properties related to the heights of the $\alpha_{i}$’s and $\beta$
(section 3). Then, assuming that Roth’s theorem is false leads to the
construction of a measurable function $\theta:S\to\mathbb{R}$ that gives the
desired contradiction on the integral estimates previously found. The crucial
point of the proofs consist in the construction of $\theta$, and this is
exactly where we need the additional technical conditions on the adelic curve.

We finally mention that section 1 is a brief review of the theory of adelic
curves introduced in [CM20], and moreover that Appendix A sketches the
construction of the interpolating polynomial employed in [Cor97].

#### Acknowledgements.

Both the authors want to express their gratitude to _Pietro Corvaja_ for
introducing them to the subject and for his enlightening explanations.
Moreover they want to thank _Lorenzo Freddi_ for the very interesting
discussions about functional analysis.

The first author was partially supported by the EPSRC grant EP/M024830/1
“Symmetries and correspondences: intra-disciplinary developments and
applications”, partially by the research grant “Higher Arakelov geometry,
Cryptography and Isogenies of Elliptic Curves” conferred by the University of
Udine, and partially by the Ben-Gurion university of the Negev.

##  1 Adelic curves

We will use the following notations throughout the whole paper:

$\log^{+}x:=\max\\{0,\log x\\}\,,\quad\log^{-}x:=\min\\{0,\log
x\\}\,;\quad\forall x\in\mathbb{R}_{>0}$

In this section we closely follow [CM20, Chapter 3]. For simplicity we
restrict to the characteristic $0$ case, but all the definitions work also in
positive characteristic.

######  Definition 1.1.

Let $\mathbb{K}$ be a field of characteristic $0$, let $M_{\mathbb{K}}$ be the
set of all absolute values of $\mathbb{K}$ and let
$\Omega=(\Omega,\mathcal{A},\mu)$ be a measure space endowed with a map

$\displaystyle\phi\colon\Omega$ $\displaystyle\to$ $\displaystyle
M_{\mathbb{K}}$ $\displaystyle\omega$ $\displaystyle\mapsto$
$\displaystyle|\cdot|_{\omega}:=\phi(\omega)\,.$

such that for any $a\in\mathbb{K}^{\times}$, the real valued function
$\omega\mapsto\log|a|_{\omega}$ lies in $L^{1}(\Omega,\mu)$. The triple
$\mathbb{X}=(\mathbb{K},\Omega,\phi)$ is called an _adelic curve_ ; $\Omega$
and $\phi$ are respectively the _parameter space_ and the _parametrization_.
We denote with $\Omega_{\infty}$ the subset of $\Omega$ made of all elements
$\omega$ such that $|\cdot|_{\omega}$ is archimedean. We set
$\Omega_{0}:=\Omega\setminus\Omega_{\infty}$.

######  Remark 1.2.

We also recall that a more general notion of adelic curve, with the
requirement that $|\cdot|_{\omega}$ is an absolute value only almost
everywhere for $\omega\in\Omega$, had been already given in [Gub97] under the
name of $M$-field.

It is easy to show that the set $\Omega_{\infty}$ is always measurable [CM20,
Proposition 3.1.1].

######  Definition 1.3.

An adelic curve $\mathbb{X}=(\mathbb{K},\Omega,\phi)$ is said to be _proper_
if for any $a\in\mathbb{K}^{\times}$:

$\int_{\Omega}\log|a|_{\omega}\,d\mu(\omega)=0\,.$ (3)

Let’s see examples of adelic curves:

######  Example 1.4.

Any field $(k,\mathcal{V}_{k})$ satisfying the product formula in the sense of
[Cor97] is a proper adelic curve. In fact in this case
$\Omega=\mathcal{V}_{k}$, $\phi$ is the identity and $\mu$ is the counting
measure.

######  Example 1.5.

An arithmetic function field $K$ with a big polarisation is a proper adelic
curve. A quick description of this has been already given in section 0.1. For
details see [CM20, 3.2.6].

######  Example 1.6.

A polarised algebraic function field (in $d\geq 1$ variables) over a field of
characteristic $0$ can be endowed with a structure of proper adelic curve. For
details see [CM20, 3.2.4].

In the remaining part of this section we study the behaviour of adelic curves
with respect to field extensions. In particular let’s fix an adelic curve
$\mathbb{X}=(\mathbb{K},\Omega_{\mathbb{K}},\phi_{\mathbb{K}})$, and let
$\mathbb{L}$ be a _finite_ extension of $\mathbb{K}$, our goal is to endow it
with a canonical structure of adelic curve “coming from $\mathbb{K}$”. In
other words, we have to define a parameter space $\Omega_{\mathbb{L}}$ and a
parametrization $\phi_{\mathbb{L}}$ in a canonical way by starting from
$\Omega_{\mathbb{K}}$ and $\phi_{\mathbb{K}}$. For any $\omega\in\Omega$ we
denote with $M_{\mathbb{L},\omega}$ the set of absolute values of $\mathbb{L}$
extending $|\cdot|_{\omega}$, so we put:

$\Omega_{\mathbb{L}}:=\bigsqcup_{\omega\in\Omega_{\mathbb{K}}}M_{\mathbb{L},\omega}$

and we have a natural projection map
$\pi_{\mathbb{L}|\mathbb{K}}:\Omega_{\mathbb{L}}\to\Omega_{\mathbb{K}}$ whose
fibres are $M_{\mathbb{L},\omega}$, for any $\omega$. The inclusion
$M_{\mathbb{L},\omega}\subset M_{\mathbb{L}}$ clearly induces a
parametrization $\phi_{\mathbb{L}}\colon\Omega_{\mathbb{L}}\to M_{\mathbb{L}}$
and for any $\nu\in\Omega_{\mathbb{L}}$ we put
$|\cdot|_{\nu}:=\phi_{\mathbb{L}}(\nu)$. We obtain the following commutative
diagram:

${\Omega_{\mathbb{L}}}$${\Omega_{\mathbb{K}}}$${M_{\mathbb{L}}}$${M_{\mathbb{K}}}$$\scriptstyle{\pi_{\mathbb{L}|\mathbb{K}}}$$\scriptstyle{\phi_{\mathbb{L}}}$$\scriptstyle{\phi_{\mathbb{K}}}$
(4)

where the bottom map is the restriction function. Note that
$\Omega_{\mathbb{L}}$ can be abstractly defined as the fibered product in the
category of sets. Now on $\Omega_{\mathbb{L}}$ we put the $\sigma$-algebra
$\mathcal{B}$ generated by $\pi_{\mathbb{L}|\mathbb{K}}$ and all the real maps
$\Omega_{\mathbb{L}}\ni\nu\mapsto|a|_{\nu}$, for any $a\in\mathbb{L}^{\times}$
(on $\mathbb{R}$ we put the standard Lebesgue measure). We want to define a
suitable measure $\eta$ on the measurable space
$(\Omega_{\mathbb{L}},\mathcal{B})$. This requires a bit of work, since in
general there is no straightforward definition of pullback measure through a
measurable map. Nevertheless, in the case of our $\pi_{\mathbb{L}|\mathbb{K}}$
we explain how it is possible to define the pullback
$\eta=\pi^{\ast}_{\mathbb{L}|\mathbb{K}}\mu$ (actually this is a construction
from measure theory which works in full generality any time we have a measure
fiberwise). Consider a fiber
$M_{\mathbb{L},\omega}\subset\Omega_{\mathbb{L}}$, then for any $\nu\in
M_{\mathbb{L},\omega}$ we can put

$P_{\omega}(\nu):=\frac{[\mathbb{L}_{\nu}:\mathbb{K}_{\omega}]}{[\mathbb{L}:\mathbb{K}]}$
(5)

where $\mathbb{L}_{\nu}$ and $\mathbb{K}_{\omega}$ denote the completions with
respect to $|\cdot|_{\nu}$ and $|\cdot|_{\omega}$ respectively. Thanks to the
well known equality $\prod_{\nu\in
M_{\mathbb{L},\omega}}[\mathbb{L}_{\nu}:\mathbb{K}_{\omega}]=[\mathbb{L}:\mathbb{K}]$
(See [Neu99, Corollary 8.4]), we conclude that equation (5) induces a
probability measure on the fibre $M_{\mathbb{L},\omega}$, with respect to the
power set. Now, for any function $f:\Omega_{\mathbb{L}}\to\mathbb{R}$, by
using the fiberwise integral along each probabilised fiber
$M_{\mathbb{L},\omega}$, we define the map
$I_{\mathbb{L}|\mathbb{K}}(f):\Omega_{\mathbb{K}}\to\mathbb{R}$ as:

$I_{\mathbb{L}|\mathbb{K}}(f):\omega\mapsto\int_{M_{\mathbb{L},\omega}}\\!\\!f\,dP_{\omega}=\\!\sum_{\nu\in
M_{\mathbb{L},\omega}}\\!P_{\omega}(\nu)f(\nu)$

######  Proposition 1.7.

The linear operator $I_{\mathbb{L}|\mathbb{K}}$ sends $\mathcal{B}$-measurable
functions to $\mathcal{A}$-measurable functions.

###### Proof.

See [CM20, Theorem 3.3.4]. ∎

At this point are ready to define the measure $\eta$. For any
$E\in\mathcal{B}$ we put:

$\eta(E):=\int_{\Omega_{\mathbb{K}}}I_{\mathbb{L}|\mathbb{K}}(\chi_{E})\,d\mu$
(6)

where $\chi_{E}$ is the characteristic function of $E$. Note that the integral
of equation (6) makes sense because of Proposition 1.7.

######  Theorem 1.8.

The following statements hold:

  * $(1)$

The map $\eta$ defined above is a measure on
$(\Omega_{\mathbb{L}},\mathcal{B})$ such that for any $\mathcal{B}$-measurable
function $f$ we have

$\int_{\Omega_{\mathbb{L}}}fd\,\eta=\int_{\Omega_{\mathbb{K}}}I_{\mathbb{L}|\mathbb{K}}(f)\,d\mu\,.$

  * $(2)$

$f\in L^{1}(\eta)$ if and only if $I_{\mathbb{L}|\mathbb{K}}(|f|)\in
L^{1}(\mu)$.

  * $(3)$

The pushforward measure of $\eta$ through $\pi_{\mathbb{L}|\mathbb{K}}$ is
$\mu$.

  * $(4)$

With the above constructions the triple
$\mathbb{Y}=(\mathbb{L},\Omega_{\mathbb{L}},\phi_{\mathbb{L}})$ is an adelic
curve. Moreover for any $b\in\mathbb{L}^{\times}$

$[\mathbb{L}:\mathbb{K}]\int_{\Omega_{\mathbb{L}}}\log|b|_{\nu}d\eta(\nu)=\int_{\Omega_{\mathbb{K}}}\log|N_{\mathbb{L}|\mathbb{K}}(b)|_{\omega}d\mu(\omega)$
(7)

and in particular if $\mathbb{X}$ is proper, then also $\mathbb{Y}$ is proper.

###### Proof.

See [CM20, Theorem 3.3.7]. ∎

At this point we study _algebraic_ extensions of adelic curves. Let’s fix the
adelic curve $\mathbb{X}=(\mathbb{K},\Omega_{K},\phi_{K})$ and let
$\mathbb{L}$ an algebraic extension of $\mathbb{K}$. We denote with
$\mathcal{F}_{\mathbb{L}|\mathbb{K}}$ the family of finite field extensions on
$\mathbb{K}$ contained in $\mathbb{L}$. Clearly
$\mathcal{F}_{\mathbb{L}|\mathbb{K}}$ is a directed set with respect to the
inclusion, and for any
$\mathbb{K}^{\prime},\mathbb{K}^{\prime\prime}\in\mathcal{F}_{\mathbb{L}|\mathbb{K}}$
such that $\mathbb{K}^{\prime}\subseteq\mathbb{K}^{\prime\prime}$ we have a
morphism of measurable spaces

$\pi_{\mathbb{K}^{\prime\prime}|\mathbb{K}^{\prime}}\colon(\Omega_{\mathbb{K}^{\prime\prime}},\mathcal{B}^{\prime\prime})\to(\Omega_{\mathbb{K}^{\prime}},\mathcal{B}^{\prime})$

and an operator $I_{\mathbb{K}^{\prime\prime}|\mathbb{K}^{\prime}}$ sending
integrable functions to integrable functions as described above in the case of
finite extensions. In other words we obtain an inverse system of measure
spaces, and we would like to define the adelic structure on $\mathbb{L}$ as “a
projective limit”. Unfortunately, in the category of measure spaces we don’t
have a straightforward notion of projective limit, therefore we need again a
bit of extra work. We can define respectively $M_{\mathbb{L},\omega}$,
$\Omega_{\mathbb{L}}$ and $\phi_{\mathbb{L}}\colon\Omega_{\mathbb{L}}\to
M_{\mathbb{L}}$ exactly as we did before in the case of finite extensions, but
we need to construct an adequate structure of measure space on
$\Omega_{\mathbb{L}}$. For any
$K^{\prime}\in\mathcal{F}_{\mathbb{L}|\mathbb{K}}$ we have a map
$\pi_{\mathbb{L}|\mathbb{K}^{\prime}}:\Omega_{\mathbb{L}}\to\Omega_{\mathbb{K}^{\prime}}$
and a square diagram like (4). It turns out that
$\pi_{\mathbb{L}|\mathbb{K}^{\prime}}$ is surjective [CM20, Proposition
3.4.5]. We endow $\Omega_{\mathbb{L}}$ with the $\sigma$-algebra $\Sigma$
generated by the maps
$\\{\pi_{\mathbb{L}|\mathbb{K}^{\prime}}\\}_{\mathbb{K}^{\prime}\in\mathcal{F}_{\mathbb{L}|\mathbb{K}}}$,
and it can be shown that $(\Omega_{\mathbb{L}},\Sigma)$ is the projective
limit of the inverse system
$\\{(\Omega_{\mathbb{K}^{\prime}},\mathcal{B}^{\prime})\\}_{\mathbb{K}^{\prime}\in\mathcal{F}_{\mathbb{L}|\mathbb{K}}}$
in the category of measurable spaces. It remains the issue of putting a
canonical measure $\lambda$ on $(\Omega_{\mathbb{L}},\Sigma)$. This process is
quite technical, but similarly to the case of finite extensions, it can be
done by using a fiberwise intergration on each $M_{L,\omega}$; all the details
are given in [CM20, 3.4.2]. What we really need is the fact that we can
construct an adelic curve $(\mathbb{L},\Omega_{\mathbb{L}},\lambda)$ which is
proper if $\mathbb{X}$ is proper and such that for any $f\in L^{1}(\mu)$ we
have that $f\circ\pi_{\mathbb{L}|\mathbb{K}}\in L^{1}(\lambda)$ with

$\int_{\Omega_{\mathbb{L}}}(f\circ\pi_{\mathbb{L}|\mathbb{K}})d\lambda=\int_{\Omega_{\mathbb{K}}}fd\mu\,.$
(8)

Below we give the notion of height for proper adelic curves:

######  Definition 1.9.

Let $\mathbb{X}=(\mathbb{K},\Omega,\phi)$ be a proper adelic curve and let
$\overline{\mathbb{K}}$ be an algebraic closure of $\mathbb{K}$. Then we have
a proper adelic curve
$\overline{\mathbb{X}}=(\overline{\mathbb{K}},\overline{\Omega},\overline{\phi})$
and the _(naive) height_ of an element $a\in\overline{\mathbb{K}}^{\times}$ is
defined as:

$h_{\mathbb{X}}(a):=\int_{\overline{\Omega}}\log^{+}|a|_{\nu}\,d\chi(\nu)\,.$

where $\nu$ denotes a generic element of $\overline{\Omega}$ and $\chi$ is the
measure on $\overline{\Omega}$. Moreover we set
$H_{\mathbb{X}}:=e^{h_{\mathbb{X}}}$.

From now on we always assume that for an adelic curve
$\mathbb{X}=(\mathbb{K},\Omega,\phi)$ we have fixed algebraic closure of
$\mathbb{K}$, therefore also $\overline{\mathbb{X}}$ is fixed and we use the
same notations of Definition 1.9. If for $\nu\in\overline{\Omega}$,
$|\cdot|_{\nu}$ is an archimedean absolute value, then by Ostrowski’s theorem
we know that there exists a real number $\varepsilon(\nu)\in\,]0,1]$ such that
$|\cdot|_{\nu}=|\cdot|^{\varepsilon(\nu)}$ where on the right we mean the
standard euclidean absolute value on $\mathbb{R}$ or $\mathbb{C}$. Thus we
have a map $\varepsilon:\overline{\Omega}_{\infty}\to]0,1]$ which can be
extended to $\varepsilon:\overline{\Omega}\to[0,1]$ by putting
$\varepsilon_{|\overline{\Omega}_{0}}:=0$. For instance, for an archimedean
$|\cdot|_{\nu}$ we have $\log|2|_{\nu}=\varepsilon(\nu)\log 2$, therefore we
obtain the explicit expression of the function $\varepsilon$ on the whole
$\overline{\Omega}$:

$\varepsilon(\nu)=\frac{\log^{+}|2|_{\nu}}{\log 2}\,.$

Clearly $\varepsilon(\nu)$ is a measurable function. We can always take a
scaling $\mu^{\prime}$ of the measure $\mu$ on $\Omega$ so that get a new
height $h^{\prime}_{\mathbb{X}}$ that satisfies
$h^{\prime}_{\mathbb{X}}(2)\leq\log 2$. Notice that if $\mathbb{X}$ is proper,
then it remains proper after any scaling of the measure $\mu$. From now on,
when we are given an adelic curve $\mathbb{X}=(\mathbb{K},\Omega,\phi)$, we
can always assume that we have performed the above mentioned scaling of the
measure $\mu$ on $\Omega$ so that $h_{\mathbb{X}}(2)\leq\log 2$.

######  Definition 1.10.

Let $P(X_{1},\cdots,X_{N})$ be a polynomial over $\overline{\mathbb{K}}$,
$\alpha\in\overline{\mathbb{K}}^{N}$ and
$\mathbf{i}=(i_{1},\cdots,i_{N})\in\mathbb{N}^{N}$. We set:

$\Delta^{\mathbf{i}}P(\alpha):=\frac{1}{i_{1}!i_{2}!\cdots
i_{N}!}\,\frac{\partial^{i_{1}+i_{2}+\ldots+i_{N}}P}{\partial
X^{i_{1}}_{1}\partial X^{i_{2}}_{2}\ldots\partial X^{i_{N}}_{N}}(\alpha).$

We can define the _local height of $P$ at $\nu\in\overline{\Omega}$_ in the
following way:

$h_{\nu}(P):=\log\left(\max_{\mathbf{i}\in\mathbb{N}^{N}}\left\\{\left|\Delta^{\mathbf{i}}P(0,\ldots,0)\right|_{\nu}\right\\}\right)$

and then we have also the notion of _global height_ of $P$:

$h_{\mathbb{X}}(P):=\int_{\overline{\Omega}}h_{\nu}(P)d\chi(\nu)$

We put $H_{\mathbb{X}}(P):=e^{h_{\mathbb{X}}(P)}$.

We conclude the section with some rather straightforward results about
heights. First of all when we want to calculate the heights of elements lying
in $\mathbb{K}^{\times}$, we don’t need to involve the algebraic closure
$\overline{\mathbb{K}}$ in the integrals:

######  Proposition 1.11.

Let $\mathbb{X}=(\mathbb{K},\Omega,\phi)$ be an adelic curve. If
$a\in\mathbb{K}^{\times}$, then:

$h_{\mathbb{X}}(a)=\int_{\Omega_{K}}\log^{+}|a|_{\omega}d\mu(\omega)\,.$

Moreover, the same result holds for the heights of polynomials in
$\mathbb{K}[X_{1},\ldots,X_{N}]$.

###### Proof.

It is an immediate consequence of equation (8). ∎

In order to simplify the notations, we often omit the subscript $\mathbb{X}$
attached to the heights when the adelic curve is fixed and there is no
confusion.

######  Proposition 1.12.

The height function of a proper adelic curve $(\mathbb{X},\Omega,\phi)$
satisfies the following properties for any
$a,b,a_{1},\ldots,a_{m}\in\mathbb{K}$ and any measurable set
$S\subseteq\Omega$

  * $(1)$

$h(a)=h(a^{-1})$

  * $(2)$

$-h(a)\leq\int_{S}\log|a|_{\omega}d\mu(\omega)\leq h(a)$

  * $(3)$

$\int_{S}\log^{-}|a|_{\omega}d\mu(\omega)\geq-2h(a)$

  * $(4)$

$h(a_{1}+\ldots+a_{m})\leq h(m)+h(a_{1})+\ldots+h(a_{m})$

  * $(5)$

$\int_{S}\log|a-b|_{\omega}d\mu(\omega)\geq-\log 2-h(a)-h(b)$

  * $(6)$

$\int_{S}\log^{-}|a-b|_{\omega}d\mu(\omega)\geq-\log 4-2h(a)-2h(b)$

###### Proof.

$(1)$ It follows from the product formula and from the fact that
$\log|a|_{\omega}=\log^{+}|a|_{\omega}-\log^{+}|\frac{1}{a}|_{\omega}$.

$(2)$ By definition $\int_{S}\log|a|_{\omega}d\mu(\omega)\leq h(a)$, so for
the other inequality it is enough to use $(1)$.

$(3)$ By the product formula we have:

$\int_{S}\log^{-}|a|_{\omega}d\mu(\omega)=-\int_{\Omega\setminus
S}\log|a|_{\omega}d\mu(\omega)-\int_{S}\log^{+}|a|_{\omega}d\mu(\omega)\,.$

Then we apply $(2)$ on the right hand side.

$(4)$ It follows from $|a_{1}+\ldots+a_{m}|_{\omega}\leq
m\max_{i}|a_{i}|_{\omega}$.

$(5)$ and $(6)$ are direct consequences of $(2)$-$(4)$ and the fact that
$h(2)\leq\log(2)$. ∎

Here we stress that the entries $(5)$-$(6)$ of Proposition 1.12 replace the
classical Liouville inequality for heights. Finally we recall an important
property of heights:

######  Definition 1.13.

A proper adelic curve $\mathbb{X}=(\mathbb{K},\Omega,\mu)$ satisfies the
_Northcott property_ if for any $C\in\mathbb{R}$ the set
$\\{\alpha\in\mathbb{K}\colon h_{\mathbb{X}}(a)\\}\leq C$ is finite.

Arithmetic function fields satisfy Northcott properties thanks to [Mor00,
Theorem 4.3].

##  2 The interpolating polynomial

We fix a proper adelic curve $\mathbb{X}=(\mathbb{K},\Omega,\phi)$ and an
algebraic closure of $\mathbb{K}$. In this section we recall the existence of
an interpolating polynomial $\delta\in\mathbb{K}[X_{1},\ldots,X_{N}]$
associated to some elements $\alpha_{1},\ldots,\alpha_{n}$ having some
explicit bounds on: the degree, the $d$-index at all the $\alpha_{j}$’s and
the height. The complete construction of $\delta$ can be found in [Cor97], and
we will recall it in appendix A.

We fix for the whole section the following data: two natural numbers $n,N\geq
2$, and a vector $d=(d_{1},\ldots,d_{N})\in\mathbb{R}^{N}_{+}$.

######  Definition 2.1.

The _$d$ -index of $P(X_{1},\dots,X_{N})$ at $\alpha\in\mathbb{R}^{N}$_ is the
real number:

$\operatorname{Ind}_{\alpha,d}(P):=\min_{\mathbf{i}\in\mathbb{N}^{N}}\left\\{\sum_{j=1}^{N}\frac{i_{j}}{d_{j}}\in\mathbb{R}_{+}\colon\Delta^{\mathbf{i}}P(\alpha)\neq
0\right\\}$

Let’s fix $t\in\mathbb{R}$ such that $0<t<N$, the following two sets will play
a central role in the theory:

$\mathcal{G}_{t}:=\left\\{\mathbf{i}\in\mathbb{N}^{N}\colon i_{j}\leq
d_{j}\;\forall j=1,\dots,N,\,{\rm{and}}\sum_{j=1}^{N}\frac{i_{j}}{d_{j}}\leq
t\right\\},$
$\mathcal{C}_{t}:=\left\\{(x_{1},\dots,x_{N})\in[0,1]^{N}\colon\sum_{j=1}^{N}x_{j}\leq
t\right\\}$

The Lebesgue measure of $\mathcal{C}_{t}$ will be denoted as $V(t)$, and for
simplicity of terminology we will refer to it simply as “volume”.

######  Lemma 2.2.

The cardinality of $\mathcal{G}_{t}$ is asymptotic to $d_{1}d_{2}\ldots
d_{N}V(t)$.

###### Proof.

See [BG06, p. 157]. ∎

Now fix some vectors $\alpha_{1},\ldots,\alpha_{n}\in\mathbb{K}^{N}$ where
$\alpha_{h}=(\alpha^{(1)}_{h},\ldots,\alpha^{(N)}_{h})$ for every
$h=1,\ldots,n$ and let $X=(X_{1},\ldots,X_{N})$ be a vector of variables.
Moreover we say that two vectors $\mathbb{K}^{N}$ are _componentwise
different_ if they differ component by component. For any two multi-indices
$\mathbf{a}=(a_{1},a_{2},\dots,a_{N})$ and
$\mathbf{i}=(i_{1},i_{2},\dots,i_{N})$ of $\mathbb{N}^{N}$ we use the
following notations:

${\mathbf{a}\choose\mathbf{i}}:={{{a_{1}}}\choose{i_{1}}}{{{a_{2}}}\choose{i_{2}}}\ldots{{{a_{N}}}\choose{i_{N}}}$
$\alpha_{h}^{\mathbf{i}}:=(\alpha_{h}^{(1)}){{}^{{i_{1}}}}(\alpha_{h}^{(2)}){{}^{{i_{2}}}}\ldots(\alpha_{h}^{(N)}){{}^{{i_{N}}}}$
$X^{\mathbf{i}}:=X^{i_{1}}_{1}X^{i_{2}}_{2}\ldots X^{i_{N}}_{N}$

with the convention ${{{p}}\choose{q}}=0$ if $0\leq p<q$ for the binomial
coefficient. Now, consider $\gamma\in\mathbb{R}$ such that
$0<\gamma<\frac{1}{2nN^{2}}$; we always assume that
$\frac{d_{j+1}}{d_{j}}\leq\gamma$ for any $j=1,\ldots,N-1$ , which means in
particular that $d_{j}=O(d_{1})$ for any $j$. We also put $\eta:=2\gamma
n<\frac{1}{N^{2}}$ and $d:=d_{1}d_{2}\ldots d_{N}$.

We have the following result about the existence of a polynomial
$\delta(X_{1},\ldots,X_{N})\in\mathbb{K}[X_{1},\ldots X_{N}]$ with some
prescribed properties. In the appendix A we will sketch Corvaja’s construction
of $\delta(X_{1},\ldots,X_{N})$ adapting it to the case of adelic curves:

######  Proposition 2.3.

Assume that in the adelic curve $\mathbb{X}$ the condition
$h_{\mathbb{X}}(2)\leq\log 2$ is satisfied and assume that the number
$\eta\in\mathbb{R}$ is chosen as explained above. Let’s fix some vectors
$\alpha_{1},\ldots,\alpha_{n},\beta\in\mathbb{K}^{N}$ that are pairwise
componentwise different. Moreover let’s choose some parameters
$s,t_{1},\dots,t_{n}\in\mathbb{R}$, with $0<s<1$ and $0<t_{h}<\frac{N}{2}$ for
$h=1,\dots,n$ such that the following condition on volumes is verified:

$(1+\eta)^{N}<V(s)+\sum_{h=1}^{n}V(t_{h})<1+2N\eta$ (9)

Then there exists a polynomial $\delta(X_{1},\ldots
X_{N})\in\mathbb{K}[X_{1},\ldots,X_{N}]$ satysfying the following properties:

  1. $(1)$

$\delta(\beta)\neq 0$;

  2. $(2)$

${\deg}_{X_{j}}\delta\leq dd_{j}V(s)+O(d)$, for $1\leq j\leq N$;

  3. $(3)$

${\rm{Ind}}_{\alpha_{h},d}(\delta)\geq
dV(s)\left(t_{h}-s-\frac{2N^{2}\eta}{V(s)}\right)$, for $1\leq h\leq n$;

  4. $(4)$

$\displaystyle h_{\mathbb{X}}(\delta)\leq d\sum_{j=1}^{N}d_{j}\left(\log
2+\sum_{h=1}^{N}V(t_{h})h(\alpha^{(j)}_{h})\right)+O(\log d)$;

###### Proof.

See [Cor97, Proposition 2.6]. ∎

##  3 Integral estimates

This technical section is “the heart” of the proof of our results since here
we will prove some integral bounds for very particular measurable functions
$\theta:S\subset\Omega\to\mathbb{R}$. We continue with all the notations fixed
in section 2 since we want to make full use of Theorem 2.3.

Consider an adelic curve $(\mathbb{K},\Omega,\phi)$ and a set of vectors
$\alpha_{1},\ldots,\alpha_{n},\beta\in\mathbb{K}^{N}$ which are componentwise
different. We construct the following matrices
$T:=T(\alpha_{1},\ldots\alpha_{n})\in M(n\times N,\mathbb{K})$ and
$T(\beta)\in M(n+1\times N,\mathbb{K})$:

$T:=\begin{pmatrix}\alpha^{(1)}_{1}&\alpha^{(2)}_{1}&\alpha^{(3)}_{1}&\ldots&\alpha^{(N)}_{1}\\\
\alpha^{(1)}_{2}&\alpha^{(2)}_{2}&\alpha^{(3)}_{2}&\ldots&\alpha^{(N)}_{2}\\\
\vdots&\vdots&\vdots&\vdots&\vdots\\\ \vdots&\vdots&\vdots&\vdots&\vdots\\\
\alpha^{(1)}_{n}&\alpha^{(2)}_{n}&\alpha^{(3)}_{n}&\ldots&\alpha^{(N)}_{n}\end{pmatrix};\quad
T(\beta)=\begin{pmatrix}\alpha^{(1)}_{1}&\alpha^{(2)}_{1}&\alpha^{(3)}_{1}&\ldots&\alpha^{(N)}_{1}\\\
\alpha^{(1)}_{2}&\alpha^{(2)}_{2}&\alpha^{(3)}_{2}&\ldots&\alpha^{(N)}_{2}\\\
\vdots&\vdots&\vdots&\vdots&\vdots\\\ \vdots&\vdots&\vdots&\vdots&\vdots\\\
\alpha^{(1)}_{n}&\alpha^{(2)}_{n}&\alpha^{(3)}_{n}&\ldots&\alpha^{(N)}_{n}\\\
\beta^{(1)}&\beta^{(2)}&\beta^{(3)}&\ldots&\beta^{(N)}\end{pmatrix}$

We denote by $\alpha^{(j)}$, $j=1,\ldots,N$ the columns of $T$, that is:

$\alpha^{(j)}=\begin{pmatrix}\alpha^{(j)}_{1}\\\ \alpha^{(j)}_{2}\\\ \vdots\\\
\vdots\\\ \alpha^{(j)}_{n}\end{pmatrix}$

and by $\alpha_{h}$, for $h\in\\{1,2,\dots,n\\}$, the rows:

$\alpha_{h}=\begin{pmatrix}\mathbb{\alpha}^{(1)}_{h},\alpha^{(2)}_{h},\alpha^{(3)}_{h},\ldots,\alpha^{(N)}_{h}\end{pmatrix}$

Note that we are asking for the matrices $T$ and $T(\beta)$ to have
componentwise different rows. Now we need to define a list of properties
depending on the aforementioned matrices:

######  Definition 3.1.

For the matrix $T(\beta)$, consider the following real numbers for any
$j=1,\ldots,N$:

$\rho_{j}:=4^{2N!}H(\beta^{(j)})\prod_{h=1}^{n}H(\alpha_{h}^{(j)})^{\frac{2N!}{n}}$
(10)
$\rho^{\prime}_{j}:=4^{N!}H(\beta^{(j+1)})\prod_{h=1}^{n}H(\alpha_{h}^{(j+1)})^{\frac{N!}{2n}}$
(11)

We say that $T(\beta)$ satisfies the _$h$ -gap condition_ is the following
inequality is satisfied:

$\frac{{\rm{log}}\,\rho_{j}}{{\rm{log}}\,\rho^{\prime}_{j}}<\frac{1}{4N^{2}N!},\qquad\forall
j=1,\ldots,N$

######  Definition 3.2.

Let’s fix a matrix $T(\beta)$. Given some mutually disjoint measurable subsets
$S_{1},\ldots,S_{n}\subset\Omega$ of finite measure such that $S:=S_{1}\sqcup
S_{2}\sqcup...\sqcup S_{n}$, we define:

$\hat{S}:=\left\\{\omega\in
S\colon\left|\alpha^{(j)}_{h}-\beta^{(j)}\right|_{\omega}<1,\;\forall
j=1,\ldots,N,\;\forall h=1\ldots,n\right\\}\,.$

Moreover we put $\hat{S}_{h}:=\hat{S}\cap S_{h}$, so that
$\hat{S}=\hat{S}_{1}\sqcup\hat{S}_{2}\sqcup...\sqcup\hat{S}_{n}$.

Note that $\hat{S}$ can be thought as the sublocus of $S$ where it happens
that $\left(\beta^{(1)},\ldots,\beta^{(N)}\right)$ is a good approximant for
all the vectors $\alpha_{h}$, moreover we emphasize the fact that $\hat{S}$
depends on $T(\beta)$, whereas $S$ is fixed. The following easy lemma will be
useful:

######  Lemma 3.3.

If $\omega\in\hat{S}_{h}$, then
$\log^{+}|\alpha^{(j)}_{h}|_{\omega}\leq\log^{+}|2|_{\omega}+\log^{+}|\beta^{(j)}|_{\omega}$
for any $j=1,\ldots,N$.

###### Proof.

The claim holds true for any couple of vectors
$\alpha,\beta\in\mathbb{K}^{N}$, so in order to ease the notations we can just
drop any reference to the index $h$. We distinguish two cases:
_$|\cdot|_{\omega}$ is ultrametric._ Then

$|\alpha^{(j)}|_{\omega}=|\alpha^{(j)}-\beta^{(j)}+\beta^{(j)}|_{\omega}\leq\max\left\\{|\alpha^{(j)}-\beta^{(j)}|_{\omega},|\beta^{(j)}|_{\omega}\right\\}<\max\left\\{1,|\beta^{(j)}|_{\omega}\right\\}\,.$

If $|\beta^{(j)}|_{\omega}\geq 1$, then clearly
$\log^{+}|\alpha^{(j)}|_{\omega}\leq\log^{+}|\beta^{(j)}|_{\omega}$. If
$|\beta^{(j)}|_{\omega}<1$, then $|\alpha^{(j)}|_{\omega}<1$ which means
$0=\log^{+}|\alpha^{(j)}|_{\omega}\leq\log^{+}|2|_{\omega}+\log^{+}|\beta^{(j)}|_{\omega}$.

_$|\cdot|_{\omega}$ is archimedean_. By Ostrowski’s theorem
$|\cdot|_{\omega}=|\cdot|^{\varepsilon(\omega)}$, therefore

$|\alpha^{(j)}|^{\frac{1}{\varepsilon(\omega)}}_{\omega}\leq|\alpha^{(j)}-\beta^{(j)}|^{\frac{1}{\varepsilon(\omega)}}_{\omega}+|\beta^{(j)}|^{\frac{1}{\varepsilon(\omega)}}_{\omega}\leq
1+|\beta^{(j)}|^{\frac{1}{\varepsilon(\omega)}}_{\omega}\,.$

This clearly means
$|\alpha^{(j)}|^{\frac{1}{\varepsilon(\omega)}}_{\omega}\leq
2\max\left\\{1,|\beta^{(j)}|^{\frac{1}{\varepsilon(\omega)}}_{\omega}\right\\}$.
After rising each side to the power of $\varepsilon(\omega)$ and applying
$\log^{+}$ we get the claim. ∎

######  Definition 3.4.

Fix $S=S_{1}\sqcup\ldots\sqcup S_{n}$ of finite measure. We say an the
integrable function $\theta\colon S\to\mathbb{R}$ is a _column bounding
function for_ $T(\beta)$ if the following inequality holds:

$-\frac{1}{\log\rho_{j}}{\log}|\alpha^{(j)}_{h}-\beta^{(j)}|_{\omega}>\theta(\omega)\qquad\forall
j=1,\ldots,N\,,\;\forall h=1,\ldots,n\,,\;\forall\omega\in\hat{S}_{h}$ (12)

######  Definition 3.5.

Fix $S=S_{1}\sqcup\ldots\sqcup S_{n}$ of finite measure. For any column
$\alpha^{(j)}\in\mathbb{K}^{N}$ of the matrix $T$ and any $b\in\mathbb{K}$ we
define the following quantity:

$\lambda(\alpha^{(j)},b):=\frac{1}{V(s)}{\rm{log}}4+h(b)+\frac{1}{V(s)}\sum_{h=1}^{n}V(t_{h})h(\alpha_{h}^{(j)})$

######  Definition 3.6.

We say that $T(\beta)$ satisfies the _$\lambda$ -gap property_ of width
$\eta\in\mathbb{R}_{+}$ if

$\max_{1\leq j\leq
N-1}\frac{\lambda(\alpha^{(j)},\beta^{(j)})}{\lambda(\alpha^{(j+1)},\beta^{(j+1)})}<\frac{\eta}{2n}$

######  Definition 3.7.

Fix a matrix $T(\beta)$. An integrable function $\theta\colon S\to\mathbb{R}$
is _$\lambda$ -bounding_ if:

$\frac{-1}{\lambda(\alpha^{(j)},\beta^{(j)})}{\log}|\alpha^{(j)}_{h}-\beta^{(j)}|_{\omega}>\theta(\omega)\qquad\forall
j=1,\dots,N;\;\forall h=1,\dots,n;\;\forall\omega\in\hat{S}_{h}$ (13)

We will need the following lemma:

######  Lemma 3.8.

Let $(\mathbb{K},|\cdot|_{\omega})$ be a field with an absolute value. Let
$P\in\mathbb{K}[X_{1},\dots X_{N}]$ such that $N\geq 1$. Then for every
$\alpha\in\mathbb{K}^{N}$

${\log}|\Delta^{{\bf i}}P(\alpha)|_{\omega}\leq
h_{\omega}(P)+\sum_{j=1}^{N}{\log}(1+\deg_{X_{j}}P)+\sum_{j=1}^{N}(\log^{+}|2|_{\omega}+{\log}^{+}|\alpha^{(j)}|_{\omega}){\deg}_{X_{j}}P$

###### Proof.

See [Cor97, Lemme pag 166]. ∎

Now we prove the analogue of [Cor97, Proposition 3.1].

######  Proposition 3.9.

Let $(\mathbb{K},\Omega,\phi)$ be a proper adelic curve. Fix a matrix
$T(\beta)$. Let $\eta,s,t_{1},\ldots,t_{n}\in\mathbb{R}$ such that all the
hypotheses of Theorem 2.3 are satisfied. Assume that $T(\beta)$ satisfies the
$\lambda$-gap property of width $\eta$ (see Definition 3.6). If $\theta\colon
S\to\mathbb{R}$ is $\lambda$-bounding for $T(\beta)$ (see Definition 3.7),
then

$\sum_{h=1}^{n}\begin{pmatrix}t_{h}-s-\frac{2\eta
N^{2}}{V(s)}\end{pmatrix}\int_{\hat{S}_{h}}\theta d\mu<N$

###### Proof.

Fix a real number $D>0$, and for every $j=1,\dots,N$, we put

$d_{j}:=\frac{D}{\lambda(\alpha^{(j)},\beta^{(j)})}$

and $d:=\prod^{N}_{j=1}d_{j}$. Note that here we use the $\lambda$-gap
property to ensure that $\frac{dj+1}{d_{j}}<\gamma$. So, the hypotheses of
Theorem 2.3 are all satisfied and we have the interpolation polynomial
$\delta$ such that $\delta(\beta)\neq 0$. We start by distinguishing two
cases.

_First case: $\omega\in\hat{S}_{h}$_. Compose the polynomial $\delta$ with the
translation of $X\mapsto X-\alpha_{h}$, then we have

$\delta(\beta)=\sum_{{\bf i}}\Delta^{{\bf
i}}\delta(\alpha_{h})\prod_{j=1}^{N}(\beta^{(j)}-\alpha_{h}^{(j)})^{i_{j}}\quad\textrm{where
}{\mathbf{i}}=(i_{1},\ldots,i_{n})$ (14)

Now take the absolute value $|\cdot|_{\omega}$ on both sides and notice that
each term

$\left|\Delta^{{\bf
i}}\delta(\alpha_{h})\prod_{j=1}^{N}(\beta^{(j)}-\alpha_{h}^{(j)})^{i_{j}}\right|_{\omega}$

is bounded from above by:

$\max_{\mathbf{i}}\left|\Delta^{{\bf
i}}\delta(\alpha_{h})\right|_{\omega}\;\max^{\ast}_{\mathbf{i}}\left|\prod_{j=1}^{N}(\beta^{(j)}-\alpha_{h}^{(j)})^{i_{j}}\right|_{\omega}\,,$

were in order to simplify the notation we put

$\max^{\ast}_{\mathbf{i}}:=\max_{\begin{subarray}{c}\mathbf{i}\\\ \Delta^{{\bf
i}}(\alpha_{h})\neq 0\end{subarray}}$

In other words from equation (14) we get

$\log|\delta(\beta)|_{\omega}\leq\sum^{N}_{j=1}\log(1+\deg_{X_{j}}\delta)+\max_{\mathbf{i}}\,\log\left|\Delta^{{\bf
i}}\delta(\alpha_{h})\right|_{\omega}+\max^{\ast}_{\mathbf{i}}\,\log\left|\prod_{j=1}^{N}(\beta^{(j)}-\alpha_{h}^{(j)})^{i_{j}}\right|_{\omega}$
(15)

The last summand of equation (15) is bounded from above by

$-{\operatorname{Ind}}_{\alpha_{h},d}(\delta)\,\min\left\\{d_{1}\log\frac{1}{|\beta^{(1)}-\alpha_{h}^{(1)}|_{\omega}},\ldots,d_{N}\log\frac{1}{|\beta^{(N)}-\alpha_{h}^{(N)}|_{\omega}}\right\\}\,.$

We use Lemma 3.8 to give an upper bound for $\log\left|\Delta^{{\bf
i}}\delta(\alpha_{h})\right|_{\omega}$ and by equation (15) we deduce:

$\int_{\hat{S}_{h}}\log|\delta(\beta)|_{\omega}d\mu(\omega)\leq\int_{\hat{S}_{h}}\left(h_{\omega}(\delta)+\sum_{j=1}^{N}\log(1+\deg_{X_{j}}\delta)+\sum_{j=1}^{N}(\log^{+}|2|_{\omega}+\log^{+}|\alpha_{h}^{(j)}|_{\omega})\deg_{X_{j}}\delta\right)d\mu(\omega)+$
$+\int_{\hat{S}_{h}}\sum_{j=1}^{N}\log(1+\deg_{X_{j}}\delta)d\mu(\omega)-\int_{\hat{S}_{h}}(\operatorname{Ind}_{\alpha_{h},d}(\delta)\,\min\left\\{d_{1}\log\frac{1}{|\beta^{(1)}-\alpha_{h}^{(1)}|_{\omega}},\ldots,d_{N}\log\frac{1}{|\beta^{(N)}-\alpha_{h}^{(N)}|_{\omega}}\right\\}d\mu(\omega)$

Now use the fact that $\hat{S}$ has finite measure, and we put

$\tau:=\sum_{j=1}^{N}\log(1+\deg_{X_{j}}\delta)\mu(\hat{S})$

So, by rearranging the terms and summing over all $h=1,\ldots,n$ we get

$\sum^{n}_{h=1}\operatorname{Ind}_{\alpha_{h},d}(\delta)\int_{\hat{S}_{h}}\min\left\\{d_{1}\log\frac{1}{|\beta^{(1)}-\alpha_{h}^{(1)}|_{\omega}},\ldots,d_{N}\log\frac{1}{|\beta^{(N)}-\alpha_{h}^{(N)}|_{\omega}}\right\\}d\mu(\omega)\leq-\int_{\hat{S}}\log|\delta(\beta)|_{\omega}d\mu(\omega)\\\
+\int_{\hat{S}}h_{\omega}(\delta)d\mu(\omega)+\tau+\sum_{h=1}^{n}\int_{\hat{S}_{h}}\sum_{j=1}^{N}(\log^{+}|2|_{\omega}+\log^{+}|\alpha_{h}^{(j)}|_{\omega})\deg_{X_{j}}\delta
d\mu(\omega)$ (16)

At this point we can use Lemma 3.3 for the following bound:

$\sum_{h=1}^{n}\int_{\hat{S}_{h}}\sum_{j=1}^{N}(\log^{+}|2|_{\omega}+\log^{+}|\alpha_{h}^{(j)}|_{\omega})\deg_{X_{j}}\\!\\!\delta
d\mu(\omega)\leq\int_{\hat{S}}\sum_{j=1}^{N}(\log^{+}|4|_{\omega}+\log^{+}|\beta_{h}^{(j)}|_{\omega})\deg_{X_{j}}\\!\\!\delta
d\mu(\omega)\,$ (17)

to obtain

$\sum^{n}_{h=1}\operatorname{Ind}_{\alpha_{h},d}(\delta)\int_{\hat{S}_{h}}\min\left\\{d_{1}\log\frac{1}{|\beta^{(1)}-\alpha_{h}^{(1)}|_{\omega}},\ldots,d_{N}\log\frac{1}{|\beta^{(N)}-\alpha_{h}^{(N)}|_{\omega}}\right\\}d\mu(\omega)\leq-\int_{\hat{S}}\log|\delta(\beta)|_{\omega}d\mu(\omega)\\\
+\int_{\hat{S}}h_{\omega}(\delta)d\mu(\omega)+\tau+\int_{\hat{S}}\sum_{j=1}^{N}(\log^{+}|4|_{\omega}+\log^{+}|\beta_{h}^{(j)}|_{\omega})\deg_{X_{j}}\\!\\!\delta
d\mu(\omega)\,.$ (18)

Second case: $\omega\not\in\hat{S}$. Consider the expression:

$\delta(\beta)=\sum_{\mathbf{i}}\Delta^{\mathbf{i}}\delta(0)\beta^{(1)_{i_{1}}}\ldots\beta^{(N)_{i_{N}}}\quad\textrm{where
}{\mathbf{i}}=(i_{1},\ldots,i_{n})$

then, similarly to equation (15) we have the bound:

$\log|\delta(\beta)|_{\omega}\leq
h_{\omega}(\delta)+\sum_{j=1}^{N}(\log^{+}|\beta^{(j)}|_{\omega}\deg_{X_{j}}\\!\\!\delta)+N\log
d$

Hence

$\int_{\Omega\setminus\hat{S}}\log|\delta(\beta)|_{\omega}d\mu(\omega)\leq\int_{\Omega\setminus\hat{S}}h_{\omega}(\delta)d\mu(\omega)+\sum_{j=1}^{N}\int_{\Omega\setminus\hat{S}}(\log^{+}|\beta^{(j)}|_{\omega}\deg_{X_{j}}\\!\\!\delta\,)d\mu(\omega)+N\log
d$ (19)

Since $(\mathbb{K},\Omega,\phi)$ is proper

$-\int_{\hat{S}}\log|\delta(\beta)|_{\omega}d\mu(\omega)=\int_{\Omega\setminus\hat{S}}\log|\delta(\beta)|_{\omega}d\mu(\omega)\,,$
(20)

The distinction of the two cases is now finished, so by using equation (20)
and estimate (19) inside (18) we get

$\sum^{n}_{h=1}\operatorname{Ind}_{\alpha_{h},d}(\delta)\int_{\hat{S}_{h}}\min\left\\{d_{1}\log\frac{1}{|\beta^{(1)}-\alpha_{h}^{(1)}|_{\omega}},\ldots,d_{N}\log\frac{1}{|\beta^{(N)}-\alpha_{h}^{(N)}|_{\omega}}\right\\}d\mu(\omega)\leq
N\log d+\tau+\\\
+\int_{\Omega}h_{\omega}(\delta)d\mu(\omega)+\sum_{j=1}^{N}\int_{\Omega\setminus\hat{S}}(\log^{+}|\beta^{(j)}|_{\omega}\deg_{X_{j}}\\!\\!\delta\,)d\mu(\omega)+\int_{\hat{S}}\sum_{j=1}^{N}(\log^{+}|4|_{\omega}+\log^{+}|\beta_{h}^{(j)}|_{\omega})\deg_{X_{j}}\\!\\!\delta
d\mu(\omega)$ (21)

Since we can always assume that $\mu$ is adequately normalised, we have
$\int_{\Omega}\log^{+}|4|_{\omega}d\mu(\omega)=h(4)\leq\log 4$, therefore:

$\sum_{j=1}^{N}\int_{\Omega\setminus\hat{S}}(\log^{+}|\beta^{(j)}|_{\omega}\deg_{X_{j}}\\!\\!\delta\,)d\mu(\omega)+\int_{\hat{S}}\sum_{j=1}^{N}(\log^{+}|4|_{\omega}+\log^{+}|\beta_{h}^{(j)}|_{\omega})\deg_{X_{j}}\\!\\!\delta
d\mu(\omega)\leq\\\ \leq\sum_{j=1}^{N}\left(h(\beta^{(j)})+\log
4\right)\deg_{X_{j}}\\!\\!\delta$

By plugging everything inside equation (21) we get

$\sum^{n}_{h=1}\operatorname{Ind}_{\alpha_{h},d}(\delta)\int_{\hat{S}_{h}}\min\left\\{d_{1}\log\frac{1}{|\beta^{(1)}-\alpha_{h}^{(1)}|_{\omega}},\ldots,d_{N}\log\frac{1}{|\beta^{(N)}-\alpha_{h}^{(N)}|_{\omega}}\right\\}d\mu(\omega)\leq
N\log d+\tau+\\\ +h(\delta)+\sum_{j=1}^{N}\left(h(\beta^{(j)})+\log
4\right)\deg_{X_{j}}\\!\\!\delta$ (22)

By the $\lambda$-bounding hypothesis for $T(\beta)$ we can write

$-\frac{D}{{\lambda(\alpha^{(j)},\beta^{(j)})}}\log|\beta^{(j)}-\alpha_{h}^{(j)}|_{\omega}>D\theta(\omega)\,.$

Then we use Proposition 2.3$(3)$, so we can conclude

$D\sum_{h=1}^{n}dV(s)\left(t_{h}-s-\frac{2N^{2}\eta}{V(s)}\right)\int_{\hat{S}_{h}}\theta
d\mu\leq\\\
\leq\sum^{n}_{h=1}\operatorname{Ind}_{\alpha_{h},d}(\delta)\int_{\hat{S}_{h}}\min\left\\{d_{1}\log\frac{1}{|\beta^{(1)}-\alpha_{h}^{(1)}|_{\omega}},\ldots,d_{N}\log\frac{1}{|\beta^{(N)}-\alpha_{h}^{(N)}|_{\omega}}\right\\}d\mu(\omega)$
(23)

Now we are going to use again Proposition 2.3 to find upper bounds for the
terms on the right hand side of equation (22). By Proposition 2.3$(2)$:

$\sum_{j=1}^{N}\left(h(\beta^{(j)})+\log
4\right)\deg_{X_{j}}\\!\\!\delta\leq\sum_{j=1}^{N}\left(h(\beta^{(j)})+\log
4\right)(dd_{j}V(s)+O(d))\,.$ (24)

Proposition 2.3$(4)$ says that,

$h(\delta)\leq d\sum_{j=1}^{N}d_{j}\left(\log
2+\sum_{h=1}^{N}V(t_{h})h(\alpha^{(j)}_{h})\right)+O(d\log d)$ (25)

So now (22) can be written in the following way:

$D\sum_{h=1}^{n}dV(s)\left(t_{h}-s-\frac{2N^{2}\eta}{V(s)}\right)\int_{\hat{S}_{h}}\theta
d\mu\leq N\log d+\tau+d\sum_{j=1}^{N}d_{j}\left(\log
2+\sum_{h=1}^{N}V(t_{h})h(\alpha^{(j)}_{h})\right)+O(d\log d)+\\\
+\sum_{j=1}^{N}\left(h(\beta^{(j)})+\log 4\right)(dd_{j}V(s)+O(d))$ (26)

By inequality (9) of Proposition 2.3 we have: $V(s)\log 4+\log 2\leq\log 4$.
Then the term

$d\sum_{j=1}^{N}d_{j}\left(\log
2+\sum_{h=1}^{N}V(t_{h})h(\alpha^{(j)}_{h})\right)+O(d{\rm{log}}d)+\sum_{j=1}^{N}\left(h(\beta^{(j)})+\log
4\right)(dd_{j}V(s)+O(d))$

is bounded by

$d\sum_{j=1}^{N}d_{j}\left(V(s)h(\beta^{(j)})+\log
4+\sum_{h=1}^{N}V(t_{h})h(\alpha^{(j)}_{h})\right)+O(d\log d)$

But by the definition of the function $\lambda$ it holds that

$dV(s)\sum_{j=1}^{N}d_{j}\left(h(\beta^{(j)})+\frac{\log
4}{V(s)}+\sum_{h=1}^{N}\frac{V(t_{h})}{V(s)}h(\alpha^{(j)}_{h})\right)=dV(s)ND\,.$
(27)

Therefore equation (26) becomes

$dDV(s)\sum_{h=1}^{n}\left(t_{h}-s-\frac{2N^{2}\eta}{V(s)}\right)\int_{\hat{S}_{h}}\theta
d\mu\leq dV(s)ND+O(d{\rm{log}}d)$

since the terms $\tau$, $O(d)$, and $N\log d$ can be put together inside
$O(d\log d)$. Finally we divide both sides for $dDV(s)$ and we take the limit
for $D\to+\infty$ (i.e. $d\to+\infty$) to conclude the proof. ∎

######  Theorem 3.10.

Let $(\mathbb{K},\Omega,\phi)$ be a proper adelic curve. Assume that the
matrix $T(\beta)\in M(n\times N,\mathbb{K})$ satisfies the h-gap property.
Moreover assume $N>\max\left\\{\frac{36}{\log 2n},9\log 2n\right\\}$. If
$\theta\colon S\to\mathbb{R}$ is a column-bounding function for $T(\beta)$,
then

$\int_{\hat{S}}\theta d\mu<2+\frac{6\sqrt{\log 2n}}{\sqrt{N}}$

###### Proof.

The proof is based on the exact calculation of the volume $V(s)$ of [Sch91,
Chap. 2, Lemma 5A, 5B]. It shows that for $N\geq 2$,
$s,\rho\in\mathbb{R}_{>0}$ with $0<s\leq 1$ and $0<\rho<\frac{N}{2}$, then

$V(s)=\frac{s^{N}}{N!}$ (28)

So, in particular

$V\left(\frac{N}{2}-\rho\right)=\frac{(\frac{N}{2}-\rho)^{N}}{N!}<e^{\frac{-\rho^{2}}{N}}$
(29)

We take $t_{1}=t_{2}=\ldots t_{n}=\frac{N}{2}-\rho$, $\eta=\frac{1}{2N^{2}N!}$
where $\rho$ is a number such that:

$V\left(\frac{N}{2}-\rho\right)=\frac{1-\eta N^{2}}{n}$

Note that $\frac{1-\eta N^{2}}{n}>\frac{1}{2n}$. Now from equation (29) we get

$\frac{-\rho^{2}}{N}>\log\left(\frac{1-\eta N^{2}}{n}\right)$

which implies easily

$\rho<\sqrt{N\log 2n}\,.$

Now let’s take $s$ as a solution of:

$(1+\eta)^{N}-1+\eta N^{2}<V(s)<2N\eta+\eta N^{2}$ (30)

Here recall that $1+\eta N^{2}=nV(t_{h})=\sum^{n}_{h=1}V(t_{h})$ (compare with
the condition (9)). Note that as a consequence we have $0<s<1$. From equation
(30) we get the following two conditions

$N!<\frac{1}{V(s)}<2N!$ $\frac{N!}{2n}<\frac{V(t_{h})}{V(s)}<\frac{2N!}{n}$

By using such conditions it is easy to see that since $T(\beta)$ satisfies
$h$-gap condition, then it satisfies the $\lambda$-gap property of width
$\eta$. Moreover, since $\theta$ is a column-bounding function for $T(\beta)$,
then it is $\lambda$-bounding for $T(\beta)$. It means that we can apply
Proposition 3.9 to get:

$\int_{\hat{S}_{h}}\theta d\mu<\frac{N}{n\left(\frac{N}{2}-\rho-s-\frac{2\eta
N^{2}}{V(s)}\right)}$

which means

$\int_{\hat{S}}\theta d\mu<\frac{N}{\left(\frac{N}{2}-\rho-s-\frac{2\eta
N^{2}}{V(s)}\right)}$ (31)

Since $s<1$, $\rho<\sqrt{N\log 2n}$ and $N^{2}\eta<V(s)$, we obtain

$\frac{N}{\frac{N}{2}-\rho-s-\frac{2\eta
N^{2}}{V(s)}}=\frac{2}{1-\frac{2(\rho+s)}{N}-\frac{4\eta
N}{V(s)}}=\frac{2}{1-\frac{2\sqrt{N\log 2n}+2+4}{N}}$

Now since $N>\frac{36}{\log 2n}$, i.e. $\sqrt{N\log 2n}>6$, we can continue
with

$\frac{2}{1-\frac{2\sqrt{N\log 2n}+2+4}{N}}<\frac{2}{1-\frac{3\sqrt{N\log
2n}}{N}}=\frac{2}{1-\frac{3\sqrt{\log 2n}}{\sqrt{N}}}=$
$=\frac{2\sqrt{N}}{\sqrt{N}-3\sqrt{\log 2n}}=\frac{2N+6\sqrt{N\log
2n}}{N-9\log 2n}=\frac{2+\frac{6\sqrt{\log 2n}}{\sqrt{N}}}{1-\frac{9\log
2n}{N}}$

At this point the assumption $N>9\log 2n$ ensures that the last term is
bounded by $2+\frac{6\sqrt{\log 2n}}{\sqrt{N}}$ as we wanted. ∎

##  4 Proof of Theorem A

######  Definition 4.1.

We say that a proper adelic curve $\mathbb{X}=(\mathbb{K},\Omega,\mu)$
satisfies the _$\mu$ -equicontinity condition_ if for any measurable set
$S\subset\Omega$ of finite measure and any real numbers $\varepsilon,\delta>0$
there exists a finite measurable cover $C_{1},\ldots,C_{m}$ of $S$ satisfying
the following conditions. For all $\beta\in\mathbb{K}^{\times}$ there exists a
measurable set $U\subset\Omega$ such that $\mu(U)<\delta$ and

$\left|-\log^{-}|\beta|_{\omega}+\log^{-}|\beta|_{\omega^{\prime}}\right|<\varepsilon
h(\beta)\,,\quad\forall\omega,\omega^{\prime}\in C_{j}\setminus U\,,\;\forall
j=1,\ldots m$

Note that the definition 4.1 resembles a condition of equicontinuity for the
family of maps $\omega\mapsto-\log^{-}|\beta|_{\omega}$ on $S\setminus U$ with
$S$ compact and $U$ arbitrary small. We stress that in [Voj21, Proposition
8.9] it is shown that the adelic curves arising from arithmetic function
fields satisfy the $\mu$-equicontinuity condition.

The following lemma will be crucial in the proof of Theorem A. It is a special
case of the deep result [Voj21, Lemma 8.10]. This can be seen as a
generalisation of the Arzelà-Ascoli theorem for measure spaces, with the
advantage that one doesn’t need to provide a uniform bound for the involved
family of functions.

######  Lemma 4.2.

Let $\mathbb{X}=(\mathbb{K},\Omega,\phi)$ a proper adelic curve satisfying the
$\mu$-equicontinuity condition. Fix $\alpha\in\mathbb{K}^{\times}$ and a
measurable subset $S\subset\Omega$ of finite measure. Let $\\{\beta_{k}\\}$ be
a sequence in $\mathbb{K}^{\times}$ such that $\beta_{k}\neq\alpha$, and
$h(\beta_{k})\to+\infty$. Then for any $\varepsilon>0$ and any $\delta>0$
there exists a subsequence $\left\\{\beta_{k_{j}}\right\\}$ together with
measurable sets $U_{j}\subset\Omega$ satisfying $\mu(U_{j})<\delta$ and such
that for any $j,j^{\prime}\in\mathbb{N}$ big enough the following holds:

$\left|-\frac{\log^{-}|\alpha-\beta_{k_{j}}|_{\omega}}{h(\beta_{k_{j}})}+\frac{\log^{-}|\alpha-\beta_{k_{j^{\prime}}}|_{\omega}}{h(\beta_{k_{j^{\prime}}})}\right|<10\varepsilon\,,\quad\forall\omega\in
S\setminus(U_{j}\cup U_{j^{\prime}})$

###### Proof.

See [Voj21, Lemma 8.10] with the following setting: $\Xi=\\{\beta_{k}\\}$,
$q=1$, $\lambda_{\beta_{k},1}=-\frac{1}{2}\log^{-}|\alpha-\beta_{k}|$,
$c_{9}=h(\alpha)+\log 2$ and $\epsilon_{10}=\varepsilon$. Notice that
hypothesis (i) of [Voj21, Lemma 8.10] in our case holds automatically true
thanks to Proposition 1.12(6). Whereas hypothesis (ii) of [Voj21, Lemma 8.10]
is ensured by our Definition 4.1. ∎

Remember that with $\overline{\mathbb{K}}_{\omega}$ we denote the algebraic
closure the completion $\mathbb{K}_{\omega}$ (with respect to the absolute
value $|\cdot|_{\omega}$). In this case $|\cdot|_{\omega}$ extends uniquely to
an absolute value of $\overline{\mathbb{K}}_{\omega}$ and by an abuse of
notation we will continue to denote this extension with the symbol
$|\cdot|_{\omega}$. Theorem A will be equivalent to the following theorem:

######  Theorem 4.3.

Let $\mathbb{X}=(\mathbb{K},\Omega,\phi)$ a proper adelic curve satisfying the
$\mu$-equicontinuity condition. Fix some distinct elements
$\alpha_{1},\dots,\alpha_{n}\in\mathbb{K}$. Let $S=S_{1}\sqcup
S_{2}\sqcup\ldots\sqcup S_{n}$ be a measurable subset of $\Omega$ with finite
measure. Then for any $\varepsilon>0$ there exists a real constant $C>0$
(depending on the fixed data) such that for any $\beta\in\mathbb{K}$ with
$h_{\mathbb{X}}(\beta)>C$ it holds that:

$\sum^{n}_{i=1}\int_{S_{i}}\log^{-}|\beta-\alpha_{i}|_{\omega}d\mu(\omega)>-(2+\varepsilon)h_{\mathbb{X}}(\beta)$

Before proving this theorem, let’s show its equivalence with Theorem A.

######  Proposition 4.4.

Theorem 4.3 is equivalent to Theorem A.

###### Proof.

One direction is obvious, so we only have to show that Theorem 4.3 implies
Theorem A. First of all we can assume that the set
$\\{\alpha_{1},\ldots,\alpha_{n}\\}$ is
$\operatorname{Gal}({\overline{\mathbb{K}}/\mathbb{K}})$ invariant, since
adding all conjugates to the set of $\alpha_{i}^{\prime}s$ actually provides a
stronger Roth inequality. Moreover put
$\mathbb{K}^{\prime}=\mathbb{K}(\alpha_{1},\ldots\alpha_{n})$ and construct
the adelic curve
$\mathbb{X}^{\prime}=(\mathbb{K}^{\prime},\Omega^{\prime},\mu^{\prime})$, but
note that $\mathbb{K}^{\prime}$ is now a finite Galois extension of
$\mathbb{K}$. For any index $i$, the embeddings $\iota_{\omega,i}$ can be then
glued together to an embedding
$\iota_{\omega}\colon\mathbb{K}^{\prime}\hookrightarrow\overline{\mathbb{K}}_{\omega}$
so that:

$\sum^{n}_{i=1}\log^{-}|\iota_{\omega,i}(\beta-\alpha_{i})|_{\omega}=\sum^{n}_{i=1}\log^{-}|\iota_{\omega}(\beta-\alpha_{i})|_{\omega}$

Now fix $i_{0}\in\\{1,\ldots,n\\}$let
$S^{\prime}_{i_{0}}:=\pi^{-1}_{\mathbb{K}^{\prime}|\mathbb{K}}(S_{i_{0}})$ and
assume that Theorem 4.3 holds for the adelic curve $\mathbb{X}^{\prime}$ and
the set $S^{\prime}$, then the following chain of equalities show that Theorem
A holds true. Notice that at a certain point we will use equation (7) and the
fact that, since $\mathbb{K}^{\prime}|\mathbb{K}$ is finite and Galois, we
have:
$N_{\mathbb{K}^{\prime}|\mathbb{K}}(\beta-\alpha_{i_{0}})=\prod^{n}_{i=1}(\beta-\alpha_{i})$:

$\int_{S^{\prime}_{i_{0}}}\log^{-}|\beta-\alpha_{i_{0}}|_{\nu}d\mu^{\prime}(\nu)=^{\textrm{eq.}(\ref{extproper})}\frac{1}{[\mathbb{K}^{\prime}:\mathbb{K}]}\int_{S_{i_{0}}}\log^{-}|N_{\mathbb{K}^{\prime}|\mathbb{K}}(\beta-\alpha_{i_{0}})|_{\omega}d\mu(\omega)=$
$=\frac{1}{[\mathbb{K}^{\prime}:\mathbb{K}]}\int_{S_{i_{0}}}\log^{-}\left|\prod^{n}_{i=1}(\beta-\alpha_{i})\right|_{\omega}d\mu(\omega)=\frac{1}{[\mathbb{K}^{\prime}:\mathbb{K}]}\int_{S_{i_{0}}}\left(\sum^{n}_{i=1}\log^{-}|(\iota_{\omega}(\beta-\alpha_{i})|_{\omega}\right)d\mu(\omega)\,.$

∎

Proof of theorem 4.3. When $n=1$, then Proposition 1.12$(6)$ says that

$\int_{S}\log^{-}|\beta-\alpha|_{\omega}d\mu(\omega)\geq-\log
4-2h(\beta)-2h(\alpha)\,,$

hence any $\beta$ such that $h(\beta)>\frac{\log 4+2h(\alpha)}{\varepsilon}$
satisfies Roth’s inequality.

Assume $n\geq 2$. We consider the following matrix of dimension $n\times N$,
where $N>\max\left\\{\frac{36}{\log 2n},9\log 2n\right\\}$ will be a big
enough fixed integer:

$T=\begin{pmatrix}\alpha_{1}&\alpha_{1}&\alpha_{1}&\ldots&\alpha_{1}\\\
\alpha_{2}&\alpha_{2}&\alpha_{2}&\ldots&\alpha_{2}\\\
\vdots&\vdots&\vdots&\vdots&\vdots\\\ \vdots&\vdots&\vdots&\vdots&\vdots\\\
\alpha_{n}&\alpha_{n}&\alpha_{n}&\ldots&\alpha_{n}\\\ \end{pmatrix}$

notice that we have repeated $N$ times the same column vector. Assume by
contradiction that the theorem is false. Namely that there exists an
$\varepsilon_{0}>0$ such that the inequality

$\sum^{n}_{i=1}\int_{S_{i}}\log^{-}|\beta_{k}-\alpha_{i}|_{\omega}d\mu(\omega)\leq-(2+\varepsilon_{0})h_{\mathbb{X}}(\beta_{k})$
(32)

is satisfied by a sequence $\\{\beta_{k}\\}_{k\in\mathbb{N}}$ in $\mathbb{K}$
with the properties that $h(\beta_{k})\to+\infty$. Pick a constant:

$L>\log\left(4^{2N!}\prod^{n}_{i=1}H(\alpha_{i})^{\frac{2N!}{n}}\right)$

with $N$ big enough such that $\varepsilon_{0}>\frac{7\sqrt{\log
2n}}{\sqrt{N}}$. Since $h(\beta_{k})\to+\infty$, by eventually passing to a
subsequence, we can assume that $\\{h(\beta_{k})\\}$ is increasing and bounded
from below by a very big value. Therefore we can assume

$\varepsilon_{0}>\frac{2L}{h(\beta_{k})}+\frac{L+h(\beta_{k})}{h(\beta_{k})}\frac{7\sqrt{\log
2n}}{\sqrt{N}}\,,\quad\forall k\in\mathbb{N}$ (33)

Choose any $N$ elements from the sequence $\\{\beta_{k}\\}$, call them
$\beta^{(1)},\ldots\beta^{(N)}$ and consider the following matrix $T(\beta$):

$T(\beta)=\begin{pmatrix}\alpha_{1}&\alpha_{1}&\alpha_{1}&\ldots&\alpha_{1}\\\
\alpha_{2}&\alpha_{2}&\alpha_{2}&\ldots&\alpha_{2}\\\
\vdots&\vdots&\vdots&\vdots&\vdots\\\ \vdots&\vdots&\vdots&\vdots&\vdots\\\
\alpha_{n}&\alpha_{n}&\alpha_{n}&\ldots&\alpha_{n}\\\
\beta^{(1)}&\beta^{(2)}&\beta^{(3)}&\ldots&\beta^{(N)}\end{pmatrix}$

Let $\hat{S}$ be the subset of $S$ associated to the above matrix $T(\beta)$
as in Definition 3.2.

We are ready to construct some functions $\theta_{k}\in L^{1}(S,\mu)$ which
will give the desired contradiction. We define them as piecewise functions by
putting for any $\omega\in\hat{S}_{i}$ and any $k\in\mathbb{N}$:

$\theta_{k}(\omega)=\frac{-\log^{-}|\beta_{k}-\alpha_{i}|_{\omega}}{L+h(\beta_{k})}$

We extend $\theta_{k}$ on the whole $S$ by setting
$\theta_{k}(S\setminus\hat{S})=0$. Now we can write

$\int_{\hat{S}}\theta_{k}d\mu=\int_{S}\theta_{k}d\mu=\sum^{n}_{i=1}\int_{S_{i}}\theta_{k}d\mu=\frac{-1}{L+h(\beta_{k})}\sum^{n}_{i=1}\int_{S_{i}}\log^{-}|\beta_{k}-\alpha_{i}|_{\omega}d\mu(\omega)\geq^{\\!\\!\textrm{eq.}(\ref{negationroth})}\frac{(2+\varepsilon_{0})h(\beta_{k})}{L+h(\beta_{k})}$
(34)

By plugging inequality (33) inside (34) and simplifying the expressions, we
finally get:

$\int_{S}\theta_{k}d\mu=\int_{\hat{S}}\theta_{k}d\mu>2+\frac{7\sqrt{\log
2n}}{\sqrt{N}}\qquad\forall k\in\mathbb{N}$ (35)

Thanks to Proposition 1.12(6):

$\int_{S_{i}}\theta_{k}<\frac{B_{i}+2h(\beta_{k})}{L+h(\beta_{k})}$

for $B_{i}\in\mathbb{R}_{>0}$. So, if $B=\max_{i}\\{B_{i}\\}$ we obtain:

$\int_{S}\theta_{k}d\mu\leq\frac{nB+2nh(\beta_{k})}{L+h(\beta_{k})}\leq 2n$

Hence, after possibly passing to a subsequence of $\\{\theta_{k}\\}$, we can
assume that $\int_{S}\theta_{k}d\mu$ admits limit and by inequality (35):

$\lim_{k\to\infty}\int_{\hat{S}}\theta_{k}d\mu\geq 2+\frac{7\sqrt{\log
2n}}{\sqrt{N}}>2+\frac{6\sqrt{\log 2n}}{\sqrt{N}}\,.$ (36)

Put by simplicity
$\lambda_{i,k}(\omega):=-\log^{-}|\beta_{k}-\alpha_{i}|_{\omega}$ so that

$\theta_{k}(\omega)=\frac{\lambda_{i,k}(\omega)}{L+h(\beta_{k})}\,.$

Then we have:

$\theta_{k}(\omega)=\frac{\lambda_{i,k}(\omega)}{h(\beta_{k})}A_{k}$

where $A_{k}:=1-\frac{L}{L+h(\beta_{k})}<1$ and $A_{k}\to 1$. We want to show
that from the sequence $\left\\{\frac{1}{A_{k}}\theta_{k}\right\\}$ we can
extract a subsequence uniformly convergent to a function $\theta$ on
$S\setminus U$ where $U$ is a measurable set of arbitrary small measure. We
apply Lemma 4.2 on $S_{i}$, for the fixed value $\alpha_{i}$, with
$0<\varepsilon<1$ and $\delta=\delta_{1}=\frac{\varepsilon}{2}$. We denote the
extracted subsequence by
$\mathcal{R}_{1}:=\left\\{\frac{1}{A_{j}}\theta_{j}\right\\}$. For $\omega\in
S_{i}$, Now pick two natural numbers $j>j^{\prime}>N_{1}$, for any $\omega\in
S_{i}\setminus U_{1}$ where $\mu(U_{1})\leq\varepsilon$:

$\left|\frac{1}{A_{j}}\theta_{j}(\omega)-\frac{1}{A_{j^{\prime}}}\theta_{j^{\prime}}(\omega)\right|=\left|\frac{\lambda_{i,j}(\omega)}{h(\beta_{j})}-\frac{\lambda_{i,j^{\prime}}(\omega)}{h(\beta_{j^{\prime}})}\right|<10\varepsilon\,.$

Now we use again Lemma 4.2 on the sequence $\mathcal{R}_{1}$ with
$\delta=\delta_{2}=\frac{\varepsilon}{4}$, to find a subsequence
$\mathcal{R}_{2}\subseteq\mathcal{R}_{1}$ such that again for
$j>j^{\prime}>N_{2}$ we get

$\left|\frac{1}{A_{j}}\theta_{j}(\omega)-\frac{1}{A_{j^{\prime}}}\theta_{j^{\prime}}(\omega)\right|<10\varepsilon$
(37)

but this time on $S_{i}\setminus U_{2}$ with
$\mu(U_{2})\leq\frac{\varepsilon}{2}$. We can iterate this process of
extracting subsequences thanks to Lemma 4.2 and we get a family of
subsequences
$\mathcal{R}_{1}\supseteq\mathcal{R}_{2}\supseteq\ldots\supseteq\mathcal{R}_{m}\supseteq\ldots$
such that equation (37) holds for elements in $\mathcal{R}_{m}$ for two
indices $j,j^{\prime}>N_{m}$ and outside from a set $U_{m}$ of measure
$\mu(U_{m})\leq\frac{\varepsilon}{2^{m-1}}$. At this point, by properly
selecting one element $\frac{1}{A_{m}}\theta_{m}$ for each
$\\{\mathcal{R}_{m}\\}$, we construct a sequence
$\left\\{\frac{1}{A_{m}}\theta_{m}\right\\}$, which is in fact a subsequence
of the initial $\left\\{\frac{1}{A_{k}}\theta_{k}\right\\}$, converging
uniformly to a function $\theta^{(i)}$ on a measurable set $S_{i}\setminus
U^{(i)}$ where $U^{(i)}=\cup_{m\geq 1}U_{m}$ and

$\mu(U^{(i)})\leq\sum^{\infty}_{m=1}\mu(U_{m})\leq 2\varepsilon\,.$

Since $S_{i}\setminus U^{(i)}$ has finite measure, by a basic argument of
measure theory we know that $\theta^{(i)}\in L^{1}(S_{i}\setminus
U^{(i)},\mu)$ and

$\int_{S_{i}\setminus
U^{(i)}}\theta^{(i)}d\mu=\lim_{m\to\infty}\int_{S_{i}\setminus
U^{(i)}}\frac{1}{A_{m}}\theta_{m}d\mu=\lim_{m\to\infty}\int_{S_{i}\setminus
U^{(i)}}\theta_{m}d\mu\,.$ (38)

Put $U=\cup^{n}_{i=1}U^{(i)}$ and note that

$\mu(U)<2n\varepsilon\,.$

We finally define $\theta\in L^{1}(S,\mu)$ in the following way: for any
$i=1,\ldots,n$ and any $\omega\in S_{i}\setminus U^{(i)}$ we set
$\theta(\omega):=\theta^{(i)}(\omega)$ and $\theta(U):=0$. Therefore we get:

$\int_{\hat{S}}\theta d\mu=\int_{S}\theta
d\mu=\sum^{n}_{i=1}\int_{S_{i}\setminus
U^{(}i)}\theta^{(i)}d\mu=\sum^{n}_{i=1}\lim_{m\to\infty}\int_{S_{i}\setminus
U^{(i)}}\theta_{m}d\mu=$
$=\sum^{n}_{i=1}\lim_{m\to\infty}\int_{S_{i}}\theta_{m}d\mu-\sum^{n}_{i=1}\lim_{m\to\infty}\int_{U^{(i)}}\theta_{m}d\mu=\lim_{m\to\infty}\int_{S}\theta_{m}d\mu-\lim_{m\to\infty}\int_{U}\theta_{m}d\mu$

where the rightmost term is finte. Now, thanks to equation (36) and the fact
that $\mu(U)$ can be taken arbitrarily small by the choice of $\varepsilon$,
we conclude that

$\int_{\hat{S}}\theta d\mu>2+\frac{6\sqrt{\log 2n}}{\sqrt{N}}\,.$ (39)

Now we show that since $h(\beta_{k})\to+\infty$, the elements
$\beta^{(1)},\ldots,\beta^{(N)}$ can be chosen, along the above subsequence
$\left\\{\beta_{m}\right\\}$, in a way that the following two conditions are
satisfied.

  * (i)

$T(\beta)$ satisfies the h-gap condition. This is obvious by choosing the
$\beta^{(j)}$ separated by big enough gaps.

  * (ii)

The function $\theta$ is column bounding for $T(\beta)$. In fact for
$\varepsilon>0$ small enough and for infinitely many big enough indexes $m$,
thanks to the choice of the constant $L$, we have the following inequalities
for any $i=1,\ldots,n$ and any $\omega\in\hat{S}_{i}$:

$\theta(\omega)<\theta_{m}(\omega)+\varepsilon<\frac{-\log^{-}|\alpha_{i}-\beta_{m}|_{\omega}}{h(\beta_{m})+\log\left(4^{2N!}\prod^{n}_{i=1}H(\alpha_{i})^{\frac{2N!}{n}}\right)}$

Therefore we can apply Theorem 3.10 to conclude that:

$\int_{\hat{S}}\theta d\mu<2+\frac{6\sqrt{\log 2n}}{\sqrt{N}}$

which contradicts inequality (39). $\square$

######  Example 4.5.

We give an example of a proper adelic curve that doesn’t satisfy the
$\mu$-equicontinuity condition of definition 4.1; for such adelic curve one
cannot apply our generalisation of Roth’s theorem. Consider
$\mathbb{X}=(\mathbb{Q},\Omega,\operatorname{id})$ naturally obtained from the
field $\mathbb{K}=\mathbb{Q}$ as in Example (1.4), we show that the adelic
curve
$\overline{\mathbb{X}}=(\overline{\mathbb{Q}},\overline{\Omega},\operatorname{id})$
doesn’t satisfy the $\mu$-equicontinuity condition. The measure on
$\overline{\Omega}$ is denoted by $\chi$, whereas the measure on $\Omega$ is
$\mu$. Consider $S=\overline{\Omega}_{\infty}$, and since
$\mu={(\pi_{\overline{\mathbb{Q}}|\mathbb{Q}})}_{\ast}\chi$ we have that
$\chi(S)=\mu(\Omega_{\infty})<+\infty$. Any element $\nu\in S$ is a complex
absolute value $|\sigma(\cdot)|$ where
$\sigma:\overline{\mathbb{Q}}\hookrightarrow\mathbb{C}$ is a field embedding
considered up to complex conjugation. One can be more precise and say that $S$
as measure space (with the restricted structure inherited by
$\overline{\Omega}$) can be identified with the following profinite group
endowed with a Haar measure:

$G=\frac{\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})}{\sim}=\varprojlim_{K}\frac{\operatorname{Gal}(K/\mathbb{Q})}{\sim}\,,$

where the limit is taken over all finite Galois extensions
$K\supset\mathbb{Q}$, and $\sim$ is the relation of complex conjugation. Take
any measurable cover $C_{1},\ldots C_{m}$ of $S$; by rearranging the terms we
can always suppose that the $C_{i}$’s are pairwise disjoint and that
$\mu(C_{i})>0$ for any $i=1,\ldots,m$. By definition of Haar measure, since
$\mu(C_{i})>0$ there exists a normal compact (so closed and of finite index)
subgroup $H_{i}\subseteq G$ such that $\mu(H_{i})>0$ and $g_{i}H_{i}\subseteq
C_{i}$ for an element $g_{i}\in G$. Consider $H=\cap^{m}_{i=1}H_{i}$, it has
positive measure and $g_{i}H\subset C_{i}$ for any $i=1,\ldots,m$. By the
fundamental theorem of Galois theory, there exists a finite Galois extension
$K\supset\mathbb{Q}$ such that
$H=\operatorname{Gal}(\overline{\mathbb{Q}}/K)$; moreover we can assume that
$1\in C_{1}$ so that $g_{1}=1$. Fix a prime $p$ such that $\sqrt{p}\not\in K$
and let $a,b\in\mathbb{Z}$ such that $a^{2}-pb^{2}=1$ and $a-b\sqrt{p}<1$. At
this point inside $H\subseteq C_{1}$ we consider the place $\omega^{\prime}$
corresponding to the identity automorphism and another place $\omega$
corresponding to an automorphism that exchanges $\sqrt{p}$ and $-\sqrt{p}$.
Therefore we have:

$\left|-\log^{-}|a+b\sqrt{p}|_{\omega}+\log^{-}|a+b\sqrt{p}|_{\omega^{\prime}}\right|=-\log(a-b\sqrt{p})$

This shows that the $\mu$-equicontinuity condition doesn’t hold.

##  5 Proof of Theorem B

######  Definition 5.1.

A proper adelic curve $\mathbb{X}=(\mathbb{K},\Omega,\mu)$ satisfies the
_uniform integrality property_ if for any $\beta\in\mathbb{K}^{\times}$ and
any $\varepsilon>0$ there exist $\delta>0$ such that if
$T\subset\Omega_{\infty}$ is a measurable subset satisfying $\mu(T)<\delta$,
then

$\int_{T}-\log^{-}|\beta|d\mu<\varepsilon h_{\mathbb{X}}(\beta)$

Vojta showed that arithmetic function fields satisfy the uniform integrality
property in [Voj21, Proposition 8.8].

######  Lemma 5.2.

Let $\mathbb{X}=(\mathbb{K},\Omega,\phi)$ a proper adelic curve satisfying the
$\mu$-equicontinuity condition, and the uniform integrability condition.
Assume that Theorem B doesn’t hold for certain $S$,
$\alpha_{1},\ldots,\alpha_{n}\in\mathbb{K}$ and
$\varepsilon_{0}\in\mathbb{R}_{>0}$. Let $N>0$ an integer,
$\varepsilon\in\,]0,\varepsilon_{0}[$, and $r_{0},r_{1}>1$ two real numbers.
Then there exist $\beta_{1},\ldots,\beta_{N}\in\mathbb{K}$ satisfying the
following conditions:

  * $(i)$

$h(\beta_{1})>r_{0}$

  * $(ii)$

$\frac{h(\beta_{k})}{h(\beta_{k-1})}>r_{1}$ for any $k=2,\ldots,N$.

  * $(iii)$

There is a partition $S=S_{1}\sqcup\ldots\sqcup S_{n}$ such that:

$\sum^{n}_{i=1}\int_{S_{i}}\,\min_{1\leq k\leq
N}\left(\frac{-\log^{-}|\beta_{k}-\alpha|_{\omega}}{h(\beta_{k})}\right)d\mu(\omega)>2+\varepsilon+\frac{1}{h(\beta_{1})}$

###### Proof.

The reader can follow line by line [Voj21, Proposition 8.12]. The proof uses
the previous [Voj21, Propositions 8.10, 8.11]. Note that in the statement of
[Voj21, Proposition 8.12] the condition $(i)$ is not explicitly mentioned, but
it can be easily deduced from the proof. ∎

The above lemma will be the main ingredient for the construction of the
desired contradiction in the following proof.

Proof of Theorem B. For $n=1$ the theorem follows trivially from the integral
Liouville inequality, i.e. Proposition 1.12$(6)$; so let’s assume that $n\geq
2$. Assume by contradiction that the theorem is false for some
$\varepsilon_{0}>0$. Hence we apply Lemma 5.2; for any $r_{0},r_{1}>1$ and
$\varepsilon<\varepsilon_{0}$ there exist
$\beta_{1},\ldots\beta_{N}\in\mathbb{K}$ such that:

  * $(i)$

$h(\beta_{1})>r_{0}$

  * $(ii)$

$\frac{h(\beta_{k})}{h(\beta_{k-1})}>r_{1}$ for any $k=2,\ldots,N$.

  * $(iii)$

There is a partition $S=S_{1}\sqcup\ldots\sqcup S_{n}$ such that:

$\sum^{n}_{i=1}\int_{S_{i}}\,\min_{1\leq k\leq
N}\left(\frac{-\log^{-}|\beta_{k}-\alpha|_{\omega}}{h(\beta_{k})}\right)d\mu(\omega)>2+\varepsilon+\frac{1}{h(\beta_{1})}$
(40)

Consider the matrix constructed with the vector
$\beta=(\beta_{1},\ldots,\beta_{N})$:

$T(\beta)=\begin{pmatrix}\alpha_{1}&\alpha_{1}&\alpha_{1}&\ldots&\alpha_{1}\\\
\alpha_{2}&\alpha_{2}&\alpha_{2}&\ldots&\alpha_{2}\\\
\vdots&\vdots&\vdots&\vdots&\vdots\\\ \vdots&\vdots&\vdots&\vdots&\vdots\\\
\alpha_{n}&\alpha_{n}&\alpha_{n}&\ldots&\alpha_{n}\\\
\beta_{1}&\beta_{2}&\beta_{3}&\ldots&\beta_{N}\end{pmatrix}$

And let $\hat{S}$ be the subset of $S$ associated to $T(\beta)$ as in
definition 3.2. Pick a constant:

$L>\log\left(4^{2N!}\prod^{n}_{i=1}H(\alpha_{i})^{\frac{2N!}{n}}\right)$

We are ready to construct a function $\theta\in L^{1}(S,\mu)$ which will give
the desired contradiction. We define it as a piecewise function by putting for
any $\omega\in S_{i}$:

$\theta|_{S_{i}}(\omega)=\min_{k}\frac{-\log^{-}|\beta_{k}-\alpha_{i}|_{\omega}}{L+h(\beta_{k})}=\min_{k}\frac{-\log^{-}|\beta_{k}-\alpha_{i}|_{\omega}}{h(\beta_{k})}A_{k}$

where $A_{k}=1-\frac{L}{L+h(\beta_{k})}$. We can choose
$h(\beta_{1})>>L>>N>>1$ in a way that we can assume:

$\varepsilon>\frac{2}{A_{1}}+\frac{6\sqrt{\log
2n}}{A_{1}\sqrt{N}}-2-\frac{1}{h(\beta_{1})}>0$ (41)

Clearly $\int_{S}\theta d\mu=\int_{\hat{S}}\theta d\mu$, so by using equations
(40) and $(\ref{theepss})$ we get:

$\int_{\hat{S}}\theta
d\mu=\sum^{n}_{i=1}\int_{S_{i}}\theta|_{S_{i}}d\mu>A_{1}\sum_{i=1}^{n}\int_{S_{i}}\min_{k}\frac{-\log^{-}|\beta_{k}-\alpha_{i}|_{\omega}}{h(\beta_{k})}>A_{1}\left(2+\varepsilon+\frac{1}{h(\beta_{1})}\right)>2+\frac{6\sqrt{\log
2n}}{\sqrt{N}}\,.$ (42)

Now notice that

  * $\bullet$

$T(\beta)$ satisfies the h-gap condition since we can choose the $\beta_{k}$
such that the heights $h(\beta_{k})$ are separated enough by $(ii)$.

  * $\bullet$

The function $\theta$ is column bounding for $T(\beta)$. Indeed for any
$k=1,\ldots,N$ we have that:

$\theta|_{S_{i}}(\omega)<\frac{-\log^{-}|\beta_{k}-\alpha_{i}|_{\omega}}{L+h(\beta_{k})}<\frac{-\log^{-}|\alpha_{i}-\beta_{k}|_{\omega}}{\log\left(4^{2N!}\prod^{n}_{i=1}H(\alpha_{i})^{\frac{2N!}{n}}\right)+h(\beta_{k})}$

Therefore we can apply theorem 31 to conclude that:

$\int_{\hat{S}}\theta d\mu<2+\frac{6\sqrt{\log 2n}}{\sqrt{N}}$

which contradicts inequality (42).

$\square$

Note that in the proof of Theorems A and B we didn’t assume Northcott property
for our adelic curve.

##  Appendix A Construction of the interpolating polynomial

In this appendix we will sketch the construction of the interpolating
polynomial $\delta$ of section 2. For all the deails the reader can check
[Cor97].

We employ the same notations of section 2 and for simplicity of notations $X$
denotes the vector of variables $(X_{1},\ldots X_{N})$. We are going to
construct a complicated matrix $A(X)$ depending on the following parameters:
$s,t_{1},\dots,t_{n}\in\mathbb{R}$, with $0<s<1$ and $0<t_{h}<\frac{N}{2}$ for
$h=1,\dots,n$. The columns of the matrix are indicized by
$\mathbf{a}\in\mathcal{G}_{N}$; the rows are indicized by
$\mathbf{i}_{h}\in\mathcal{G}_{t_{h}}$ (for any $h=1,\ldots,n$) and moreover
we put $\mathbf{i}_{n+1}\in\mathcal{G}_{s}$. The order on multi-indices is the
lexicographic one.

$A(X):=\begin{pmatrix}{{\mathbf{a}}\choose{\mathbf{i}}_{h}}\alpha_{h}^{{\mathbf{a}}-{{\mathbf{i}}_{h}}}\\\
\\\
{{\mathbf{a}}\choose{\mathbf{i}}_{n+1}}X^{{\mathbf{a}}-{\mathbf{i}}_{n+1}}\end{pmatrix}_{\mathbf{i}_{h},\mathbf{a}}\,\,$
(43)

Note that $A(X)$ has $\\#(\mathcal{G}_{s})+\sum_{h}\\#(\mathcal{G}_{t_{h}})$
rows and $\\#(\mathcal{G}_{N})=\prod^{n}_{h=1}\lfloor d_{h}+1\rfloor$ columns,
and moreover the last $\\#(\mathcal{G}_{s})$ rows are monomials. Clearly we
can always choose the paramenters $s,t_{1},\dots,t_{n}\in\mathbb{R}$ in order
to obtain a matrix $A(X)$ with more rows than columns. Let’s see an explicit
condition that tells us when this can be achieved: the number of rows is
greater than the number of columns if

$\\#(\mathcal{G}_{s})+\sum^{n}_{h=1}\\#(\mathcal{G}_{t_{h}})>\prod^{n}_{h=1}\lfloor
d_{h}+1\rfloor\,,$

therefore thanks to Lemma 2.2, for $d_{1},\ldots,d_{N}$ very big, it is enough
to have the following conditions on volumes:

$V(s)+\sum_{h=1}^{n}V(t_{h})>1.$ (44)

The volumes $V(s),V(t_{1}),\ldots V(t_{n})$ heavily determine the algebraic
properties of the matrix $A(X)$, in fact we will now present a stronger
condition on the quantity $V(s)+\sum_{h=1}^{n}V(t_{h})$ ensuring that $A(X)$
has maximal rank.

######  Proposition A.1.

Let $d_{1},\ldots,d_{N}$ be big enough and let $d_{1}>d_{2}>\ldots>d_{N}$.
Moreover assume that

$\alpha^{(j)}_{h}\neq\alpha^{(j)}_{k},\,\,\forall\,j=1,\dots
N,\,\,\,\forall\,h,k=1,\dots,n,\,\,h\neq k\,.$

If for $s,t_{1},\dots,t_{n}\in\mathbb{R}$, with $0<s<1$ and
$0<t_{h}<\frac{N}{2}$ we have that

$V(s)+\sum_{h=1}^{n}V(t_{h})>\prod_{j=1}^{N-1}\left(1+(n-1)\sum_{i=j+1}^{N}\frac{d_{j}}{d_{j+1}}\right)$
(45)

then, for any
$\beta=(\beta^{(1)},\beta^{(2)},\dots,\beta^{(N)})\in\mathbb{K}^{N}$ such that

$\beta^{(j)}\neq\alpha^{(j)}_{h}\,,\,\,\forall\,j=1,\dots
N,\,\,\,\forall\,h=1,\dots,n,\,,$

the rank of $A(\beta)$ is maximal and equal to the number of columns.

###### Proof.

See [Cor97, Proposition 2.1] and notice that it uses a version of Dyson’s
lemma for polynomials in many variables proved in [EV84]. ∎

One can always assume that the parameter $s,t_{1},\ldots,t_{n}$ are chosen in
a way that we always get

$(1+\eta)^{N}<V(s)+\sum_{h=1}^{n}V(t_{h})<1+2N\eta$ (46)

It is not difficult to see (check [Cor97, page 159]) that equation (46)
implies (45). Therefore thanks to Proposition A.1 we can extract from $A(X)$ a
square submatrix $M(X)$ of dimension

$r:=\\#(\mathcal{G}_{N})=\prod^{n}_{h=1}\lfloor d_{h}+1\rfloor\,,$

which is the number of columns of $A(X)$, such that $M(X)$ has maximal rank
for any $\beta$ componentwise different from any $\alpha_{h}$. Moreover one
can choose $M(X)$ in a way that it is contained in the last
$\\#(\mathcal{G}_{s})$ rows of $A(X)$, since they are linearly independent for
any choice of $\beta\in\mathbb{R}^{N}$. The polynomial $\delta(X)$ is the
determinant of the matrix $M(X)$. At this point all the arguments used in
[Cor97] to prove the bounds about $\delta$ can be applied verbatim in our
setting with the only difference that in order to estimate
$h_{\mathbb{X}}(\delta)$ one has to take the integral over $\Omega$ instead of
the summation over all places.

## References

  * [BG06] E. Bombieri and W. Gubler.  Heights in diophantine geometry.  Cambridge University Press, Cambridge New York, 2006.
  * [BvdP88] E. Bombieri and A. J. van der Poorten.  Some quantitative results related to Roth’s theorem.  J. Austral. Math. Soc. Ser. A, 45(2):233–248, 1988.
  * [BvdP90] E. Bombieri and A. J. van der Poorten.  Corrigenda: “Some quantitative results related to Roth’s theorem”.  J. Austral. Math. Soc. Ser. A, 48(1):154–155, 1990.
  * [CM20] H. Chen and A. Moriwaki.  Arakelov geometry over Adelic curves.  Springer, Singapore, 2020.
  * [Cor95] P. Corvaja.  An explicit version of the theorem of Roth-Ridout.  Rend. Sem. Mat. Univ. Politec. Torino, 53(3):253–260, 1995.
  * [Cor97] P. Corvaja.  Autour du théorème de Roth.  Monatsh. Math., 124(2):147–175, 1997.
  * [EV84] H. Esnault and E. Viehweg.  Dyson’s lemma for polynomials in several variables (and the theorem of Roth).  Invent. Math., 78(3):445–490, 1984.
  * [Eve97] J.-H. Evertse.  The number of algebraic numbers of given degree approximating a given algebraic number.  In Analytic number theory (Kyoto, 1996), volume 247 of London Math. Soc. Lecture Note Ser., pages 53–83. Cambridge Univ. Press, Cambridge, 1997.
  * [Gub97] W. Gubler.  Heights of subvarieties over $M$-fields.  In Arithmetic geometry (Cortona, 1994), Sympos. Math., XXXVII, pages 190–227. Cambridge Univ. Press, Cambridge, 1997.
  * [Lan83] S. Lang.  Fundamentals of diophantine geometry.  Springer-Verlag, New York, 1983.
  * [Mor00] A. Moriwaki.  Arithmetic height functions over finitely generated fields.  Invent. Math., 140(1):101–142, 2000.
  * [Neu99] J. Neukirch.  Algebraic number theory.  Springer, Berlin New York, 1999.
  * [Rot55] K. F. Roth.  Rational approximations to algebraic numbers.  Matematika, 2(3), 1955.
  * [Sch91] W. Schmidt.  Diophantine approximations and diophantine equations.  Springer-Verlag, Berlin New York, 1991.
  * [Voj21] P. Vojta.  Roth’s Theorem over arithmetic function fields.  Algebra $\&$ Number Theory, 15(8):1943–2017, 2021.