# KAM, Lyapunov exponents, and the Spectral Dichotomy for typical one-
frequency Schrödinger operators

Artur Avila  Institut für Mathematik, Universität Zürich, Winterthurerstrasse
190, CH-8057 Zürich, Switzerland and IMPA, Estrada Dona Castorina, 110, Rio de
Janeiro, 22460-320, Brazil<EMAIL_ADDRESS>
###### Abstract.

We show that a one-frequency analytic $\mathrm{SL}(2,\mathbb{R})$ cocycle with
Diophantine rotation vector is analytically linearizable if and only if the
Lyapunov exponent is zero through a complex neighborhood of the circle. More
generally, we show (without any arithmetic assumptions) that regularity
implies almost reducibility, i.e., the range of validity of the perturbative
analysis near constants is specified by a condition on the Lyapunov exponents.
Together with our previous work, this establishes a Spectral Dichotomy for
typical one-frequency Schrödinger operators: they can be written as a direct
sum of large-like and small-like operators. In particular, the typical
operator has no singular continuous spectrum.

This work was partially conducted during the period the author served as a
Clay Research Fellow. It was supported by an ERC Starting Grant and a grant
from the SNSF

##  1\. Introduction

This paper has two main purposes. From the dynamical side we will show that
for analytic $\mathrm{SL}(2,\mathbb{R})$ cocycles over a translation of
$\mathbb{R}/\mathbb{Z}$, the “domain of applicability of KAM techniques” is
given precisely by a condition on the Lyapunov exponent of complexifications,
which closes a chapter opened in [H2]. From the spectral side, it provides the
last step in our program to prove the Spectral Dichotomy for typical one-
frequency Schrödinger operators.

###  1.1. “…et quelques examples montrant le caractère local d’un théorème
d’Arnold et de Moser sur le tore de dimension deux”

This is the second half of the title of Michael Herman’s celebrated paper
[H2], where he introduced his famous “subharmonic method” for minoration of
the Lyapunov exponents of cocycles, which he then uses to make explicit the
“lack of globality” of KAM theory on the two-dimensional torus (in striking
contrast with the one-dimensional situation by the remarkable Herman-Yoccoz
rigidity theory [H1], [Y]).

The main class of “counterexamples to global KAM” in the two-torus discussed
in [H2] is given by one-frequency $\mathrm{SL}(2,\mathbb{R})$ cocycles
homotopic to a constant. Those have the form
$(\alpha,A):(x,y)\mapsto(x+\alpha,A(x)\cdot y)$, where
$\alpha\in\mathbb{R}\smallsetminus\mathbb{Q}$ and $A(x)$ is a continuous
$\mathrm{SL}(2,\mathbb{R})$ function of $x$, understood to act projectively in
the second coordinate. In the case

(1) $A(x)=A^{(E-\lambda v)}(x)=\left(\begin{matrix}E-\lambda v(x)&-1\\\
1&0\end{matrix}\right),$

where $\lambda>0$ is some coupling constant and $v$ is a trigonometric
polynomial, the argument of Herman can be summarized as follows. For general
reasons, $(\alpha,A)$ has always a well defined rotation vector
$(\alpha,\rho)$ with $\rho\in[0,1]$, and all values of $\rho$ can be obtained
as $E$ changes. In particular, if $\alpha\in{\mathrm{DC}}$, i.e., it satisfies
a Diophantine condition, then one can always choose $E$ such that the vector
$(\alpha,\rho)$ satisfies some fixed Diophantine condition. Thus if $\lambda$
is suitably small, one can apply KAM theory (this idea in fact dates back to
Dinaburg-Sinai [DS]) to show that the torus dynamics is in fact quasiperiodic.
On the other hand, he showed that if $\lambda$ is large then the cocycle
defined by $(\alpha,A)$ has a positive Lyapunov exponent

(2) $L=\lim\frac{1}{n}\int_{\mathbb{R}/\mathbb{Z}}\ln\|A(x+(n-1)\alpha)\cdots
A(x)\|dx>0,$

which implies that the dynamics on the two-torus has exactly two ergodic
invariant measures with one zero Lyapunov exponent corresponding to the basis,
and one non-zero Lyapunov exponent corresponding to the fiber, which is either
$2L$ or $-2L$ according to the measure. In particular the dynamics is not
quasiperiodic, even though the rotation vector can be chosen to satisfy a
Diophantine condition.

Around 1990, two key developments complemented Herman’s theory. On the large
$\lambda$ side, Sorets-Spencer [SS] showed that a minoration of the Lyapunov
exponent can be proved for real analytic functions (and not only trigonometric
polynomials). On the small $\lambda$ side, Eliasson [E] proved that a non-
standard KAM algorithm allows to “control the dynamics” for all values of $E$,
introducing the concept of almost reducibility in this context. More recently,
non-KAM methods [AJ] led to the following result: for any fixed real analytic
$v$, if $\lambda$ is small then for every $\alpha\in{\mathrm{DC}}$ and any
$E\in\mathbb{R}$, the dynamics is almost reducible, that is, up to coordinate
changes, it can be made arbitrarily close to a product dynamics
$(x,y)\mapsto(x+\alpha,A_{*}\cdot y)$ (as an analytic dynamical system), that
is, $A_{*}$ does not depend on $x$.

Clearly the powerful notion of almost reducibility captures precisely what it
means to “belong to the domain of KAM techniques”. So the natural follow up to
the title of this section is: What are then the obstructions to almost
reducibility? As Herman pointed out, the behavior of the Lyapunov exponent is
part of the answer. We will show that it is in fact the whole answer.

Notice that it is not exactly the positivity of the Lyapunov exponent that is
to blame for the failure of almost reducibility: indeed a cocycle may very
well be conjugate to $(\alpha,A_{*})$ with $A_{*}$ a hyperbolic matrix. It is
only by looking at the complexification (the main idea introduced in [H2])
that one gets to a key feature of constant cocycles that must be shared by
almost reducible cocycles, called regularity:111The notion of regularity was
introduced in [A1] for $\mathrm{SL}(2,\mathbb{C})$ valued cocycles, but
coincides with the one given here when restricted to
$\mathrm{SL}(2,\mathbb{R})$ cocycles. The Lyapunov exponent remains constant
under small perturbations in the imaginary direction. In other words the value
of $L$ in (2) must remain unchanged when the integration is taken over
$\mathbb{R}/\mathbb{Z}+\epsilon i$.

The Almost Reducibility Conjecture (ARC) states precisely that regularity is
equivalent to almost reducibility.222We note that the ARC is usually stated
just in the case of zero Lyapunov exponents, since the positive Lyapunov
exponent case is known. It lies clearly beyond the scope of local reducibility
theory, since it assumes no proximity to a well understood model.333While non-
local reducibility results exist, based on renormalization, the required a
priori bounds currently depend on no average growth of the cocycle (see [AK1],
[AK2]), while a hypothesis on the Lyapunov exponent can only give slow growth
at most. We have previously established the ARC under an exponentially
Liouville hypothesis on the frequency [A2]. Here we treat the complementary
case when $\beta=\limsup\frac{\ln q_{n+1}}{q_{n}}=0$, where $q_{n}$ are the
denominators of the continued fraction approximants of $\alpha$.

######  Theorem 1.

Let $\alpha\in\mathbb{R}\smallsetminus\mathbb{Q}$ and $A\in
C^{\omega}(\mathbb{R}/\mathbb{Z},\mathrm{SL}(2,\mathbb{R}))$ be such that
$(\alpha,A)$ is regular. Then $(\alpha,A)$ is almost reducible.

######  Corollary 2.

Let $\alpha\in\mathbb{R}\smallsetminus\mathbb{Q}$ and $A\in
C^{\omega}(\mathbb{R}/\mathbb{Z},\mathrm{SL}(2,\mathbb{R}))$ be such that
$(\alpha,A)$ has a Diophantine rotation vector. Then $(\alpha,A)$ is regular
if and only if $(\alpha,A)$ is analytically conjugate to a constant
translation.

###### Proof.

Almost reducibility reduces the analysis to the local case which was done in
[AJ]. ∎

However nice it is to establish a conceptual connection between Lyapunov
exponents and perturbative theory, one is still justified to wonder what is to
be gained in practice. The answer lies in the recent advances in [A1] that
introduced the tools to analyze the parameter dependence of the Lyapunov
exponent of complexifications. Combined with the results of this paper, one
does get a remarkably precise description of the phase transitions as one
consider the global situation, that is, how one moves from KAM-like to non-
uniformly hyperbolic dynamics, the most striking consequences of which are
seen in the application to the theory of one-frequency Schrödinger operators
to which we now turn.

###  1.2. Global theory of one-frequency Schrödinger operators

Cocycles of the form $(\alpha,A^{(E-v)})$ arise in the consideration of one-
frequency Schrödinger operators with frequency $\alpha$ and analytic potential
$v$, i.e., bounded self-adjoint operators $H=H_{\alpha,v}$ on
$\ell^{2}(\mathbb{Z})$ of the form444It is common to consider a third
parameter, a phase $\theta$, in the definition of $H$. This corresponds to
considering the shifted potential $v_{\theta}(\cdot)=v(\theta+\cdot)$.

(3) $(Hu)_{n}=u_{n+1}+u_{n-1}+v(n\alpha)u_{n}.$

While such operators have been an important topic of research since the
1970’s, the understanding was mostly restricted to two local regimes,
corresponding to operators with large-like and small-like potentials. Only
very recently an approach towards a global theory has emerged [A1].

One of the most remarkable accomplishments of the local theories was a good
understanding of the spectral measures in the two respective regimes:
typically point spectrum for large-like potentials [BG] and absolutely
continuous spectrum for small-like potentials ([E], [BJ2], [AJ], [A2]). In
[A1] we set as one of the goals of the global theory to prove that a typical
(in a measure-theoretical sense) operator $H$ has no singular continuous
spectrum. This is achieved here:

######  Theorem 3.

For a typical Schrödinger operator (3), the spectral measures have no singular
continuous component.

In fact we get a much more precise description of the spectrum. Recall that in
[A1], it is shown that any energy in the spectrum can be classified into one
of three classes, subcritical, critical and supercritical, so that the
critical locus is at most “codimension-one”. From the point of view of this
classification, in [A1], we achieved a best possible description of the phase
transition behavior for a typical quasiperiodic operator: it is shown that for
every $\alpha\in\mathbb{R}\smallsetminus\mathbb{Q}$ and for a typical $v\in
C^{\omega}(\mathbb{R}/\mathbb{Z},\mathbb{R})$, the corresponding operator is
acritical, that is, the spectrum $\Sigma=\Sigma_{\alpha,v}$ splits as a
disjoint union of compact sets $\Sigma_{-}$ and $\Sigma_{+}$ corresponding to
subcritical and supercritical energies. Moreover, any small perturbation of
$H$ (which may involve both the frequency and the potential) is acritical as
well, and the corresponding splitting of $\Sigma_{\alpha,v}$ depends
continuously on the perturbation.

The definition of criticality in [A1] is based on the behavior of the Lyapunov
exponent of phase complexifications of the cocycle $(\alpha,A^{(E-v)})$
associated to the Schrödinger equation $Hu=Eu$: for $E$ in the spectrum,
subcritical corresponds to regularity, supercritical to positivity of the
Lyapunov exponent, and critical to the complement. The immediate question such
classification raises is whether the non-critical regimes do behave according
to the local theories, as the terminology suggests.

One of the nice properties of acritical operators is that the Lyapunov
exponent is bounded from below on $\Sigma_{+}$. The behavior for energies with
bounded from below Lyapunov exponent was extensively analyzed in the non-
perturbative theory of Bourgain, Goldstein and Schlag, in particular:

  1. 1.

Under a full measure arithmetic condition on the frequency, the integrated
density of states (i.d.s.) is locally Hölder [GS1],

  2. 2.

Up to a typical small perturbation of the frequency, there is Anderson
localization (pure point spectrum with exponentially decaying eigenfunctions)
[BG] and the i.d.s. is absolutely continuous [GS2].

The analysis of the behavior at $\Sigma_{-}$ is much more recent, in fact the
notion which would later be called subcriticality first appears in passing in
[AJ], where it is first speculated to be connected to almost reducibility. The
corresponding region of the spectrum $\Sigma_{ar}$ (which is open by [A2]) can
be analyzed by either KAM theory [E], [AFK] or Aubry duality [AJ], in
particular

  1. 1.

The spectral measures are absolutely continuous [AJ], [A2],

  2. 2.

Under a full measure arithmetic condition on the frequency, the i.d.s. is
locally $1/2$-Hölder [AJ], [Am].

Our main theorem implies that $\Sigma_{-}=\Sigma_{ar}$. Combined with the
prevalence of acriticality, we obtain the following Spectral Dichotomy: a
typical operator $H$ can be written as a direct sum of a large-like operator
$H_{+}$ and a small-like operator $H_{-}$ with disjoint spectra (just define
$H_{\pm}$ using the spectral projections associated to $\Sigma_{\pm}$).

As a consequence of our understanding of $\Sigma_{+}$ and $\Sigma_{ar}$, we
get Theorem 3 (with Anderson localization in $\Sigma_{+}$ and absolutely
continuous spectrum in $\Sigma_{-}$). Moreover, we also have:

######  Theorem 4.

For a typical Schrödinger operator (3), the integrated density of states is
Hölder and absolutely continuous.

######  Remark 1.

Absolute continuity of the i.d.s. implies that the spectrum is “everywhere
thick” in the sense that any non-empty intersection with an open set must have
positive Lebesgue measure. Let us note for completeness that it was known that
the spectrum of a typical operator $H$ is a Cantor set. This is a direct
consequence of [GS3] (which shows that up to a typical perturbation of
$\alpha$, $\Sigma_{+}$ has empty interior) and [A3] (which shows that up to a
typical perturbation of $v$, $\Sigma\smallsetminus\Sigma_{+}$ has empty
interior).

######  Remark 2.

As discussed above, before our work the supercritical theory could be
considered to be much more developed than the subcritical one. This is
reverted with the ARC. Indeed the supercritical analysis is still by no means
complete in crucial places, expecially since many results prove typical
properties by a parameter exclusion argument on the frequency. While some
exclusion is necessary (particularly for localization), it would be desirable
for it to be given in terms of explicit arithmetic conditions (with any
further parameter exclusion being relegated to the potential, preferably just
through a phase shift). On the other hand, in the subcritical region we have
now a far better understanding, with many results (such as absolute continuity
of the spectral measures [A2]) not depending at all on parameter exclusion.

######  Remark 3.

As we hoped, the Global Theory and the ARC have found many applications. It is
used for instance in joint work with Jiangong You and Qi Zhou to solve the Dry
Ten Martini problem in the non-critical case [AYZ]. Another exciting direction
of progress is the use of duality techniques in this context, which gives new
information about the supercritical region as well (addressing some of the
outstanding issues discussed in the previous remark), as presented by
Jitomirskaya in her Plenary Address at the 2022 ICM.

Acknowledgements: The main result of this paper was proved in 2009 (in the
Diophantine case) and 2012 (in the more general $\beta=0$ case). Descriptions
were given in a course in the Fields institute in 2011, and during a meeting
in Oberwolfach in 2012. I thank the community for their continuous
encouragement to provide a written account. To present accurately the history,
the introduction and the bibliography reflect my understanding of the subject
in 2012, with the exception of Remark 3. I thank my PhD students Fernando
Argentieri, Pedram Safaee and Andrea Ulliana for the detailed reading of this
text.

##  2\. What is happening in the proof

The following is an explanation more than an outline. Since the proof is short
we recommend the reader to first read it and then to come back here for the
narrative.

The argument is mostly based on complex analysis, and is mostly self
contained. The preparatory section 3 in particular collects the main outside
ingredients (coming from [A1] and [BJ1]), their use is pointed out below.

An $\mathrm{SL}(2,\mathbb{R})$ cocycle naturally gives rise to a one-parameter
family of perturbations obtained by composition with rigid rotations
$R_{\theta}$. For imaginary parameters $i\theta$, such cocycles are uniformly
hyperbolic at real $z$ (by hyperbolic geometry). Uniformly hyperbolic cocycles
can be diagonalized by straightening out the unstable and stable directions
$u_{\theta}$ and $s_{\theta}$. If we have good enough bounds on this conjugacy
$B$ (as the perturbation becomes smaller), it will also (complex) conjugate
the initial cocycle close to diagonal. Indeed if
$\|B\|^{2}=\theta^{-1+\kappa-o(1)}$ then closeness is $\theta^{\kappa-o(1)}$.
Getting $\kappa$ non-negative is already non-obvious, but we need a positive
$\kappa$ (and there is not much room, since $\kappa$ can not go beyond $1/2$
when starting with a parabolic cocycle).

The first step is to show that the uniform hyperbolicity of the perturbed
matrices does persist for non-real $z$ since the hyperbolic geometry argument
only works at the real line. This follows from regularity through a central
result of the first part of [A1].

Given the existence of the extensions, it is possible to show that the size of
the conjugacy $\|B\|^{2}$ is basically the inverse of the angle between the
unstable and stable directions. This is actually subtle because we need to
obtain holomorphic solutions with optimal behavior regarding some real
geometric constraint, see section 5.

The estimate on the angle (section 4) is our main concern. The basic idea is
that if the angle is small then the Lyapunov exponent $L_{\theta}$ is small.
But if the Lyapunov exponent is small ($L_{\theta}=\theta^{\kappa(\theta)}$
with $\kappa(\theta)$ large) then the angle must be often large
($\theta^{1-\kappa(\theta)}$) at real $z$ (this is a hyperbolic geometry
estimate basic to Kotani theory, and shows in particular that
$0<\kappa(\theta)<1$). These two estimates work together to deal with non-real
$z$ by subharmonicity. Notice that the number $1/2$ appears when combining
$\kappa(\theta)$ and $1-\kappa(\theta)$ and this is essentially sharp as
discussed before.

Showing the basic estimate relating angle and the Lyapunov exponent is done in
two rounds. First we need an a priori bound in order to know that the angle is
not terrible. This depends on a quantification of the argument of [A1] which
establishes uniform hyperbolicity within the domain of cocycles which are
regular and have positive Lyapunov exponent. It shows that weak hyperbolicity
(small angle) can only arise near the boundary of this (infinite dimensional)
domain, so we need to control the distance to the boundary. Regularity is not
an issue, since it is stable, but the Lyapunov exponent is relatively close to
$0$. Here we use a continuity estimate on the Lyapunov exponent (due to [BJ1])
to control how fast it could drop to $0$. In the second round we show that the
angle satisfies the Lyapunov exponent bound in a large (equidistributed) set,
and in the presence of the a priori bound this allows us to get this better
bound everywhere through a Brownian motion argument.

With the angle estimate done we can complex conjugate the cocycle close to
rotations (section 6). This provides in particular good polynomial growth
estimates for the initial cocycle. More crucially, it gives good bounds on
vector representations $U_{\theta}=B^{-1}\cdot\begin{pmatrix}i\\\
1\end{pmatrix}$ and $S_{\theta}=B^{-1}\cdot\begin{pmatrix}-i\\\
1\end{pmatrix}$ of the unstable and stable directions of the perturbations. We
also derive a better upper bound on $L_{\theta}$, $\kappa(\theta)>1/2-o(1)$.

The conjugacy obtained so far was built on the basis of the unstable and
stable directions of complex perturbations of the cocycle. Since the complex
perturbations are not real symmetric, neither are the conjugacies $B$. In
order to fix this issue (section 7), we pair $u_{\theta}$ with its
symmetrization $u^{\prime}_{\theta}$ and try to straighten then out with a
real symmetric $B_{r}$. The issue is again to get a lower bound on the angle.
But now it is easier, since we have vector representations $U_{\theta}$ and
$U^{\prime}_{\theta}$ we can use the determinant $\Delta$ to get information
on the angle. If the angle is small somewhere (at some $z_{*}$) this must
propagate through an equidistributed set (of translates of $z_{*}$) parallel
to the real line. This makes the determinant drop on this set, and since the
determinant is a nice function a Lagrange interpolation argument shows that it
satisfy a similar bound on the real line. But on the real line we have the
hyperbolic geometry lower bound for the angle (now improved due to the new
upper bound on $L_{\theta}$), and too small determinant means that
$U_{\theta}$ must be small. If this does not happen, we get the angle
estimate, as shows that the angle could not have been small at any $z_{*}$. If
$U_{\theta}$ is indeed small, we just observe that $S_{\theta}$ can not be
small at the same time (the determinant of $U_{\theta}$ and $S_{\theta}$
having been normalized as $2i$ by definition), so we obtain an angle estimate
for the pairing of $s_{\theta}$ and its symmetrization $s^{\prime}_{\theta}$.

##  3\. Some ingredients

Below, once $\alpha\in\mathbb{R}$ and $A\in
C^{\omega}(\mathbb{R}/\mathbb{Z},\mathrm{SL}(2,\mathbb{C}))$ are fixed, we let
$A_{n}(x)=A(x+(n-1)\alpha)\cdots A(x)$,
$L_{n}=\frac{1}{n}\int\ln\|A_{n}(x)\|dx$ and $L=\lim L_{n}$.

For $\epsilon>0$, we write $C^{\omega}_{\epsilon}(\mathbb{R}/\mathbb{Z},*)$
for the space of analytic functions to the target $*$ (typically $\mathbb{R}$,
$\mathrm{SL}(2,\mathbb{R})$,…) admitting a bounded holomorphic extension to
$|\Im z|<\epsilon$, and we let $\|\cdot\|_{\epsilon}$ denote the corresponding
$L^{\infty}$ norm of the extension. We also let $\|\cdot\|_{0}$ denote the
$L^{\infty}$ norm on $\Im z=0$.

We identify $\mathbb{P}\mathbb{C}^{2}$ with the Riemann sphere
$\overline{\mathbb{C}}$ in the familiar way $(x,y)\mapsto\frac{x}{y}$. The
action of $\mathrm{SL}(2,\mathbb{C})$ on $\overline{\mathbb{C}}$ is then given
by $\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\cdot z=\frac{az+b}{cz+d}$, so
$\mathrm{SL}(2,\mathbb{R})$ consists of matrices preserving the upper half
plane ${\mathbb{H}}$. We denote $R_{\theta}=\begin{pmatrix}\cos
2\pi\theta&-\sin 2\pi\theta\\\ \sin 2\pi\theta&\cos 2\pi\theta\end{pmatrix}$.

We let $d(u,s)$ be the absolute value of the sine of the angle between $u$ and
$s$ on $\mathbb{P}\mathbb{C}^{2}$, which is invariant under
${\mathrm{SU}}(2)$. We note that $-\ln d$ is plurisubharmonic in the
complement of the diagonal in
$\mathbb{P}\mathbb{C}^{2}\times\mathbb{P}\mathbb{C}^{2}$.

Sometimes it is convenient to move from ${\mathbb{H}}$ to the disk
$\mathbb{D}$, so we denote $\dot{z}=\frac{z-i}{z+i}$.

###  3.1. Matrix growth

The next is Lemma 7 of Bourgain-Jitomirskaya [BJ1].

######  Lemma 4.

For every $\xi,\Xi>0$, there exist constants $C>0$, $c>0$, $c^{\prime}>0$ with
the following properties. Let $A\in
C^{\omega}_{\xi}(\mathbb{R}/\mathbb{Z},\mathrm{SL}(2,\mathbb{C}))$ be such
that $\|A\|_{\xi}<\Xi$. Let $\alpha$ and $q$ be such that
$|\alpha-\frac{p}{q}|<\frac{1}{q^{2}}$. Let $0<\kappa<1$, let
$N>C\kappa^{-2}q$ and let $N^{\prime}$ be a multiple of $N$ with
$\frac{N^{\prime}}{N}<e^{e^{c\kappa q}}$. If $L_{2N}>\frac{9}{10}L_{N}$ and
$L_{N}>100\kappa$ then

(4) $|L_{N^{\prime}}+L_{N}-2L_{2N}|<e^{-c^{\prime}\kappa
q}+C\frac{N}{N^{\prime}}.$

######  Remark 5.

The result is originally stated for (real) Schrödinger cocycles, but the proof
extends without change. The original statement also assumes $L>100\kappa$, but
the proof only uses the weaker condition discussed here.

######  Lemma 6.

There exists $M>1$ with the following property. Let
$\alpha\in\mathbb{R}\smallsetminus\mathbb{Q}$ be such that $\beta=0$. Then for
every $\xi,\Xi>0$, if $q=q_{n}$ is sufficiently large and $A\in
C^{\omega}_{\xi}(\mathbb{R}/\mathbb{Z},\mathrm{SL}(2,\mathbb{C}))$ is such
that $\|A\|_{\xi}\leq\Xi$ and $L=0$ then $L_{q^{M}}<q^{-1/M}$.

###### Proof.

Fix some small $\sigma>0$. Assume that $L_{q^{M}}\geq q^{-1/M}$ with $M$
large. Then there exists some $q^{5}<N<q^{M}$ such that $L_{N}>q^{-\sigma}$
and $L_{2N}>\frac{9}{10}L_{N}$ (for instance, $N$ a suitable power of $2$).
Let $q^{\prime}=q_{n^{\prime}}$ be minimal with $q^{\prime}>e^{q^{1/2}}$, and
let $N^{\prime}>q^{\prime 5}$ be the smallest multiple of $N$. Then
$N^{\prime}/N=e^{o(e^{q^{1/2}})}<e^{e^{c\kappa q}}/2$, where
$\kappa=q^{-\sigma}/100$ (here we use $\beta=0$). It follows from Lemma 4 that
$L_{N^{\prime}}$ and $L_{2N^{\prime}}$ are $e^{-c^{\prime}\kappa
q}+Cq^{M}e^{-5q^{1/2}}<e^{-4q^{1/2}}$ close to $2L_{2N}-L_{N}$. This allows us
to iterate the procedure starting with $N^{\prime}$ instead of $N$. We
conclude that $L>L_{N}/2>0$. ∎

###  3.2. Complex perturbations

The result below is of a quite general nature, holding for continuous cocycles
over a homeomorphism of a compact metric space.

######  Lemma 7.

Let $\alpha\in\mathbb{R}\smallsetminus\mathbb{Q}$ and $A\in
C^{\omega}(\mathbb{R}/\mathbb{Z},\mathrm{SL}(2,\mathbb{R}))$. Then for every
$\theta>0$, $(\alpha,R_{-i\theta}A)$ is uniformly hyperbolic, and the unstable
direction satisfies $u(x)\in{\mathbb{H}}$ and $|\dot{u}(x)|<e^{-4\pi\theta}$.
Moreover

(5)
$L(\alpha,R_{-i\theta}A)=2\pi\theta+\frac{1}{2}\int_{\mathbb{R}/\mathbb{Z}}\ln\frac{1-|\dot{u}(x)|^{2}}{1-e^{8\pi\theta}|\dot{u}(x)|^{2}}dx\geq
2\pi\theta.$

###### Proof.

It is obvious to check that the action of $R_{-i\theta}A$ takes ${\mathbb{H}}$
into the precompact subset
${\mathbb{H}}_{\theta}=\\{|\dot{x}|<e^{-4\pi\theta}\\}\subset{\mathbb{H}}$,
thus uniform hyperbolicity follows by the usual cone field criterium. The
Lyapunov exponent can then be computed as $-\frac{1}{2}$ the average rate of
projective contraction of the inclusion ${\mathbb{H}}_{\theta}\to{\mathbb{H}}$
at the unstable direction, calculated with respect to any conformal metric:
the given formula corresponds to the choice of the Poincaré metric on
${\mathbb{H}}$. ∎

###  3.3. Robustness of uniform hyperbolicity and angle estimate

The next result is a quantification of a basic estimate in the proof of
Theorem 6 of [A1].

######  Theorem 5.

Let $\alpha\in\mathbb{R}\smallsetminus\mathbb{Q}$ and let $A\in
C^{\omega}_{\epsilon_{0}}(\mathbb{R}/\mathbb{Z},\mathrm{SL}(2,\mathbb{C}))$ be
such that $(\alpha,A(x+ti))$ is uniformly hyperbolic for every
$|t|<\epsilon_{0}$ and let $u$ and $s$ be the unstable and stable directions.
Let $0<\epsilon_{1}<\epsilon_{0}$,
$0<\epsilon_{2}<\min\\{\epsilon_{1},\epsilon_{0}-\epsilon_{1}\\}$ and
$0<\delta<1$ be such that for both $t=\epsilon_{1}$ and $t=-\epsilon_{1}$, for
every $\tilde{A}\in
C^{\omega}_{\epsilon_{2}}(\mathbb{R}/\mathbb{Z},\mathrm{SL}(2,\mathbb{C}))$
such that $\|\tilde{A}(x)-A(x+ti)\|_{\epsilon_{2}}<\delta$ the cocycle
$(\alpha,\tilde{A})$ is uniformly hyperbolic. Then
$\inf_{x\in\mathbb{R}/\mathbb{Z}}d(u(x),s(x))\geq C^{-1}\delta$, where
$C=C(\epsilon_{0},\epsilon_{1},\epsilon_{2},\|A\|_{\epsilon_{0}})$.

###### Proof.

Given two distinct directions $\mu,\nu\in\mathbb{P}\mathbb{C}^{2}$, let
$\begin{pmatrix}a&b\\\ c&d\end{pmatrix}\in\mathrm{SL}(2,\mathbb{C})$ be any
matrix whose columns are aligned with $\mu$ and $\nu$. Then
$Q_{2}=Q_{2}(\mu,\nu)=-ab$ and $Q_{3}=Q_{3}(\mu,\nu)=cd$ are well defined and
depend holomorphically on $\mu$ and $\nu$. We notice that $\eta\leq
d(\mu,\nu)^{-1}\leq\sqrt{5}\eta$ where
$\eta=\max\\{1,|Q_{2}|,|Q_{3}|\\}$.555Obviously $d(\mu,\nu)^{-2}\geq 1$, and
since $d(\mu,\nu)^{-2}=(|a|^{2}+|c^{2}|)(|b|^{2}+|d|^{2})$, we have the
estimate
$Q_{2}^{2}+Q_{3}^{2}\leq(|a|^{2}+|c^{2}|)(|b|^{2}+|d|^{2})\leq(|ab|+|cd|)^{2}+(|ad|-|bc|)^{2}\leq(Q_{2}+Q_{3})^{2}+1\leq
5\eta^{2}$.

For $j=2,3$, let $Q_{j}(z)=Q_{j}(u(z),s(z))$ be defined on $|\Im
z|<\epsilon_{0}$. We need to show that $\|Q_{j}\|_{0}\leq C\delta^{-1}$,
$j=2,3$.

For $w\in C^{\omega}(\mathbb{R}/\mathbb{Z},\mathbb{C})$ let
$\mathcal{E}^{w}_{2}=\begin{pmatrix}1&w\\\
0&1\end{pmatrix},\mathcal{E}^{w}_{3}=\begin{pmatrix}1&0\\\ w&1\end{pmatrix}\in
C^{\omega}(\mathbb{R}/\mathbb{Z},\mathrm{SL}(2,\mathbb{C}))$.

By Lemma 9 of [A1], for any $|t|<\epsilon_{0}$, and for $j=2,3$, the
derivative of $\lambda\mapsto L(\alpha,A(x+ti)\mathcal{E}^{\lambda w}_{j})$ at
$\lambda=0$ is given by $\Re\int_{\mathbb{R}/\mathbb{Z}}Q_{j}(x+ti)w(x)dx$.
The hypothesis implies that for $t=\pm\epsilon_{1}$, the Lyapunov exponent is
a bounded pluriharmonic function on a ball of radius $\delta$ around $A(x+ti)$
in
$C^{\omega}_{\epsilon_{2}}(\mathbb{R}/\mathbb{Z},\mathrm{SL}(2,\mathbb{C}))$,
implying

(6)
$\left|\Re\int_{\mathbb{R}/\mathbb{Z}}Q_{j}(x\pm\epsilon_{1}i)w(x)dx\right|\leq
C^{\prime}\delta^{-1}\|w\|_{\epsilon_{2}},$

for $j=2,3$. By considering monomials $w=\gamma e^{-2\pi ikx}$ with
$k\in\mathbb{Z}$ and $|\gamma|=1$, this shows that the $k$-th Fourier
coefficient of $Q_{j}$ must be bounded by
$C^{\prime}\delta^{-1}e^{(\epsilon_{2}-\epsilon_{1})|k|}$ as desired. ∎

##  4\. Angle estimate

Through this section, we fix $\alpha\in\mathbb{R}\smallsetminus\mathbb{Q}$ and
$A\in
C^{\omega}_{\epsilon_{0}}(\mathbb{R}/\mathbb{Z},\mathrm{SL}(2,\mathbb{R}))$
such that $L(\alpha,A(x+ti))=0$ for $|t|<\epsilon_{0}$. By stability of
regularity [A1], for every $0<\epsilon<\epsilon_{0}$, there exists
$\theta(\epsilon)>0$ such that for every $0<\theta<\theta(\epsilon)$,
$L(\alpha,R_{-i\theta}A(x+ti))$ is a constant $L_{\theta}$ for $|t|<\epsilon$.
Since $L_{\theta}\geq 2\pi\theta$, by Lemma 7, we can write
$L_{\theta}=\theta^{\kappa(\theta)}$ with $\kappa(\theta)\leq 1$. Since
regularity with positive Lyapunov exponent implies uniform hyperbolicity [A1],
$(\alpha,R_{-i\theta}A(x+ti))$ is uniformly hyperbolic for
$0<\theta<\theta(\epsilon)$ and $|t|<\epsilon$. For
$0<\theta<\theta(\epsilon)$, let $u_{\theta},s_{\theta}$ be the unstable and
stable directions, which are holomorphic functions on $|\Im x|<\epsilon$ with
values in $\mathbb{P}\mathbb{C}^{2}$.

The proof of the following result will take the remaining of this section:

######  Theorem 6.

Assume that $\beta=0$, let $0<\epsilon<\epsilon_{0}$ and let
$0<\kappa<\min\\{\frac{1}{2},1-\frac{\epsilon}{\epsilon_{0}}\\}$. Then for
every $\theta>0$ sufficiently small we have

(7) $d(u_{\theta}(x),s_{\theta}(x))\geq\theta^{1-\kappa},\quad|\Im
x|<\epsilon.$

Moreover we also have

(8) $d(u_{\theta}(x),s_{\theta}(x))\geq\theta^{\kappa(\theta)+o(1)},\quad|\Im
x|<\epsilon_{0}-o(1).$

Let $\rho_{\theta}(z)=-\ln d(u_{\theta}(z),s_{\theta}(z))$. Let $M$ be as in
Lemma 6.

######  Lemma 8.

If $\beta=0$, for every $0<\epsilon<\epsilon_{0}$, for every $\theta>0$
sufficiently small, if $q=q_{n}$ is such that $\theta>q^{-1/M}$ then
$\rho_{\theta}(z)<q^{M}$ through $|\Im z|<\epsilon$.

###### Proof.

Note that $\frac{1}{q}\ln\|(R_{-i\theta}A)_{q}\|$ is small if $\theta$ is
small. If $\tilde{A}$ is $e^{-q^{M}/2}$ close to $R_{-i\theta}A$ at $\\{\Im
z=t\\}$, then as in [AK1],

(9)
$\|\tilde{A}_{q^{M}}^{-1}(R_{-i\theta}A)_{q^{M}}-\mathit{id}\|=o(1),\quad\Im
z=t,$

uniformly on $|t|<\epsilon^{\prime}$ for any given
$\epsilon^{\prime}<\epsilon_{0}$. This gives

(10) $\displaystyle\frac{1}{q^{M}}\int\ln\|\tilde{A}_{q^{M}}(x+ti)\|dx$
$\displaystyle\geq\frac{1}{q^{M}}\int\ln\|(R_{-i\theta}A)_{q^{M}}(x+it)\|dx-o(q^{-M})$
$\displaystyle\geq L_{\theta}-o(\theta)\geq 2\pi\theta-o(\theta)>q^{-1/M}.$

By Lemma 6, if $\tilde{A}$ has a well behaved analytic extension around
$\\{\Im z=t\\}$, then the Lyapunov exponent for $(\alpha,\tilde{A}(x+ti))$ is
positive. By stability of regularity [A1], $(\alpha,\tilde{A}(x+ti))$ is
regular as well, so it is uniformly hyperbolic. By Theorem 5, this gives the
estimate $\rho_{\theta}<\frac{q^{M}}{2}+C$, and the result follows. ∎

######  Lemma 9 (Brownian motion argument).

Assume that $\rho_{\theta}$ is defined and bounded through $|\Im z|<\epsilon$.
Then for $0<\epsilon^{\prime}<\epsilon$ and each $q$ we have

(11) $\sup_{|\Im z|<\epsilon^{\prime}}\rho_{\theta}\leq
C+e^{-C_{0}(\epsilon-\epsilon^{\prime})q}\sup_{|\Im
z|<\epsilon}\rho_{\theta}-\ln L_{\theta}+\sup_{0\leq n\leq
q-1}2\ln\|(R_{-i\theta}A)_{n}\|_{\epsilon},$

where $C_{0}$ is an absolute constant and $C$ depends only on $\sup_{|\Im
z|<\epsilon}\|R_{-i\theta}A(z)\|$.

###### Proof.

The function $\rho_{\theta}(x)$ is bounded and subharmonic on $|\Im
x|<\epsilon$. Let $K\subset\\{|\Im z|<\epsilon\\}$ be the set of all $z$ such
that $\rho_{\theta}(z)=\min_{\Im w=\Im z}\rho_{\theta}(w)$.

Note that for each $|t|<\epsilon$, $2L_{\theta}$ is the integral over $|\Im
z|=t$ of $\ln\frac{\|R_{-i\theta}A(z)\cdot
u_{\theta}(z)\|}{\|R_{-i\theta}A(z)\cdot s_{\theta}(z)\|}$, where we abuse
notation denoting $u_{\theta}$ and $s_{\theta}$ unit vectors in the
corresponding directions. But $\frac{\|R_{-i\theta}A\cdot
u_{\theta}\|}{\|R_{-i\theta}A\cdot s_{\theta}\|}\leq
1+d(u_{\theta},s_{\theta})\frac{\|R_{-i\theta}A\|}{\|R_{-i\theta}A\cdot
s_{\theta}\|}$. It follows that

(12) $2L_{\theta}\leq C^{\prime}\int d(u_{\theta}(z+x),s_{\theta}(z+x))dx.$

In particular, for $z\in K$ we have $d(u_{\theta}(z),s_{\theta}(z))\geq
2L_{\theta}/C^{\prime}$, which gives $\rho_{\theta}(z)\leq-\ln
L_{\theta}+\ln\frac{C^{\prime}}{2}$.

Notice that since $(R_{-i\theta}A)_{k}(z)$ takes
$(u_{\theta}(z),s_{\theta}(z))$ to
$(u_{\theta}(z+k\alpha),s_{\theta}(z+k\alpha))$, the corresponding angles can
only differ by a factor of $\|(R_{-i\theta}A)_{k}(z)\|^{2}$ (up to absolute
constant). Thus for $z\in\tilde{K}=\bigcup_{0\leq k\leq q-1}K+k\alpha$ we have
the estimate

(13) $\rho_{\theta}(z)\leq-\ln L_{\theta}+\sup_{0\leq n\leq
q-1}2\ln\|(R_{-i\theta}A)_{n}\|_{\epsilon}+C.$

Let us now consider a point $z_{0}$ with $|\Im z_{0}|<\epsilon^{\prime}$, and
let us run the Brownian motion starting at $z_{0}$ up to the moment it either
touches $\tilde{K}$ or $\\{|\Im z|=\epsilon\\}$. Since $\rho_{\theta}$ is
subharmonic, $\rho_{\theta}(z_{0})$ is bounded by the expectation of its value
at the endpoint of the Brownian motion. Thus

(14) $\rho_{\theta}(z_{0})\leq p\sup_{|\Im
z|<\epsilon}\rho_{\theta}(z)+(1-p)\sup_{z\in\tilde{K}}\rho_{\theta}(z),$

where $p$ is the probability that the Brownian motion reaches $|\Im
z|=\epsilon$ without hitting $\tilde{K}$.

Thus it is enough to show that $p<e^{-C_{0}(\epsilon-\epsilon^{\prime})q}$. In
order to see this, it is enough to show that for some absolute constant
$C_{2}>0$, the probability that the Brownian motion started at a point $z$
with $|\Im z|<\epsilon-2q^{-1}$ reaches the boundary of the square $Q(z)$
centered on $z$ and of side $4q^{-1}$ without touching $\tilde{K}$ is at most
$e^{-C_{2}}$. But since $q$ is an approximant of $\alpha$,
$\\{k\alpha\\}_{0\leq k\leq q-1}$ is $2q^{-1}$ dense on the circle. So
$\tilde{K}\cap Q$ intersects each horizontal segment
$J_{s}=\\{z+t+is\\}_{t\in[-q^{-1},q^{-1}]}$ for $s\in[-2q^{-1},2q^{-1}]$. Such
a set is clearly a definite obstacle for the Brownian motion (after rescaling
by $q$, it has definite logarithmic capacity) and is hit with positive
probability. ∎

######  Lemma 10.

If $\beta=0$, then $\rho_{\theta}\leq-\ln L_{\theta}+o(-\ln\theta)$ over $|\Im
z|<\epsilon$.

###### Proof.

Fix $R>0$ large. Below the $q_{(j)}$ are specific choices of continued
fraction denominators.

Let $q_{(0)}$ be least with $\theta>q_{(0)}^{-1/2M}$. By Lemma 8,
$\rho_{\theta}\leq q_{(0)}^{M}$ through a band bigger than $\epsilon$. Take
$q_{(1)}$ minimal with $R\ln q_{(0)}\leq q_{(1)}<q_{(0)}$ (use $\beta=0$).
Apply the Brownian motion argument with $q_{(1)}$. We get an improved estimate

(15) $\rho_{\theta}\leq e^{-cq_{(1)}}q_{(0)}^{M}-\ln
L_{\theta}+c_{\theta}q_{(1)}\leq-\ln L_{\theta}+c_{\theta}q_{(1)}+o(1)$

where $c_{\theta}\to 0$ as $\theta\to 0$. If $q_{(1)}=O(-\ln\theta)$, we are
done.

Otherwise, repeat the procedure taking $R\ln q_{(1)}\leq q_{(2)}<q_{(1)}$
minimal. We get a new estimate $\rho_{\theta}\leq-\ln
L_{\theta}+c^{\prime}_{\theta}q_{(2)}$. If $q_{(2)}=O(-\ln\theta)$, we are
done.

Otherwise, repeat the procedure taking $R\ln q_{(2)}\leq q_{(3)}<q_{(2)}$
minimal. We get a new estimate $\rho_{\theta}\leq-\ln
L_{\theta}+c^{\prime\prime}_{\theta}q_{(3)}$. Notice that $q_{(3)}<R\ln
q_{(1)}$ and $\theta\leq q_{(1)}^{-1/2M}$ so $q_{(3)}=O(-\ln\theta)$, so we
are done. ∎

So far we have seen that a smaller $\kappa(\theta)$ (so larger $L_{\theta}$)
yields better angle estimates. However a large $\kappa(\theta)$ turns out to
help at $\Im z=0$:

######  Lemma 11.

For a set of probability $1-o(1)$ in $\Im z=0$,
$|\dot{u}_{\theta}(z)|<1-\theta^{1-\kappa(\theta)+o(1)}$.

###### Proof.

Direct application of Lemma 7. ∎

Proof of Theorem 6. Fix $\epsilon<\epsilon^{\prime}<\epsilon_{0}$ close to
$\epsilon_{0}$. By Lemma 11, the set of $x\in\mathbb{R}/\mathbb{Z}$ such that
$\rho_{\theta}(x)<(-1+\kappa(\theta)-o(1))\ln\theta$ has probability at least
$1-o(1)$. By Lemma 10, we also have
$\rho_{\theta}(x)<(-\kappa(\theta)-o(1))\ln\theta$ over any $|\Im
x|<\epsilon^{\prime}$, which implies the second claim and also the first claim
if $\kappa(\theta)<1/2$. Otherwise, since $\rho_{\theta}$ is subharmonic, it
follows that

(16) $\sup_{|\Im
x|<\epsilon}\rho_{\theta}(x)\leq-\ln\theta\left(\frac{\epsilon}{\epsilon^{\prime}}(\kappa(\theta)+o(1))+\left(1-\frac{\epsilon}{\epsilon^{\prime}}\right)(1-\kappa(\theta)+o(1))\right),$

which implies the first claim in the case $\kappa(\theta)\geq 1/2$. ∎

##  5\. Choosing a third point

We call $B\in\mathrm{SL}(2,\mathbb{C})$ a minimizer (or minimizing) if

(17) $\left\|B^{-1}\cdot\left(\begin{matrix}i\\\
1\end{matrix}\right)\right\|=\left\|B^{-1}\cdot\left(\begin{matrix}-i\\\
1\end{matrix}\right)\right\|.$

Notice that $B$ is minimizing if and only if
$B^{-1}\cdot\overline{\mathbb{R}}$ is the great circle $H_{B^{-1}\cdot
i,B^{-1}\cdot-i}$ of points $z$ which are at the same distance from both
$B^{-1}\cdot i$ and $B^{-1}\cdot-i$.

Let $x,y\in\overline{\mathbb{C}}$ be two distinct points. We let
$K_{x,y}\subset\mathrm{SL}(2,\mathbb{C})$ be the set of all $B$ such that
$B\cdot x=i$ and $B\cdot y=-i$. We let $k(x,y)=\inf_{B\in K_{x,y}}\|B\|^{2}$.
It is easy to see that the infimum is attained, and this happens precisely at
the set of minimizing elements of $K_{x,y}$.

Note that $k(x,y)$ is a decreasing function of $d(x,y)$ and so it satisfies
the maximal principle. Indeed, up to an ${\mathrm{SU}}(2)$ change of
coordinates we may assume that $x,y$ are such that $x=-y=\epsilon i$ with
$0<\epsilon\leq 1$. Then $d(x,y)=\frac{2\epsilon}{1+\epsilon^{2}}$ and a
minimizing matrix $B\in K_{x,y}$ is of the form
$R_{\lambda}D_{\epsilon^{-1/2}}$, where $\lambda\in\mathbb{R}$ and $D_{\mu}$
is the diagonal matrix $\begin{pmatrix}\mu&0\\\ 0&\mu^{-1}\end{pmatrix}$, so
that $k(x,y)=\epsilon^{-1}$. In particular, $1\leq d(x,y)k(x,y)\leq 2$.

Notice that at a minimizing matrix $B\in K_{x,y}$ we have

(18) $\left\|B^{-1}\cdot\left(\begin{matrix}\pm i\\\
1\end{matrix}\right)\right\|^{2}=k(x,y)+k(x,y)^{-1}.$

######  Lemma 12.

Let $u,s:\mathbb{D}\to\overline{\mathbb{C}}$ be holomorphic functions such
that $k_{\mathbb{D}}(u,s)=\sup_{z\in\mathbb{D}}k(u(z),s(z))<\infty$. Then
there exists $B:\mathbb{D}\to\mathrm{SL}(2,\mathbb{C})$ holomorphic with
$B(z)\in K_{u(z),s(z)}$,
$k_{\mathbb{D}}(u,s)=\|B\|^{2}_{\mathbb{D}}=\sup_{z\in\mathbb{D}}\|B(z)\|^{2}$,
and such that the non-tangential limits $B(z_{0})$,
$z_{0}\in\partial\mathbb{D}$ are minimizers.

###### Proof.

Let $U\in{\mathrm{SU}}(2)$ be such that $U\cdot\infty=i$ and $U\cdot 0=-i$.
For $t<1$, let $u_{t}(z)=u(tz)$, $s_{t}(z)=s(tz)$. Choose $B_{t}(z)\in
K_{u_{t}(z),s_{t}(z)}$ depending holomorphically on $z$ in a neighborhood of
$\overline{\mathbb{D}}$.666Write $u_{t}=a_{t}/c_{t}$ as a quotient of
holomorphic functions without common zeros, and similarly write
$s_{t}=b_{t}/d_{t}$. Then one can define $B_{t}$ so that
$B_{t}^{-1}U=\frac{1}{(a_{t}d_{t}-b_{t}c_{t})^{1/2}}\left(\begin{matrix}a_{t}&b_{t}\\\
c_{t}&d_{t}\end{matrix}\right)$. Notice that
$U^{-1}B_{t}(z)H_{u_{t}(z),s_{t}(z)}$ is a circle centered on $0$. Let
$r_{t}(z)$ be its radius. Using the Hilbert transform, let
$\nu_{t}(z):\mathbb{D}\to\mathbb{C}$ be a holomorphic function on
$\mathbb{D}$, smooth up to the boundary, such that $\Re\nu_{t}(z)=-\ln
r_{t}(z)$ for $z\in\partial\mathbb{D}$. Then
$B^{\prime}_{t}=UD_{e^{\nu_{t}/2}}U^{-1}B_{t}$ is such that
$B^{\prime}_{t}(z)\in K_{u_{t}(z),s_{t}(z)}$ for $z\in\mathbb{D}$ and
$B^{\prime}_{t}(z)$ takes $H_{u_{t}(z),s_{t}(z)}$ to $\overline{\mathbb{R}}$
for $z\in\partial\mathbb{D}$. We conclude that

(19) $\|B^{\prime}_{t}(z)\|^{2}=k(u_{t}(z),s_{t}(z))\quad\mathrm{for}\quad
z\in\partial\mathbb{D},$

so that $\sup_{z\in\partial\mathbb{D}}\|B^{\prime}_{t}(z)\|^{2}\leq
k_{\mathbb{D}}(u,s)$. Taking a limit of $B^{\prime}_{t}$ as $t\to 1$, we get
some $B:\mathbb{D}\to\mathrm{SL}(2,\mathbb{C})$ holomorphic such that
$\|B\|_{\mathbb{D}}^{2}\leq k_{\mathbb{D}}(u,s)$. The existence of the non-
tangential limits of $B$ implies the existence of the non-tangential limits of
$u$ and $s$. Then we get

(20) $\int_{0}^{1}\|B(e^{2\pi ix})\|^{2}dx\leq\limsup_{t\to
1}\int_{0}^{1}\|B^{\prime}_{t}(e^{2\pi ix})\|^{2}dx=\int_{0}^{1}k(u(e^{2\pi
ix}),s(e^{2\pi ix}))dx.$

So the minimizing property $\|B(z)\|^{2}\geq k(u(z),s(z))$ gives
$\|B(z)\|^{2}=k(u(z),s(z))$ for almost every $z\in\partial\mathbb{D}$, that
is, $B(z)$ is a minimizer for almost every $z\in\partial\mathbb{D}$. ∎

######  Lemma 13.

Let $u$ and $s$ be as in Lemma 12, and let
$B,\tilde{B}:\mathbb{D}\to\mathrm{SL}(2,\mathbb{C})$ be holomorphic with
$B(z),\tilde{B}(z)\in K_{u(z),s(z)}$,
$k_{\mathbb{D}}(u,s)=\|B\|^{2}_{\mathbb{D}}=\|\tilde{B}\|_{\mathbb{D}}^{2}$
and the non-tangential limits of $B$ and $\tilde{B}$ are minimizers. Then
$\tilde{B}=R_{\lambda}B$ for some $\lambda\in\mathbb{R}$.

###### Proof.

Since $B(z),\tilde{B}(z)\in K_{u(z),s(z)}$, $\tilde{B}(z)=R_{\theta(z)}B(z)$
for some holomorphic function $\lambda:\mathbb{D}\to\mathbb{C}$. Since
$R_{\theta(z)}$ is bounded, $\Im\lambda(z)$ is bounded as well, so the non-
tangential limits $\lambda(z)$ exist for almost every
$z\in\partial\mathbb{D}$. For $z\in\partial\mathbb{D}$ we must have
$\lambda(z)\in\mathbb{R}$. Since $\Im\lambda$ is a bounded harmonic function
in $\mathbb{D}$ which vanishes almost everywhere in $\partial\mathbb{D}$, we
conclude that $\Im\theta$ vanishes over $\mathbb{D}$, so $\lambda(z)$ is a
real constant. ∎

######  Lemma 14.

Let $u,s:\mathbb{R}/\mathbb{Z}\to\overline{\mathbb{C}}$ have holomorphic
extensions to $|\Im z|<\delta$ with $k_{\delta}(u,s)=\sup_{|\Im
z|<\delta}k(u(z),s(z))<\infty$. Then there exists $B\in
C^{\omega}_{\delta}(\mathbb{R}/\mathbb{Z},\mathrm{SL}(2,\mathbb{C}))$ such
that $B(z)\in K_{u(z),s(z)}$ for $|\Im z|<\delta$ and $\|B\|_{\delta}^{2}\leq
C_{\delta}k_{\delta}(u,s)$. Moreover, for almost every $z_{0}$ with $|\Im
z_{0}|=\delta$ we have

(21)
$C_{\delta}^{-1}k(u(z_{0}),s(z_{0}))^{1/2}\leq\left\|B(z_{0})^{-1}\cdot\left(\begin{matrix}\pm
i\\\ 1\end{matrix}\right)\right\|\leq C_{\delta}k(u(z_{0}),s(z_{0}))^{1/2},$

where $u,s,B$ are defined at $\\{|\Im z_{0}|=\delta\\}$ through non-tangential
limits. Moreover, if $u(z)\in{\mathbb{H}}$ for $\Im z=0$ and
$s(z)=\overline{u(\overline{z})}$ then $B$ can be taken real-symmetric.

###### Proof.

Let $B^{\prime}$ be a holomorphic function with $B^{\prime}(z)\in
K_{u(z),s(z)}$ defined on the strip $\\{|\Im z|<\delta\\}\subset\mathbb{C}$,
which is given by Lemma 12 after changing coordinates from the disk to the
strip. It satisfies $\|B^{\prime}\|_{\delta}=k_{\delta}(u,s)$. Then $z\mapsto
B^{\prime}(z+1)$ has the same properties, so by the previous lemma we have
$B^{\prime}(z+1)=R_{\lambda}B^{\prime}(z)$ for some $\lambda\in[-1/2,1/2)$.
Then $B(z)=R_{-\lambda z}B^{\prime}(z)$ is $1$-periodic holomorphic and
satisfies $B(z)\in K_{u(z),s(z)}$ and $\|B\|_{\delta}^{2}\leq
C_{\delta}k_{\delta}(u,s)$.

Since $B^{\prime}(z_{0})$ is minimizing at the boundary, (21) follows.

For the last statement, note that (still by the previous lemma)
$\overline{B^{\prime}(\overline{z})}=R_{\lambda^{\prime}}B^{\prime}(z)$ for
some $\lambda^{\prime}\in[-1/2,1/2)$. Moreover, if $\Im z=0$ we have
$B^{\prime}(z)=R_{\lambda^{\prime\prime}(z)}B^{\prime\prime}(z)$, where
$B^{\prime\prime}(z)$ is any $\mathrm{SL}(2,\mathbb{R})$ matrix sending $u(z)$
to $i$ and $s(z)$ to $-i$. Then for $\Im z=0$ we have
$R_{\overline{\lambda}^{\prime\prime}(z)}=R_{\lambda^{\prime}}R_{\lambda^{\prime\prime}(z)}$,
so $\Im\lambda^{\prime\prime}(z)=0$ and $\lambda^{\prime}=0$, that is
$B^{\prime}$ is real-symmetric, and the resulting $B$ will be real-symmetric
as well. ∎

##  6\. Complex conjugacies

######  Theorem 7.

Let $\alpha\in\mathbb{R}\smallsetminus\mathbb{Q}$ be such $\beta=0$ and let
$A\in
C^{\omega}_{\epsilon_{0}}(\mathbb{R}/\mathbb{Z},\mathrm{SL}(2,\mathbb{R}))$ be
such that $L(\alpha,A(x+ti))=0$ for $|t|<\epsilon_{0}$. Let
$0<\epsilon<\epsilon_{0}$ and let
$\kappa_{0}=\min\\{\frac{1}{2},1-\frac{\epsilon}{\epsilon_{0}}\\}$. Then for
every $\theta>0$ sufficiently small, there exists $B\in
C^{\omega}_{\epsilon}(\mathbb{R}/\mathbb{Z},\mathrm{SL}(2,\mathbb{C}))$ such
that $\|B\|_{\epsilon}^{2}\leq\theta^{-1+\kappa_{0}-o(1)}$ and
$B(x+\alpha)R_{-i\theta}A(x)B(x)^{-1}=R_{\lambda}$ for some
$\lambda\in\mathbb{C}$ with
$\Im\lambda=-\frac{\theta^{\kappa(\theta)}}{2\pi}$. Moreover,
$\kappa(\theta)>\frac{1}{2}-o(1)$, so $\tilde{A}(x)=B(x+\alpha)A(x)B(x)^{-1}$
satisfies
$\|\tilde{A}-R_{\Re\lambda}\|_{\epsilon}\leq\theta^{\kappa_{0}-o(1)}$.

###### Proof.

Fix $0<\kappa<\kappa^{\prime}<\kappa_{0}$ and
$0<\epsilon<\epsilon^{\prime}<\epsilon_{0}$ such that
$\kappa^{\prime}<1-\frac{\epsilon^{\prime}}{\epsilon_{0}}$. For $\theta>0$
small, $(\alpha,R_{-i\theta}A(x+ti))$ is uniformly hyperbolic for
$|t|<\epsilon^{\prime}$ and by the first claim of Theorem 6 the unstable and
stable directions $u_{\theta},s_{\theta}$ form an angle at least
$\theta^{1-\kappa^{\prime}}$. By Lemma 14, there exists $B^{\prime}\in
C^{\omega}_{\epsilon^{\prime}}(\mathbb{R}/\mathbb{Z},\mathrm{SL}(2,\mathbb{C}))$
with $\|B^{\prime}\|_{\epsilon^{\prime}}^{2}\leq
C\theta^{-1+\kappa^{\prime}}$, such that $B^{\prime}\cdot u_{\theta}=i$ and
$B^{\prime}\cdot s_{\theta}=-i$. Let
$A^{\prime}(x)=B^{\prime}(x+\alpha)R_{-i\theta}A(x)B^{\prime}(x)^{-1}$.

Since the angle between $u_{\theta}$ and $s_{\theta}$ at points $x$ and
$x+\alpha$ is of the same order, (21) yields $\|A^{\prime}(x)\|\leq C$ at
$|\Im x|=\epsilon^{\prime}$ (take the vectors $\begin{pmatrix}\pm i\\\
1\end{pmatrix}$ and apply $A^{\prime}(x)$). Since $A^{\prime}\cdot i=i$ and
$A^{\prime}\cdot-i=-i$ we have that $A^{\prime}(x)=R_{\psi(x)}$ for some
function $\psi:\mathbb{R}/\mathbb{Z}\to\mathbb{C}/\mathbb{Z}$ with a
holomorphic extension to $|\Im x|<\epsilon^{\prime}$ such that $\Im\psi$ is
bounded. Thus for some $k\in\mathbb{Z}$, $\psi(x)=kx+\phi(x)$ for some
$\phi:\mathbb{R}/\mathbb{Z}\to\mathbb{C}$ which admits a bounded holomorphic
extension to $|\Im x|<\epsilon^{\prime\prime}$, where
$\epsilon<\epsilon^{\prime\prime}<\epsilon^{\prime}$. Since
$L(\alpha,R_{-i\theta}A(x+ti))=L(\alpha,A^{\prime}(x+ti))$ is constant
$L_{\theta}=\theta^{\kappa(\theta)}$, it follows that $k=0$ and
$\Im\lambda=-\frac{\theta^{\kappa(\theta)}}{2\pi}$, where $\lambda$ is the
average of $\phi$. Using that $\beta=0$ we can solve the cohomological
equation $\phi(x)=w(x+\alpha)-w(x)+\lambda$ with $\|w\|_{\epsilon}=O(1)$. Let
$B(x)=R_{-wx}B^{\prime}(x)$. Then
$B(x+\alpha)R_{-i\theta}A(x)B(x)^{-1}=R_{\lambda}$ with
$\|B\|_{\epsilon}^{2}=o(\theta^{-1+\kappa})$.

It remains to show that $\kappa(\theta)>\frac{1}{2}-o(1)$. Assume that
$\kappa(\theta)<\frac{1}{2}-\delta$. Then we can argue using the second claim
of Theorem 6 to produce an improved $B$ with the bound
$\|B\|_{\epsilon}^{2}\leq\theta^{-\kappa(\theta)-o(1)}$ conjugating
$R_{-i\theta}A$ to some $R_{\lambda}$ with
$\Im\lambda=-\frac{\theta^{\kappa(\theta)}}{2\pi}$. Then
$\tilde{A}(x)=B(x+\alpha)A(x)B(x)^{-1}$ satisfies
$\|\tilde{A}-R_{\lambda}\|_{\epsilon}\leq\theta^{\frac{1}{2}+\delta-o(1)}=o(-\Im\lambda)$.
It follows that $\tilde{A}$ sends ${\mathbb{H}}$ into a precompact subset of
${\mathbb{H}}$. So $(\alpha,\tilde{A})$ is uniformly hyperbolic, which is
impossible since it is conjugated to $(\alpha,A)$. ∎

Let us note the following consequence:

######  Lemma 15.

Let $\alpha\in\mathbb{R}\smallsetminus\mathbb{Q}$ be such that $\beta=0$ and
let $A\in
C^{\omega}_{\epsilon_{0}}(\mathbb{R}/\mathbb{Z},\mathrm{SL}(2,\mathbb{R}))$ be
such that $L(\alpha,A(x+ti))=0$ for $|t|<\epsilon_{0}$. Let
$0<\epsilon<\epsilon_{0}$ and let
$\kappa_{0}=\min\\{\frac{1}{2},1-\frac{\epsilon}{\epsilon_{0}}\\}$. Then

(22) $\|A_{n}\|_{\epsilon}\leq n^{\frac{1}{\kappa_{0}}-1+o(1)}.$

###### Proof.

Let $0<\kappa<\kappa_{0}$ and, take $\theta=n^{-1/\kappa}$ and use Theorem 7
to get $\|\tilde{A}_{n}\|=O(1)$ so that $\|A_{n}\|=O(\theta^{-1+\kappa})$. ∎

##  7\. Real conjugacies

Let $\theta>0$ be small and let $u=u_{\theta}$ and $s=s_{\theta}$ be the
unstable and stable directions of $(\alpha,R_{-i\theta}A)$, defined through a
band of size $\epsilon_{0}-o(1)$.

Take $0<\epsilon<\epsilon_{0}$ and let $B$ be given by Lemma 14, so that
$B\cdot u=i$ and $B\cdot s=-i$. Then
$\|B\|_{\epsilon_{0}-o(1)}^{2}\leq\theta^{-\kappa(\theta)-o(1)}$ by Theorem 6.
Let $u^{\prime}(z)=\overline{u(\overline{z})}$ and
$s^{\prime}(z)=\overline{s(\overline{z})}$.

We have previously constructed conjugacies by straightening up the pair
$(u,s)$, but this was not real-symmetric. In order to do a real symmetric
construction, we will make use of the additional directions $u^{\prime}$ and
$s^{\prime}$.

Let $U=B^{-1}\cdot\begin{pmatrix}i\\\ 1\end{pmatrix}$ and
$S=B^{-1}\cdot\begin{pmatrix}-i\\\ 1\end{pmatrix}$.
$U^{\prime}(z)=\overline{U(\overline{z})}$ and
$S^{\prime}(z)=\overline{S(\overline{z})}$. Let $\Delta$ be the determinant of
the matrix with columns $U$ and $U^{\prime}$. Let also
$\omega(z)=\|U(z)\|\|U^{\prime}(z)\|$ so that
$|\Delta(z)|/\omega(z)=d(u(z),u^{\prime}(z))$. Note that
$\omega(z)\leq\theta^{-\kappa(\theta)-o(1)}$ over $|\Im z|<\epsilon_{0}-o(1)$.

We note that at $\Im z=0$, $u$ and $u^{\prime}$ must have distance at least of
order $\theta$, by Lemma 7. This can be improved for many $z$: by Lemma 11,
$d(u(z),u^{\prime}(z))>\theta^{1-\kappa(\theta)+o(1)}$ over a set of
probability $1-o(1)$. (Recall that by Theorem 7, we have $\kappa(\theta)\geq
1/2-o(1)$.)

For non real $z$ we have the following estimate:

######  Lemma 16.

Let $z_{*}$ be such that $|\Im z_{*}|<\epsilon$. Then

(23) $\sup_{\Im z=\Im
z_{*}}|\Delta(z)|\leq\theta+\max\\{\theta,d(u(z_{*}),u^{\prime}(z_{*}))\\}\theta^{-o(1)}\sup_{\Im
z=\Im z_{*}}\omega(z).$

###### Proof.

Notice that both $u$ and $u^{\prime}$ are $\theta$-almost invariant under $A$,
since we have $R_{-i\theta}A(z)\cdot u(z)=u(z+\alpha)$ and
$R_{i\theta}A(z)\cdot u^{\prime}(z)=u^{\prime}(z+\alpha)$. Fix $\delta>0$
small. By Lemma 15,

(24)
$d(u(z_{*}+k\alpha),u^{\prime}(z_{*}+k\alpha))\leq\theta^{-\delta}d(u(z_{*}),u^{\prime}(z_{*}))+k\theta^{1-\delta},\quad\mathrm{for}\quad
0\leq k\leq\theta^{-c\delta}$

for some $0<c<1$ depending on $\epsilon$ and $\epsilon_{0}$.777We have
$d(A_{k}(z_{*})\cdot u(z_{*}),A_{k}(z_{*})\cdot u^{\prime}(z_{*}))\leq
C\|A_{k}(z_{*})\|^{2}d(u(z_{*}),u^{\prime}(z_{*}))$. By almost invariance
$d(A_{k-j+1}(z_{*}+(j-1)\alpha)\cdot
u(z_{*}+(j-1)\alpha),A_{k-j}(z_{*}+j\alpha)\cdot u(z_{*}+j\alpha))\leq
C\|A_{k-j}(z_{*}+j\alpha)\|^{2}\theta$ and
$d(A_{k-j+1}(z_{*}+(j-1)\alpha)\cdot
u^{\prime}(z_{*}+(j-1)\alpha),A_{k-j}(z_{*}+j\alpha)\cdot
u^{\prime}(z_{*}+j\alpha))\leq C\|A_{k-j}(z_{*}+j\alpha)\|^{2}\theta$.

We now use a Lagrange interpolation argument. Let
$\Delta_{\theta}(x)=\sum_{|k|\leq N_{\theta}}\hat{\Delta}_{k}e^{2\pi ikx}$
where $N_{\theta}\geq-\ln\theta$ is chosen minimal so that
$\|\Delta-\Delta_{\theta}\|_{\epsilon}<\theta^{2}$. Since
$\|\Delta\|_{\epsilon_{0}-o(1)}\leq\theta^{-\kappa(\theta)-o(1)}$, we have
$N_{\theta}=O(-\ln\theta)$, so $2N_{\theta}\leq\theta^{-c\delta}$. In
particular (24) implies

(25)
$|\Delta(z_{*}+k\alpha)|\leq(\theta^{-2\delta}d(u(z_{*}),u^{\prime}(z_{*}))+\theta^{1-2\delta})\sup_{\Im
z=\Im z_{*}}\omega(z).$

Let $q_{n}$ be maximal with $q_{n}\leq 5N_{\theta}$, and let $r\geq 1$ be
minimal such that $rq_{n}\geq 2N_{\theta}+2$. In particular $rq_{n}<q_{n+1}$.
By Theorem 6.1 of [AJ], we have

(26) $\sup_{\Im z=\Im z_{*}}|\Delta_{\theta}(z)|\leq Cq_{n+1}^{Cr}\sup_{0\leq
k\leq 2N_{\theta}}|\Delta_{\theta}(z_{*}+k\alpha)|.$

Since $\beta=0$, $Cq_{n+1}^{Cr}\leq Cq_{n+1}^{C\frac{5N_{\theta}}{q_{n}}}\leq
e^{o(N_{\theta})}\leq\theta^{-o(1)}$. It follows that

(27) $\sup_{\Im z=\Im
z_{*}}|\Delta_{\theta}(z)|\leq\theta^{-o(1)}(\theta^{2}+\sup_{0\leq k\leq
2N_{\theta}}|\Delta(z_{*}+k\alpha)|),$

so that

(28) $\sup_{\Im z=\Im
z_{*}}|\Delta(z)|\leq\theta^{2}+\theta^{-o(1)}(\theta^{2}+\sup_{0\leq k\leq
2N_{\theta}}|\Delta(z_{*}+k\alpha)|)$

which together with (25) implies the desired estimate. ∎

######  Lemma 17.

For $0<\epsilon^{\prime}<\epsilon$, if $\inf_{|\Im
z|<\epsilon}d(u(z),u^{\prime}(z))<\theta^{l}$ with $0\leq l\leq 1$ then
$\|U\|^{2}_{0}\leq\max\\{\theta^{\kappa(\theta)-o(1)},\theta^{-(1-\frac{\epsilon}{\epsilon^{\prime}})\kappa(\theta)-\frac{\epsilon}{\epsilon^{\prime}}(1-l)-o(1)}\\}$.
In particular, if $\|U\|_{0}^{2}\geq 2$ we have $l\leq
1-(1-\frac{\epsilon^{\prime}}{\epsilon})\kappa(\theta)+o(1)$.

###### Proof.

Define $\gamma_{t}$, $0\leq t\leq\epsilon$ so that
$\theta^{-\gamma_{t}}=\sup_{|\Im z|\leq t}\omega(z)$. In particular
$\|U\|_{0}^{2}=\theta^{-\gamma_{0}}$. We have
$\gamma_{\epsilon}\leq\kappa(\theta)+o(1)$ and subharmonicity gives
$\gamma_{\epsilon^{\prime}}\leq\frac{\epsilon^{\prime}}{\epsilon}\kappa(\theta)+(1-\frac{\epsilon^{\prime}}{\epsilon})\gamma_{0}+o(1)$.
Noting that $i\Delta$ is real-symmetric, Lemma 16 gives
$\|\Delta\|_{0}\leq\theta+\theta^{l-\gamma_{\epsilon^{\prime}}-o(1)}$. By
Lemma 11, we conclude that

(29)
$\omega(z)\leq\theta^{-1+\kappa(\theta)-o(1)}|\Delta(z)|\leq\theta^{\kappa(\theta)-o(1)}+\theta^{l-\gamma_{\epsilon^{\prime}}-1+\kappa(\theta)-o(1)}$

for $\Im z=0$ with probability $1-o(1)$. Since there is also an upper bound
$\omega(z)\leq\theta^{-\kappa(\theta)-o(1)}$ through $|\Im z|<\epsilon$, this
implies that (29) in fact holds for all $\Im z=0$ (subharmonicity). Thus
$\gamma_{0}\leq\max\\{-\kappa(\theta),\gamma_{\epsilon^{\prime}}+1-l-\kappa(\theta)\\}+o(1)$.
In particular, if $\gamma_{0}\geq-\kappa(\theta)+o(1)$ we have
$\gamma_{0}\leq\frac{\epsilon^{\prime}}{\epsilon}\kappa(\theta)+(1-\frac{\epsilon^{\prime}}{\epsilon})\gamma_{0}+1-l-\kappa(\theta)+o(1)$,
so that
$\gamma_{0}\leq(1-\frac{\epsilon}{\epsilon^{\prime}})\kappa(\theta)+\frac{\epsilon}{\epsilon^{\prime}}(1-l)+o(1)$.
∎

######  Theorem 8.

Let $\alpha\in\mathbb{R}\smallsetminus\mathbb{Q}$ be such $\beta=0$ and let
$A\in
C^{\omega}_{\epsilon_{0}}(\mathbb{R}/\mathbb{Z},\mathrm{SL}(2,\mathbb{R}))$ be
such that $L(\alpha,A(x+ti))=0$ for $|t|<\epsilon_{0}$. Let
$0<\epsilon^{\prime}<\epsilon_{0}$ and let
$\kappa^{\prime}_{0}=\frac{1}{2}(1-\frac{\epsilon^{\prime}}{\epsilon_{0}})$.
Then for every $\theta>0$ sufficiently small, there exists $B_{r}\in
C^{\omega}_{\epsilon^{\prime}}(\mathbb{R}/\mathbb{Z},\mathrm{SL}(2,\mathbb{R}))$
such that
$\|B_{r}\|^{2}_{\epsilon^{\prime}}\leq\theta^{-1+\kappa^{\prime}_{0}-o(1)}$
and $\tilde{A}_{r}(x)=B_{r}(x+\alpha)A(x)B_{r}(x)^{-1}$ satisfies
$\|\tilde{A}_{r}-R_{\lambda}\|_{\epsilon^{\prime}}\leq\theta^{\kappa^{\prime}_{0}-o(1)}$
for some $\lambda\in\mathbb{R}$.

###### Proof.

Fix $0<\kappa^{\prime}<\kappa^{\prime}_{0}$, take $\epsilon^{\prime\prime}$
slightly larger than $\epsilon^{\prime}$ and $\kappa^{\prime\prime}$ slightly
larger than $\kappa^{\prime}$. Take $0<\epsilon<\epsilon_{0}$ close to
$\epsilon_{0}$. Assume first that $\|U\|_{0}^{2}\geq 2$. Since
$\kappa(\theta)\geq\frac{1}{2}-o(1)$, Lemma 17 implies that
$d(u(z),u^{\prime}(z))\geq\theta^{1-\kappa^{\prime\prime}}$ for $|\Im
z|<\epsilon^{\prime\prime}$.

We can then conclude following the argument of the proof of Theorem 7.
Consider $B^{\prime}_{r}\in
C^{\omega}_{\epsilon^{\prime\prime}}(\mathbb{R}/\mathbb{Z},\mathrm{SL}(2,\mathbb{R}))$
such that $B^{\prime}_{r}\cdot u=i$ and $B^{\prime}_{r}\cdot u^{\prime}=-i$.
This matrix will satisfy the bound
$\|B_{r}^{\prime}\|_{\epsilon^{\prime\prime}}^{2}=O(\theta^{-1+\kappa^{\prime\prime}})$.
Since $u$ and $u^{\prime}$ are $\theta$-almost invariant under $A$,
$A^{\prime}(z)=B_{r}^{\prime}(z+\alpha)A(z)B_{r}(z)^{-1}$ takes $i$ and $-i$
somewhere $\theta^{\kappa^{\prime\prime}}$ close to themselves. So
$A^{\prime}(z)=A^{\prime\prime}(z)R_{\psi(z)}$ where $A^{\prime\prime}$ and
$\psi$ (which takes values in $\mathbb{C}/\mathbb{Z}$) are real-symmetric and
$\|A^{\prime\prime}-\mathit{id}\|=O(\theta^{\kappa^{\prime\prime}})$. Write
$\psi(x)=kx+\phi(x)$ with $k\in\mathbb{Z}$ and $\phi$ taking values in
$\mathbb{C}$. Solve the cohomological equation
$\phi(x)=w(x+\alpha)-w(x)+\lambda$ with $w$ real-symmetric and bounded. Then
$R_{w(z+\alpha)}A^{\prime\prime}(z)R_{\psi(z)}R_{-w(z)}=R_{w(z+\alpha)}A^{\prime\prime}(z)R_{-w(z+\alpha)}R_{kz+\lambda}$.
Since the Lyapunov exponent vanishes through the band, $k=0$. The result then
follows by taking $B_{r}(z)=R_{w(z)}B^{\prime}_{r}(z)$.

Assume now that $\|U\|^{2}_{0}<2$. Then $\|S\|^{2}_{0}\geq 2$ (since $B$ has
determinant $1$), so we can do the whole argument with $s$ and $s^{\prime}$
instead of $u$ and $u^{\prime}$ to conclude. ∎

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To illustrate the function approximation via LRE, we place a set of “grid
points” on a Cartesian mesh in $[-0.5:0.5]^{3}$ with a spacing of $\Delta_{\rm
g}=0.07$. Our “particles” are originally placed on a Cartesian grid between
$[-0.9:0.9]\times[-0.9:0.9]\times[-0.9:0.9]$ with a spacing of $\Delta_{\rm
p}=0.1$, but then slightly randomized by adding to each particle position
component a random number in $[-0.01,0.01]$. This regular but not perfect
distribution is meant to mimmick an SPH particle distribution. Each of the
particles is assigned a smoothing length of $h=0.1$ and a function value
according to

$f(x,y,z)=10-\sqrt{x^{2}+y^{2}+z^{2}}.$ (99)

We then calculate the LRE approximations of a specified polynomial order at
each grid point and we measure at the grid points the relative error of the
LRE estimate compared with the original function $f$, see Figure A1, left
panel.

Figure A1: Relative errors of the function approximation (left) and the
gradient of the function approximation (right) as function of radius of the
grid points.

The lowest polynomial order already provides estimates below $1\%$ error
(black), but with a large spread of accuracies, while the linear order
provides estimates of the same order, but with substantially less noise (red).
The transition from linear to quadratic polynomials (blue) provides a serious
enhancement of the accuracy, while cubic (orange) and quartic polynomials
(magenta) only marginally improve the results further.
The situation is similar for the gradient estimates, where we use

${\rm err}\equiv\frac{|\;|(\nabla f)_{\rm num}|-|(\nabla f)_{\rm
theo}|\;|}{|(\nabla f)_{\rm theo}|}.$ (100)

as error measure and $(\nabla f)_{\rm num}$ is the numerically calculated
gradient estimate while $(\nabla f)_{\rm theo}$ is the exact result. Also
here, the overall scale between the lowest (linear) and next-to-lowest order
(quadratic) is similar, but the quadratic results are much less noisy. Again
we find a substantial improvement going to the next order (cubic), but no more
substantial gain when further increasing the polynomial order.

Figure A2: Condition number for (not) re-scaled basis functions for polynomial
orders from one to four. Note that for order four the re-scaling improves the
condition number by as much as five orders of magnitude.

We also want to briefly illustrate the effect of re-scaling the moment matrix
with appropriate powers of the length $l_{p}=(\nu_{p}/N_{p})^{1/3}$ to ensure
that all matrix elements have the same dimension. In practice, this is
achieved by de-dimensionalizing the shifted basis functions, e.g. $\Delta^{\rm
G}x\rightarrow\Delta^{\rm G}x/l_{p}$, $\Delta^{\rm G}x\Delta^{\rm
G}y\rightarrow\Delta^{\rm G}x\Delta^{\rm G}y/l_{p}^{2}$ etc. We show in Figure
A2 the condition numbers, $\mathcal{C}\equiv||MM^{-1}||$, a measure for how
close a matrix is to being singular, for the above experiment, once for the
straight forward and once for the re-scaled version. The condition numbers in
the re-scaled case improve dramatically, e.g. by five orders of magnitude in
the case of quartic basis functions (orange curves in Figure A2). We do,
however, not see any noteworthy improvement of the accuracy which we interpret
as a success of the well-working Singular Value Decomposition. Nevertheless,
since the re-scaling is essentially free of computational cost, we always use
re-scaled basis functions.


# Truncated second main theorem for non-Archimedean meromorphic maps

Si Duc Quang  Department of Mathematics
Hanoi National University of Education
136-Xuan Thuy, Cau Giay, Hanoi, Vietnam<EMAIL_ADDRESS>
###### Abstract.

Let ${\mathbb{F}}$ be an algebraically closed field of characteristic $p\geq
0$, which is complete with respect to a non-Archimedean absolute value. Let
$V$ be a projective subvariety of ${\mathbb{P}}^{M}({\mathbb{F}})$. In this
paper, we will prove some second main theorems for non-Archimedean meromorphic
maps of ${\mathbb{F}}^{m}$ into $V$ intersecting a family of hypersurfaces in
$N-$subgeneral position with truncated counting functions.

###### Key words and phrases:

non-Archimedean, second main theorem, meromorphic mapping, Nevanlinna,
hypersurface, subgeneral position

###### 2020 Mathematics Subject Classification:

Primary 11S80, 11J97; Secondary 32H30

##  1\. Introduction and Main results

Let ${\mathbb{F}}$ be an algebraically closed field of characteristic $p\geq
0$, which is complete with respect to a non-Archimedean absolute value. Let
$N\geq n$ and $q\geq N+1.$ Let $H_{1},\ldots,H_{q}$ be hyperplanes in
${\mathbb{P}}^{n}(\mathbb{F}).$ The family of hyperplanes
$\\{H_{1}\\}_{i=1}^{q}$ is said to be in $N$-subgeneral position in
${\mathbb{P}}^{n}(\mathbb{F})$ if $H_{j_{0}}\cap\cdots\cap
H_{j_{N}}=\varnothing$ for every $1\leq j_{0}<\cdots<j_{N}\leq q.$

In 2017, Yan [6] proved a truncated second main theorem for a non-Archimedean
meromorphic map into ${\mathbb{P}}^{n}(\mathbb{F})$ with a family of
hyperplanes in subgeneral position. With the standart notations on the
Nevanlinna theory for non-Archimedean meromorphic maps, his result is stated
as follows.

Theorem A (cf. [6, Theorem 4.6]) Let ${\mathbb{F}}$ be an algebraically closed
field of characteristic $p\geq 0$, which is complete with respect to a non-
Archimedean absolute value. Let
$f:{\mathbb{F}}^{m}\rightarrow{\mathbb{P}}^{n}({\mathbb{F}})$ be a linearly
non-degenerate non-Archimedean meromorphic map with index of independence $s$
and $\mathrm{rank}f=k$. Let $H_{1},\ldots,H_{q}$ be hyperplanes in
${\mathbb{P}}^{n}({\mathbb{F}})$ in $N$-subgeneral position $(N\geq n)$. Then,
for all $r\geq 1$,

$(q-2N+n-1)T_{f}(r)\leq\sum_{i=1}^{q}N^{(a)}_{f}(H_{i},r)-\frac{N+1}{n+1}\log
r+O(1),$

where

$a=\begin{cases}p^{s-1}(n-k+1)&\text{ if }p>0,\\\ n-k+1&\text{ if
}p=0.\end{cases}$

Here, the index of independence $s$ and the $\mathrm{rank}f$ are defined in
Section 2 (Definition 3).

Also, in 2017, An and Quang [2] proved a truncated second main theorem for
meromorphic mappings from ${\mathbb{C}}^{m}$ into a projective variety
$V\subset{\mathbb{P}}^{M}({\mathbb{C}})$ with hypersurfaces. Motivated by the
methods of Yan [6] and An-Quang [2], our aim in this article is to generalize
Theorem A to the case where the map $f$ is from ${\mathbb{F}}^{m}$ into an
arbitrary projective variety $V$ of dimension $n$ in
${\mathbb{P}}^{M}({\mathbb{F}})$ and the hyperplanes are replaced by
hypersurfaces of ${\mathbb{P}}^{M}({\mathbb{F}})$ in $N$-subgeneral position
with respect to $V$.

Firstly, we give the following definitions.

Definition B. Let $V$ be a projective subvariety of
${\mathbb{P}}^{M}(\mathbb{F})$ of dimension $n\ (n\leq M)$. Let
$Q_{1},\ldots,Q_{q}\ (q\geq n+1)$ be $q$ hypersurfaces in
${\mathbb{P}}^{M}(\mathbb{F})$. The family of hypersurfaces
$\\{Q_{i}\\}_{i=1}^{q}$ is said to be in $N$-subgeneral position with respect
to $V$ if

$V\cap(\bigcap_{j=1}^{N+1}Q_{i_{j}})=\varnothing\ \text{ for any }1\leq
i_{1}<\cdots<i_{N+1}\leq q.$

If $N=n$, we just say $\\{Q_{i}\\}_{i=1}^{q}$ is in general position with
respect to $V.$

Now, let $V$ be as above and let $d$ be a positive integer. We denote by
$I(V)$ the ideal of homogeneous polynomials in
$\mathbb{F}[x_{0},\ldots,x_{M}]$ defining $V$ and by $H_{d}$ the
$\mathbb{F}$-vector space of all homogeneous polynomials in
$\mathbb{F}[x_{0},\ldots,x_{M}]$ of degree $d.$ Define

$I_{d}(V):=\dfrac{H_{d}}{I(V)\cap H_{d}}\text{ and
}H_{V}(d):={\mathrm{dim}}_{{\mathbb{F}}}I_{d}(V).$

Then $H_{V}(d)$ is called the Hilbert function of $V$. Each element of
$I_{d}(V)$ which is an equivalent class of an element $Q\in H_{d},$ will be
denoted by $[Q]$,

Definition C. Let $f:{\mathbb{F}}^{m}\rightarrow V$ be a non-Archimedean
meromorphic map with a reduced representation
${\mathbf{f}}=(f_{0},\ldots,f_{M})$. We say that $f$ is degenerate over
$I_{d}(V)$ if there is $[Q]\in I_{d}(V)\setminus\\{0\\}$ such that
$Q({\mathbf{f}})\equiv 0.$ Otherwise, we say that $f$ is non-degenerate over
$I_{d}(V)$.

We will generalize Theorem A to the following.

######  Theorem 1.

Let $V$ be a projective subvariety of ${\mathbb{P}}^{M}({\mathbb{F}})$ of
dimension $n\ (n\leq M)$. Let $\\{Q_{i}\\}_{i=1}^{q}$ be hypersurfaces of
${\mathbb{P}}^{M}({\mathbb{F}})$ in $N$-subgeneral position with respect to
$V$ with $\deg Q_{i}=d_{i}\ (1\leq i\leq q)$. Let $d$ be the least common
multiple of $d_{i}^{\prime}$s. Let $f$ be a non-Archimedean meromorphic map of
${\mathbb{F}}^{m}$ into $V$, which is non-degenerate over $I_{d}(V)$ with the
$d^{th}$-index of non-degeneracy $s$ and $\mathrm{rank}f=k$. Then, for all
$r\geq 1$,

$\left(q-\dfrac{(2N+n-1)H_{d}(V)}{n+1}\right)T_{f}(r)\leq\sum_{i=1}^{q}\dfrac{1}{d_{i}}N^{(\kappa_{0})}_{f}(Q_{i},r)-\frac{N(H_{d}(V)-1)}{nd}\log
r+O(1),$

where

$\kappa_{0}=\begin{cases}p^{s-1}(H_{d}(V)-k)&\text{ if }p>0,\\\
H_{d}(V)-k&\text{ if }p=0.\end{cases}$

Here, the $d^{th}$-index of non-degeneracy $s$ is defined in Section 2
(Definition 3). Note that, in the case where
$V={\mathbb{P}}^{n}({\mathbb{C}}),d=1,H_{d}(V)=n+1$, our result will give back
Theorem A.

For the case of counting function without truncation level, we will prove the
following.

######  Theorem 2.

Let $V$ be a arbitrary projective subvariety of
${\mathbb{P}}^{M}({\mathbb{F}})$. Let $\\{Q_{i}\\}_{i=1}^{q}$ be hypersurfaces
of ${\mathbb{P}}^{M}({\mathbb{F}})$ in $N$-subgeneral position with respect to
$V$. Let $f$ be a non-constant non-Archimedean meromorphic map of
${\mathbb{F}}^{m}$ into $V$. Then, for any $r>0$,

$(q-N)T_{f}(r)\leq\sum_{i=1}^{q}\dfrac{1}{\deg Q_{i}}N_{f}(Q_{i},r)+O(1),$

where the quantity $O(1)$ depends only on $\\{Q_{i}\\}_{i=1}^{q}$.

We see that, the above result is a generalization of the previous results in
[1, 5].

##  2\. Basic notions and auxiliary results

In this section, we will recall some basic notions from Nevanlinna theory for
non-Archimedean meromorphic maps due to Cherry-Ye [3] and Yan [6].

2.1. Non-Archimedean meromorphic function. Let ${\mathbb{F}}$ be an
algebraically closed field of characteristic $p$, complete with respect to a
non-Archimedean absolute value $|\ |$. We set $\|z\|=\max_{1\leq i\leq
m}|z_{i}|$ for $z=(z_{1},\dots,z_{m})\in{\mathbb{F}}^{m}$ and define

$\displaystyle{\mathbb{B}}^{m}(r):=\\{z\in{\mathbb{F}}^{m};\|z\|<r\\}.$

For a multi-index
$\gamma=(\gamma_{1},\ldots,\gamma_{m})\in{\mathbb{Z}}^{m}_{\geq 0}$, define

$z^{\gamma}=z_{1}^{\gamma_{1}}\cdots z_{m}^{\gamma_{m}},\
|\gamma|=\gamma_{1}+\cdots+\gamma_{m},\ \gamma!=\gamma!\cdots\gamma_{m}!.$

For an analytic function $f$ on ${\mathbb{F}}^{m}$ (i.e., entire function)
given by a formal power series

$f=\sum_{\gamma}a_{\gamma}z^{\gamma}$

with $a_{\gamma}\in{\mathbb{F}}$ such that
$\lim_{|\gamma|\rightarrow\infty}|a_{\gamma}|r^{|\gamma|}=0\ (\forall
r\in{\mathbb{F}}^{*}={\mathbb{F}}\setminus\\{0\\})$, define

$|f|_{r}=\mathrm{sup}_{\gamma}|a_{\gamma}|r^{|\gamma|}.$

We denote by $\mathcal{E}_{m}$ the ring of all analytic functions on
${\mathbb{F}}^{m}$.

We define a meromorphic function $f$ on ${\mathbb{F}}^{m}$ to be the quotient
of two analytic functions $g,h\in\mathcal{E}_{m}$ such that $g$ and $h$ have
no common factors in $\mathcal{E}_{m},$ i.e., $f=\frac{g}{h}$. We define

$|f|_{r}=\frac{|g|_{r}}{|h|_{r}}.$

We denote by $\mathcal{M}_{m}$ the field of all meromorphic functions on
${\mathbb{F}}^{m}$, which is the fractional field of $\mathcal{E}_{m}$.

2.2. Derivatives and Hasse derivatives. For a meromorphic function
$f\in\mathcal{M}_{m}$ and a multi-index
$\gamma=(\gamma_{1},\ldots,\gamma_{m})$, we set

$\partial^{\gamma}f=\frac{\partial^{|\gamma|}f}{\partial
z_{1}^{\gamma_{1}}\cdots\partial z_{m}^{\gamma_{m}}}.$

Let $\alpha=(\alpha_{1},\ldots,\alpha_{m})$ and
$\beta=(\beta_{1},\ldots,\beta_{m})$ be multi-indices. We say that
$\alpha\geq\beta$ if $\alpha_{i}\geq\beta_{i}$ for all $i=1,.\ldots,m.$ If
$\alpha\geq\beta$, we define

$\alpha-\beta=(\alpha_{1}-\beta_{1},\ldots,\alpha_{m}-\beta_{m}),\binom{\alpha}{\beta}=\binom{\alpha_{1}}{\beta_{1}}\cdots\binom{\alpha_{m}}{\beta_{m}}.$

For an analytic function $f=\sum_{\alpha}a_{\alpha}z^{\alpha}$ and a multi-
index $\gamma$, we define the Hasse derivative of multi-index $\gamma$ of $f$
by

$D^{\gamma}f=\sum_{\alpha\geq\gamma}\binom{\alpha}{\gamma}a_{\alpha}z^{\alpha-\gamma}.$

We may verify that
$D^{\alpha}D^{\beta}f=\binom{\alpha+\beta}{\beta}D^{\alpha+\beta}$ for all
$f\in\mathcal{E}_{m}$. Therefore, the Hasse derivative $D$ can be extended to
meromorphic functions in the following way:

  * •

For a multi-index
$e_{i}=(0,\ldots,0,\underset{j^{th}-position}{1},0,\ldots,0)$, we set
$D^{k}_{j}f:=D^{ke_{i}}(f)$.

  * •

For a meromorphic function $f=\frac{g}{h}\ (g,h\in\mathcal{E}_{m})$, we define

$D^{e_{i}}=D^{1}_{j}f:=\frac{hD^{1}_{i}g-gD^{1}_{i}h}{h^{2}},\ j=1,\ldots,m.$

  * •

For $\gamma=(\gamma_{1},\ldots,\gamma_{m})$, we may choose a sequence of
multi-indices $\gamma=\alpha^{1}>\alpha^{2}>\cdots>\alpha^{|\gamma|}$ such
that $\alpha^{i}=\alpha^{i+1}+e_{j_{i}}\ ({j_{i}}\in\\{1,\ldots,m\\})$ for
$1\leq i\leq|\gamma|-1$ and $\alpha^{|\gamma|}=e_{j_{|\gamma|}}\
(j_{|\gamma|}\in\\{1,\ldots,m\\})$ and define

$D^{\alpha_{i}}h=\frac{1}{\binom{\alpha_{i+1}+e_{j_{i}}}{\alpha_{i+1}}}D^{e_{j_{i}}}D^{\alpha_{i+1}}h,\forall
i=|\gamma|-1,|\gamma|-2,\ldots,1.$

We summarize here the fundamental properties of the Hasse derivative from [6]
as follows:

(i) $D^{\gamma}(f+g)=D^{\gamma}f+D^{\gamma}g,\ f,g\in\mathcal{M}_{m}.$

(ii) $D^{\gamma}(fg)=\sum_{\alpha,\beta}D^{\alpha}fD^{\beta}g,\
f,g\in\mathcal{M}_{m}.$

(iii) $D^{\alpha}D^{\beta}f=\binom{\alpha+\beta}{\beta}D^{\alpha+\beta}f,\
f\in\mathcal{M}_{m}$

(iv) (Lemma on the logarithmic derivative) For $f\in\mathcal{E}_{m}$,

$|D^{\gamma}f|_{r}\leq\frac{|f|_{r}}{r^{|\gamma|}},\
|\partial^{\gamma}f|_{r}\leq\frac{|f|_{r}}{r^{|\gamma|}}.$

(v) For $f\in\mathcal{E}_{m}$ and a multi-index $\gamma$, let $P$ be an
irreducible element of $\mathcal{E}_{m}$ that divides $f$ with exact
multiplicity $e$. If $e>|\gamma|$, then $P^{e-|\gamma|}$ divides
$D^{\gamma}f$.

For each integer $k\geq 2$, let

$\mathcal{M}_{m}[k]=\\{Q\in\mathcal{M}_{m}:D^{i}_{j}Q\equiv 0\text{ for all
}0<i<k\text{ and }1\leq j\leq m\\}.$

If $F$ has characteristic $0$, then $\mathcal{M}_{m}[k]={\mathbb{F}}$ for all
$k\geq 2$. If ${\mathbb{F}}$ has characteristic $p>0$ and if $s\geq 1$ is an
integer, then $\mathcal{M}_{m}[p^{s}]$ is the fraction field of
$\mathcal{E}_{m}$, where
$\mathcal{E}_{m}[p^{s}]=\\{g^{p^{s}}:g\in\mathcal{E}_{m}\\}$ is a subring of
$\mathcal{E}_{m}$. Moreover,

$\mathcal{M}_{m}[p^{s-1}+1]=\mathcal{M}_{m}[p^{s}].$

2.3. Non-Archimedean Nevanlinna’s function.

Let $f=\sum_{\gamma}a_{\gamma}z^{\gamma}\in\mathcal{E}_{m}$ be an holomorphic
function. The counting function of zeros of $f$ is defined as follows:

$N_{f}(0,r)=n_{f}(0,0)\log r+\int_{0}^{r}(n_{f}(0,t)-n_{f}(0,0))\frac{dt}{t}\
(r>0),$

where

$n_{f}(0,r)=\mathrm{sup}\\{|\gamma|;|a_{\gamma}|r^{|\gamma|}=|f|_{r}\\}\text{
and }n_{f}(0,0)=\min\\{|\gamma|;a_{\gamma}\neq 0\\}.$

Let $f$ be a meromorphic function on ${\mathbb{F}}^{m}$. Assume that
$f=\frac{g}{h}$, where $g,h$ are holomorphic functions without common factors.
We define

$N_{f}(0,r)=N_{g}(0,r)\text{ and }N_{f}(\infty,r)=N_{h}(0,r).$

The Poisson-Jensen-Green formula (see [3, Theorem 3.1]) states that

$N_{f}(0,r)-N_{f}(\infty,r)=\log|f|_{r}+C_{f}\ \text{ for all }r>0,$

where $C_{f}$ is a constant depending on $f$ but not on $r$.

Suppose that $f\not\equiv a$ for $a\in{\mathbb{F}}$. The counting function of
$f$ with respect to the point $a$ is defined by

$N_{f}(a,r)=N_{f-a}(0,r).$

The proximity functions of $f$ with respect to $\infty$ and $a$ are defined
respectively as follows

$m_{f}(\infty,r)=\max\\{0,\log|f|_{r}\\}=\log^{+}|f|_{r}\text{ and
}m_{f}(a,r)=m_{1/(f-a)}(\infty,r).$

The characteristic function of $f$ is defined by

$T_{f}(r)=m_{f}(\infty,r)+N_{f}(\infty,r).$

Note that, if $f=\frac{g}{h}$ as above then
$T_{f}(r)=\max\\{\log|g|_{r},\log|h|_{r}\\}+O(1)$.

The first main theorem is stated as follows:

$T_{f}(r)=m_{f}(a,r)+N_{f}(a,r)+O(1)\ (\forall r>0).$

2.4. Truncated counting function.

Let $f\in\mathcal{E}_{m}$. For $j=1,\ldots,m,$ define

$g_{j}=\mathrm{gcd}(f,D^{1}_{j}(f))\text{ and }h_{j}=\frac{f}{g_{j}}.$

The radical $R(f)$ of $f$ is defined to be the least common multiple of
$h_{j}$’s.

Case 1: ${\mathbb{F}}$ has characteristic $p=0$. The truncated counting
function of zeros of $f$ is defined by

$N^{(l)}_{f}(0,r)=N_{\mathrm{gcd}(f,R(f)^{l})}(0,r).$

In particular,

$N^{(1)}_{f}(0,r)=N_{R(f)}(0,r).$

Case 2: ${\mathbb{F}}$ has characteristic $p>0$. We define $R_{p^{s}}(f)$ by
induction in $s=0,1,\ldots$ For $s=0$, set $R_{p^{0}}(f)=R(f)$. For $s\geq 1$,
assume that $R_{p^{s-1}}(f)$ has been defined. We set

$\overline{f}=\frac{f}{\mathrm{gcd}(f,R_{p^{s-1}}(f)^{p^{s}})},\
g_{i}=\mathrm{gcd}(\overline{f},D_{i}^{p^{s}}\overline{f}),\
h_{i}=\frac{\overline{f}}{g_{i}}$

for $i=1,\ldots,m$. Let $H$ be the least common multiple of $h_{i}$’s, and set

$G=\frac{H}{\mathrm{gcd}(H,R_{p^{s-1}}(H)^{p^{s-1}})},$

which is a $p^{s}$th power. Let $R$ be the $p^{s}$th root of $G$ and define
the higher $p^{s}$-radical $R_{p^{s}}(f)$ of $f$ to be the least common
multiple of $R_{p^{s-1}}(f)$ and $R$.

Take a sequence $\\{r_{j}\\}_{i\in\mathbb{N}}\subset|{\mathbb{F}}^{*}|$ such
that $r_{j}\rightarrow\infty.$ Take $s_{j}$ such that if $P\in\mathcal{E}_{m}$
is irreducible such that $P|f$ and $P$ is not unit on
${\mathbb{B}}^{m}(r_{j})$ then $P|R_{p^{s}}(f)$ for $s>s_{j}$. Let $u_{j}$ be
a unit on ${\mathbb{B}}^{m}(r_{j})$ such that

$R_{p^{s_{j}}}(f)=u_{j}R_{p^{s_{j+1}}}(f).$

Define $v_{j}=\prod_{l=j}^{\infty}u_{j}$, which is unit on
${\mathbb{B}}^{m}(r_{j})$, and

$S(f)=\lim_{j\rightarrow\infty}\frac{R_{p^{s_{j}}}(f)}{v_{j}}\in\mathcal{E}_{m},$

which is called the square free part of $f$. The truncated (to level $l$)
counting function of zeros of $f$ is defined by

$N^{(l)}_{f}(0,r)=N_{\mathrm{gcd}(f,S(f)^{l})}(0,r).$

2.5. Non-Archimedean meromorphic maps and family of hypersurfaces.

Let $V$ be a projective subvariety of ${\mathbb{P}}^{M}(\mathbb{F})$ of
dimension $n\ (n\leq M)$. For a positive integer $d$, take a basis
$\\{[A_{1}],\ldots,[A_{H_{d}(V)}]\\}$ of $I_{d}(V)$, where
$A_{i}\in\mathcal{H}_{d}[x_{0},\ldots,x_{M}]$. Let
$f:{\mathbb{F}}^{m}\rightarrow{\mathbb{P}}^{M}({\mathbb{F}})$ be a non-
Archimedean meromorphic map with a reduced representation
${\mathbf{f}}=(f_{0},\ldots,f_{M})$, which is non-degenerate over $I_{d}(V)$.
We have the following definition.

######  Definition 3.

Assume that ${\mathbb{F}}$ has the character $p>0$. Denote by $s$ the smallest
integer such that any subset of
$\\{A_{1}({\mathbf{f}}),\ldots,A_{H_{d}(V)}({\mathbf{f}})\\}$ linearly
independent over ${\mathbb{F}}$ remains linearly independent over
$\mathcal{M}_{m}[p^{s}]$. We call $s$ is the $d^{th}$-index of non-degeneracy
of $f$.

We see that the above definition does not depend on the choice of the basis
$\\{[A_{i}];1\leq i\leq H_{d}(V)\\}$ and the choice of the reduced
representation ${\mathbf{f}}$. If $V={\mathbb{P}}^{M}({\mathbb{F}})$ and $d=1$
then $s$ is also called the index of independence of $f$ (see [6, Definition
4.1]).

The following three lemmas are proved in [2] for the case of
${\mathbb{F}}={\mathbb{C}}$ and the canonical absolute value. However, with
the same proof, they also hold for arbitrary algebraic closed field
${\mathbb{F}}$ of character $p\geq 0$ and complete with an arbitrary absolute
value. We state them here without the proofs.

Throughout this paper, we sometimes identify each hypersurface in a projective
variety with its defining homogeneous polynomial. The following lemma of An-
Quang [2] may be considered as a generalization of the lemma on Nochka weights
in [4].

######  Lemma 4 (cf. [2, Lemma 3]).

Let $V$ be a projective subvariety of ${\mathbb{P}}^{M}(\mathbb{F})$ of
dimension $n\ (n\leq M)$. Let $Q_{1},\ldots,Q_{q}$ be $q\ (q>2N-k+1)$
hypersurfaces in ${\mathbb{P}}^{M}(\mathbb{F})$ in $N$-subgeneral position
with respect to $V$ of the common degree $d.$ Then there are positive rational
constants $\omega_{i}\ (1\leq i\leq q)$ satisfying the following:

i) $0<\omega_{i}\leq 1,\ \forall i\in\\{1,\ldots,q\\}$,

ii) Setting $\tilde{\omega}=\max_{j\in Q}\omega_{j}$, one gets

$\sum_{j=1}^{q}\omega_{j}=\tilde{\omega}(q-2N+n-1)+n+1.$

iii) $\dfrac{n+1}{2N-n+1}\leq\tilde{\omega}\leq\dfrac{n}{N}.$

iv) For $R\subset\\{1,\ldots,q\\}$ with $\sharp R=N+1$, then $\sum_{i\in
R}\omega_{i}\leq n+1$.

v) Let $E_{i}\geq 1\ (1\leq i\leq q)$ be arbitrarily given numbers. For
$R\subset\\{1,\ldots,q\\}$ with $\sharp R=N+1$, there is a subset
$R^{o}\subset R$ such that $\sharp
R^{o}=\mathrm{rank}_{{\mathbb{F}}}\\{[Q_{i}];i\in R^{o}\\}=n+1$ and

$\prod_{i\in R}E_{i}^{\omega_{i}}\leq\prod_{i\in R^{o}}E_{i}.$

Let $Q$ be a hypersurface in ${\mathbb{P}}^{n}({\mathbb{F}})$ of degree $d$
defined by $\sum_{I\in\mathcal{I}_{d}}a_{I}x^{I}=0,$ where
$\mathcal{I}_{d}=\\{(i_{0},\ldots,i_{M})\in\mathbb{N}_{0}^{M+1}\ :\
i_{0}+\cdots+i_{M}=d\\}$, $I=(i_{0},\ldots,i_{M})\in\mathcal{I}_{d},$
$x^{I}=x_{0}^{i_{0}}\cdots x_{M}^{i_{M}}$ and $(x_{0}:\cdots:x_{M})$ is
homogeneous coordinates of ${\mathbb{P}}^{M}({\mathbb{F}})$. Let $f$ be an
non-Archimedean meromorphic map from ${\mathbb{F}}^{m}$ into a projective
subvariety $V$ of ${\mathbb{P}}^{M}({\mathbb{F}})$ with a reduced
representation ${\mathbf{f}}=(f_{0},\ldots,f_{M})$. We define

$Q({\mathbf{f}})=\sum_{I\in\mathcal{I}_{d}}a_{I}f^{I},$

where $f^{I}=f_{0}^{i_{0}}\cdots f_{n}^{i_{n}}$ for $I=(i_{0},\ldots,i_{n})$.
We have the following lemma.

######  Lemma 5 (cf. [2, Lemma 4]).

Let $\\{Q_{i}\\}_{i\in R}$ be a set of hypersurfaces in
${\mathbb{P}}^{n}({\mathbb{F}})$ of the common degree $d$ and let $f$ be a
meromorphic mapping of ${\mathbb{F}}^{m}$ into
${\mathbb{P}}^{n}({\mathbb{F}})$ with a reduced representation
${\mathbf{f}}=(f_{0},\ldots,f_{M})$. Assume that $\bigcap_{i\in R}Q_{i}\cap
V=\varnothing$. Then, there exist positive constants $\alpha$ and $\beta$ such
that

$\alpha\|{\mathbf{f}}\|_{r}^{d}\leq\max_{i\in
R}|Q_{i}({\mathbf{f}})|_{r}\leq\beta\|{\mathbf{f}}\|_{r}^{d}\text{ for any
}r>0.$

######  Lemma 6 (cf. [2, Lemma 5]).

Let $\\{Q_{i}\\}_{i=1}^{q}$ be a set of $q$ hypersurfaces in
${\mathbb{P}}^{M}({\mathbb{F}})$ of the common degree $d$. Then there exist
$(H_{V}(d)-n-1)$ hypersurfaces $\\{T_{i}\\}_{i=1}^{H_{V}(d)-n-1}$ in
${\mathbb{P}}^{M}({\mathbb{F}})$ such that for any subset
$R\in\\{1,\ldots,q\\}$ with $\sharp
R=\mathrm{rank}_{{\mathbb{F}}}\\{[Q_{i}];i\in R\\}=n+1,$ we get
$\mathrm{rank}_{{\mathbb{F}}}\\{\\{[Q_{i}];i\in R\\}\cup\\{[T_{i}];1\leq i\leq
H_{d}(V)-n-1\\}\\}=H_{V}(d).$

2.5. Value distribution theory for non-Archimedean meromorphic maps.

Let $f:{\mathbb{F}}^{m}\rightarrow V\subset{\mathbb{P}}^{M}({\mathbb{F}})$ be
a non-Archimedean meromorphic map with a reduced representation
${\mathbf{f}}=(f_{0},\dots,f_{N})$. The characteristic function of $f$ is
defined by

$\displaystyle T_{f}(r)=\log\|{\mathbf{f}}\|_{r},$

where $\|{\mathbf{f}}\|_{r}=\max_{1\leq 0\leq n}|f_{i}|_{r}$. This definition
is well-defined upto a constant.

Let $Q$ be a hypersurface in ${\mathbb{P}}^{n}({\mathbb{F}})$ of degree $d$
defined by $\sum_{I\in\mathcal{I}_{d}}a_{I}x^{I}=0,$ where
$a_{I}\in{\mathbb{F}}\ (I\in\mathcal{I}_{d})$ and are not all zeros. If
$Q({\mathbf{f}})\not\equiv 0$ then we define the proximity function of $f$
with respect to $Q$ by

$m_{f}(Q,r)=\log\frac{\|{\mathbf{f}}\|^{d}_{r}\cdot\|Q\|}{|Q({\mathbf{f}})|_{r}},$

where $\|Q\|:=\max_{I\in\mathcal{I}_{d}}|a_{I}|$. We see that the definition
of $m_{f}(Q,r)$ does not depend on the choices of the presentations of $f$ and
$Q$.

The truncated (to level $l$) counting function of $f$ with respect to $Q$ is
defined by

$N_{f}^{(l)}(Q,r):=N^{(l)}_{Q({\mathbf{f}})}(0,r).$

For simplicity, we will omit the character (l) if $l=\infty$.

The first main theorem for non-Archimedean meromorphic maps states that

$dT_{f}(r)=m_{f}(Q,r)+N_{f}(Q,r)+O(1).$

######  Proposition 7 (cf. [6, Propositions 4.3, 4.4]).

Let $p$ be the character of ${\mathbb{F}}$. Assume that
$f:{\mathbb{F}}_{m}\rightarrow{\mathbb{P}}^{n}({\mathbb{F}})$ is a non-
Achimedean meromorphic map, which is linearly non-degenerate over
${\mathbb{F}}$, with a reduced representation
${\mathbf{f}}=(f_{0},\ldots,f_{n})$. Then there exist multi-indices
$\gamma^{0}=(0,\ldots,0),\gamma^{1},\ldots,\gamma^{n}$ with

$|\gamma^{0}|\leq\cdots\leq|\gamma^{n}|\leq\kappa_{0}\leq\begin{cases}p^{s-1}(n-k+1)&\text{
if }p>0,\\\ n-k+1&\text{ if }p=0\end{cases}$

where $s$ is the index of independence of $f$ and $k=\mathrm{rank}f$, such
that the generalized Wronskian

$W_{\gamma^{0},\ldots,\gamma^{n}}(f_{0},\ldots,f_{n})=\det\left(D^{\gamma^{i}}f_{j}\right)_{0\leq
i,j\leq n}\not\equiv 0.$

Here $\mathrm{rank}f$ is defined by

$\mathrm{rank}f=\mathrm{rank}_{\mathcal{M}_{m}}\\{(D^{\gamma}f_{0},\ldots,D^{\gamma}f_{n});|\gamma|\leq
1\\}-1.$

##  3\. Proof of main theorems

###### Proof of Theorem 1.

By replacing $Q_{i}$ with $Q_{i}^{d/d_{i}}$ if necessary, we may assume that
all $Q_{i}\ (i=1,\ldots,q)$ do have the same degree $d$. It is easy to see
that there is a positive constant $\beta$ such that
$\beta\|{\mathbf{f}}\|^{d}\geq|Q_{i}({\mathbf{f}})|$ for every $1\leq i\leq
q.$ Set $Q:=\\{1,\cdots,q\\}$. Let $\\{\omega_{i}\\}_{i=1}^{q}$ be as in Lemma
4 for the family $\\{Q_{i}\\}_{i=1}^{q}$. Let
$\\{T_{i}\\}_{i=1}^{H_{d}(V)-n-1}$ be $(H_{d}(V)-n-1)$ hypersurfaces in
${\mathbb{P}}^{M}({\mathbb{F}})$, which satisfy Lemma 6.

Take a ${\mathbb{F}}$-basis $\\{[A_{i}]\\}_{i=1}^{H_{V}(d)}$ of $I_{d}(V)$,
where $A_{i}\in H_{d}$. Since $f$ is non-degenerate over $I_{d}(V)$, it
implies that $\\{A_{i}({\mathbf{f}});1\leq i\leq H_{V}(d)\\}$ is linearly
independent over ${\mathbb{F}}$. By Proposition 7, there multi-indices
$\\{\gamma^{1}=(0,\ldots,0),\gamma^{2}\cdots,\gamma^{H_{V}(d)}\\}\subset\mathbb{Z}_{+}^{m}$
such that $|\gamma^{0}|\leq\cdots\leq|\gamma^{H_{d}(V)}|\leq\kappa_{0},$ where

$\kappa_{0}\leq\begin{cases}p^{s-1}(H_{V}(d)-k)&\text{ if }p>0,\\\
H_{d}(V)-k&\text{ if }p=0\end{cases}$

and the generalized Wronskian

$W=\det\left(D^{\gamma^{i}}A_{j}({\mathbf{f}})\right)_{1\leq i,j\leq
H_{d}(V)}\not\equiv 0.$

Here, we note that

$\displaystyle k$
$\displaystyle=\mathrm{rank}_{\mathcal{M}_{m}}\\{(D^{\gamma}f_{0},\ldots,D^{\gamma}f_{M});|\gamma|\leq
1\\}-1$
$\displaystyle=\mathrm{rank}_{\mathcal{M}_{m}}\left\\{\left(D^{\gamma}\bigl{(}\frac{f_{1}}{f_{0}}\bigl{)},\ldots,D^{\gamma}\bigl{(}\frac{f_{M}}{f_{0}}\bigl{)}\right);|\gamma|\leq
1\right\\}$
$\displaystyle\leq\mathrm{rank}_{\mathcal{M}_{m}}\left\\{\left(D^{\gamma}\bigl{(}\frac{A_{2}({\mathbf{f}})}{A_{1}({\mathbf{f}})}\bigl{)},\ldots,D^{\gamma}\bigl{(}\frac{A_{H_{d}(V)}({\mathbf{f}})}{A_{1}({\mathbf{f}})}\bigl{)}\right);|\gamma|\leq
1\right\\}$
$\displaystyle=\mathrm{rank}_{\mathcal{M}_{m}}\left\\{(D^{\gamma}(A_{1}({\mathbf{f}})),\ldots,D^{\gamma}(A_{H_{d}(V)}({\mathbf{f}})));|\gamma|\leq
1\right\\}-1.$

For each $R^{o}=\\{r^{0}_{1},\ldots,r^{0}_{n+1}\\}\subset\\{1,\ldots,q\\}$
with $\mathrm{rank}_{{\mathbb{F}}}\\{Q_{i}\\}_{i\in R^{o}}=\sharp R^{o}=n+1$,
set

$W_{R^{o}}\equiv\det\bigl{(}D^{\gamma^{j}}Q_{r^{0}_{v}}({\mathbf{f}})(1\leq
v\leq n+1),D^{\gamma^{j}}T_{l}({\mathbf{f}})(1\leq l\leq
H_{V}(d)-n-1)\bigl{)}_{1\leq j\leq H_{V}(d)}.$

Since $\mathrm{rank}_{{\mathbb{F}}}\\{[Q_{r^{0}_{v}}](1\leq v\leq
n+1),[T_{l}](1\leq l\leq H_{V}(d)-n-1)\\}=H_{V}(d)$, there exists a nonzero
constant $C_{R^{o}}\in{\mathbb{F}}$ such that $W_{R^{o}}=C_{R^{o}}\cdot W$.

We denote by $\mathcal{R}^{o}$ the family of all subsets $R^{o}$ of
$\\{1,\ldots,q\\}$ satisfying

$\mathrm{rank}_{{\mathbb{F}}}\\{[Q_{i}];i\in R^{o}\\}=\sharp R^{o}=n+1.$

For each $r>0$, there exists $\bar{R}\subset Q$ with $\sharp\bar{R}=N+1$ such
that $|Q_{i}({\mathbf{f}})|_{r}\leq|Q_{j}({\mathbf{f}})|_{r},\forall
i\in\bar{R},j\not\in\bar{R}$. We choose $R^{o}\subset R$ such that
$R^{o}\in\mathcal{R}^{o}$ and $R^{o}$ satisfies Lemma 4(v) with respect to
numbers
$\bigl{\\{}\dfrac{\beta\|{\mathbf{f}}\|_{r}^{d}}{|Q_{i}({\mathbf{f}})|_{r}}\bigl{\\}}_{i=1}^{q}$.
Since $\bigcap_{i\in\bar{R}}Q_{i}=\varnothing$, by Lemma 5, there exists a
positive constant $\alpha^{\bar{R}}$ such that

$\alpha^{\bar{R}}\|{\mathbf{f}}\|_{r}^{d}\leq\max_{i\in\bar{R}}|Q_{i}({\mathbf{f}})|_{r}.$

Then, we get

$\displaystyle\dfrac{\|{\mathbf{f}}\|_{r}^{d(\sum_{i=1}^{q}\omega_{i})}|W|_{r}}{|Q_{1}({\mathbf{f}})|_{r}^{\omega_{1}}\cdots|Q_{q}({\mathbf{f}})|_{r}^{\omega_{q}}}$
$\displaystyle\leq\dfrac{|W|_{r}}{\alpha^{q-N-1}_{\bar{R}}\beta^{N+1}}\prod_{i\in\bar{R}}\left(\dfrac{\beta\|{\mathbf{f}}\|_{r}^{d}}{|Q_{i}({\mathbf{f}})|_{r}}\right)^{\omega_{i}}$
$\displaystyle\leq
A_{\bar{R}}\dfrac{|W|_{r}\cdot\|{\mathbf{f}}\|_{r}^{d(n+1)}}{\prod_{i\in\bar{R}^{o}}|Q_{i}({\mathbf{f}})|_{r}}$
$\displaystyle\leq
B_{\bar{R}}\dfrac{|W_{\bar{R}^{o}}|_{r}\cdot\|{\mathbf{f}}\|_{r}^{dH_{V}(d)}}{\prod_{i\in\bar{R}^{o}}|Q_{i}({\mathbf{f}})|_{r}\prod_{i=1}^{H_{V}(d)-n-1}|T_{i}({\mathbf{f}})|_{r}},$

where $A_{\bar{R}},B_{\bar{R}}$ are positive constants.

Therefore, for every $r>0$,

$\displaystyle\log\dfrac{\|{\mathbf{f}}\|_{r}^{d(\sum_{i=1}^{q}\omega_{i}-H_{d}(V)}|W|_{r}}{|Q_{1}({\mathbf{f}})|_{r}^{\omega_{1}}\cdots|Q_{q}({\mathbf{f}})|_{r}^{\omega_{q}}}$
$\displaystyle\leq\max_{R}\log\dfrac{|W_{R}|_{r}}{\prod_{i\in
R}|Q_{i}({\mathbf{f}})|_{r}\prod_{i=1}^{H_{V}(d)-n-1}|T_{i}({\mathbf{f}})|_{r}}+O(1)$
$\displaystyle\leq-\sum_{j=1}^{H_{d}(V)}|\gamma^{j}|\log r+O(1),$

where the maximum is taken over all subsets $R\subset\\{1,\ldots,q\\}$ such
that $\sharp R=n+1$ and $\mathrm{rank}_{{\mathbb{F}}}\\{[Q_{i}];i\in
R\\}=n+1$. Here, the last inequality comes from the lemma on logarithmic
derivative. By the Poisson-Jensen-Green formula, the definitions of the
approximation function and the characteristic function, we have

$\sum_{i=1}^{q}\omega_{i}m_{f}(Q_{i},r)-dH_{d}(V)T_{f}(r)-N_{W}(0,r)\leq-(H_{d}(V)-1)\log
r+O(1),$

(note that $\sum_{i=1}^{H_{d}(V)}|\gamma^{i}|\leq H_{d}(V)-1$). Then, by the
first main theorem, we obtain

(8)
$\displaystyle(\sum_{i=1}^{q}\omega_{i}-H_{d}(V))dT_{f}(r)\leq\sum_{i=1}^{q}\omega_{i}N_{f}(Q_{i},r)-N_{W}(0,r)-(H_{d}(V)-1)\log
r+O(1).$

Claim.
$\sum_{i=1}^{q}\omega_{i}N_{f}(Q_{i},r)-N_{W}(0,r)\leq\sum_{i=1}^{q}\omega_{i}N^{(\kappa_{0})}_{f}(Q_{i},r)+O(1)$.

Indeed, set
$\tilde{G}_{j}=\mathrm{gcd}(Q_{j}({\mathbf{f}}),S(Q_{j}({\mathbf{f}}))^{\kappa_{0}})$.
Since $\omega_{i}\ (1\leq i\leq q)$ are rational numbers, there exists an
integer $A$ such that $\tilde{\omega}_{i}=A\omega_{i}\ (1\leq i\leq q)$ are
integers.

Let $P\in\mathcal{E}_{m}$ be an irreducible element with
$P|\prod_{i=1}^{q}Q_{i}({\mathbf{f}})^{\tilde{\omega}_{i}}$. There exists a
subset $R$ of $\\{1,\ldots,q\\}$ with $\sharp R=N+1$ such that $P$ is not a
division of $Q_{i}({\mathbf{f}})$ for any $i\not\in R$. Denote by $e_{i}$ the
largest integer such that $P^{e_{i}}|Q_{i}({\mathbf{f}})$ for each $i\in R$.
Then, there is a subset $R^{o}\subset R$ with $\sharp R^{o}=n+1$,
$W_{R^{o}}\not\equiv 0$ and

$\sum_{i\in R}\omega_{i}\max\\{0,e_{i}-\kappa_{0}\\}\leq\sum_{i\in
R^{o}}\max\\{0,e_{i}-\kappa_{0}\\}.$

Also, since $W=C_{R^{o}}\cdot W_{R^{o}}$, it clear that $P$ divides $W$ with
multiplicity at least

$\displaystyle\min_{\\{j_{1},\ldots,j_{n+1}\\}\subset\\{1,\ldots,H_{d}(V)\\}}\sum_{i\in
R^{0}}\min\\{0,e_{i}-|\gamma^{j_{i}}|\\}$ $\displaystyle\geq\sum_{i\in
R^{0}}\min\\{0,e_{i}-\kappa_{0}\\}$ $\displaystyle\geq\sum_{i\in
R}\omega_{i}\max\\{0,e_{i}-\kappa_{0}\\}$ $\displaystyle=\sum_{i\in
R}\omega_{i}(e_{i}-\min\\{e_{i},\kappa_{0}\\}).$

This implies that

$P^{\sum_{i\in R}\tilde{\omega}_{i}e_{i}}|W^{A}\cdot P^{\sum_{i\in
R}\tilde{\omega}_{i}\min\\{e_{i},\kappa_{0}\\}}.$

We note that
$P^{\tilde{\omega}_{i}\min\\{e_{i},\kappa_{0}\\}}|G_{i}^{\tilde{\omega}_{i}}$.
Therefore,

$P^{\sum_{i\in R}\tilde{\omega}_{i}e_{i}}|W^{A}\cdot\prod_{i\in
R}G_{i}^{\tilde{\omega}_{i}}.$

This holds for every such irreducible element $P$. Then it yields that

$\prod_{i=1}^{q}Q_{i}({\mathbf{f}})^{\tilde{\omega}_{i}}|W^{A}\cdot\prod_{i=1}^{q}G_{i}^{\tilde{\omega}_{i}}.$

Hence,

$\sum_{i=1}^{q}N_{f}(Q_{i},r)\leq
N_{W}(0,r)+\sum_{i=1}^{q}N^{(\kappa_{0})}_{f}(Q_{i},r).$

The claim is proved.

From the claim, Lemma 4(ii) and the inequality (8), we obtain

$\displaystyle(\tilde{\omega}(q-2N+n-1)$ $\displaystyle-
H_{d}(V)+n+1)dT_{f}(r)$
$\displaystyle\leq\sum_{i=1}^{q}\omega_{i}N^{(\kappa_{0})}_{f}(Q_{i},r)-(H_{d}(V)-1)\log
r+O(1).$

Note that, $\omega_{i}\leq\tilde{\omega}(1\leq i\leq q)$ and
$\dfrac{n+1}{2N-n+1}\leq\tilde{\omega}\leq\dfrac{n}{N}$. Then, the above
inequality implies that

$\left(q-\frac{(2N-n+1)H_{d}(V)}{n+1}\right)\leq\sum_{i=1}^{q}\dfrac{1}{d}N^{(\kappa_{0})}_{f}(Q_{i},r)-\dfrac{N(H_{d}(V)-1)}{nd}\log
r+O(1).$

The theorem is proved. ∎

###### Proof of Theorem 2.

For $r>0$, without loss of generality, we may assume that

$|Q_{1}({\mathbf{f}})|_{r}^{1/\deg Q_{1}}\leq|Q_{2}({\mathbf{f}})|_{r}^{1/\deg
Q_{2}}\leq\cdots\leq|Q_{q}({\mathbf{f}})|_{r}^{1/\deg Q_{N+1}}.$

Since $\bigcap_{i=1}^{N+1}Q_{i}=\varnothing$, by Lemma 5, there exists a
positive constant $C$ such that

$C\|{\mathbf{f}}\|_{r}\leq\max_{1\leq i\leq
N+1}|Q_{i}({\mathbf{f}})|_{r}^{1/\deg
Q_{i}}=|Q_{N+1}({\mathbf{f}})|_{r}^{1/\deg Q_{N+1}}.$

Then, we get

$\displaystyle\sum_{i=1}^{q}\dfrac{m_{f}(Q_{i},r)}{\deg Q_{i}}$
$\displaystyle=\log\dfrac{\|{\mathbf{f}}\|_{r}^{q}}{|Q_{1}({\mathbf{f}})|_{r}^{1/\deg
Q_{1}}\cdots|Q_{q}({\mathbf{f}})|_{r}^{1/\deg Q_{q}}}+O(1)$
$\displaystyle\leq\log\prod_{i=1}^{N}\dfrac{\|{\mathbf{f}}\|_{r}}{|Q_{i}({\mathbf{f}})|_{r}^{1/\deg
Q_{i}}}+O(1)$ $\displaystyle=\sum_{i=1}^{N}\dfrac{m_{f}(Q_{i},r)}{\deg
Q_{i}}+O(1)$ $\displaystyle\leq N\cdot T_{f}(r)+O(1).$

Therefore,

$(q-N)T_{f}(r)\leq\sum_{i=1}^{q}\dfrac{1}{\deg Q_{i}}N_{f}(Q_{i},r)+O(1)\ \
(r>0).$

The theorem is proved. ∎

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