# Non-perturbative localization on the strip and Avila’s almost reducibility
conjecture

Rui Han  Department of Mathematics
Louisiana State University
Baton Rouge, LA 70803, USA<EMAIL_ADDRESS>and  Wilhelm Schlag  Department of
Mathematics
Yale University
New Haven, CT 06511, USA<EMAIL_ADDRESS>
###### Abstract.

We prove non-perturbative Anderson localization and almost localization for a
family of quasi-periodic operators on the strip. As an application we
establish Avila’s almost reducibility conjecture for Schrödinger operators
with trigonometric potentials and all Diophantine frequencies, whose proof for
analytic potentials was announced in 2015 in [Av3]. As part of our analysis,
we derive a non-selfadjoint version of Haro and Puig’s formula connecting
Lyapunov exponents of the dual model to those of the original operator.

R. Han is partially supported by NSF DMS-2143369. W. Schlag is partially
supported by NSF grant DMS-1902691.

##  1\. Introduction

In the first part of this paper, we prove non-perturbative Anderson
localization for a family of one-dimensional finite-range quasi-periodic
Schrödinger operators, which can also be viewed as an operator on the strip
$\mathbb{Z}\times\\{1,...,d\\}$ or block-valued quasi-periodic Jacobi
matrices. Here non-perturbative refers to the positive Lyapunov exponent (half
of exponents of the corresponding symplectic cocycles are positive) regime. We
also prove almost localization all $\theta$ and any Diophantine $\alpha$
(those satisfy $\beta(\alpha)=0$, see (2.21)). This together with the dual
(almost) reducibility techniques developed by Avila-Jitomirskaya in [AJ2],
proves Avila’s almost reducibility conjecture (ARC) in [Av3] for Schrödinger
operators with trigonometric potentials and all Diophantine $\alpha$’s.
Combined with Avila’s proof of ARC for Liouville $\alpha$ (those satisfying
$\beta(\alpha)>0$), this gives a complete proof of ARC for trigonometric
potentials.

Let

(1.1)
$\displaystyle(H_{\alpha,\theta,\varepsilon}u)_{n}=u_{n+1}+u_{n-1}+v(\theta+n\alpha+i\varepsilon)u_{n},$

$v:\mathbb{T}=\mathbb{R}/\mathbb{Z}\to\mathbb{R}$ analytic, be a quasi-
periodic Schrödinger operator on $\mathbb{Z}$. Let

(1.2) $\displaystyle
A_{E,\varepsilon}(\theta)=\left(\begin{matrix}E-v(\theta+i\varepsilon)&-1\\\
1&0\end{matrix}\right)$

be the transfer matrix, and $(\alpha,A_{E,\varepsilon})$ be the associated
cocycle. Let $L(\alpha,A_{E,\varepsilon})$ be the Lyapunov exponent. In his
seminal work [Av3], Avila introduced the acceleration, defined as:

(1.3) $\displaystyle\kappa(\alpha,A_{E,\varepsilon})=\lim_{\delta\to
0^{+}}\frac{L(\alpha,A_{E,\varepsilon+\delta})-L(\alpha,A_{E,\varepsilon})}{2\pi\delta}.$

Furthermore, he proved the acceleration is quantized, meaning
$\kappa(\alpha,A_{E,\varepsilon})\in\mathbb{Z}$. Using the quantized
acceleration, Avila divides the spectrum
$\sigma(H_{\alpha,\theta,0})=\sigma(H_{\alpha})$ (we restrict to the
irrational $\alpha$ case, for which the spectrum is independent of $\theta$)
into three regimes: the supercritical regime

$\Sigma_{\alpha,+}:=\\{E\in\sigma(H_{\alpha}):\,L(\alpha,A_{E,0})>0,\text{ and
}\kappa(\alpha,A_{E,0})>0\\},$

the subcritical regime

$\Sigma_{\alpha,0}:=\\{E\in\sigma(H_{\alpha}):\,L(\alpha,A_{E,0})=0,\text{ and
}\kappa(\alpha,A_{E,0})=0\\},$

and the critical regime

$\Sigma_{\alpha,c}:=\\{E\in\sigma(H_{\alpha}):\,L(\alpha,A_{E,0})=0,\text{ and
}\kappa(\alpha,A_{E,0})>0\\}.$

In the supercritical regime, non-perturbative proofs of Anderson localization
have been developed for the Harper operator with all
$\alpha\in\mathrm{DC}$111$\mathrm{DC}=\bigcup_{c>0}\bigcup_{b>1}\\{\alpha\in\mathbb{T}:\,\|n\alpha\|_{\mathbb{T}}\geq
c\,|n|^{-b},n\neq 0\\}\subsetneq\\{\alpha\in\mathbb{T}:\,\beta(\alpha)=0\\}$
in [J1] (where $v(\theta)=2\lambda\cos(2\pi\theta)$ with $|\lambda|>1$), for
general analytic potentials with a.e. $\alpha$ in [BG], and for all
$\alpha\in\mathrm{DC}$ in the first supercritical stratum (where
$\kappa(\alpha,A_{E,0})=1$) in [HS]. In the critical regime, the spectrum is
conjectured to be singular continuous (see Conjecture 3.1 in [AJM]). This
conjecture has been proved for the critical Harper operator (where
$v(\theta)=2\cos(2\pi\theta)$) in [Av1, AJM, J2].

In the subcritical regime, Avila formulated the almost reducibility conjecture
(ARC), which we shall elaborate on below. A cocycle $(\alpha,A)$ is called
(analytically) reducible if there exists $D\in
C^{\omega}(\mathbb{T},\mathrm{PSL}(2,\mathbb{R}))$ such that the analytic
conjugacy $D^{-1}(\theta+\alpha)A(\theta)D(\theta)$ is a constant. A cocycle
is called almost reducible if the closure of its analytic conjugacies contains
a constant.

######  Conjecture 1.1 (ARC by Avila, [Av3]).

Subcritical cocycles are almost reducible.

As remarked by Avila in [Av3], “Proving this almost reducibility conjecture
would at once provide an almost complete understanding of subcriticality.”
This conjecture has thus far been proved in two different cases:

  * •

for Diophantine $\alpha$’s: (i) for general analytic potentials in the small
coupling regime (ii) for the entire subcritical regime of the almost Mathieu
(or Harper) operator; both in [AJ2]. See also [GYZ].

  * •

for Liouville $\alpha$’s (those $\beta(\alpha)>0$) in [Av2].

In this paper, we prove the following non-perturbative quantitative ARC for
Schrödinger cocycles with trigonometric potentials and all Diophantine
$\alpha$. In the rest of the paper, we assume that $v(x)=\sum_{1\leq|k|\leq
d}\hat{v}_{k}e^{2\pi ikx}$ is a real-valued trigonometric polynomial, which
implies for $1\leq k\leq d$, $\hat{v}_{k}=\overline{\hat{v}_{-k}}$. Without
loss of generality we assume that $\hat{v}_{0}=0$.

######  Theorem 1.2.

Let $\beta(\alpha)=0$, and $E\in\Sigma_{\alpha,0}$. Let
$0<\varepsilon_{0}<\frac{\hat{L}_{d}(\alpha,\hat{A}_{E})}{30\hat{L}^{d}(\hat{A}_{E})+100C_{3}d}$
where $C_{3}$ is the absolute constant in Lemma 3.10, and
$\delta=\min(\frac{\hat{L}_{d}(\alpha,\hat{A}_{E})}{5000},\frac{\varepsilon_{0}}{25})$.
Let $\theta=\theta(E)$ be given by Theorem A.1. Let $\\{n_{j}\\}$ be the
sequence of $\varepsilon_{0}$-resonances of $\theta$ and $L_{j}:=\|2\theta-
n_{j}\alpha\|_{\mathbb{T}}$. Then there exists an absolute constant $C>0$ such
that for $j>j_{0}(\alpha,v,E,\varepsilon_{0})$ there exists $W\in
C^{\omega}_{\frac{\delta}{2\pi}}(\mathbb{T},\mathrm{SL}(2,\mathbb{R}))$ such
that $|\mathrm{deg}(W)|\leq C|n_{j}|$, $\|W\|_{\frac{\delta}{2\pi}}\leq
L_{j}^{-C}$ and

(1.4)
$\displaystyle\|W(x+\alpha)^{-1}A_{E}(x)W(x)-R_{-\theta}\|_{\frac{\delta}{2\pi}}\leq
e^{-\frac{\hat{L}_{d}(\alpha,\hat{A}_{E})}{300}|n_{j+1}|}.$

Above, $\hat{L}_{d}(\alpha,\hat{A}_{E})>0$ is the $d$-th Lyapunov exponent of
the dual operator, see Corollary 1.12. The core of the proof is to establish
the almost localization for the dual model (Theorem 3.8 below). Once
established, one can combine it with the quantitative duality arguments in
[AJ2] to prove Theorem 1.2. We include the duality argument in Appendix A in
order to keep the paper self-contained.

One example to which Theorem 1.2 applies is the perturbation of the
subcritical almost Mathieu operator, which is also subcritical due to the
upper semi-continuity of the acceleration.

######  Example 1.3.

Let $H^{\mathrm{PAMO}}_{\alpha,\theta,\nu,w}$ be the operator with
$v(\theta)=2\lambda\cos(2\pi\theta)+\nu\cdot w(\theta)$ and $\varepsilon=0$,
where $w(\theta)$ is a trigonometric polynomial. For $|\lambda|<1$ and
sufficiently small $|\nu|<\nu_{0}(\alpha,\lambda,w)$,
$\sigma(H^{\mathrm{PAMO}}_{\alpha,\theta,\nu,w})=\Sigma_{\alpha,0}$.

######  Theorem 1.4.

For $|\lambda|<1$, $\beta(\alpha)=0$, and sufficiently small
$|\nu|<\nu_{0}(\alpha,\lambda,w)$, for any
$E\in\sigma(H^{\mathrm{PAMO}}_{\alpha,\theta,\nu,w})$, the cocycle
$(\alpha,A_{E}^{\mathrm{PAMO}})$ associated to
$H^{\mathrm{PAMO}}_{\alpha,\theta,\nu,w}$ is quantitatively almost reducible.

######  Remark 1.5.

It is well-known that for
$E\notin\sigma(H^{\mathrm{PAMO}}_{\alpha,\theta,\nu,w})$,
$(\alpha,A_{E}^{\mathrm{PAMO}})$ is $\mathrm{PSL}(2,\mathbb{R})$ reducible,
see e.g. Theorem 2.1 of [AJ2].

As a consequence of Theorem 1.4, we can prove the dry Ten Martini problem for
$H^{\mathrm{PAMO}}_{\alpha,\theta,\nu,w}$.

######  Theorem 1.6.

For $|\lambda|<1$, $\beta(\alpha)=0$, and for sufficiently small
$|\nu|<\nu_{0}(\alpha,\lambda,w)$, all the spectral gaps of
$H^{\mathrm{PAMO}}_{\alpha,\theta,\nu w}$ as predicted by the gap labelling
theorem are open.

This is an analogue of Theorem 1.1 of [AJ2], we include its proof in the
appendix for completeness. Next, we introduce the dual model of
$H_{\alpha,\theta,\varepsilon}$:

(1.5)
$\displaystyle(\hat{H}_{\alpha,\theta,\varepsilon}\phi)_{n}=\sum_{0<|k|\leq
d}\hat{v}_{k}\phi_{n-k}e^{-2\pi
k\varepsilon}+2\cos(2\pi(\theta+n\alpha))\phi_{n}.$

This long-range operator can also be written in terms of block-valued Jacobi
matrix, acting on $\ell^{2}(\mathbb{Z},\mathbb{C}^{d})$, as follows.

(1.6)
$\displaystyle(\hat{H}_{\alpha,\theta,\varepsilon}\Phi)_{n}=B_{\varepsilon}\Phi_{n+1}+\tilde{B}_{\varepsilon}\Phi_{n-1}+C_{\varepsilon}(\theta+nd\alpha)\Phi_{n},$

where
$\Phi_{n}=(\phi_{nd+d-1},...,\phi_{nd+1},\phi_{nd})^{T}\in\mathbb{C}^{d}$, and
$B_{\varepsilon},\tilde{B}_{\varepsilon}$ and $C_{\varepsilon}$ are defined in
Section 2.3. The transfer matrix formulation of the block-valued Hamiltonian
eigenvalue equation $\hat{H}_{\alpha,\theta,\varepsilon}\Phi=E\Phi$ is

(1.7) $\displaystyle\left(\begin{matrix}B_{\varepsilon}\Phi_{n+1}\\\
\Phi_{n}\end{matrix}\right)=M_{E}^{\varepsilon}(\theta+nd\alpha)\left(\begin{matrix}B_{\varepsilon}\Phi_{n}\\\
\Phi_{n-1}\end{matrix}\right),$

where

(1.8) $\displaystyle
M_{E}^{\varepsilon}(\theta)=\left(\begin{matrix}(E-C_{\varepsilon}(\theta))B_{\varepsilon}^{-1}&-\tilde{B}_{\varepsilon}\\\
B_{\varepsilon}^{-1}&0\end{matrix}\right).$

We denote its Lyapunov exponents by
$\hat{L}_{1}(\alpha,M_{E}^{\varepsilon})\geq...\geq\hat{L}_{2d}(\alpha,M_{E}^{\varepsilon})$.
The transfer matrix $M_{E}^{\varepsilon}$ can also be viewed as $d$-step
transfer matrix for the scalar-valued Hamiltonian, where

(1.9) $\displaystyle
M_{E}^{\varepsilon}(\theta)=\mathrm{diag}(B_{\varepsilon},I_{d})\cdot\hat{A}_{d,E}^{\varepsilon}(\alpha,\theta)\cdot\mathrm{diag}(B_{\varepsilon}^{-1},I_{d}),$

in which the one step transfer matrix $\hat{A}_{E}^{\varepsilon}$ is as in
(2.47). It is easy to see by (1.9) that for $1\leq j\leq 2d$,

(1.10)
$\displaystyle\hat{L}_{j}(d\alpha,M_{E}^{\varepsilon})=d\cdot\hat{L}_{j}(\alpha,\hat{A}_{E}^{\varepsilon}).$

The transfer matrix $M_{E}^{0}$ at an arbitrary energy $E\in\mathbb{R}$ is
complex symplectic, see (2.38). Hence

(1.11)
$\displaystyle\hat{L}_{j}(d\alpha,M_{E}^{0})=-\hat{L}_{2d+1-j}(d\alpha,M_{E}^{0}),\text{
for }1\leq j\leq 2d.$

Proving Anderson localization in the non-perturbative
$\hat{L}_{d}(\alpha,M_{E}^{0})>0$ regime has been a long standing problem in
quasi-periodic Schrödinger operators, while the Anderson model counterpart has
been known for over 40 years, see [G, L, KLS, MS]. Resolving this problem is
one of the main accomplishments of this paper.

######  Theorem 1.7.

Let $\beta(\alpha)=0$ and $\theta$ be such that

(1.12)
$\displaystyle\gamma(\alpha,\theta):=\lim_{n\to\infty}\left(-\frac{\log\|2\theta+n\alpha\|_{\mathbb{T}}}{|n|}\right)=0,$

Then $\hat{H}_{\alpha,\theta,0}$ has pure point spectrum with exponentially
decaying eigenfunctions in

$\sigma(\hat{H}_{\alpha,\theta,0})\cap\\{E:\,\hat{L}_{d}(d\alpha,M_{E}^{0})>0\\}.$

We briefly discuss the difficulties. We write $\hat{L}_{j}(d\alpha,M_{E}^{0})$
as $\hat{L}_{j}$. The proof of Anderson localization in the literature relies
on the exponential decay of the Green’s function (with Dirichlet boundary
condition). In fact, let $\hat{H}_{\alpha,n}(\theta)$ be the finite volume
operator restricted to $[0,nd-1]$, see (2.58). Let
$\tilde{G}_{E,n}(\theta):=(\hat{H}_{\alpha,n}(\theta)-E)^{-1}$ be the
resolvent. The key to prove localization is to prove exponential decay of
$|G_{E,n}(\theta+j\alpha;1,x)|$ for $x\sim n/2$. By Cramer’s rule, one has

$G_{E,n}(\theta+j\alpha;1,x)=\tilde{\mu}_{n,1,x}(\theta+j\alpha)/\tilde{f}_{E,n}(\theta+j\alpha),$

where $\tilde{f}_{E,n}(\theta)=\det(\hat{H}_{\alpha,n}(\theta)-E)$.

In the scalar-valued case, $d=1$, the numerator
$\hat{\mu}_{n,1,x}(\theta+j\alpha)$ is equal to
$\tilde{f}_{E,n-x}(\theta+(x+j)\alpha)$, which is an element (out of four in
total) of $M_{n-x,E}^{0}(\theta+(x+j)\alpha)$, and can thus be easily
controlled by

$\|M_{n-x,E}^{0}(\theta+(x+j)\alpha)\|\leq\exp((n-x)(\hat{L}^{1}+\varepsilon)).$

The key for the $d=1$ case is to find a lower bound for the denominator
$|\tilde{f}_{E,n}(\theta+j\alpha)|$ for a properly chosen $j$. It was proved
by Jitomirskaya [J1] that $\tilde{f}_{E,n}(\theta)$ is a polynomial in
$\cos(2\pi(\theta+(n-1)\alpha/2))$ of degree $n$, hence if it is uniformly
small at $(n+1)$ uniformly distributed $\theta+j\alpha$’s and at three
consecutive scales $n_{0}-2,n_{0}-1,n_{0}$, then

$\sup_{n\in\\{n_{0}-2,n_{0}-1,n_{0}\\}}\sup_{\theta\in\mathbb{T}}|\tilde{f}_{E,n}(\theta)|\leq\exp(n(\hat{L}_{1}-\varepsilon)).$

However in (and only in) the scalar-valued case, we have

$\sup_{\theta\in\mathbb{T}}\|M_{n_{0},E}^{0}(\theta)\|\leq\max_{n\in\\{n_{0}-2,n_{0}-1,n_{0}\\}}\sup_{\theta\in\mathbb{T}}|\tilde{f}_{E,n}(\theta)|\leq\exp(n(\hat{L}_{1}-\varepsilon)),$

which would lead to a contradiction.

In the block-valued case $d\geq 1$, there are significant difficulties in
estimating both the numerator and the denominator. First of all the numerator
can no longer be identified with a determinant $\tilde{f}_{E,n}$. In [BJ, K]
and [AJ2]222In [AJ2], $d=\infty$, the numerators are estimated through a
random walk type expansion, where the expansion is only efficient in the large
coupling regime (replacing $2\cos$ with $2\lambda^{-1}\cos$ and $\lambda$
small). In the non-perturbative regime, a much tighter upper bound is need, in
fact we prove

$|\mu_{n,1,x}(\theta)|\leq|\hat{v}_{d}|^{nd}\cdot\max\left(e^{d^{-1}x(\hat{L}^{d-1}+\varepsilon)+(n-d^{-1}x)(\hat{L}^{d}+\varepsilon)},e^{d^{-1}x(\hat{L}^{d}+\varepsilon)+(n-d^{-1}x)(\hat{L}^{d+1}+\varepsilon)}\right)$

uniformly in $\theta$ and $\mu_{n,1,x}(\theta)$ is the numerator of the
Green’s function for the finite volume operator $P_{\alpha,n}(\theta)$ with
periodic boundary condition (see (3.62)). We can prove a similar bound for
$\tilde{\mu}_{n,1,x}$ as well. The reason why we choose the periodic instead
Dirichlet boundary condition lies in the estimate of the denominator which we
elaborate on below.

For the denominator, we only know that
$|\tilde{f}_{E,n}(\theta)|/|\hat{v}_{d}|^{nd}$ is the determinant of the upper
left $d\times d$ corner of $M_{n,E}^{0}(\theta)$, hence it is unclear why
$\sup_{\theta}|\tilde{f}_{E,n}(\theta)|/|\hat{v}_{d}|^{nd}$ is close to

$\sup_{\theta}\|\wedge^{d}M_{n,E}^{0}(\theta)\|=\exp(n(\hat{L}^{d}+o(1))).$

One might attempt to argue that

(1.13) $\displaystyle\frac{1}{nd}\log|\tilde{f}_{E,n}(\theta)|=$
$\displaystyle\int_{\mathbb{R}}\log|E-E^{\prime}|\,\hat{\mu}_{n,\alpha,\theta}(\mathrm{d}E^{\prime})\longrightarrow$
(1.14)
$\displaystyle\qquad\qquad\qquad\int_{\mathbb{R}}\log|E-E^{\prime}|\,\widehat{\mathcal{N}}(\mathrm{d}E^{\prime})=\hat{L}^{d}(\alpha,\hat{A}_{E}^{0})+\log|\hat{v}_{d}|,$

where $\hat{\mu}_{n,\alpha,\theta}$ is the finite-scale density of states
measure (see (2.59)) and the second line is the Thouless formula (2.61) that
was proved in the $d>1$ case in [CS2, KS, HP]. However the finite volume
density of states measure $\hat{\mu}_{n,\alpha,\theta}$ only converges in the
weak-$*$ sense (when integrated against continuous function of compact
support) to the density of states measure $\widehat{\mathcal{N}}$, which does
not apply when integrated against $f(\cdot)=\log|E-\cdot|$ for
$E\in\mathbb{R}$. Therefore the approach (1.13) does not lead to the desired
result when $E\in\mathbb{R}$. See Remark 3.4, too. We resolve this difficulty
by studying the Green’s function
$G_{E,n}(\theta)=(P_{\alpha,n}(\theta)-E)^{-1}$ with a periodic boundary
condition. Let $f_{E,n}(\theta)=\det(P_{\alpha,n}(\theta)-E)$ be the
denominator of the Green’s function $G_{E,n}(\theta)$. We prove that

$\log|f_{E,n}(\theta)|=nd\log|\hat{v}_{d}|+\log|\det(M^{0}_{n,E}(\theta)-I_{2d})|.$

We then show that for an admissible sequence of $n$, on a large measure set of
$\theta$,

$\log|\det(M^{0}_{n,E}(\theta)-I_{2d})|=n(\hat{L}^{d}+o(1)).$

The choice of the admissible sequence relies crucially on Lemma 3.2 of [GS3],
which guarantees that the most expanding $d$-planes of $M_{n,E}^{0}$,
respectively of $(M_{n,E}^{0})^{*}$, are not entirely orthogonal to each
other. Our lower bound Lemma 3.2 for the denominator is robust, and can be
applied to establish Anderson localization for operators
$\hat{H}_{\alpha,\theta}$ on strips with general analytic potentials defined
on higher-dimensional tori. We will address this problem in an upcoming work.
For the cosine potential which arises in this paper, there is a more
elementary argument using Herman’s subharmonic argument, see Lemma 3.5.

A stronger version of almost localization holds for arbitrary $\theta$, see
Theorem 3.8. The link between Theorem 3.8 and Theorem 1.2 is provided by
Corollary 1.12 that if $E\in\Sigma_{\alpha,0}$, then the dual Lyapunov
exponent satisfies $\hat{L}_{d}(d\alpha,M_{E}^{0})>0$. This was announced in
Jitomirskaya’s 2022 ICM talk, and is indeed a consequence of a non-selfadjoint
variant of Haro and Puig’s result on dual Lyapunov exponents, which we
elaborate on below.

In [HP], Haro and Puig proved the following remarkable theorem concerning
Lyapunov exponents of dual models. The techniques in [HP] are inspired by
Johnson’s parametrization of the stable and unstable subspaces in the
uniformly hyperbolic setting via the Weyl-Kodaira $M$-matrix, see [Joh, JPS,
JONNF].

######  Theorem 1.8.

[HP]*Theorem 1.3 and Corollary 1.6 For any irrational $\alpha$,

(1.15) $\displaystyle
L^{1}(\alpha,A_{E,0})=\hat{L}^{d}(\alpha,\hat{A}_{E}^{0})+\log|\hat{v}_{-d}|=\sum_{j:\hat{L}_{j}(\alpha,\hat{A}_{E}^{0})\geq
0}\hat{L}_{j}(\alpha,\hat{A}_{E}^{0})+\log|\hat{v}_{-d}|.$

The proof is built on two key ingredients. One is the Thouless formula, that
links the Lyapunov exponent to the density of states measure, for both
$H_{\alpha,\theta,\varepsilon=0}$ (the selfadjoint Schrödinger operator) and
$\hat{H}_{\alpha,\theta,\varepsilon=0}$ (the dual model on the strip, which is
also selfadjoint). The second is the fact that the two dual selfadjoint
operators share the same density of states measure.

The following non-selfadjoint version of the Haro, Puig formula was announced
in Jitomirskaya’s ICM talk in 2022.

######  Theorem 1.9.

[Jitomirskaya, ICM talk 2022] For any irrational $\alpha$,

(1.16) $\displaystyle
L^{1}(\alpha,A_{E,\varepsilon})=L^{1}(\alpha,A_{E,0})-\sum_{j:0\leq\hat{L}_{j}(\alpha,\hat{A}_{E}^{0})<2\pi\varepsilon}\hat{L}_{j}(\alpha,\hat{A}_{E}^{0})+\\#\\{j:0\leq\hat{L}_{j}(\alpha,\hat{A}_{E}^{0})<2\pi\varepsilon\\}\cdot(2\pi\varepsilon).$

Noting that
$\hat{L}_{j}(\alpha,\hat{A}_{E}^{\varepsilon})=\hat{L}_{j}(\alpha,\hat{A}_{E}^{0})-2\pi\varepsilon$,
see (2.50), and taking Haro and Puig’s formula (which holds at
$\varepsilon=0$) into account, we see that Theorem 1.9 in the same as a non-
selfadjoint version of Haro and Puig’s formula, see Theorem 1.10 below.

Since the proof of Theorem 1.9 does not seem available in the literature, and
because it appears in our resolution of the almost reducibility conjecture, we
provide a proof in Section 6 for the reader’s convenience.

######  Theorem 1.10.

For any irrational $\alpha$,

(1.17) $\displaystyle
L^{1}(\alpha,A_{E,\varepsilon})=\sum_{j:\hat{L}_{j}(\alpha,\hat{A}_{E}^{\varepsilon})\geq
0}\hat{L}_{j}(\alpha,\hat{A}_{E}^{\varepsilon})+\log|\hat{v}_{-d}e^{2\pi
d\varepsilon}|.$

A major difficulty in extending Haro and Puig’s formula to the non-selfadjoint
case (i.e., when $\varepsilon\neq 0$) is the lack of spectral theorems. Hence
there is no density of states measure, which played a crucial role in Haro and
Puig’s original proof. We overcome this problem by establishing the Aubry
duality, by means of the characters on cyclic groups, for finite scale
operators with periodic boundary conditions and rational frequencies. These
finite-scale operators naturally have (finite scale) density of states
measures via the normalized counting measures of the eigenvalues. We then
prove Thouless formulas for the finite scale rational-frequency operators,
which after combining with the duality, lead to (1.17) with rational $\alpha$.
Finally, (1.17) for irrational $\alpha$ follows from rational approximation
and continuity of the Lyapunov exponents at irrational frequencies.

It was also addressed by Jitomirskaya in her ICM talk that Theorem 1.9 leads
to the following characterization of Avila’s acceleration.

######  Corollary 1.11.

For irrational $\alpha$, and any $\varepsilon\geq 0$

(1.18)
$\displaystyle\kappa(\alpha,A_{E,\varepsilon})=\\#\\{j:0\leq\hat{L}_{j}(\alpha,\hat{A}_{E}^{\varepsilon})\leq
2\pi\varepsilon\\}.$

We give a proof of this in Section 6. Clearly, Corollary 1.11 and (2.50) lead
to the following.

######  Corollary 1.12.

If the acceleration of $H_{\alpha,\theta,\varepsilon}$ satisfies
$\kappa(\alpha,A_{E,\varepsilon})\equiv 0$ for
$|\varepsilon|\leq\varepsilon_{1}$, then

(1.19)
$\displaystyle\hat{L}_{1}(\alpha,\hat{A}_{E}^{0})\geq...\geq\hat{L}_{d}(\alpha,\hat{A}_{E}^{0})\geq
2\pi\varepsilon_{1}>0.$

In the rest of the paper, we shall omit the dependence on $\varepsilon$ in
various notations in the $\varepsilon=0$ case.

The paper is organized as follows: some preliminary knowledge is recalled in
Sec. 2, the proofs of Theorem 1.7 and almost localization are given in Sec. 3
assuming the numerator bound of the Green’s function established in Sec. 4 and
the denominator bound in Sec. 5. Finally we present the proof of Theorem 1.10
in Sec. 6.

Notations: for $x\in\mathbb{R}$, let
$\|x\|_{\mathbb{T}}:=\mathrm{dist}(x,\mathbb{Z})$ be the torus norm, and
$\lfloor x\rfloor=\sup\\{m\in\mathbb{Z}:\,m\leq x\\}$ be the integer part of
$x$. If $a\leq Cb$ for some constant $C>0$, we shall write $a=O(b)$. For a
matrix $M\in\mathbb{C}^{d_{1}\times d_{2}}$, and the sets of indicies
$B_{1}\subset[1,d_{1}]\cap\mathbb{Z},B_{2}\subset[1,d_{2}]\cap\mathbb{Z}$, let
$M_{B_{1},B_{2}}$ be the submatrix of $M$ with row numbers in $B_{1}$ and
column numbers in $B_{2}$.

##  2\. Preliminaries

###  2.1. Continued fraction expansion

Give $\alpha\in(0,1)$, let $[a_{1},a_{2},...]$ be the continued fraction
expansion of $\alpha$. For $n\geq 1$, let
$p_{n}/q_{n}=[a_{1},a_{2},...,a_{n}]$ be the continued fraction approximants.
The following property is well-known for $n\geq 1$,

(2.20) $\displaystyle\|q_{n}\alpha\|_{\mathbb{T}}=\min_{1\leq
k<q_{n+1}}\|k\alpha\|_{\mathbb{T}},$

The $\beta(\alpha)$ exponent measures the exponential closeness of $\alpha$ to
rational numbers:

(2.21) $\displaystyle\beta(\alpha):=\limsup_{n\to\infty}\frac{\log
q_{n+1}}{q_{n}}=\limsup_{n\to\infty}\left(-\frac{\log\|n\alpha\|_{\mathbb{T}}}{n}\right).$

###  2.2. Lyapunov exponents

Let
$(\alpha,A)\in(\mathbb{T},C^{\omega}(\mathbb{T},\mathrm{Mat}(k,\mathbb{C})))$.
Let

(2.22) $\displaystyle A_{n}(\alpha,\theta)=A(\theta+(n-1)\alpha)\cdots
A(\theta).$

Let the finite-scale and infinite-scale Lyapunov exponents be defined as

(2.23) $\displaystyle
L_{j,(n)}(\alpha,A):=\frac{1}{n}\int_{\mathbb{T}}\log\sigma_{j}(A_{n}(\alpha,\theta))\,\mathrm{d}\theta,\text{
for }1\leq j\leq k,$

where $\sigma_{j}(A)$ is the $j$-th singular value of $A$, and the $j$-th
Lyapunov exponent

(2.24) $\displaystyle L_{j}(\alpha,A)=\lim_{n\to\infty}L_{j,(n)}(\alpha,A).$

It is easy to see that for $1\leq j\leq k$,

(2.25) $\displaystyle
L^{j}_{(n)}(\alpha,A):=\sum_{\ell=1}^{j}L_{\ell,(n)}(\alpha,A)=\frac{1}{n}\int_{\mathbb{T}}\log\|\wedge^{j}A_{n}(\alpha,\theta)\|\,\mathrm{d}\theta,$

where $\wedge^{j}A$ is the $j$-th exterior power of $A$. Similarly
$L^{j}(\alpha,A)=\sum_{\ell=1}^{j}L_{\ell}(\alpha,A)$. If
$\alpha=p/q\in\mathbb{Q}$, we define

(2.26) $\displaystyle
L^{j}(p/q,A,\theta)=\lim_{n\to\infty}\frac{1}{n}\log\|\wedge^{j}A_{n}(p/q,\theta)\|=\frac{1}{q}\log(\rho(\wedge^{j}A_{q}(p/q,\theta))),$

where $\rho(A)$ is the spectral radius of $A$. Let
$L_{1}(p/q,A,\theta)=L^{1}(p/q,A,\theta)$ and

(2.27) $\displaystyle
L_{j}(p/q,A,\theta)=L^{j}(p/q,A,\theta)-L^{j-1}(p/q,A,\theta),\text{ for
}j\geq 2,$

and for $1\leq j\leq k$,

(2.28) $\displaystyle
L^{j}(p/q,A)=\int_{\mathbb{T}}L^{j}(p/q,A,\theta)\,\mathrm{d}\theta,\text{ and
}L_{j}(p/q,A)=\int_{\mathbb{T}}L_{j}(p/q,A,\theta)\,\mathrm{d}\theta.$

###  2.3. Transfer matrix of $\hat{H}_{\alpha,\theta,\varepsilon}$

As we mentioned in the introduction, $\hat{H}_{\alpha,\theta,\varepsilon}$ can
be viewed as a block-valued Jacobi matrix:

(2.29)
$\displaystyle(\hat{H}_{\alpha,\theta,\varepsilon}\Phi)_{n}=B_{\varepsilon}\Phi_{n+1}+\tilde{B}_{\varepsilon}\Phi_{n-1}+C_{\varepsilon}(\theta+nd\alpha)\Phi_{n},$

where
$\Phi_{n}=(\phi_{nd+d-1},...,\phi_{nd+1},\phi_{nd})^{T}\in\mathbb{C}^{d}$, the
upper triangular matrix

(2.30) $\displaystyle B_{\varepsilon}=\left(\begin{matrix}\hat{v}_{-d}e^{2\pi
d\varepsilon}&\hat{v}_{-d+1}e^{2\pi(d-1)\varepsilon}\ &...\
&\hat{v}_{-1}e^{2\pi\varepsilon}\\\ &\hat{v}_{-d}e^{2\pi
d\varepsilon}&&\hat{v}_{-2}e^{4\pi\varepsilon}\\\ \\\ &&\ddots\\\ &&\\\
&&&\hat{v}_{-d}e^{2\pi d\varepsilon}\end{matrix}\right)=e^{2\pi
d\varepsilon}F_{\varepsilon}B_{0}F_{\varepsilon}^{-1},$

where
$F_{\varepsilon}=\mathrm{diag}(e^{2\pi\varepsilon},e^{4\pi\varepsilon},...,e^{2\pi
d\varepsilon})$. The lower triangular matrix $\tilde{B}_{\varepsilon}=e^{-2\pi
d\varepsilon}F_{\varepsilon}B_{0}^{*}F_{\varepsilon}^{-1}$, and

(2.31) $\displaystyle C_{\varepsilon}(\theta)=$
$\displaystyle\left(\begin{matrix}2\cos
2\pi(\theta+(d-1)\alpha)&\hat{v}_{1}e^{-2\pi\varepsilon}&\hat{v}_{2}e^{-4\pi\varepsilon}&...&&\hat{v}_{d-1}e^{-2\pi(d-1)\varepsilon}\\\
\hat{v}_{-1}e^{2\pi\varepsilon}&2\cos
2\pi(\theta+(d-2)\alpha)&\hat{v}_{1}e^{-2\pi\varepsilon}&...&&\hat{v}_{d-2}e^{-2\pi(d-2)\varepsilon}\\\
\hat{v}_{-2}e^{4\pi\varepsilon}&\hat{v}_{-1}e^{2\pi\varepsilon}&\ddots&\ddots&&\vdots\\\
\vdots&&\ddots\\\ &&&&&\hat{v}_{1}e^{-2\pi\varepsilon}\\\
\hat{v}_{-d+1}e^{2\pi(d-1)\varepsilon}&...&&&\hat{v}_{-1}e^{2\pi\varepsilon}&2\cos
2\pi\theta\end{matrix}\right)$ (2.32)
$\displaystyle=F_{\varepsilon}C_{0}(\theta)F_{\varepsilon}^{-1}.$

In the rest of the paper, we shall write $C_{\varepsilon=0}(\theta)$ and
$B_{\varepsilon=0}$ as $C(\theta)$ and $B$, and $M_{E}^{\varepsilon=0}=M_{E}$,
$\hat{A}_{E}^{\varepsilon=0}=\hat{A}_{E}$, respectively. The eigenvalue
equation $\hat{H}_{\alpha,\theta,\varepsilon}\phi=E\phi$ can be reformulated
as:

(2.33) $\displaystyle\left(\begin{matrix}B_{\varepsilon}\Phi_{n+1}\\\
\Phi_{n}\end{matrix}\right)=M_{E}^{\varepsilon}(\theta+nd\alpha)\left(\begin{matrix}B_{\varepsilon}\Phi_{n}\\\
\Phi_{n-1}\end{matrix}\right),$

where

(2.34) $\displaystyle M_{E}^{\varepsilon}(\theta)=$
$\displaystyle\left(\begin{matrix}(E-C_{\varepsilon}(\theta))B_{\varepsilon}^{-1}&-\tilde{B}_{\varepsilon}\\\
B_{\varepsilon}^{-1}&0\end{matrix}\right)$ (2.35) $\displaystyle=$
$\displaystyle\left(\begin{matrix}e^{-2\pi
d\varepsilon}F_{\varepsilon}(E-C(\theta))B^{-1}F_{\varepsilon}^{-1}&-e^{-2\pi
d\varepsilon}F_{\varepsilon}B^{*}F_{\varepsilon}^{-1}\\\ e^{-2\pi
d\varepsilon}F_{\varepsilon}B^{-1}F_{\varepsilon}^{-1}&0\end{matrix}\right)$
(2.36) $\displaystyle=$ $\displaystyle e^{-2\pi
d\varepsilon}\mathrm{diag}(F_{\varepsilon},F_{\varepsilon})\cdot
M_{E}(\theta)\cdot\mathrm{diag}(F_{\varepsilon}^{-1},F_{\varepsilon}^{-1}).$

Clearly this implies that for any $1\leq j\leq 2d$,

(2.37) $\displaystyle
L_{j}(d\alpha,M_{E}^{\varepsilon})=L_{j}(d\alpha,M_{E})-2\pi d\varepsilon.$

It is easy to see that

(2.38) $\displaystyle M_{E}^{*}\cdot\Omega\cdot M_{E}=\Omega,$

where

(2.39) $\displaystyle\Omega:=\left(\begin{matrix}0&I_{d}\\\
-I_{d}&0\end{matrix}\right).$

Hence $M_{E}^{0}$ is complex symplectic matrix. This implies that for any
$1\leq j\leq 2d$,

(2.40)
$\displaystyle\hat{L}_{j}(d\alpha,M_{E})=-\hat{L}_{2d+1-j}(d\alpha,M_{E}).$

Let

(2.43) $\displaystyle
M_{n,E}(\theta)=\left(\begin{array}[]{c|c}M_{n,E}^{UL}(\theta)&M_{n,E}^{UR}(\theta)\\\
\hline\cr M_{n,E}^{LL}(\theta)&M_{n,E}^{LR}(\theta)\end{array}\right),$

where each $M_{n,E}^{\star}$ is a $d\times d$ block, $\star=UL,UR,LL,LR$. The
following recursive relations hold: for $n=1$,

(2.44) $\displaystyle\begin{cases}M_{1,E}^{UL}(\theta)=-(C(\theta)-E)B^{-1}\\\
M_{1,E}^{UR}(\theta)=-B^{*}\\\ M_{1,E}^{LL}(\theta)=B^{-1}\\\
M_{1,E}^{LR}(\theta)=0\end{cases}$

and for $n\geq 2$,

(2.45)
$\displaystyle\begin{cases}M_{n,E}^{UL}(\theta)=-M_{n-1,E}^{UL}(\theta+d\alpha)(C(\theta)-E)B^{-1}+M_{n-1,E}^{UR}(\theta+d\alpha)B^{-1}\\\
M_{n,E}^{UR}(\theta)=-M_{n-1,E}^{UL}(\theta+d\alpha)B^{*}\\\
M_{n,E}^{LL}(\theta)=-M_{n-1,E}^{LL}(\theta+d\alpha)(C(\theta)-E)B^{-1}+M_{n-1,E}^{LR}(\theta+d\alpha)B^{-1}\\\
M_{n,E}^{LR}(\theta)=-M_{n-1,E}^{LL}(\theta+d\alpha)B^{*}\end{cases}$

The eigenvalue equation $\hat{H}_{\alpha,\theta,\varepsilon}\phi=E\phi$ (when
viewed as a scalar-valued Hamiltonian) can also be rewritten as:

(2.46) $\displaystyle\left(\begin{matrix}\phi_{n+d}\\\ \vdots\\\ \phi_{n+1}\\\
\phi_{n}\\\ \vdots\\\
\phi_{n-d+1}\end{matrix}\right)=\hat{A}_{E}^{\varepsilon}(\theta+n\alpha)\left(\begin{matrix}\phi_{n+d-1}\\\
\vdots\\\ \phi_{n}\\\ \phi_{n-1}\\\ \vdots\\\ \phi_{n-d}\end{matrix}\right),$

where

(2.47)
$\begin{split}&\hat{A}_{E}^{\varepsilon}(\theta):=\frac{1}{\hat{v}_{-d}e^{2\pi
d\varepsilon}}\\\
&\left(\begin{matrix}-\hat{v}_{1-d}e^{2\pi(d-1)\varepsilon}&\cdots&-\hat{v}_{-1}e^{2\pi\varepsilon}&E-2\cos(2\pi\theta)&|&-\hat{v}_{1}e^{-2\pi\varepsilon}&\cdots&-\hat{v}_{d-1}e^{-2\pi(d-1)\varepsilon}&-\hat{v}_{d}e^{-2\pi
d\varepsilon}\\\ \hat{v}_{-d}e^{2\pi d\varepsilon}&&&&|\\\ &\ddots&&&|&\\\
&&\hat{v}_{-d}e^{2\pi d\varepsilon}&&|&\\\ \hline\cr&&&\hat{v}_{-d}e^{2\pi
d\varepsilon}&|&\\\ &&&&|&\hat{v}_{-d}e^{2\pi d\varepsilon}\\\
&&&&|&&\ddots\\\ &&&&|&&&\hat{v}_{-d}e^{2\pi
d\varepsilon}\end{matrix}\right),\end{split}$

is the 1-step transfer matrix. By (2.46) and (2.33) we have

(2.48) $\displaystyle
M_{E}^{\varepsilon}(\theta)=\mathrm{diag}(B_{\varepsilon},I_{d})\cdot\hat{A}_{d,E}^{\varepsilon}(\alpha,\theta)\cdot\mathrm{diag}(B_{\varepsilon}^{-1},I_{d}).$

Recall that $\hat{A}_{d}$ signifies a product of $d$ factors of $\hat{A}$
matrices. Hence for any $1\leq m\leq 2d$,

(2.49)
$\displaystyle\hat{L}_{m}(d\alpha,M_{E}^{\varepsilon})=d\cdot\hat{L}_{m}(\alpha,\hat{A}_{E}^{\varepsilon}).$

Combining this with (2.37) yields

(2.50)
$\displaystyle\hat{L}_{m}(\alpha,\hat{A}_{E}^{\varepsilon})=\hat{L}_{m}(\alpha,\hat{A}_{E})-2\pi\varepsilon.$

###  2.4. Avila’s acceleration

Let $(\alpha,A)\in(\mathbb{T},C^{\omega}(\mathrm{SL}(2,\mathbb{R})))$, and
$A_{\varepsilon}(\theta):=A(\theta+i\varepsilon)$. The (top) Lyapunov exponent
$L_{1}(\alpha,A_{\varepsilon})$ is a convex and even function in
$\varepsilon$. Avila defined the acceleration to be the right-derivative as
follows:

(2.51) $\displaystyle\kappa(\alpha,A_{\varepsilon}):=\lim_{\delta\to
0^{+}}\frac{L_{1}(\alpha,A_{\varepsilon+\delta})-L_{1}(\alpha,A_{\varepsilon})}{2\pi\delta}.$

As a cornerstone of his Global theory [Av3], he showed that for
$A\in\mathrm{SL}(2,\mathbb{R})$ and irrational $\alpha$,
$\kappa(\alpha,A_{\varepsilon})\in\mathbb{Z}$ is always quantized.

###  2.5. Uniform upper semi-continuity

The following lemma is an easy corollary of the arguments, essentially a
subadditivity argument, in the proof of Lemma 5.1 in [AJS].

######  Lemma 2.1.

For any small $\varepsilon>0$, there exists $N=N(\varepsilon,\alpha,v,E,d)$
such that for any $1\leq j\leq 2d$ and $n\geq N$,

(2.52)
$\displaystyle\frac{1}{n}\log\left\|\wedge^{j}M_{n,E}(\theta)\right\|\leq\hat{L}_{1}(d\alpha,\wedge^{j}M_{E})+\varepsilon=\hat{L}^{j}(d\alpha,M_{E})+\varepsilon,$

uniformly in $\theta\in\mathbb{T}$.

###  2.6. Large deviation estimates

The following large deviation estimates play a crucial role in Section 5. For
Diophantine $\alpha$, a stronger form (with a deviation of size $n^{-\sigma}$)
of the following estimate for $\mathrm{SL}(2,\mathbb{R})$ cocycles was first
proved in Lemma 1.1 of [BG] by Bourgain and Goldstein, and further developed
in sharp forms in [GS1, GS2] by Goldstein and Schlag. In this paper we need
the following weaker form, but it holds for any $\alpha$ such that
$\beta(\alpha)=0$.

######  Lemma 2.2.

[HZ]*Theorem 1.1 Let

(2.53)
$\displaystyle\mathcal{B}_{n,E,\varepsilon}^{d}:=\left\\{\theta:\,\frac{1}{n}\log\|\wedge^{d}M_{n,E}(\theta)\|\leq(1-\varepsilon)\cdot\hat{L}^{d}(d\alpha,M_{E})\right\\}$

be the large deviation set of $\|\wedge^{d}M_{n,E}(\theta)\|$. For $\alpha$
such that $\beta(\alpha)=0$, for any small $\varepsilon>0$, there exists
$N=N(\varepsilon,\alpha,\hat{L}_{d}(d\alpha,M_{E}),v,d)>0$ and
$c_{1}=c_{1}(v,d)>0$, such that for $n\geq N$,

(2.54) $\displaystyle\mathrm{mes}(\mathcal{B}_{n,E,\varepsilon}^{d})\leq
e^{-c_{1}n\varepsilon^{2}\cdot\hat{L}^{d}(d\alpha,M_{E})}.$

######  Remark 2.3.

The above large deviation lemma was stated and proved in [HZ] for
$\mathrm{SL}(2,\mathbb{R})$ cocycles, but it is clear that the same proof
applies to $(d\alpha,\wedge^{d}M_{E})$.

###  2.7. Avalanche Principle

The Avalanche Principle was first introduced for $\mathrm{SL}(2,\mathbb{R})$
cocycles by Goldstein and Schlag in [GS1]. Later it was extended to higher-
dimensional cocycles in [Sch1, DK]. The following form is from [DK].

######  Theorem 2.4.

[DK]*Proposition 2.42 There exists $c_{0},\tilde{C}_{0}>0$ such that given
$0<\varepsilon<1$, $0<\kappa<c_{0}\varepsilon^{2}$ and
$g_{0},g_{1},...,g_{n-1}\in\mathrm{Mat}(m,\mathbb{R})\setminus\\{0\\}$, if

(2.55)
$\displaystyle\frac{\sigma_{1}(g_{j})}{\sigma_{2}(g_{j})}>\frac{1}{\kappa},\text{
for all }0\leq j\leq n-1$ (2.56)
$\displaystyle\frac{\|g_{j}g_{j-1}\|}{\|g_{j}\|\|g_{j-1}\|}>\varepsilon,\text{
for all }1\leq j\leq n-1$

then

(2.57) $\displaystyle\left|\log\|g_{n-1}\cdots
g_{0}\|+\sum_{j=1}^{n-2}\log\|g_{j}\|-\sum_{j=1}^{n-1}\log\|g_{j}g_{j-1}\|\right|\leq\tilde{C}_{0}n\frac{\kappa}{\varepsilon^{2}}.$

###  2.8. Integrated density of states

Let

(2.58)
$\displaystyle\hat{H}_{\alpha,n}(\theta):=\chi_{[0,nd-1]}\hat{H}_{\alpha,\theta,\varepsilon=0}\,\chi_{[0,nd-1]}$

be the restriction of the operator to the interval $[0,nd-1]$ with Dirichlet
boundary condition. Let $E_{j}(\theta)$, $j=0,1,...,nd-1$ be the eigenvalues
of $\hat{H}_{\alpha,n}(\theta)$. Consider the sequence of measures

(2.59)
$\displaystyle\hat{\mu}_{n,\alpha,\theta}(\mathrm{d}E):=\frac{1}{nd}\sum_{j=0}^{nd-1}\delta_{E_{j}(\theta)}(\mathrm{d}E).$

It is well-known the weak limit exists and is independent of $\theta$ (up to
measure zero):

(2.60)
$\displaystyle\lim_{n\to\infty}\hat{\mu}_{n,\alpha,\theta}(\mathrm{d}E)=\widehat{\mathcal{N}}(\mathrm{d}E).$

The distribution function $\widehat{\mathcal{N}}(E)$ is called the integrated
density of states.

The integrated density of states $\mathcal{N}$ are defined analogously for
$H_{\alpha,\theta,\varepsilon=0}$ (indeed as the $d=1$ special case of the
above). It is well-known that $\mathcal{N}=\widehat{\mathcal{N}}$, see e.g.
[HP]

The Thouless formula relates the Lyapunov exponent to the integrated density
of states:

(2.61)
$\displaystyle\hat{L}^{d}(\alpha,\hat{A}_{E})+\log|\hat{v}_{d}|=\int_{\mathbb{R}}\log|E^{\prime}-E|\,\widehat{\mathcal{N}}(\mathrm{d}E^{\prime}).$

The scalar case $d=1$ was proved in [AS, CS1], and the $d>1$ case was proved
in [CS2, KS, HP].

##  3\. Anderson localization and almost localization

###  3.1. Green’s function

The key to prove Anderson localization is to estimate the decay of Green’s
functions. We consider the finite volume Hamiltonian with periodic boundary
conditions. To distinguish $\hat{H}_{\alpha,n}(\theta)$ (see (2.58)), which
exhibits Dirichlet boundary conditions, from the operator with periodic ones,
we denote the latter by $P_{\alpha,n}(\theta)$. Moreover, for simplicity we
shall omit $\alpha$ in the notation in the rest of the section. Thus,

(3.62) $\displaystyle
P_{n}(\theta)=\left(\begin{matrix}C(\theta+(n-1)d\alpha)&B^{*}&&&B\\\
B&C(\theta+(n-2)d\alpha)&\ddots\\\ &\ddots&\ddots&\ddots\\\
&&\ddots&\ddots&B^{*}\\\ B^{*}&&&B&C(\theta)\end{matrix}\right).$

Let

(3.63) $\displaystyle f_{E,n}(\theta):=\det(P_{n}(\theta)-E),$

and

(3.64) $\displaystyle G_{E,n}(\theta):=(P_{n}(\theta)-E)^{-1}$

be the resolvent, which is also called the finite volume Green’s function. It
follows from Cramer’s rule that

(3.65) $\displaystyle
G_{E,n}(\theta;x,y)=\frac{\mu_{n,x,y}(\theta)}{f_{E,n}(\theta)},$

where $\mu_{n,x,y}(\theta)$ is the determinant of the submatrix of
$P_{n}(\theta)-E$ obtained by deleting the $x$-th row and $y$-th column. Let
$\hat{u}$ be a solution to the eigenvalue equation
$\hat{H}_{\alpha,\theta}\hat{u}=E\hat{u}$. For any $k\in\mathbb{Z}$ the
following Poisson formula holds for all $k\leq m\leq k+nd-1$:

(3.66) $\displaystyle\hat{u}_{m}=$
$\displaystyle\sum_{y_{1}=0}^{d-1}G_{E,n}(\theta+k\alpha;m-k,y_{1})\left(B^{*}\cdot\left(\begin{matrix}\hat{u}_{k+nd-1}-\hat{u}_{k+d-1}\\\
\vdots\\\ \hat{u}_{k+(n-1)d}-\hat{u}_{k}\end{matrix}\right)\right)_{y_{1}}$
(3.67)
$\displaystyle\qquad\qquad+\sum_{y_{2}=(n-1)d}^{nd-1}G_{E,n}(\theta+k\alpha;m-k,y_{2})\left(B\cdot\left(\begin{matrix}\hat{u}_{k+d-1}-\hat{u}_{k+nd-1}\\\
\vdots\\\
\hat{u}_{k}-\hat{u}_{k+(n-1)d}\end{matrix}\right)\right)_{y_{2}-(n-1)d},$

in which $(M)_{y}$ refers to the element of vector $M$ in row $y$. This
implies

(3.68) $\displaystyle|\hat{u}_{m}|\leq$ $\displaystyle
4d^{3/2}\|B\|\cdot\max_{y\in\\{0,...,d-1\\}\cup\\{(n-1)d,...,nd-1\\}}|G_{E,n}(\theta+k\alpha;m-k,y)|\cdot$
(3.69)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\cdot\max_{\ell\in\\{0,...,d-1\\}\cup\\{(n-1)d,...,nd-1\\}}|\hat{u}_{k+\ell}|$

For the remainder of this section, we shall write
$\hat{L}^{k}(d\alpha,M_{E}),\hat{L}_{k}(d\alpha,M_{E})$ as
$\hat{L}^{k},\hat{L}_{k}$, respectively. We split the estimate of the Green’s
function into Lemma 3.1 for the numerator, and Lemma 3.2 for the denominator.

######  Lemma 3.1.

Let $3d\leq y\leq(n-1)d-1$ and $0\leq x\leq d-1$ or $(n-1)d\leq x\leq nd-1$.
Then for any $\varepsilon>0$, and uniformly in $\theta\in\mathbb{T}$,

(3.70) $\displaystyle|\mu_{n,x,y}(\theta)|\leq C_{d}\|B^{-1}\|^{2}\cdot|\det
B|^{n}\cdot\max\left(e^{\ell(1+\varepsilon)\hat{L}^{d-1}+(n-\ell)(1+\varepsilon)\hat{L}^{d}}+e^{\ell(1+\varepsilon)\hat{L}^{d}+(n-\ell)(1+\varepsilon)\hat{L}^{d-1}}\right),$

provided $n>N(\varepsilon,\alpha,E,d,v)$ is large enough. Here $C_{d}$ is a
constant depending only on $d$.

We defer the proof to Section 4.

######  Lemma 3.2.

Let $N_{0}=N_{0}(\varepsilon,\alpha,E,d)$ and
$\kappa_{0}=\kappa_{0}(\varepsilon,\alpha,E,d,N_{0})$ be defined as in Lemma
5.3 and $\mathcal{N}:=\\{n\geq
8N_{0}:\|nd\alpha\|_{\mathbb{T}}\leq\kappa_{0}\\}$ be the
$\kappa_{0}$-admissible sequence. For any $E\in\mathbb{R}$, for any small
$0<\varepsilon<\hat{L}_{d}/(11\hat{L}^{d})$, and for $n\geq
N(\varepsilon,\alpha,E,d,v)$ and $n\in\mathcal{N}$, we have

(3.71)
$\displaystyle\mathrm{mes}\left\\{\theta\in\mathbb{T}:\frac{1}{n}\log|f_{E,n}(\theta)|<|\det
B|+(1-8\varepsilon)\hat{L}^{d}\right\\}<\frac{1}{10}.$

We present the proof of Lemma 3.2 in Section 5. Although Lemma 3.2 does not
hold for arbitrary $n$, it holds for all $\kappa_{0}$-admissible $n$’s, which
form a positive density subsequence. Indeed, there exists
$C_{\kappa_{0},d,\alpha}$ such that for any $n\in\mathbb{N}$, there is a
$\kappa_{0}$-admissible $\tilde{n}$ such that $|n-\tilde{n}|\leq
C_{\kappa_{0},d,\alpha}$. Hence Lemma 3.2 yields the following.

######  Corollary 3.3.

For any $E\in\mathbb{R}$, for any small
$0<\varepsilon<\hat{L}_{d}/(11\hat{L}^{d})$, let $N_{0},\kappa_{0}$ be as in
Lemma 3.2. There exists $C_{\kappa_{0},d,\alpha}>0$ such that for any $n\geq
N(\varepsilon,\alpha,E,d,v)$, there exists $\kappa_{0}$-admissible
$\tilde{n}$, $|\tilde{n}-n|\leq C_{\kappa_{0},d,\alpha}$, so that

(3.72)
$\displaystyle\mathrm{mes}\left\\{\theta\in\mathbb{T}:\frac{1}{\tilde{n}}\log|f_{E,\tilde{n}}(\theta)|<|\det
B|+(1-8\varepsilon)\hat{L}^{d}\right\\}<\frac{1}{10}.$

######  Remark 3.4.

The reason why we consider $P_{\alpha,n}(\theta)$ with periodic boundary
conditions, instead of the finite volume operator $\hat{H}_{\alpha,n}$ on
$[0,nd-1]$ with Dirichlet boundary conditions, lies with the differences
between the denominators of their respective Green’s functions. In fact, as we
will show in Lemma 5.1

$|f_{E,n}(\theta)|=|\det B|^{n}\cdot|\det(M_{n,E}(\theta)-I_{2d})|,$

while

$|\det(\hat{H}_{\alpha,n}(\theta)-E)|=|\det B|^{n}\cdot|\det
M_{n,E}^{UL}(\theta)|,$

see [BGV]. To prove non-perturbative Anderson localization, it is crucial to
obtain a lower bound

$|\det(M_{n,E}(\theta)-I_{2d})|\gtrsim\exp(n(\hat{L}^{d}-\varepsilon))$

for certain $\theta$’s, which we establish in Lemma 5.13 along the
$\kappa_{0}$-admissible sequence. It appears to be more difficult to obtain a
similar result for $|\det M_{n,E}^{UL}(\theta)|$. One may attempt to derive
the inequality

$|\det M_{n,E}^{UL}(\theta)|\gtrsim\exp(n(\hat{L}^{d}-\varepsilon))$

via Thouless’ formula. For $E\notin\mathbb{R}$, such a lower bound for $|\det
M_{n,E}^{UL}(\theta)|$ can indeed be deduced directly from the Thouless
formula (proved for scalar case $(d=1)$ in [AS, CS1], and operators on the
strip in [CS2, KS, HP]). In fact, if $E\notin\mathbb{R}$, then
$g(E^{\prime})=\log|E-E^{\prime}|$ is a continuous function on $\mathbb{R}$.
By the $\sigma^{*}$-convergence of the finite volume density of states measure
to the (infinite volume) density states measure $\widehat{\mathcal{N}}$, it
follows that

(3.73) $\displaystyle\lim_{n\to\infty}\frac{1}{nd}(\log\left|\det
M_{n,E}^{UL}(\theta)\right|+n\log|\det B|)$
$\displaystyle=\lim_{n\to\infty}\frac{1}{nd}\log|\det(\hat{H}_{\alpha,n}(\theta)-E)|$
(3.74)
$\displaystyle=\int_{\mathbb{T}}\log|E-E^{\prime}|\,\widehat{\mathcal{N}}(\mathrm{d}E^{\prime})=\frac{1}{d}\hat{L}^{d}+\log|\hat{v}_{d}|$

for a.e. $\theta$. However this simple proof does not apply to
$E\in\mathbb{R}$, since $g$ is not continuous. See also (1.13).

We emphasize the robustness of the lower bound in Lemma 3.2 for the
denominator. It can be applied to operators $\hat{H}_{\alpha,\theta}$ on the
strip with general analytic potentials on higher-dimensional tori. We will use
it in an upcoming work to address non-perturbative Anderson localization in
the general case. For the special case of cosine potential, a simpler lower
bound can be obtained via Herman’s subharmonic argument.

######  Lemma 3.5.

For any $n\geq 1$, we have

(3.75)
$\displaystyle\frac{1}{n}\int_{\mathbb{T}}\log|f_{E,n}(\theta)|\,\mathrm{d}\theta\geq
0=\hat{L}^{d}+d\log|\hat{v}_{d}|.$

We present the proof at the end of Section 5. Note the equality sign above is
exactly Haro-Puig’s theorem (see Theorem 1.8) combined with the subcritical
assumption on the Schrödinger cocycle.

###  3.2. Almost localization

The proof of almost localization follows the scheme of [AJ2], combined with
our Lemmas 3.1 and 3.2. A minor adjustment is that [AJ2] deals with
$\alpha\in\mathrm{DC}$ while we assume a slightly weaker condition
$\beta(\alpha)=0$. The almost localization for the $\beta(\alpha)=0$ case was
previously addressed for the extended Harper’s model in [H]. It should be
sufficient to assume that $0\leq\beta(\alpha)<c$ for some small constant
$c>0$, but we do not pursue this improvement here.

Let $C_{3}>0$ be the universal constant in Lemma 3.10 below. Fix

(3.76) $\displaystyle
0<\varepsilon_{0}<\frac{\hat{L}_{d}(\alpha,\hat{A}_{E})}{30\hat{L}^{d}(\alpha,\hat{A}_{E})+100C_{3}d}<\frac{\hat{L}_{d}}{11\hat{L}^{d}}.$

######  Definition 3.1 ($\varepsilon_{0}$-resonance of $\theta$).

Let $\alpha,\theta\in\mathbb{T}$ and constant $\varepsilon_{0}>0$. We say
$k\in\mathbb{Z}$ is an $\varepsilon_{0}$-resonance of $\theta$ if
$\|2\theta-k\alpha\|_{\mathbb{T}}\leq e^{-\varepsilon_{0}|k|}$ and
$\|2\theta-k\alpha\|_{\mathbb{T}}=\min_{|j|\leq
k}\|2\theta-j\alpha\|_{\mathbb{T}}$.

Since $\beta(\alpha)=0$, there exists $C_{0}=C_{0}(\alpha,\varepsilon_{1})>0$
such that for $n\neq 0$,

(3.77) $\displaystyle\|n\alpha\|_{\mathbb{T}}\geq
C_{0}e^{-\varepsilon_{1}|n|}.$

In the following we shall fix $\varepsilon_{1}=\varepsilon_{0}/1000$. This
implies the following.

######  Lemma 3.6.

If $|k|$ is large enough, and $\|2\theta-k\alpha\|_{\mathbb{T}}\leq
e^{-\varepsilon_{0}|k|}$, then $\|2\theta-k\alpha\|_{\mathbb{T}}=\min_{|j|\leq
k}\|2\theta-j\alpha\|_{\mathbb{T}}$.

###### Proof.

In fact, for $|j|\leq k$ and $j\neq k$,

(3.78) $\displaystyle\|2\theta-j\alpha\|_{\mathbb{T}}\geq$
$\displaystyle\|(j-k)\alpha\|_{\mathbb{T}}-\|2\theta-k\alpha\|_{\mathbb{T}}$
(3.79) $\displaystyle\geq$ $\displaystyle
C_{0}e^{-\varepsilon_{0}|j-k|/1000}-e^{-\varepsilon_{0}|k|}$ (3.80)
$\displaystyle\geq$ $\displaystyle
C_{0}e^{-\varepsilon_{0}|k|/500}-e^{-\varepsilon_{0}|k|}$ (3.81)
$\displaystyle\geq$ $\displaystyle
e^{-\varepsilon_{0}|k|}\geq\|2\theta-k\alpha\|_{\mathbb{T}},$

in which we used (3.77) to estimate $\|(j-k)\alpha\|_{\mathbb{T}}$. ∎

######  Definition 3.2.

Let $0=|n_{0}|\leq|n_{1}|\leq|n_{2}|\leq...$ be the
$\varepsilon_{0}$-resonances of $\theta$. If this sequence is infinite, we say
$\theta$ is $\varepsilon_{0}$-resonant, otherwise we say it is
$\varepsilon_{0}$-non-resonant. Furthermore, if $\theta$ is
$\varepsilon_{0}$-non-resonant with a finite sequence of resonances
$0=|n_{0}|\leq|n_{1}|\leq...\leq|n_{j}|$, we let $n_{j+1}=\infty$.

Due to $\beta(\alpha)=0$, we can show the following growth of the infinite
sequence of $\theta$ resonances.

######  Lemma 3.7.

Let $\beta(\alpha)=0$, then there exists
$N_{0}=N_{0}(\alpha,\varepsilon_{0})$, such that for $|n_{j}|>N_{0}$,

(3.82) $\displaystyle|n_{j+1}|>250|n_{j}|$

###### Proof.

By (3.77),

(3.83) $\displaystyle e^{-\varepsilon_{0}|n_{j}|}\geq\|2\theta-
n_{j}\alpha\|_{\mathbb{T}}\geq$
$\displaystyle\|(n_{j+1}-n_{j})\alpha\|_{\mathbb{T}}-\|2\theta-
n_{j+1}\alpha\|_{\mathbb{T}}$ (3.84) $\displaystyle\geq$ $\displaystyle
C_{0}e^{-\varepsilon_{0}|n_{j+1}-n_{j}|/1000}-e^{-\varepsilon_{0}|n_{j+1}|}$
(3.85) $\displaystyle\geq$ $\displaystyle
C_{0}e^{-\varepsilon_{0}|n_{j+1}|/500}-e^{-\varepsilon_{0}|n_{j+1}|}$ (3.86)
$\displaystyle\geq$
$\displaystyle\frac{C_{0}}{2}e^{-\varepsilon_{0}|n_{j+1}|/500},$

which implies the desired bound. ∎

######  Definition 3.3 (Almost localization).

We say the family $\\{\hat{H}_{\alpha,\theta}\\}_{\theta\in\mathbb{T}}$ is
$(C_{1},C_{2},\varepsilon_{2})$-almost localized for some constant
$C_{1},C_{2},\varepsilon_{2}>0$ if for every solution $\hat{u}$ of
$\hat{H}_{\alpha,\theta}\hat{u}=E\hat{u}$ for some energy $E\in\mathbb{R}$,
satisfying $\hat{u}_{0}=1$ and $|\hat{u}_{k}|\leq 1+|k|$, and for every
$C_{1}(1+|n_{j}|)<|k|<C_{1}^{-1}|n_{j+1}|$, the bound $|\hat{u}_{k}|\leq
C_{2}e^{-\varepsilon_{2}|k|}$ holds, where the $n_{j}$’s are the
$\varepsilon_{0}$-resonances of $\theta$.

We will prove the following.

######  Theorem 3.8.

There exists a constant $C_{2}=C_{2}(v,d)>0$ such that
$\\{\hat{H}_{\alpha,\theta}\\}_{\theta\in\mathbb{T}}$ is
$(3,C_{2},\hat{L}_{d}/10)$-almost localized.

###### Proof.

Let $f_{E,N}(\theta)$ be defined as in (3.63), and we omit the dependence on
$E$ for simplicity. The following observation was made for the Harper operator
(with Dirichlet boundary conditions) in [J1], and extended to the long-range
setting in [AJ2]. The extension to the periodic boundary condition setting is
immediate.

######  Fact 3.9.

$f_{N}(\theta)=Q_{dN}(\cos(2\pi(\theta+\frac{dN-1}{2}\alpha)))$, where
$Q_{k}(x)$ denotes a real polynomial in $x$ of degree $k$.

Next we introduce of definition of uniformity, introduced in [J1], that
measures the evenness of distribution of a set in $\mathbb{T}$.

######  Definition 3.4 (Uniformity).

For some $\kappa>0$, we say that a set
$\\{\theta_{1},\theta_{2},...,\theta_{m+1}\\}$ is $\kappa$-uniform if

$\displaystyle\max_{z\in[0,1]}\max_{j=1,\cdots,m+1}\Bigg{|}\prod_{\begin{subarray}{c}\ell=1\\\
\ell\neq
j\end{subarray}}^{m+1}\frac{z-\cos(2\pi\theta_{\ell})}{\cos(2\pi\theta_{j})-\cos(2\pi\theta_{\ell})}\Bigg{|}<e^{\kappa
m}.$

Without loss of generality, assume $3(|n_{j}|+1)<y<|n_{j+1}|/3$ and
$|n_{j}|>N_{0}(\alpha,\varepsilon_{0})$. The case of negative $y$ can be
handled similarly. Choose $n$ such that $dq_{n}\leq y/8<dq_{n+1}$ and let $s$
be the largest positive integer satisfying $sdq_{n}\leq y/8$. Define intervals
$I_{1},I_{2}\subset\mathbb{Z}$ as follows:

$\displaystyle I_{1}=[-2sdq_{n},0]\cap\mathbb{Z}\text{\ \ and\ \
}I_{2}=[y-2sdq_{n}+1,y+2sdq_{n}]\cap\mathbb{Z},\,\,$
$\displaystyle\mathrm{if}\ n_{j}<0$ $\displaystyle
I_{1}=[-1,2sdq_{n}-1]\cap\mathbb{Z}\text{\ \ and\ \
}I_{2}=[y-2sdq_{n}+1,y+2sdq_{n}]\cap\mathbb{Z},\,\,$
$\displaystyle\mathrm{if}\ n_{j}\geq 0$

The key in choosing $I_{1},I_{2}$ is to guarantee that $-n_{j}\notin(I_{1}\cup
I_{2})+(I_{1}\cup I_{2})$. Let $6sq_{n}=N$ whence
$|I_{1}|+|I_{2}|=6dsq_{n}+1=dN+1$.

######  Lemma 3.10.

The set

$\left\\{\theta+j\alpha-\left\lfloor\frac{dN-1}{2}\right\rfloor\alpha+\frac{dN-1}{2}\alpha\right\\}_{j\in
I_{1}\cup I_{2}}\subset\mathbb{T}$

is $(C_{3}d\,\varepsilon_{0})$-uniform for some absolute constant $C_{3}\geq
1$ and $y>y_{0}(\alpha,\varepsilon_{0},C_{\kappa_{0}})$.

For $d=1$, when $N-1$ is even, this is Lemma 4.3 of [H]. In the general case,
the proof requires a slight modification, which we present below.

###### Proof.

We treat the $n_{j}>0$ case in detail and leave the other case to the reader.
Note that for $y$ large, $n$ is also large. Set $z=\cos(2\pi\xi)$, and denote

(3.87)
$\displaystyle\tilde{\theta}_{j}:=\theta+j\alpha-\left\lfloor\frac{dN-1}{2}\right\rfloor\alpha+\frac{dN-1}{2}\alpha.$

Note that since

(3.88) $\displaystyle\max_{z\in[0,1]}\max_{\begin{subarray}{c}j\in I_{1}\cup
I_{2}\\\
j\neq-1\end{subarray}}\left|\frac{z-\cos(2\pi\tilde{\theta}_{-1})}{\cos(2\pi\tilde{\theta}_{j})-\cos(2\pi\tilde{\theta}_{-1})}\right|$
(3.89) $\displaystyle\leq$ $\displaystyle 2\max_{\begin{subarray}{c}j\in
I_{1}\cup I_{2}\\\
j\neq-1\end{subarray}}|\cos(2\pi\tilde{\theta}_{j})-\cos(2\pi\tilde{\theta}_{-1})|^{-1}$
(3.90) $\displaystyle\leq$ $\displaystyle C\max_{\begin{subarray}{c}j\in
I_{1}\cup I_{2}\\\
j\neq-1\end{subarray}}\left(\|\tilde{\theta}_{j}-\tilde{\theta}_{-1}\|_{\mathbb{T}}^{-1}\cdot\|\tilde{\theta}_{j}+\tilde{\theta}_{-1}\|_{\mathbb{T}}^{-1}\right)$

We use (3.77) with $\varepsilon_{1}=\varepsilon_{0}/10$ to estimate

(3.91)
$\displaystyle\|\tilde{\theta}_{j}-\tilde{\theta}_{-1}\|_{\mathbb{T}}=\|(j+1)\alpha\|_{\mathbb{T}}\geq
C_{\varepsilon_{0}}e^{-\frac{\varepsilon_{0}}{10}|j+1|}\geq
C_{\varepsilon_{0}}e^{-\frac{\varepsilon_{0}}{10}(y+2sdq_{n}+1)}\geq
C_{\varepsilon_{0}}e^{-\frac{3}{10}\varepsilon_{0}dN},$

where we used $y+2sdq_{n}\leq 18sdq_{n}=3dN$. Regarding the term

(3.92) $\displaystyle\|\tilde{\theta}_{j}+\tilde{\theta}_{-1}\|_{\mathbb{T}}=$
$\displaystyle\|2\theta+(j-1)\alpha-2\left\lfloor(dN-1)/2\right\rfloor\alpha+(dN-1)\alpha\|_{\mathbb{T}}$

we distinguish into two cases.

Case 1. Suppose
$\|2\theta+(j-1)\alpha-2\left\lfloor(dN-1)/2\right\rfloor\alpha+(dN-1)\alpha\|_{\mathbb{T}}<e^{-\varepsilon_{0}|j-1-2\left\lfloor(dN-1)/2\right\rfloor+(dN-1)|}$.
Then by Lemma 3.6, we have $-n_{j}<-1\leq
j-1-2\left\lfloor(dN-1)/2\right\rfloor+(dN-1)=-n_{p}$ for some $p\leq n-1$.
This implies

(3.93)
$\displaystyle\|2\theta+(j-1)\alpha-2\left\lfloor(dN-1)/2\right\rfloor\alpha+(dN-1)\alpha\|_{\mathbb{T}}=$
$\displaystyle\|2\theta-n_{p}\alpha\|_{\mathbb{T}}$ (3.94) $\displaystyle\geq$
$\displaystyle\|(n_{j}-n_{p})\alpha\|_{\mathbb{T}}-\|2\theta-
n_{j}\alpha\|_{\mathbb{T}}$ (3.95) $\displaystyle\geq$ $\displaystyle
C_{\varepsilon_{0}}e^{-\frac{\varepsilon_{0}}{10}|n_{j}-n_{p}|}-e^{-\varepsilon_{0}|n_{j}|}$
(3.96) $\displaystyle\geq$ $\displaystyle
C_{\varepsilon_{0}}e^{-\frac{\varepsilon_{0}}{5}|n_{j}|}-e^{-\varepsilon_{0}|n_{j}|}$
(3.97) $\displaystyle\geq$ $\displaystyle
C_{\varepsilon_{0}}e^{-\frac{\varepsilon_{0}}{4}|n_{j}|}$ (3.98)
$\displaystyle\geq$ $\displaystyle
C_{\varepsilon_{0}}e^{-\frac{\varepsilon_{0}}{4}dN},$

where in the last inequality we used $|n_{j}|<y/3\leq dN$.

Case 2. Suppose
$\|2\theta+(j-1)\alpha-2\left\lfloor(dN-1)/2\right\rfloor\alpha+(dN-1)\alpha\|_{\mathbb{T}}\geq
e^{-\varepsilon_{0}|j-1-2\left\lfloor(dN-1)/2\right\rfloor+(dN-1)|}$. Then we
directly bound

(3.99)
$\displaystyle\|2\theta+(j-1)\alpha-2\left\lfloor(dN-1)/2\right\rfloor\alpha+(dN-1)\alpha\|_{\mathbb{T}}\geq$
$\displaystyle
e^{-\varepsilon_{0}|j-1-2\left\lfloor(dN-1)/2\right\rfloor+(dN-1)|}$ (3.100)
$\displaystyle\geq$ $\displaystyle e^{-3\varepsilon_{0}dN},$

where we used $j-1-2\left\lfloor(dN-1)/2\right\rfloor+(dN-1)\leq
y+2sdq_{n}\leq 3dN$. Taking (3.88), (3.91), (3.92), (3.93) and (3.99) into
account, it is enough to prove the claimed result with $I_{1}=[-1,2sdq_{n}-1]$
replaced with $I_{1}=[0,2sdq_{n}-1]$.

######  Lemma 3.11.

[AJ1] Let $\alpha\in\mathbb{R}\backslash\mathbb{Q}$, $x\in\mathbb{R}$ and
$0\leq j_{0}\leq q_{n}-1$ be such that $|\sin\pi(x+j_{0}\alpha)|=\inf_{0\leq
j\leq q_{n}-1}|\sin\pi(x+j\alpha)|$, then for some absolute constant
$C_{4}>0$,

$-C_{4}\log q_{n}\leq\sum_{0\leq j\leq q_{n}-1,j\neq
j_{0}}\log|\sin\pi(x+j\alpha)|+(q_{n}-1)\log 2\leq C_{4}\log q_{n}$

For any $j\in I_{1}\cup I_{2}$, it suffices to estimate

$\displaystyle\sum_{\ell\in I_{1}\cup I_{2},\ \ell\neq
j}\left(\log{|\cos(2\pi\xi)-\cos(2\pi\tilde{\theta}_{\ell})|}-\log{|\cos(2\pi\tilde{\theta}_{j})-\cos(2\pi\tilde{\theta}_{\ell})|}\right)=:\sum_{1}-\sum_{2}.$

Lemma 3.11 reduces this problem to estimating the minimal terms. We first
bound $\sum_{1}$:

$\displaystyle\sum_{1}$ $\displaystyle=\sum_{\ell\in I_{1}\cup I_{2},\ell\neq
j}\log|\cos(2\pi\xi)-\cos(2\pi\tilde{\theta}_{\ell})|$
$\displaystyle=\sum_{\ell\in I_{1}\cup I_{2},\ell\neq
j}\log|\sin{(\pi(\xi+\tilde{\theta}_{\ell}))}|+\sum_{\ell\in I_{1}\cup
I_{2},\ell\neq
j}\log|\sin(\pi(\xi-\tilde{\theta}_{\ell}))|+(|I_{1}|+|I_{2}|-1)\log 2$
(3.101) $\displaystyle=:\sum_{1,+}+\sum_{1,-}+dN\log 2.$

Dividing $I_{1}\cup I_{2}=\bigcup_{k=1}^{6sd}T_{k}$ into $6sd$ small intervals
of integers, each of length $q_{n}$, and denoting

(3.103)
$\displaystyle|\sin(\pi(\xi+\tilde{\theta}_{\ell_{k}}))|:=\min_{\ell\in
T_{k}}|\sin(\pi(\xi+\tilde{\theta}_{\ell}))|,$

one has

$\displaystyle\sum_{\ell\in I_{1}\cup I_{2},\ell\neq
j}\log|\sin(\pi(\xi+\tilde{\theta}_{\ell}))|\leq$
$\displaystyle\sum_{k=1}^{6sd}\sum_{\ell\in
T_{k},\ell\neq\ell_{k}}\log|\sin(\pi(\xi+\tilde{\theta}_{\ell}))|$ (3.104)
$\displaystyle\leq$ $\displaystyle 6sd(C_{4}\log q_{n}-(q_{n}-1)\log 2),$

where we used Lemma 3.11 in the last inequality. Similarly, replacing $\xi$
with $-\xi$,

(3.105) $\displaystyle\sum_{1,-}\leq 6sq(C_{4}\log q_{n}-(q_{n}-1)\log 2).$

Combining (3.2) with (3.2) and (3.105), we arrive at

$\displaystyle\sum_{1}\leq$ $\displaystyle 12sd(C_{4}\log q_{n}-(q_{n}-1)\log
2)+dN\log 2$ (3.106) $\displaystyle\leq$ $\displaystyle-6sdq_{n}\log
2+\varepsilon_{0}sdq_{n},$

for $n$ large. Next, we estimate $\sum_{2}$:

$\displaystyle\sum_{2}=$ $\displaystyle\sum_{\ell\in I_{1}\cup I_{2},\ell\neq
j}\log|\cos(2\pi\tilde{\theta}_{j})-\cos(2\pi\tilde{\theta}_{\ell})|$
$\displaystyle=$ $\displaystyle\sum_{\ell\in I_{1}\cup I_{2},\ell\neq
j}\log|\sin(\pi(\tilde{\theta}_{j}+\tilde{\theta}_{\ell}))|+\sum_{\ell\in
I_{1}\cup I_{2},\ell\neq
j}\log|\sin(\pi(\tilde{\theta}_{j}-\tilde{\theta}_{\ell}))|+dN\log 2$ (3.107)
$\displaystyle=:$ $\displaystyle\sum_{2,+}+\sum_{2,-}+dN\log 2.$

Let $T_{1}=[0,q_{n}-1]\cap\mathbb{Z}$ be the left-most small interval in
$I_{1}\cup I_{2}=\cup_{k=1}^{6sd}T_{k}$, and for $k\geq 1$, let $T_{k+1}$ be
on the right-hand side of $T_{k}$. Then $\bigcup_{k=1}^{2sd}T_{k}=I_{1}$ and
$\bigcup_{k=2sd+1}^{6sd}T_{k}=I_{2}$. We also group $T_{k}$’s into intervals
of length $sq_{n}$, for $1\leq h\leq 6d$, by defining

(3.108) $\displaystyle R_{h}=\bigcup_{k=(h-1)s+1}^{hs}T_{k}.$

Denote the smallest term in each $T_{k}$, for $1\leq k\leq 6sd$, by:

(3.109)
$\displaystyle|\sin(\pi(\tilde{\theta}_{j}+\tilde{\theta}_{\ell_{k}^{+}}))|=\min_{\ell\in
T_{k}}|\sin(\pi(\tilde{\theta}_{j}+\tilde{\theta}_{\ell}))|,$

and

(3.110)
$\displaystyle|\sin(\pi(\tilde{\theta}_{j}-\tilde{\theta}_{\ell_{k}^{-}}))|=\min_{\ell\in
T_{k}}|\sin(\pi(\tilde{\theta}_{j}-\tilde{\theta}_{\ell}))|.$

Denote the smallest term in each $R_{h}$, for $1\leq h\leq 6d$, by:

(3.111)
$\displaystyle|\sin(\pi(\tilde{\theta}_{j}+\tilde{\theta}_{\tilde{\ell}_{h}^{+}}))|=\min_{\ell\in
R_{h}}|\sin(\pi(\tilde{\theta}_{j}+\tilde{\theta}_{\ell}))|,$

and

(3.112)
$\displaystyle|\sin(\pi(\tilde{\theta}_{j}-\tilde{\theta}_{\tilde{\ell}_{h}^{-}}))|=\min_{\ell\in
R_{h}}|\sin(\pi(\tilde{\theta}_{j}-\tilde{\theta}_{\ell}))|.$

We now establish the following estimate for $\sum_{2,+}$.

######  Lemma 3.12.

For $n$ large,

(3.113)
$\displaystyle\sum_{2,+}\geq-300\varepsilon_{0}d^{2}sq_{n}-6sdq_{n}\log 2.$

###### Proof.

We bound

(3.114) $\displaystyle\sum_{2,+}=$
$\displaystyle\sum_{k=1}^{6sd}\sum_{\begin{subarray}{c}\ell\in T_{k}\\\
\ell\neq
j\end{subarray}}\log|\sin(\pi(\tilde{\theta}_{j}+\tilde{\theta}_{\ell}))|$
(3.115) $\displaystyle\geq$ $\displaystyle\sum_{k=1}^{6sd}\sum_{\ell\in
T_{k}}\log|\sin(\pi(\tilde{\theta}_{j}+\tilde{\theta}_{\ell}))|$ (3.116)
$\displaystyle\geq$
$\displaystyle\sum_{k=1}^{6sd}\log|\sin(\pi(\tilde{\theta}_{j}+\tilde{\theta}_{\ell_{k}^{+}}))|-(6sd)(C_{4}\log
q_{n}+(q_{n}-1)\log 2)$ (3.117) $\displaystyle\geq$
$\displaystyle\sum_{k=1}^{6sd}\log(\|\tilde{\theta}_{j}+\tilde{\theta}_{\ell_{k}^{+}}\|_{\mathbb{T}})-6sdq_{n}\log
2-\varepsilon_{0}sdq_{n},$

for $n$ large, where we used Lemma 3.11 in the second to last inequality, and
$|\sin(\pi x)|\geq 2|x|$ in the final one. As far as the minimal terms at
$\ell_{k}^{+}$, $1\leq k\leq 6sd$ are concerned, we now establish the
following result.

######  Lemma 3.13.

(3.118)
$\displaystyle\min_{k=1,\ldots,6sd}\;\|\tilde{\theta}_{j}+\tilde{\theta}_{\ell_{k}^{+}}\|_{\mathbb{T}}\geq
e^{-36\varepsilon_{0}dsq_{n}}.$

###### Proof.

Within the proof, for $1\leq k\leq 6sd$, let

(3.119) $\displaystyle m_{k}:=j+\ell_{k}^{+}-2\lfloor(dN-1)/2\rfloor+(dN-1)$

for simplicity. Then
$\tilde{\theta}_{j}+\tilde{\theta}_{\ell_{k}^{+}}=2\theta+m_{k}\alpha$. Note
that since $j,\ell_{k}^{+}\in I_{1}\cup I_{2}$, one has

(3.120) $\displaystyle-n_{j}<-3\leq m_{k}\leq
2y+4sdq_{n}\leq\begin{cases}\frac{5}{2}y<\frac{5}{6}|n_{j+1}|\\\
16(s+1)dq_{n}+4sdq_{n}\leq 36sdq_{n}\end{cases}$

Next, we prove Lemma 3.13 by contradiction. Suppose for some $k$ that

(3.121)
$\displaystyle\|\tilde{\theta}_{j}+\tilde{\theta}_{\ell_{k}^{+}}\|_{\mathbb{T}}=\left\|2\theta+m_{k}\alpha\right\|_{\mathbb{T}}\leq
e^{-36\varepsilon_{0}dsq_{n}}$

Then by (3.120), we conclude that

(3.122) $\displaystyle\left\|2\theta+m_{k}\alpha\right\|_{\mathbb{T}}\leq
e^{-36\varepsilon_{0}dsq_{n}}\leq e^{-\varepsilon_{0}|m_{k}|}.$

Furthermore, for any $0\leq|m|\leq 6sdq_{n}$, by (3.120)

(3.123) $\displaystyle|m-m_{k}|\leq 42sdq_{n}.$

Hence by (3.77) with $\varepsilon_{1}=\varepsilon_{0}/10$, one obtains

(3.124) $\displaystyle\|2\theta+m\alpha\|_{\mathbb{T}}\geq$
$\displaystyle\left\|(m-m_{k})\alpha\right\|_{\mathbb{T}}-\|2\theta+m_{k}\alpha\|_{\mathbb{T}}$
(3.125) $\displaystyle\geq$ $\displaystyle
C_{\varepsilon_{0}}e^{-\frac{21}{5}\varepsilon_{0}sdq_{n}}-e^{-36\varepsilon_{0}dsq_{n}}$
(3.126) $\displaystyle>$ $\displaystyle e^{-36\varepsilon_{0}dsq_{n}}$ (3.127)
$\displaystyle\geq$ $\displaystyle\|2\theta+m_{k}\alpha\|_{\mathbb{T}},$

for $n$ large enough. Combining (3.122) with (3.124), we see that $(-m_{k})$
must be an $\varepsilon_{0}$-resonance.

Since $3|n_{j}|<y<8(s+1)dq_{n}\leq 16sdq_{n}$, we obtain $|n_{j}|\leq
6sdq_{n}$. Hence by (3.124) with $m=-n_{j}$, we infer that

(3.128) $\displaystyle\|2\theta-
n_{j}\alpha\|_{\mathbb{T}}>\|2\theta+m_{k}\alpha\|_{\mathbb{T}}.$

This implies $-m_{k}=n_{p}$ for some $p\geq j+1$. However, this leads to a
contradiction with (3.120) via $|m_{k}|<|n_{j+1}|$. ∎

With Lemma 3.13, we conclude the following.

######  Lemma 3.14.

For any $1\leq h\leq 6d$,

(3.129) $\displaystyle\sum_{\ell_{k}^{+}\in
R_{h}}\log(\|\tilde{\theta}_{j}+\tilde{\theta_{\ell_{k}^{+}}}\|)\geq-40\varepsilon_{0}dsq_{n}.$

###### Proof.

We divide the argument into two cases.

Case 1. Suppose
$2\|\tilde{\theta}_{j}+\tilde{\theta}_{\tilde{\ell}_{h}^{+}}\|_{\mathbb{T}}\leq|\sin(\pi(\tilde{\theta}_{j}+\tilde{\theta}_{\tilde{\ell}_{h}^{+}}))|\leq
e^{-2\varepsilon_{0}q_{n}}$. For $\ell_{k}^{+},\tilde{\ell}_{h}^{+}\in R_{h}$
and $\ell_{k}^{+}\neq\tilde{\ell}_{h}^{+}$, we have
$|\ell_{k}^{+}-\tilde{\ell}_{h}^{+}|\leq sq_{n}$. Since $sdq_{n}\leq
y/8<dq_{n+1}$, in particular $sq_{n}<q_{n+1}$, it follows by (2.20) and (3.77)
with $\varepsilon_{1}=\varepsilon_{0}$ that

(3.130)
$\displaystyle\|(\ell_{k}^{+}-\tilde{\ell}_{h}^{+})\alpha\|_{\mathbb{T}}\geq\|q_{n}\alpha\|_{\mathbb{T}}\geq
C_{\varepsilon_{0}}e^{-\varepsilon_{0}q_{n}}.$

This implies, by the triangle inequality, that

(3.131)
$\displaystyle\|\tilde{\theta}_{j}+\tilde{\theta}_{\ell_{k}^{+}}\|_{\mathbb{T}}\geq$
$\displaystyle\|(\ell_{k}^{+}-\tilde{\ell}_{h}^{+})\alpha\|_{\mathbb{T}}-\|\tilde{\theta}_{j}+\tilde{\theta}_{\tilde{\ell}_{h}^{+}}\|_{\mathbb{T}}$
(3.132) $\displaystyle\geq$ $\displaystyle
C_{\varepsilon_{0}}e^{-\varepsilon_{0}q_{n}}-\frac{1}{2}e^{-2\varepsilon_{0}q_{n}}$
(3.133) $\displaystyle\geq$ $\displaystyle e^{-2\varepsilon_{0}q_{n}},$

for $n$ large. Combining (3.131) with Lemma 3.13, yields

(3.134) $\displaystyle\sum_{\ell_{k}^{+}\in
R_{h}}\log(\|\tilde{\theta}_{j}+\tilde{\theta}_{\ell_{k}^{+}}\|_{\mathbb{T}})\geq$
$\displaystyle s\log
2+(s-1)\log(e^{-2\varepsilon_{0}q_{n}})+\log(e^{-36\varepsilon_{0}dsq_{n}})$
(3.135) $\displaystyle\geq$ $\displaystyle-40\varepsilon_{0}dsq_{n}.$

Case 2. Assume
$\pi\|\tilde{\theta}_{j}+\tilde{\theta}_{\tilde{\ell}_{h}^{+}}\|_{\mathbb{T}}\geq|\sin(\pi(\tilde{\theta}_{j}+\tilde{\theta}_{\tilde{\ell}_{h}^{+}}))|>e^{-2\varepsilon_{0}q_{n}}$.
Then for any $\ell_{k}^{+}\in R_{h}$,

$\pi\|\tilde{\theta}_{j}+\tilde{\theta}_{\ell_{k}^{+}}\|_{\mathbb{T}}\geq|\sin(\pi(\tilde{\theta}_{j}+\tilde{\theta}_{\ell_{k}^{+}}))|\geq
e^{-2\varepsilon_{0}q_{n}}.$

This implies

(3.136) $\displaystyle\sum_{\ell_{k}^{+}\in
R_{h}}\log(\|\tilde{\theta}_{j}+\tilde{\theta}_{\ell_{k}^{+}}\|_{\mathbb{T}})\geq$
$\displaystyle s\log
2+s\log(e^{-2\varepsilon_{0}q_{n}})\geq-40\varepsilon_{0}dsq_{n}.$

The claimed result follows by combining (3.134) with (3.136). ∎

Lemma 3.14 implies

(3.137)
$\displaystyle\sum_{k=1}^{6sd}\log(\|\tilde{\theta}_{j}+\tilde{\theta}_{\ell_{k}^{+}}\|_{\mathbb{T}})\geq-240\varepsilon_{0}d^{2}sq_{n}.$

Lemma 3.12 follows by plugging (3.137) into (3.114), whence

(3.138)
$\displaystyle\sum_{2,+}\geq-300\varepsilon_{0}d^{2}sq_{n}-6dsq_{n}\log 2,$

as claimed. ∎

It remains to estimate $\sum_{2,-}$. Similar to (3.114), one has

(3.139)
$\displaystyle\sum_{2,-}\geq\sum_{k=1}^{6sd}\log(\|\tilde{\theta}_{j}-\tilde{\theta}_{\ell_{k}^{-}}\|_{\mathbb{T}})-6sdq_{n}\log
2-\varepsilon_{0}sdq_{n}.$

We will prove the following.

######  Lemma 3.15.

For $n$ large,

(3.140)
$\displaystyle\sum_{2,-}\geq-300\varepsilon_{0}d^{2}sq_{n}-6sdq_{n}\log 2.$

###### Proof.

We have

######  Lemma 3.16.

(3.141) $\displaystyle\sum_{\ell_{k}^{-}\in R_{h},\ell_{k}^{-}\neq
j}\log(\|\tilde{\theta}_{j}-\tilde{\theta}_{\ell_{k}^{-}}\|_{\mathbb{T}})\geq-40\varepsilon_{0}sdq_{n}.$

###### Proof.

We split the argument into two different cases depending on whether or not
$j\in R_{h}$.

Case 1. If $j\in R_{h}$, then $\tilde{\ell}_{h}^{-}=j$.

For $\ell_{k}^{-}\in R_{h}$ and $\ell_{k}^{-}\neq j$, we note that
$0\leq|\ell_{k}^{-}-j|\leq sq_{n}<q_{n+1}$. Hence by (2.20) and (3.77),

(3.142)
$\displaystyle\|\tilde{\theta}_{j}-\tilde{\theta}_{\ell_{k}^{-}}\|_{\mathbb{T}}=\|(\ell_{k}^{-}-j)\alpha\|_{\mathbb{T}}\geq\|q_{n}\alpha\|_{\mathbb{T}}\geq
C_{\varepsilon_{0}}e^{-\varepsilon_{0}q_{n}}.$

This implies that

(3.143) $\displaystyle\sum_{\ell_{k}^{-}\in R_{h},\ell_{k}^{-}\neq
j}\log(\|\tilde{\theta}_{j}-\tilde{\theta}_{\ell_{k}^{-}}\|_{\mathbb{T}})\geq(s-1)\log(C_{\varepsilon_{0}}e^{-\varepsilon_{0}q_{n}})\geq-40dsq_{n}.$

Case 2. Assume $j\notin R_{h}$.

Case 2.1. Assume further that
$2\|\tilde{\theta}_{j}-\tilde{\theta}_{\tilde{\ell}_{h}^{-}}\|\leq|\sin(\pi(\tilde{\theta}_{j}-\tilde{\theta}_{\tilde{\ell}_{h}^{-}}))|\leq
e^{-2\varepsilon_{0}q_{n}}$.

For $\ell_{k}^{-}\in R_{h}$ and $\ell_{k}^{-}\neq\tilde{\ell}_{h}^{-}$,
analogously to (3.142) one has

(3.144)
$\displaystyle\|\tilde{\theta}_{j}-\tilde{\theta}_{\ell_{k}^{-}}\|_{\mathbb{T}}\geq$
$\displaystyle\|(\ell_{k}^{-}-\tilde{\ell}_{h}^{-})\alpha\|_{\mathbb{T}}-e^{-2\varepsilon_{0}q_{n}}$
(3.145) $\displaystyle\geq$ $\displaystyle
C_{\varepsilon_{0}}e^{-\varepsilon_{0}q_{n}}-e^{-2\varepsilon_{0}q_{n}}$
(3.146) $\displaystyle\geq$ $\displaystyle e^{-2\varepsilon_{0}q_{n}},$

for $n$ large. Also since $|j-\tilde{\ell}_{h}^{-}|\leq y+2sdq_{n}+1\leq
20dsq_{n}$, we infer by (3.77) that

(3.147)
$\displaystyle\|\tilde{\theta}_{j}-\tilde{\theta}_{\tilde{\ell}_{h}^{-}}\|_{\mathbb{T}}\geq\|(j-\tilde{\ell}_{h}^{-})\alpha\|_{\mathbb{T}}\geq
C_{\varepsilon_{0}}e^{-20\varepsilon_{0}dsq_{n}}.$

This implies that

(3.148) $\displaystyle\sum_{\ell_{k}^{-}\in
R_{h}}\log(\|\tilde{\theta}_{j}-\tilde{\theta}_{\ell_{k}^{-}}\|_{\mathbb{T}})\geq(s-1)\log(e^{-2\varepsilon_{0}q_{n}})+\log(e^{-20\varepsilon_{0}dsq_{n}})\geq-40\varepsilon_{0}dsq_{n}.$

Case 2.2. Suppose
$\pi\|\tilde{\theta}_{j}-\tilde{\theta}_{\tilde{\ell}_{h}^{-}}\|\geq|\sin(\pi(\tilde{\theta}_{j}-\tilde{\theta}_{\tilde{\ell}_{h}^{-}}))|>e^{-2\varepsilon_{0}q_{n}}$.
Thus, for any $\ell_{k}^{-}\in R_{h}$,

$\pi\|\tilde{\theta}_{j}-\tilde{\theta}_{\ell_{k}^{-}}\|\geq|\sin(\pi(\tilde{\theta}_{j}-\tilde{\theta}_{\ell_{k}^{-}}))|\geq
e^{-2\varepsilon_{0}q_{n}}.$

This implies

(3.149) $\displaystyle\sum_{\ell_{k}^{-}\in
R_{h}}\|\tilde{\theta}_{j}-\tilde{\theta}_{\ell_{k}^{-}}\|_{\mathbb{T}}\geq
s\log(\pi^{-1}e^{-2\varepsilon_{0}q_{n}})>-40dsq_{n}.$

Combining the cases above, we have proved Lemma 3.16. ∎

Lemma 3.15 follows from plugging Lemma 3.16 into (3.139). ∎

Finally, combining (3.2), (3.2) with Lemmas 3.12 and 3.15, we obtain

(3.150) $\displaystyle\sum_{\ell\in I_{1}\cup I_{2},\ \ell\neq
j}\left(\log{|\cos(2\pi\xi)-\cos(2\pi\tilde{\theta}_{\ell})|}-\log{|\cos(2\pi\tilde{\theta}_{j})-\cos(2\pi\tilde{\theta}_{\ell})|}\right)\leq
650\varepsilon_{0}d^{2}sq_{n}.$

This establishes Lemma 3.10 with $C_{3}=110$. ∎

Lemma 3.10 implies the following. Recall that
$\hat{L}^{d}=\hat{L}(d\alpha,M_{E})=d\cdot\hat{L}^{d}(\alpha,\hat{A}_{E})$.

######  Lemma 3.17.

If $y>y_{1}(\alpha,\varepsilon_{0},v,E)$, then there exists $j\in I_{1}\cup
I_{2}$ such that

(3.151)
$\displaystyle\frac{1}{dN}\log\left|f_{N}\left(\theta+j\alpha-\left\lfloor\frac{dN-1}{2}\right\rfloor\alpha\right)\right|\geq\hat{L}^{d}(\alpha,\hat{A}_{E})+\log|\hat{v}_{d}|-2C_{3}d\,\varepsilon_{0}.$

###### Proof.

Assuming the lemma is false, let $\tilde{\theta}_{j}$ be defined as in (3.87).
If (3.151) fails for all $j\in I_{1}\cup I_{2}$, then

(3.152)
$\displaystyle\frac{1}{dN}\log\left|f_{N}\left(\theta+j\alpha-\left\lfloor\frac{dN-1}{2}\right\rfloor\alpha\right)\right|$
(3.153)
$\displaystyle\qquad\qquad=\frac{1}{dN}\log\left|Q_{dN}(\cos(2\pi\tilde{\theta}_{j}))\right|<\hat{L}^{d}(\alpha,\hat{A}_{E})+\log|\hat{v}_{d}|-2C_{3}d\,\varepsilon_{0}.$

Note that $Q_{dN}(x)$ is a polynomial in $x$ of degree $dN\leq
dN+1=|I_{1}|+|I_{2}|$. Hence by the Lagrange interpolation theorem, for any
$x\in[0,1]$,

(3.154) $\displaystyle Q_{dN}(x)=\sum_{j\in I_{1}\cup
I_{2}}Q_{dN}(\cos(2\pi\tilde{\theta}_{j}))\cdot\prod_{\begin{subarray}{c}m\in
I_{1}\cup I_{2}\\\ m\neq
j\end{subarray}}\frac{x-\cos(2\pi\tilde{\theta}_{m})}{\cos(2\pi\tilde{\theta}_{j})-\cos(2\pi\tilde{\theta}_{m})}.$

By Lemma 3.10, we infer that

(3.155) $\displaystyle\max_{x\in[0,1]}\max_{j\in I_{1}\cup
I_{2}}\left|\prod_{\begin{subarray}{c}m\in I_{1}\cup I_{2}\\\ m\neq
j\end{subarray}}\frac{x-\cos(2\pi\tilde{\theta}_{m})}{\cos(2\pi\tilde{\theta}_{j})-\cos(2\pi\tilde{\theta}_{m})}\right|\leq
e^{C_{3}\varepsilon_{0}d^{2}N}.$

Combining (3.154) with (3.152) and (3.155), we conclude that

(3.156)
$\displaystyle\max_{\theta\in\mathbb{T}}|f_{N}(\theta)|=\max_{x\in[0,1]}|Q_{dN}(x)|\leq
e^{dN(\hat{L}^{d}(\alpha,\hat{A}_{E})+\log|\hat{v}_{d}|-C_{3}d\,\varepsilon_{0})}.$

However, combining Lemma 3.5 with (2.49), there exists $\theta$ such that

(3.157)
$\displaystyle|f_{N}(\theta)|\geq|\hat{v}_{d}|^{dN}e^{dN(\hat{L}^{d}(\alpha,\hat{A}_{E})-\frac{C_{3}d\,\varepsilon_{0}}{2})},$

which is a contradiction. ∎

Lemma 3.17 implies the following result.

######  Corollary 3.18.

If $y>y_{1}(\alpha,\varepsilon_{0},v,E)$, then there exists $j\in I_{2}$ such
that

(3.158)
$\displaystyle\frac{1}{dN}\log\left|f_{N}\left(\theta+j\alpha-\left\lfloor\frac{dN-1}{2}\right\rfloor\alpha\right)\right|\geq\hat{L}^{d}(\alpha,\hat{A}_{E})+\log|\hat{v}_{d}|-2C_{3}d\,\varepsilon_{0}.$

###### Proof.

It suffices to show that (3.158) does not hold for any $j\in I_{1}$. Thus,
assume that (3.158) does hold for some $j_{1}\in I_{1}$. Let

(3.159)
$\displaystyle\begin{cases}x_{1}:=j_{1}-\left\lfloor\frac{dN-1}{2}\right\rfloor\\\
x_{2}:=x_{1}+dN-1\end{cases}$

We estimate the Green’s function,

(3.160) $\displaystyle
G_{N}\left(\theta+x_{1}\alpha;-x_{1},m\right)=\frac{\mu_{N,-x_{1},m}(\theta+x_{1}\alpha)}{f_{N}(\theta+x_{1}\alpha)},$

for $0\leq m\leq d-1$. By Lemma 3.1, and using $|\det B|=|\hat{v}_{d}|^{dN}$,
one has

(3.161) $\displaystyle|\mu_{N,-x_{1},m}(\theta+x_{1}\alpha)|\leq
C_{d}\|B^{-1}\|\cdot|\hat{v}_{d}|^{dN}\cdot$ (3.162)
$\displaystyle\qquad\cdot\max\left(e^{|x_{1}|(1+\varepsilon_{0})\hat{L}^{d-1}(\alpha,\hat{A}_{E})+|x_{2}|(1+\varepsilon_{0})\hat{L}^{d}(\alpha,\hat{A}_{E})},e^{|x_{1}|(1+\varepsilon_{0})\hat{L}^{d}(\alpha,\hat{A}_{E})+|x_{2}|(1+\varepsilon_{0})\hat{L}^{d-1}(\alpha,\hat{A}_{E})}\right)$

Combining this with the lower bound in (3.158) for $j=j_{1}$ yields

(3.163) $\displaystyle|G_{N}(\theta+x_{1}\alpha;-x_{1},m)|$ (3.164)
$\displaystyle\qquad\leq
C_{d}\|B^{-1}\|\max\left(e^{-|x_{1}|\hat{L}_{d}(\alpha,\hat{A}_{E})},e^{-|x_{2}|\hat{L}_{d}(\alpha,\hat{A}_{E})}\right)\cdot
e^{\varepsilon_{0}(\hat{L}^{d}(\alpha,\hat{A}_{E})+2C_{3}d)dN}.$

Similarly, for any $0\leq m\leq d-1$,

(3.165) $\displaystyle|G_{N}(\theta+x_{1}\alpha;-x_{1},dN-1-m)|$ (3.166)
$\displaystyle\qquad\leq
C_{d}\|B^{-1}\|\max\left(e^{-|x_{1}|\hat{L}_{d}(\alpha,\hat{A}_{E})},e^{-|x_{2}|\hat{L}_{d}(\alpha,\hat{A}_{E})}\right)\cdot
e^{\varepsilon_{0}(\hat{L}^{d}(\alpha,\hat{A}_{E})+2C_{3}d)dN}.$

Using the Poisson formula (3.68), and bounding $|\hat{u}_{k}|\leq 1+|k|$ and
$\|B\|\leq C_{v}$, $\|B^{-1}\|\leq C_{v}|\hat{v}_{d}|^{-1}$, we infer that

(3.167) $\displaystyle|\hat{u}_{0}|\leq$ $\displaystyle
C_{d,v}|\hat{v}_{d}|^{-2}\max\left(e^{-|x_{1}|\hat{L}_{d}(\alpha,\hat{A}_{E})},e^{-|x_{2}|\hat{L}_{d}(\alpha,\hat{A}_{E})}\right)\cdot$
(3.168) $\displaystyle\qquad\qquad\qquad\cdot\max(|x_{1}|,|x_{2}|)\cdot
e^{\varepsilon_{0}(\hat{L}^{d}(\alpha,\hat{A}_{E})+2C_{3}d)dN}$

Clearly,

$dsq_{n}\leq\min(|x_{1}|,|x_{2}|)\leq 5dsq_{n}.$

Hence, we can further bound (3.167) by

(3.169) $\displaystyle|\hat{u}_{0}|$ $\displaystyle\leq
C_{d,v}\,|\hat{v}_{d}|^{-2}e^{-dsq_{n}(\hat{L}_{d}(\alpha,\hat{A}_{E})-6\varepsilon_{0}\hat{L}^{d}(\alpha,\hat{A}_{E})-20C_{3}d\,\varepsilon_{0})}.$

With the choice of $\varepsilon_{0}$ in (3.76), we arrive a contradiction with
the assumption that $\hat{u}_{0}=1$ by sending $n\to\infty$. ∎

Corollary 3.18 implies the following bound:

(3.170) $\displaystyle|\hat{u}_{y}|\leq
C_{d,v}\,|\hat{v}_{d}|^{-2}e^{-\frac{y}{8}(\hat{L}_{d}(\alpha,\hat{A}_{E})-6\varepsilon_{0}\hat{L}^{d}(\alpha,\hat{A}_{E})-20C_{3}d\,\varepsilon_{0})}\leq
C_{d,v}|\hat{v}_{d}|^{-2}e^{-\frac{y}{10}\hat{L}_{d}(\alpha,\hat{A}_{E})},$

which we used (3.76) in the last inequality. The proof is similar to arguments
in the proof of Corollary 3.18, after replacing the role of $\hat{u}_{0}$ with
$\hat{u}_{y}$ and using $y\geq 8dsq_{n}$. This finishes the proof of almost
localization. ∎

###  3.3. Localization

Theorem 1.7 follows from Theorem 3.8 by noting that any $\theta$ with
$\gamma(\alpha,\theta)=0$ is $\varepsilon_{0}$-nonresonant for any
$\varepsilon_{0}>0$.

######  Remark 3.19.

For $\alpha$ satisfying a stronger Diophantine condition

(3.171)
$\displaystyle\alpha\in\mathrm{DC}:=\bigcup_{c>0}\bigcup_{b>1}\left\\{\alpha\in\mathbb{T}:\|n\alpha\|_{\mathbb{T}}\geq\frac{c}{|n|^{b}}\right\\}$

one can give another proof of localization by combining the large deviation
theorem with the zero count of $\det(P_{\alpha,n}(\theta)-E)$ as in [HS]. This
approach will allow us to prove a non-perturbative localization for a.e.
$\alpha\in\mathrm{DC}$ for a general class of operators obtained by replacing
the cosine potential in $\hat{H}_{\alpha,\theta}$ with arbitrary analytic
potentials. We will address this problem in an upcoming work.

##  4\. Numerator upper bound for periodic boundary conditions

This section is devoted to the proof of Lemma 3.1. We consider the finite
volume Hamiltonian with periodic boundary conditions. To distinguish it from
$\hat{H}_{\alpha,n}(\theta)$, which satisfies Dirichlet boundary conditions,
we denote the periodic Hamiltonian by $P_{\alpha,n}(\theta)$. We shall omit
$\alpha$ from the notation for simplicity, and denote $C(\theta+jd\alpha)-E$
by $C_{j}$. Thus,

(4.172) $\displaystyle
P_{n}(\theta)-E=\left(\begin{matrix}C_{n-1}&B^{*}&&&B\\\ B&C_{n-2}&\ddots\\\
&\ddots&\ddots&\ddots\\\ &&\ddots&\ddots&B^{*}\\\
B^{*}&&&B&C_{0}\end{matrix}\right).$

Let $P_{n}(\theta;x,y)$ be the submatrix of $P_{n}(\theta)-E$ obtained by
deleting the $x-$th row and $y-$th column. We treat the case $0\leq x\leq d-1$
in detail, and briefly go through the other case at the end of this section.

Case 1. Suppose $0\leq x\leq d-1$ and $3d\leq y\leq(n-1)d-1$. We write $y=\ell
d+r$ for some $\ell\in[3,n-2]$ and $r\in[0,d-1]$. Let

(4.175) $\displaystyle R_{x,y}:=\left(\begin{array}[]{c|c}P_{n}(\theta)-E&{\bf
e}_{dn,x}\\\ \hline\cr{\bf e}_{dn,y}^{*}&0\end{array}\right),$

in which ${\bf e}_{m,j}^{*}=(\delta_{j}(m-1),...,\delta_{j}(1),\delta_{j}(0))$
and $\delta_{j}(k)$ is the Kronecker delta. Clearly

(4.176) $\displaystyle|\mu_{n,x,y}(\theta)|=|\det(P_{n}(\theta;x,y)-E)|=|\det
R_{x,y}|$

Next, we carry out elementary row operations to reduce $R_{x,y}$. These
operations do not alter $|\det R_{x,y}|$. We have

(4.187) $\displaystyle\qquad R_{x,y}=$
$\displaystyle\left(\begin{array}[]{c|c|c|c|c|c|c|c|c|c}C_{n-1}&B^{*}&&&&&&&B&\\\
\hline\cr B&\ddots&\ddots&&&&&&&\\\ \hline\cr&\ddots&\ddots&\ddots&&&&&&\\\
\hline\cr&&\ddots&\ddots&B^{*}&&&&&\\\ \hline\cr&&&B&C_{\ell}&B^{*}&&&&\\\
\hline\cr&&&&B&C_{\ell-1}&\ddots&&&\\\ \hline\cr&&&&&\ddots&\ddots&\ddots&&\\\
\hline\cr&&&&&&\ddots&\ddots&B^{*}&\\\ \hline\cr B^{*}&&&&&&&B&C_{0}&{\bf
e}_{d,x}\\\ \hline\cr&&&&{\bf e}_{d,r}^{*}&&&&&\end{array}\right)$ (4.188)
$\displaystyle=:$ $\displaystyle\left(\begin{matrix}\text{Row}_{1}\\\
\text{Row}_{2}\\\ \vdots\\\ \text{Row}_{n+1}\end{matrix}\right)$

In the following we also denote $M_{E,k}(\theta+jd\alpha)$ by $M_{k}(j)$ for
simplicity. We use the following row operations.

### Row operations on $\text{Row}_{1}$

Replacing the block-valued $\text{Row}_{1}$ in (4.187),

$\text{Row}_{1}\longrightarrow\text{Row}_{1}-C_{n-1}\cdot
B^{-1}\cdot\text{Row}_{2},$

yields a new row as follows

$\text{Row}_{1}^{(1)}=(0,B^{*}-C_{n-1}B^{-1}C_{n-2},-C_{n-1}B^{-1}B^{*},0,...,0,B,0).$

Appealing to (2.44) and (2.45), it is easy to see that

$\text{Row}_{1}^{(1)}=(0,-M_{2}^{UL}(n-2)B,\,-M_{2}^{UR}(n-2),0,...,0,B,0).$

Replacing

$\text{Row}_{1}^{(1)}\longrightarrow\text{Row}_{1}^{(1)}+M_{2}^{UL}(n-2)\cdot\text{Row}_{3},$

yields the new row

$\text{Row}_{1}^{(2)}=(0,0,-M_{2}^{UR}(n-2)+M_{2}^{UL}(n-2)C_{n-3},\,M_{2}^{UL}(n-2)B^{*},0,...,0,B,0).$

Appealing to (2.44) and (2.45), it is easy to see that

$\text{Row}_{1}^{(2)}=(0,0,-M_{3}^{UL}(n-3)B,\,-M_{3}^{UR}(n-3),0,...,0,B,0).$

One can iterate this process, and after replacing

$\text{Row}_{1}^{(n-3)}\longrightarrow\text{Row}_{1}^{(n-3)}+M_{n-2}^{UL}(2)\cdot\text{Row}_{n-1},$

we arrive at

$\text{Row}_{1}^{(n-2)}=(0,...,0,-M_{n-1}^{UL}(1)B,\,B-M_{n-1}^{UR}(1),0).$

### Row operations on $\text{Row}_{2}$

Replacing

$\text{Row}_{2}\longrightarrow\text{Row}_{2}-C_{n-2}\cdot
B^{-1}\cdot\text{Row}_{3},$

yields the following new row

(4.189) $\displaystyle\text{Row}_{2}^{(1)}=$
$\displaystyle(B,0,B^{*}-C_{n-2}B^{-1}C_{n-3},-C_{n-2}B^{-1}B^{*},0,...,0)$
(4.190) $\displaystyle=$
$\displaystyle(B,0,-M_{2}^{UL}(n-3)B,-M_{2}^{UR}(n-3),0,...,0).$

Repeating this process, and after replacing

$\text{Row}_{2}^{(n-4)}\longrightarrow\text{Row}_{2}^{(n-4)}+M_{n-3}^{UL}(2)\cdot\text{Row}_{n-1},$

we arrive at

(4.191)
$\displaystyle\text{Row}_{2}^{(n-3)}=(B,0,...,0,-M_{n-2}^{UL}(1)B,-M_{n-2}^{UR}(1),0).$

Next, we perform row reduction on $\text{Row}_{n-\ell+1}$. Recall that
$3\leq\ell\leq n-2$.

### Row operations on $\text{Row}_{n-\ell+1}$ (the row containing
$C_{\ell-1}$).

Replacing

$\text{Row}_{n-\ell+1}\longrightarrow\text{Row}_{n-\ell+1}-C_{\ell-1}B^{-1}\cdot\text{Row}_{n-\ell+2},$

yields a new row as follows

(4.192) $\displaystyle\text{Row}_{n-\ell+1}^{(1)}=$
$\displaystyle(0,...,0,B,0,B^{*}-C_{\ell-1}B^{-1}C_{\ell-2},-C_{\ell-1}B^{-1}B^{*},0,...,0)$
(4.193) $\displaystyle=$
$\displaystyle(0,...,0,B,0,-M_{2}^{UL}(\ell-2)B,\,-M_{2}^{UR}(\ell-2),0,...,0).$

where the matrix $B$ is the $(n-\ell)-$th entry (from the left) of the block-
valued vector above, and we have used (2.44) and (2.45) to obtain the second
line. Repeating this process, and after replacing

$\text{Row}_{n-\ell+1}^{(\ell-3)}\longrightarrow\text{Row}_{n-\ell+1}^{(\ell-3)}+M_{\ell-2}^{UL}(2)\cdot\text{Row}_{n-1},$

we arrive at

$\text{Row}_{n-\ell+1}^{(\ell-2)}=(0,...,0,B,0,...,0,-M_{\ell-1}^{UL}(1)B,-M_{\ell-1}^{UR}(1),0).$

At this point, $R_{x,y}$ becomes

(4.207) $\displaystyle
R^{(1)}_{x,y}=\left(\begin{array}[]{c|c|c|c|c|c|c|c|c|c|c|c|c}0&0&0&\cdots&&&&&\cdots&0&-M_{n-1}^{UL}(1)B&B-M_{n-1}^{UR}(1)&0\\\
\hline\cr B&0&0&\cdots&&&&&\cdots&0&-M_{n-2}^{UL}(1)B&-M_{n-2}^{UR}(1)&0\\\
\hline\cr 0&B&C_{n-3}&B^{*}&&&&&&&&&\\\
\hline\cr\vdots&&\ddots&\ddots&\ddots&&&&&&&&\\\
\hline\cr&&&\ddots&\ddots&\ddots&&&&&&&\\\
\hline\cr\vdots&&&&B&C_{\ell}&B^{*}&&&&&&\\\ \hline\cr
0&\cdots&&\cdots&0&B&0&0&\cdots&0&-M_{\ell-1}^{UL}(1)B&-M_{\ell-1}^{UR}(1)&0\\\
\hline\cr\vdots&&&&&0&B&C_{\ell-2}&B^{*}&&&0&0\\\
\hline\cr&&&&&&&\ddots&\ddots&\ddots&&\vdots&\vdots\\\
\hline\cr\vdots&&&&&&&&\ddots&\ddots&\ddots&0&\vdots\\\ \hline\cr
0&&&&&&&&&\ddots&\ddots&B^{*}&0\\\ \hline\cr
B^{*}&0&\cdots&&&&&&\cdots&0&B&C_{0}&{\bf e}_{d,x}\\\ \hline\cr
0&\cdots&&\cdots&0&{\bf e}_{d,r}^{*}&0&\cdots&&\cdots&0&0&0\end{array}\right)$

The first, second, $(n-\ell+1)$-th, $n$-th, and $(n+1)$-th rows are

(4.213) $\displaystyle\left(\begin{matrix}\text{Row}_{1}^{(n-2)}\\\
\text{Row}_{2}^{(n-3)}\\\ \text{Row}_{n-\ell+1}^{(\ell-2)}\\\
\text{Row}_{n}\\\
\text{Row}_{n+1}\end{matrix}\right)=\left(\begin{array}[]{ccccccccccc}0&0&\cdots&0&0&0&\cdots&0&-M_{n-1}^{UL}(1)B&B-M_{n-1}^{UR}(1)&0\\\
B&0&\cdots&0&0&0&\cdots&0&-M_{n-2}^{UL}(1)B&-M_{n-2}^{UR}(1)&0\\\
0&0&\cdots&0&B&0&\cdots&0&-M_{\ell-1}^{UL}(1)B&-M_{\ell-1}^{UR}(1)&0\\\
B^{*}&0&\cdots&0&0&0&\cdots&0&B&C_{0}&{\bf e}_{d,x}\\\ 0&0&\cdots&0&{\bf
e}_{d,r}^{*}&0&\cdots&0&0&0&0\end{array}\right),$

in which only columns $1$, $n-\ell$, $n-1$, $n$, $n+1$ are non-vanishing. Let

(4.219) $\displaystyle
S_{1}:=\left(\begin{array}[]{ccccc}0&0&-M_{n-1}^{UL}(1)B&B-M_{n-1}^{UR}(1)&0\\\
B&0&-M_{n-2}^{UL}(1)B&-M_{n-2}^{UR}(1)&0\\\
0&B&-M_{\ell-1}^{UL}(1)B&-M_{\ell-1}^{UR}(1)&0\\\ B^{*}&0&B&C_{0}&{\bf
e}_{d,x}\\\ 0&{\bf e}^{*}_{d,r}&0&0&0\end{array}\right)$

be the non-vanishing $(4d+1)\times(4d+1)$ submatrix of rows $1$, $2$,
$n-\ell+1$, $n$ and $n+1$. Let

(4.236) $\displaystyle
S_{2}:=\left(\begin{array}[]{cccccccccccccccc}B&C_{n-3}&B^{*}&&&&&&&&&&&&&\\\
&B&C_{n-4}&\ddots&&&&&&&&&&&&\\\ &&\ddots&\ddots&&&&&&&&&&&&\\\ \\\
&&&&&\ddots&\ddots&&&&&&&&&\\\ &&&&&\ddots&C_{\ell+2}&B^{*}&&&&&&&&\\\
&&&&&&B&C_{\ell+1}&0&&&&&&&\\\ &&&&&&&B&B^{*}&0&&&&&&\\\
&&&&&&&&B&C_{\ell-2}&B^{*}&&&&&\\\ &&&&&&&&&B&\ddots&\ddots&&&&\\\
&&&&&&&&&&\ddots&&&&&\\\ \\\ &&&&&&&&&&&&\ddots&\ddots&\ddots&\\\
&&&&&&&&&&&&&B&C_{3}&B^{*}\\\ &&&&&&&&&&&&&&B&C_{2}\\\
&&&&&&&&&&&&&&&B\end{array}\right),$

which is the submatrix of $R^{(1)}_{x,y}$ after deleting rows $1$, $2$,
$n-\ell+1$, $n$, $n+1$ and columns $1$, $n-\ell$, $n-1$, $n$, $n+1$. Hence

(4.237) $\displaystyle|\det R^{(1)}_{x,y}|=|\det S_{1}|\cdot|\det S_{2}|=|\det
B|^{n-4}\cdot|\det S_{1}|,$

where we used $\det S_{2}=(\det B)^{n-4}$. It suffices to compute $\det
S_{1}$. Expanding determinants and using that
$M_{k-2}^{UL}(1)=BM_{k-1}^{LL}(1)$, $M_{k-2}^{UR}(1)=BM_{k-1}^{LR}(1)$, we
have

(4.243) $\displaystyle|\det S_{1}|=$
$\displaystyle\left|\det\left(\begin{array}[]{ccccc}0&0&-M_{n-1}^{UL}(1)B&B-M_{n-1}^{UR}(1)&0\\\
B&0&-BM_{n-1}^{LL}(1)B&-BM_{n-1}^{LR}(1)&0\\\
0&B&-M_{\ell-1}^{UL}(1)B&-M_{\ell-1}^{UR}(1)&0\\\ B^{*}&0&B&C_{0}&{\bf
e}_{d,x}\\\ 0&{\bf e}^{*}_{d,r}&0&0&0\end{array}\right)\right|$ (4.249)
$\displaystyle=$ $\displaystyle|\det
B|\cdot\left|\det\left(\begin{array}[]{ccccc}0&0&-M_{n-1}^{UL}(1)&B-M_{n-1}^{UR}(1)&0\\\
B&0&-BM_{n-1}^{LL}(1)&-BM_{n-1}^{LR}(1)&0\\\
0&B&-M_{\ell-1}^{UL}(1)&-M_{\ell-1}^{UR}(1)&0\\\ B^{*}&0&I_{d}&C_{0}&{\bf
e}_{d,x}\\\ 0&{\bf e}^{*}_{d,r}&0&0&0\end{array}\right)\right|$

Expanding along the first column this simplifies further to

(4.255) $\displaystyle|\det S_{1}|=$ $\displaystyle|\det B|^{2}\cdot|\det
B^{*}|\cdot\left|\det\left(\begin{array}[]{ccccc}0&0&-M_{n-1}^{UL}(1)&B-M_{n-1}^{UR}(1)&0\\\
I_{d}&0&-M_{n-1}^{LL}(1)&-M_{n-1}^{LR}(1)&0\\\
0&B&-M_{\ell-1}^{UL}(1)&-M_{\ell-1}^{UR}(1)&0\\\
I_{d}&0&(B^{*})^{-1}&(B^{*})^{-1}C_{0}&(B^{*})^{-1}{\bf e}_{d,x}\\\ 0&{\bf
e}^{*}_{d,r}&0&0&0\end{array}\right)\right|$ (4.262) $\displaystyle=$
$\displaystyle|\det
B|^{3}\cdot\left|\det\left(\begin{array}[]{cccccc}0&0&0&-M_{n-1}^{UL}(1)&B-M_{n-1}^{UR}(1)&0\\\
0&I_{d}&0&-M_{n-1}^{LL}(1)&-M_{n-1}^{LR}(1)&0\\\
0&0&B&-M_{\ell-1}^{UL}(1)&-M_{\ell-1}^{UR}(1)&0\\\ I_{d}&0&0&0&-B&0\\\
0&I_{d}&0&(B^{*})^{-1}&(B^{*})^{-1}C_{0}&(B^{*})^{-1}{\bf e}_{d,x}\\\ 0&0&{\bf
e}^{*}_{d,r}&0&0&0\end{array}\right)\right|$ (4.267) $\displaystyle=$
$\displaystyle|\det
B|^{3}\cdot\left|\det\left(\begin{array}[]{c|c|c|c}\left(\begin{matrix}0&0\\\
0&I_{d}\end{matrix}\right)&&\left(\begin{matrix}0&B\\\
0&0\end{matrix}\right)-M_{n-1}(1)&\\\ \hline\cr&B&-M_{\ell-1}^{U}(1)&\\\
\hline\cr I_{2d}&&-M_{1}^{-1}(0)&\left(\begin{matrix}0\\\
I_{d}\end{matrix}\right)(B^{*})^{-1}{\bf e}_{d,x}\\\ \hline\cr&{\bf
e}^{*}_{d,r}&&\end{array}\right)\right|$ (4.268) $\displaystyle=:$
$\displaystyle|\det B|^{3}\cdot|\det S_{3}|,$

in which
$M_{\ell-1}^{U}(1)=(M_{\ell-1}^{UL}(1),M_{\ell-1}^{UR}(1))=(I_{d},0)\cdot
M_{\ell-1}(1)$. Empty spaces are to be filled in by zeros. We carry out the
following row operations on $S_{3}$ that do not alter the determinant of
$S_{3}$. Below $\text{row}_{j}$ refers to the $j$-th (block-valued) row of
$S_{3}$. Replacing

$\text{row}_{1}\longrightarrow\text{row}_{1}-\left(\begin{matrix}0&0\\\
0&I_{d}\end{matrix}\right)\cdot\text{row}_{3},$

and

$\text{row}_{4}\longrightarrow\text{row}_{4}-{\bf
e}_{d,r}^{*}B^{-1}\cdot\text{row}_{2},$

we arrive at

(4.273) $\displaystyle
S_{3}^{(1)}=\left(\begin{array}[]{c|c|c|c}&&\left(\begin{matrix}0&B\\\
0&0\end{matrix}\right)-M_{n-1}(1)+\left(\begin{matrix}0&0\\\
0&I_{d}\end{matrix}\right)M_{1}^{-1}(0)&-\left(\begin{matrix}0\\\
I_{d}\end{matrix}\right)(B^{*})^{-1}{\bf e}_{d,x}\\\
\hline\cr&B&-(I_{d},0)\cdot M_{\ell-1}(1)&\\\ \hline\cr
I_{2d}&&-M_{1}^{-1}(0)&\left(\begin{matrix}0\\\
I_{d}\end{matrix}\right)(B^{*})^{-1}{\bf e}_{d,x}\\\ \hline\cr&&{\bf
e}_{d,r}^{*}B^{-1}(I_{d},0)\cdot M_{\ell-1}(1)&\end{array}\right).$

Next, multiplying $\text{col}_{3}$ of $S_{3}^{(1)}$ on the right by $M_{1}(0)$
(for which $\det M_{1}(0)=1$, so $\det S_{3}^{(1)}$ is invariant), yields

(4.278) $\displaystyle
S_{3}^{(2)}=\left(\begin{array}[]{c|c|c|c}&&\left(\begin{matrix}0&B\\\
0&0\end{matrix}\right)M_{1}(0)-M_{n}(0)+\left(\begin{matrix}0&0\\\
0&I_{d}\end{matrix}\right)&-\left(\begin{matrix}0\\\
I_{d}\end{matrix}\right)(B^{*})^{-1}{\bf e}_{d,x}\\\
\hline\cr&B&-(I_{d},0)\cdot M_{\ell}(0)&\\\ \hline\cr
I_{2d}&&-I_{d}&\left(\begin{matrix}0\\\
I_{d}\end{matrix}\right)(B^{*})^{-1}{\bf e}_{d,x}\\\ \hline\cr&&{\bf
e}_{d,r}^{*}B^{-1}(I_{d},0)\cdot M_{\ell}(0)&\end{array}\right).$

Expanding the determinant along the first two columns, and noting that
$\left(\begin{matrix}0&B\\\
0&0\end{matrix}\right)M_{1}(0)=\left(\begin{matrix}I_{d}&0\\\
0&0\end{matrix}\right)$, yields

(4.281) $\displaystyle|\det S_{3}^{(2)}|$ $\displaystyle=|\det
B|\cdot\left|\det\left(\begin{array}[]{c|c}-M_{n}(0)+I_{2d}&-\left(\begin{matrix}0\\\
I_{d}\end{matrix}\right)(B^{*})^{-1}{\bf e}_{d,x}\\\ \hline\cr{\bf
e}_{d,r}^{*}B^{-1}(I_{d},0)\cdot M_{\ell}(0)&0\end{array}\right)\right|$
(4.282) $\displaystyle=:|\det B|\cdot|\det S_{3}^{(3)}|.$

We expand $S_{3}^{(3)}$ as the following product:

(4.289) $\displaystyle S_{3}^{(3)}=$
$\displaystyle\left(\begin{array}[]{c|c}I_{2d}&\\\ \hline\cr&{\bf
e}_{d,r}^{*}B^{-1}(I_{d},0)\end{array}\right)_{(2d+1)\times
4d}\cdot\left(\begin{array}[]{c|c}-M_{n-\ell}(\ell)&I_{2d}\\\ \hline\cr
I_{2d}&\end{array}\right)\cdot\left(\begin{array}[]{c|c}M_{\ell}(0)&\\\
\hline\cr&I_{2d}\end{array}\right)\cdot$ (4.292)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\cdot\left(\begin{array}[]{c|c}I_{2d}&\\\
\hline\cr I_{2d}&-\left(\begin{matrix}0\\\
I_{d}\end{matrix}\right)(B^{*})^{-1}{\bf
e}_{d,x}\end{array}\right)_{4d\times(2d+1)}$

We bound

(4.297) $\displaystyle|\det S_{3}^{(3)}|=$
$\displaystyle\left|\left(\wedge^{2d+1}\left(\begin{array}[]{c|c}I_{2d}&\\\
\hline\cr&{\bf
e}_{d,r}^{*}B^{-1}(I_{d},0)\end{array}\right)\right)\cdot\left(\wedge^{2d+1}\left(\begin{array}[]{c|c}-M_{n-\ell}(\ell)&I_{2d}\\\
\hline\cr I_{2d}&\end{array}\right)\right)\cdot\right.$ (4.302)
$\displaystyle\left.\qquad\cdot\left(\wedge^{2d+1}\left(\begin{array}[]{c|c}M_{\ell}(0)&\\\
\hline\cr&I_{2d}\end{array}\right)\right)\cdot\left(\wedge^{2d+1}\left(\begin{array}[]{c|c}I_{2d}&\\\
\hline\cr I_{2d}&-\left(\begin{matrix}0\\\
I_{d}\end{matrix}\right)(B^{*})^{-1}{\bf
e}_{d,x}\end{array}\right)\right)\right|$ (4.305) $\displaystyle\leq$
$\displaystyle\sum_{y=2d+1}^{3d}\sum_{\begin{subarray}{c}1\leq
j_{1}<...<j_{2d+1}\leq 4d\\\ 1\leq k_{1}<...<k_{2d+1}\leq 4d\\\
1\leq\ell_{1}<...<\ell_{2d+1}\leq
4d\end{subarray}}\left|\det\left(\begin{array}[]{c|c}I_{2d}&\\\ \hline\cr&{\bf
e}_{d,r}^{*}B^{-1}(I_{d},0)\end{array}\right)_{(1,...,2d,2d+1),(1,...,2d,y)}\right|$
(4.308)
$\displaystyle\qquad\qquad\qquad\cdot\det\left|\left(\begin{array}[]{c|c}-M_{n-\ell}(\ell)&I_{2d}\\\
\hline\cr
I_{2d}&\end{array}\right)_{(1,...,2d,y),(j_{1},...,j_{2d+1})}\right|$ (4.311)
$\displaystyle\qquad\qquad\qquad\cdot\left|\det\left(\begin{array}[]{c|c}M_{\ell}(0)&\\\
\hline\cr&I_{2d}\end{array}\right)_{(j_{1},...,j_{2d+1}),(k_{1},...,k_{2d+1})}\right|$
(4.314)
$\displaystyle\qquad\qquad\qquad\cdot\left|\det\left(\begin{array}[]{c|c}I_{2d}&\\\
\hline\cr I_{2d}&-\left(\begin{matrix}0\\\
I_{d}\end{matrix}\right)(B^{*})^{-1}{\bf
e}_{d,x}\end{array}\right)_{(k_{1},...,k_{2d+1}),(\ell_{1},...,\ell_{2d+1})}\right|.$

Suppose that

$k_{1}<...<k_{m_{0}}\leq 2d<k_{m_{0}+1}<...<k_{2d+1}.$

In order for

(4.317) $\displaystyle\det\left(\begin{array}[]{c|c}M_{\ell}(0)&\\\
\hline\cr&I_{2d}\end{array}\right)_{(j_{1},...,j_{2d+1}),(k_{1},...,k_{2d+1})},$

to be non-vanishing, necessarily

(4.318) $\displaystyle j_{1}<...<j_{m_{0}}\leq 2d\text{\ \ and\ \
}j_{m}=k_{m}\text{ for }m_{0}+1\leq m\leq 2d+1,$

If it is non-vanishing, then

(4.321) $\displaystyle\left|\det\left(\begin{array}[]{c|c}M_{\ell}(0)&\\\
\hline\cr&I_{2d}\end{array}\right)_{(j_{1},...,j_{2d+1}),(k_{1},...,k_{2d+1})}\right|$
(4.322) $\displaystyle=$
$\displaystyle\left|\det(M_{\ell}(0))_{\\{j_{1},...,j_{m_{0}}\\},\\{k_{1},...,k_{m_{0}}\\}}\right|$
(4.323) $\displaystyle\leq$ $\displaystyle\|\wedge^{m_{0}}M_{\ell}(0)\|\leq
e^{\ell(1+\varepsilon)\cdot\hat{L}^{m_{0}}},$

where we used Lemma 2.1 to bound $\|\wedge^{m_{0}}M_{\ell}(0)\|$. Taking
(4.318) into account, one has

(4.326)
$\displaystyle\left|\det\left(\begin{array}[]{c|c}-M_{n-\ell}(\ell)&I_{2d}\\\
\hline\cr
I_{2d}&\end{array}\right)_{(1,...,2d,y),(j_{1},...,j_{2d+1})}\right|$ (4.327)
$\displaystyle=$
$\displaystyle\left|\det(M_{n-\ell}(\ell))_{\\{1,...,2d\\}\setminus\\{j_{m_{0}+1}-2d,...,j_{2d+1}-2d\\},\\{j_{1},...,j_{m_{0}}\\}\setminus\\{y-2d\\}}\right|$
(4.328) $\displaystyle\leq$
$\displaystyle\|\wedge^{m_{0}-1}M_{n-\ell}(\ell)\|\leq
e^{(n-\ell)(1+\varepsilon)\cdot\hat{L}^{m_{0}-1}},$

where we used Lemma 2.1 again. The following estimates are easy to verify:

(4.331) $\displaystyle\left|\det\left(\begin{array}[]{c|c}I_{2d}&\\\
\hline\cr&{\bf
e}_{d,r}^{*}B^{-1}(I_{d},0)\end{array}\right)_{(1,...,2d,2d+1),(1,...,2d,y)}\right|\leq
C\|B^{-1}\|,\text{ and }$ (4.334)
$\displaystyle\left|\det\left(\begin{array}[]{c|c}I_{2d}&\\\ \hline\cr
I_{2d}&-\left(\begin{matrix}0\\\ I_{d}\end{matrix}\right)(B^{*})^{-1}{\bf
e}_{d,x}\end{array}\right)_{(k_{1},...,k_{2d+1}),(\ell_{1},...,\ell_{2d+1})}\right|\leq
C\|B^{-1}\|.$

Combining (4.321), (4.326) with (4.331), one has

(4.335) $\displaystyle|\det S_{3}^{(3)}|\leq$ $\displaystyle
C_{d}\|B^{-1}\|^{2}\sum_{m_{0}}e^{\ell(1+\varepsilon)\cdot\hat{L}^{m_{0}}+(n-\ell)(1+\varepsilon)\cdot\hat{L}^{m_{0}-1}}$
(4.336) $\displaystyle\leq$ $\displaystyle
C_{d}\|B^{-1}\|^{2}\max\left(e^{\ell(1+\varepsilon)\cdot\hat{L}^{d}+(n-\ell)(1+\varepsilon)\cdot\hat{L}^{d-1}},e^{\ell(1+\varepsilon)\cdot\hat{L}^{d+1}+(n-\ell)(1+\varepsilon)\cdot\hat{L}^{d}}\right),$

where we used that $\hat{L}^{1}<...<\hat{L}^{d-1}<\hat{L}^{d}$ and
$\hat{L}^{d}>\hat{L}^{d+1}>...>\hat{L}^{2d}$, due to
$\hat{L}_{d}>0>\hat{L}_{d+1}$. Combining the above with (4.237),(4.255) and
(4.281) yields the claimed result for Case 1.

Case 2. Suppose $(n-1)d\leq x\leq nd-1$. Again, we write $y=\ell d+r$ with
$3\leq\ell\leq n-2$ and $0\leq r\leq d-1$. We also denote
$x-(n-1)d=\tilde{x}$. In analogy to (4.187), we have
$|\mu_{n,x,y}(\theta)|=|\det\tilde{R}_{x,y}|$, where

(4.347) $\displaystyle\qquad\tilde{R}_{x,y}=$
$\displaystyle\left(\begin{array}[]{c|c|c|c|c|c|c|c|c|c}C_{n-1}&B^{*}&&&&&&&B&{\bf
e}_{d,\tilde{x}}\\\ \hline\cr B&\ddots&\ddots&&&&&&&\\\
\hline\cr&\ddots&\ddots&\ddots&&&&&&\\\ \hline\cr&&\ddots&\ddots&B^{*}&&&&&\\\
\hline\cr&&&B&C_{\ell}&B^{*}&&&&\\\ \hline\cr&&&&B&C_{\ell-1}&\ddots&&&\\\
\hline\cr&&&&&\ddots&\ddots&\ddots&&\\\ \hline\cr&&&&&&\ddots&\ddots&B^{*}&\\\
\hline\cr B^{*}&&&&&&&B&C_{0}&\\\ \hline\cr&&&&{\bf
e}_{d,r}^{*}&&&&&\end{array}\right)$

Applying exactly the same row operations on $\text{Row}_{1}$, $\text{Row}_{2}$
and $\text{Row}_{n-\ell+1}$ as in Case 1, reduces $\tilde{R}_{x,y}$ to the
following shape:

(4.361)
$\displaystyle\tilde{R}^{(1)}_{x,y}=\left(\begin{array}[]{c|c|c|c|c|c|c|c|c|c|c|c|c}0&0&0&\cdots&&&&&\cdots&0&-M_{n-1}^{UL}(1)B&B-M_{n-1}^{UR}(1)&{\bf
e}_{d,\tilde{x}}\\\ \hline\cr
B&0&0&\cdots&&&&&\cdots&0&-M_{n-2}^{UL}(1)B&-M_{n-2}^{UR}(1)&0\\\ \hline\cr
0&B&C_{n-3}&B^{*}&&&&&&&&&\vdots\\\
\hline\cr\vdots&&\ddots&\ddots&\ddots&&&&&&&&\\\
\hline\cr&&&\ddots&\ddots&\ddots&&&&&&&\\\
\hline\cr\vdots&&&&B&C_{\ell}&B^{*}&&&&&&\vdots\\\ \hline\cr
0&\cdots&&\cdots&0&B&0&0&\cdots&0&-M_{\ell-1}^{UL}(1)B&-M_{\ell-1}^{UR}(1)&0\\\
\hline\cr\vdots&&&&&0&B&C_{\ell-2}&B^{*}&&&0&0\\\
\hline\cr&&&&&&&\ddots&\ddots&\ddots&&\vdots&\vdots\\\
\hline\cr\vdots&&&&&&&&\ddots&\ddots&\ddots&0&\\\ \hline\cr
0&&&&&&&&&\ddots&\ddots&B^{*}&\vdots\\\ \hline\cr
B^{*}&0&\cdots&&&&&&\cdots&0&B&C_{0}&0\\\ \hline\cr 0&\cdots&&\cdots&0&{\bf
e}_{d,r}^{*}&0&\cdots&&\cdots&0&0&0\end{array}\right)$

This implies, similarly to (4.237), that

(4.362) $\displaystyle|\det\tilde{R}_{x,y}|=|\det\tilde{R}^{(1)}_{x,y}|=|\det
B|^{n-4}\cdot|\det\tilde{S}_{1}|,$

where in analogy with (4.255) and (4.281) one has

(4.368) $\displaystyle|\det\tilde{S}_{1}|=$
$\displaystyle\left|\det\left(\begin{array}[]{ccccc}0&0&-M_{n-1}^{UL}(1)B&B-M_{n-1}^{UR}(1)&{\bf
e}_{d,\tilde{x}}\\\ B&0&-BM_{n-1}^{LL}(1)B&-BM_{n-1}^{LR}(1)&0\\\
0&B&-M_{\ell-1}^{UL}(1)B&-M_{\ell-1}^{UR}(1)&0\\\ B^{*}&0&B&C_{0}&0\\\ 0&{\bf
e}^{*}_{d,r}&0&0&0\end{array}\right)\right|$ (4.374) $\displaystyle=$
$\displaystyle|\det
B|\cdot\left|\det\left(\begin{array}[]{ccccc}0&0&-M_{n-1}^{UL}(1)&B-M_{n-1}^{UR}(1)&{\bf
e}_{d,\tilde{x}}\\\ B&0&-BM_{n-1}^{LL}(1)&-BM_{n-1}^{LR}(1)&0\\\
0&B&-M_{\ell-1}^{UL}(1)&-M_{\ell-1}^{UR}(1)&0\\\ B^{*}&0&I_{d}&C_{0}&0\\\
0&{\bf e}^{*}_{d,r}&0&0&0\end{array}\right)\right|$ (4.380) $\displaystyle=$
$\displaystyle|\det B|^{2}\cdot|\det
B^{*}|\cdot\left|\det\left(\begin{array}[]{ccccc}0&0&-M_{n-1}^{UL}(1)&B-M_{n-1}^{UR}(1)&{\bf
e}_{d,\tilde{x}}\\\ I_{d}&0&-M_{n-1}^{LL}(1)&-M_{n-1}^{LR}(1)&0\\\
0&B&-M_{\ell-1}^{UL}(1)&-M_{\ell-1}^{UR}(1)&0\\\
I_{d}&0&(B^{*})^{-1}&(B^{*})^{-1}C_{0}&0\\\ 0&{\bf
e}^{*}_{d,r}&0&0&0\end{array}\right)\right|$

Rewriting this in block form yields

(4.385) $\displaystyle|\det\tilde{S}_{1}|=$ $\displaystyle|\det
B|^{3}\cdot\left|\det\left(\begin{array}[]{c|c|c|c}\left(\begin{matrix}0&0\\\
0&I_{d}\end{matrix}\right)&&\left(\begin{matrix}0&B\\\
0&0\end{matrix}\right)-M_{n-1}(1)&\left(\begin{matrix}I_{d}\\\
0\end{matrix}\right){\bf e}_{d,\tilde{x}}\\\
\hline\cr&B&-M_{\ell-1}^{U}(1)&\\\ \hline\cr I_{2d}&&-M_{1}^{-1}(0)&\\\
\hline\cr&{\bf e}^{*}_{d,r}&&\end{array}\right)\right|$ (4.390)
$\displaystyle=$ $\displaystyle|\det
B|^{3}\cdot\left|\det\left(\begin{array}[]{c|c|c|c}\left(\begin{matrix}0&0\\\
0&I_{d}\end{matrix}\right)&&\left(\begin{matrix}0&B\\\
0&0\end{matrix}\right)M_{1}(0)-M_{n}(0)&\left(\begin{matrix}I_{d}\\\
0\end{matrix}\right){\bf e}_{d,\tilde{x}}\\\
\hline\cr&B&-(I_{d},0)\,M_{\ell}(0)&\\\ \hline\cr I_{2d}&&-I_{2d}&\\\
\hline\cr&{\bf e}^{*}_{d,r}&&\end{array}\right)\right|$ (4.395)
$\displaystyle=$ $\displaystyle|\det
B|^{3}\cdot\left|\det\left(\begin{array}[]{c|c|c|c}&&I_{2d}-M_{n}(0)&\left(\begin{matrix}I_{d}\\\
0\end{matrix}\right){\bf e}_{d,\tilde{x}}\\\
\hline\cr&B&-(I_{d},0)\,M_{\ell}(0)&\\\ \hline\cr I_{2d}&&-I_{2d}&\\\
\hline\cr&{\bf e}^{*}_{d,r}&&\end{array}\right)\right|$ (4.400)
$\displaystyle=$ $\displaystyle|\det
B|^{3}\cdot\left|\det\left(\begin{array}[]{c|c|c|c}&&I_{2d}-M_{n}(0)&\left(\begin{matrix}I_{d}\\\
0\end{matrix}\right){\bf e}_{d,\tilde{x}}\\\
\hline\cr&B&-(I_{d},0)\,M_{\ell}(0)&\\\ \hline\cr I_{2d}&&-I_{2d}&\\\
\hline\cr&&{\bf
e}_{d,r}^{*}B^{-1}(I_{d},0)\,M_{\ell}(0)&\end{array}\right)\right|$ (4.405)
$\displaystyle=$ $\displaystyle|\det
B|^{3}\cdot\left|\det\left(\begin{array}[]{c|c|c|c}&&I_{2d}-M_{n}(0)&\left(\begin{matrix}I_{d}\\\
0\end{matrix}\right){\bf e}_{d,\tilde{x}}\\\
\hline\cr&B&-(I_{d},0)\,M_{\ell}(0)&\\\ \hline\cr I_{2d}&&-I_{2d}&\\\
\hline\cr&{\bf e}^{*}_{d,r}&&\end{array}\right)\right|$ (4.408)
$\displaystyle=$ $\displaystyle|\det
B|^{3}\cdot\left|\det\left(\begin{array}[]{c|c}I_{2d}-M_{n}(0)&\left(\begin{matrix}I_{d}\\\
0\end{matrix}\right){\bf e}_{d,\tilde{x}}\\\ \hline\cr{\bf
e}_{d,r}^{*}B^{-1}(I_{d},0)\,M_{\ell}(0)&\end{array}\right)\right|$ (4.409)
$\displaystyle=:$ $\displaystyle|\det B|^{4}\cdot|\det\tilde{S}_{(3)}|,$

where

(4.416) $\displaystyle\tilde{S}_{3}^{(3)}=$
$\displaystyle\left(\begin{array}[]{c|c}I_{2d}&\\\ \hline\cr&{\bf
e}_{d,r}^{*}B^{-1}(I_{d},0)\end{array}\right)_{(2d+1)\times
4d}\cdot\left(\begin{array}[]{c|c}-M_{n-\ell}(\ell)&I_{2d}\\\ \hline\cr
I_{2d}&\end{array}\right)\cdot\left(\begin{array}[]{c|c}M_{\ell}(0)&\\\
\hline\cr&I_{2d}\end{array}\right)\cdot$ (4.419)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\cdot\left(\begin{array}[]{c|c}I_{2d}&\\\
\hline\cr I_{2d}&-\left(\begin{matrix}I_{d}\\\ 0\end{matrix}\right){\bf
e}_{d,\tilde{x}}\end{array}\right)_{4d\times(2d+1)}$

The remainder of the argument is exactly the same as in Case 1. ∎

##  5\. Denominator lower bound for periodic boundary condition

In this section we prove Lemma 3.2. Throughout, we denote $B_{0}$ by $B$, and
$C_{0}(\theta+jd\alpha)-E$ by $C_{j}$. First, we relate the determinant
$f_{E,n}(\theta)=\det(P_{\alpha,n}(\theta)-E)$ to the transfer matrix
$M_{n,E}$.

######  Lemma 5.1.

The following identity holds pointwise in $\theta$

(5.420) $\displaystyle|f_{E,n}(\theta)|=|\det
B|^{n}\cdot|\det(M_{n,E}(\theta)-I_{2d})|.$

###### Proof.

The proof is similar to, and is in fact simpler than, the one of Lemma 3.1.
Indeed, following exactly the same row operations (that do not alter
$f_{E,n}$) applied to $\text{Row}_{1}$, $\text{Row}_{2}$ and
$\text{Row}_{n-\ell+1}$ of $P_{\alpha,n}(\theta)-E$, instead of those on
$R_{x,y}$ in Section 4 (note these operations do not involve the terms ${\bf
e}_{d,x}$ or ${\bf e}_{d,r}^{*}$), in analogy to (4.207) we arrive at the
equality

(5.433) $\displaystyle
P^{(1)}_{n}(\theta)=\left(\begin{array}[]{c|c|c|c|c|c|c|c|c|c|c|c}0&0&0&\cdots&&&&&\cdots&0&-M_{n-1}^{UL}(1)B&B-M_{n-1}^{UR}(1)\\\
\hline\cr B&0&0&\cdots&&&&&\cdots&0&-M_{n-2}^{UL}(1)B&-M_{n-2}^{UR}(1)\\\
\hline\cr 0&B&C_{n-3}&B^{*}&&&&&&&&\\\
\hline\cr\vdots&&\ddots&\ddots&\ddots&&&&&&&\\\
\hline\cr&&&\ddots&\ddots&\ddots&&&&&&\\\
\hline\cr\vdots&&&&B&C_{\ell}&B^{*}&&&&&\\\ \hline\cr
0&\cdots&&\cdots&0&B&0&0&\cdots&0&-M_{\ell-1}^{UL}(1)B&-M_{\ell-1}^{UR}(1)\\\
\hline\cr\vdots&&&&&0&B&C_{\ell-2}&B^{*}&&&0\\\
\hline\cr&&&&&&&\ddots&\ddots&\ddots&&\vdots\\\
\hline\cr\vdots&&&&&&&&\ddots&\ddots&\ddots&0\\\ \hline\cr
0&&&&&&&&&\ddots&\ddots&B^{*}\\\ \hline\cr
B^{*}&0&\cdots&&&&&&\cdots&0&B&C_{0}\end{array}\right)$

This implies, similarly to (4.237), that

(5.434) $\displaystyle|f_{E,n}(\theta)|=|\det P_{n}^{(}1)(\theta)|=|\det
B|^{n-4}\cdot|\det\tilde{S}_{1}|,$

where

(5.439)
$\displaystyle\tilde{S}_{1}:=\left(\begin{array}[]{cccc}0&0&-M_{n-1}^{UL}(1)B&B-M_{n-1}^{UR}(1)\\\
B&0&-M_{n-2}^{UL}(1)B&-M_{n-2}^{UR}(1)\\\
0&B&-M_{\ell-1}^{UL}(1)B&-M_{\ell-1}^{UR}(1)\\\
B^{*}&0&B&C_{0}\end{array}\right).$

Furthermore, in place of (4.255), we now have

(5.443) $\displaystyle|\det\tilde{S}_{1}|=$ $\displaystyle|\det
B|^{2}\cdot\left|\det\left(\begin{array}[]{cccc}0&-M_{n-1}^{UL}(1)&B-M_{n-1}^{UR}(1)\\\
B&-M_{n-2}^{UL}(1)&-M_{n-2}^{UR}(1)\\\
B^{*}&I_{d}&C_{0}\end{array}\right)\right|$ (5.447) $\displaystyle=$
$\displaystyle|\det B|^{2}\cdot|\det
B^{*}|\cdot\left|\det\left(\begin{array}[]{cccc}0&-M_{n-1}^{UL}(1)&B-M_{n-1}^{UR}(1)\\\
B&-M_{n-2}^{UL}(1)&-M_{n-2}^{UR}(1)\\\
I_{d}&(B^{*})^{-1}&(B^{*})^{-1}C_{0}\end{array}\right)\right|$ (5.452)
$\displaystyle=$ $\displaystyle|\det
B|^{4}\cdot\left|\det\left(\begin{array}[]{cccc}0&0&-M_{n-1}^{UL}(1)&B-M_{n-1}^{UR}(1)\\\
0&I_{d}&-M_{n-1}^{LL}(1)&-M_{n-1}^{LR}(1)\\\ I_{d}&0&0&-B\\\
0&I_{d}&(B^{*})^{-1}&(B^{*})^{-1}C_{0}\end{array}\right)\right|$ (5.455)
$\displaystyle=$ $\displaystyle|\det
B|^{4}\cdot\left|\det\left(\begin{array}[]{c|c}\left(\begin{matrix}0&0\\\
0&I_{d}\end{matrix}\right)&\left(\begin{matrix}0&B\\\
0&0\end{matrix}\right)-M_{n-1}(1)\\\ \hline\cr
I_{2d}&-M_{1}^{-1}(0)\end{array}\right)\right|$

Rewriting this in block form yields

(5.458) $\displaystyle|\det\tilde{S}_{1}|=$ $\displaystyle|\det
B|^{4}\cdot\left|\det\left(\begin{array}[]{c|c}\left(\begin{matrix}0&0\\\
0&I_{d}\end{matrix}\right)&\left(\begin{matrix}0&B\\\
0&0\end{matrix}\right)M_{1}(0)-M_{n}(0)\\\ \hline\cr
I_{2d}&-I_{2d}\end{array}\right)\right|$ (5.461) $\displaystyle=$
$\displaystyle|\det
B|^{4}\cdot\left|\det\left(\begin{array}[]{c|c}\left(\begin{matrix}0&0\\\
0&I_{d}\end{matrix}\right)&I_{2d}-M_{n}(0)\\\ \hline\cr
I_{2d}&0\end{array}\right)\right|$ (5.462) $\displaystyle=:$
$\displaystyle|\det B|^{4}\cdot|\det M_{n}(0)-I_{2d}|.$

The claimed result follows by combining (5.434) with (5.458). ∎

To estimate $\det(M_{n,E}(\theta)-I_{2d})$, we need the following elementary
lemma concerning the vector spaces associated with the singular value
decomposition of a symplectic matrix. We include the proof for the sake of
completeness.

######  Lemma 5.2.

Let $M\in\mathbb{C}^{2d\times 2d}$ be a complex symplectic matrix, i.e.,
$M^{*}\Omega M=\Omega$. Let $\sigma_{1}\geq...\geq\sigma_{2d}$ be the singular
values of $M$. Let $\mu_{1}>...>\mu_{k}$ be the non-repeated singular values
and $V_{j}$, $1\leq j\leq k$, be the singular vector space of $M$ associated
to $\mu_{j}$. Similarly, let $W_{j}$ be the singular vector space of $M^{*}$
associated to $\mu_{j}$. Then

(5.463) $\displaystyle\Omega V_{j}=V_{k+1-j},\text{ and }\Omega
W_{j}=W_{k+1-j}.$

###### Proof.

Let $v_{j}\in V_{j}$ be a unit vector, and $w_{j}$ be the unit vector such
that

(5.464) $\displaystyle Mv_{j}=\mu_{j}w_{j}.$

This implies

(5.465) $\displaystyle\begin{cases}M^{*}w_{j}=\mu_{j}v_{j}\\\
(M^{*})^{-1}v_{j}=\mu_{j}^{-1}w_{j}\\\
M^{-1}w_{j}=\mu_{j}^{-1}v_{j}\end{cases}$

Since $M$ is symplectic, we also have

(5.466) $\displaystyle\begin{cases}M^{*}\Omega M=\Omega\\\ \Omega
M^{*}\Omega=-M^{-1}\\\ \Omega M\Omega=-(M^{*})^{-1}\end{cases}$

Combining (5.464), (5.465) with (5.466), yields

(5.467) $\displaystyle M^{*}M\Omega v_{j}=-M^{*}\Omega^{2}M\Omega
v_{j}=M^{*}\Omega(M^{*})^{-1}v_{j}=M^{*}\Omega\mu_{j}^{-1}w_{j}=$
$\displaystyle-\mu_{j}^{-1}\Omega^{2}M^{*}\Omega w_{j}$ (5.468)
$\displaystyle=$ $\displaystyle\mu_{j}^{-1}\Omega M^{-1}w_{j}$ (5.469)
$\displaystyle=$ $\displaystyle\mu_{j}^{-2}\Omega v_{j}.$

This implies that $\Omega v_{j}$ is an eigenvector of $M^{*}M$ associated to
the eigenvalue $\mu_{j}^{-2}=\mu_{k+1-j}^{2}$, whence $\Omega v_{j}\in
V_{k+1-j}$. Therefore, $\Omega V_{j}\subset V_{k+1-j}$ for any $1\leq j\leq
k$, which implies that $\Omega V_{j}=V_{k+1-j}$. The claim concerning $W_{j}$
can be proved analogously. ∎

Next, we will apply the previous lemma to $M_{n,E}(\theta)$. For simplicity,
we shall write $\hat{L}^{j}(d\alpha,M_{E})$, $\hat{L}_{j}(d\alpha,M_{E})$ as
$\hat{L}^{j}$ and $\hat{L}_{j}$, respectively. Recall that the large deviation
sets $\mathcal{B}_{n,E,\varepsilon}^{d}$ was defined in (2.53). By Lemma 2.2,
this set is exponentially small for large enough $n$. We fix
$0<\varepsilon<\frac{\hat{L}_{d}}{11\hat{L}^{d}}$. Note that for such
$\varepsilon$,

(5.470)
$\displaystyle(1-10\varepsilon)\hat{L}^{d}>(1+\varepsilon)\hat{L}^{d-1}$

Let $\theta\in(\mathcal{B}^{d}_{n,E,\varepsilon})^{c}$, and $v_{j,n}(\theta)$
be the normalized singular vector of $M_{n,E}(\theta)$ corresponding to the
singular value $\sigma_{j}(M_{n,E}(\theta))$. Suppose $w_{j,n}(\theta)$ is the
normalized singular vector of $M_{n,E}^{*}(\theta)$ such that

$M_{n,E}(\theta)v_{j,n}(\theta)=\sigma_{j}(M_{n,E}(\theta))w_{j,n}(\theta).$

According to Lemmas 2.1 and 2.2, for
$\theta\in(\mathcal{B}_{n,E,\varepsilon}^{d})^{c}$, one has

(5.471)
$\displaystyle\prod_{j=1}^{d}\sigma_{j}(M_{n,E}(\theta))=\|\wedge^{d}M_{n,E}(\theta)\|\geq(1-\varepsilon)\cdot\hat{L}^{d}\geq(1+\varepsilon)\cdot\hat{L}^{d+1}\geq\prod_{j=1}^{d+1}\sigma_{j}(M_{n,E}(\theta)).$

It follows that

$\sigma_{d}(M_{n,E}(\theta))>1>\sigma_{d+1}(M_{n,E}(\theta))=(\sigma_{d}(M_{n,E}(\theta)))^{-1}.$

Let $V_{j,n}(\theta)$ and $W_{j,n}(\theta)$ be as defined in Lemma 5.2 with
$M$ replaced with $M_{n,E}(\theta)$. Clearly $k$ is even. By that lemma, for
any $1\leq j\leq k/2$,

(5.472) $\displaystyle\Omega V_{j,n}(\theta)=V_{k+1-j,n}(\theta),\text{ and
}\Omega W_{j,n}(\theta)=W_{k+1-j,n}(\theta).$

This implies the following relations between the most expanding and most
contracting $d$-planes:

(5.473) $\displaystyle(\wedge^{d}\Omega)(v_{1,n}(\theta)\wedge\cdots\wedge
v_{d,n}(\theta))=\pm v_{d+1,n}(\theta)\wedge\cdots\wedge
v_{2d,n}(\theta),\text{ and }$ (5.474)
$\displaystyle(\wedge^{d}\Omega)(w_{1,n}(\theta)\wedge\cdots\wedge
w_{d,n}(\theta))=\pm w_{d+1,n}(\theta)\wedge\cdots\wedge w_{2d,n}(\theta),$

and in particular

(5.475) $\displaystyle\left|\langle v_{1,n}(\theta)\wedge\cdots\wedge
v_{d,n}(\theta),w_{1,n}(\theta)\wedge\cdots\wedge
w_{d,n}(\theta)\rangle\right|$ (5.476)
$\displaystyle\qquad\qquad\qquad=\left|\langle
v_{d+1,n}(\theta)\wedge\cdots\wedge
v_{2d,n}(\theta),w_{d+1,n}(\theta)\wedge\cdots\wedge
w_{2d,n}(\theta)\rangle\right|$

The following result will play a crucial role in our argument, cf. [GS3]*Lemma
3.2. It will enable us to show that for a subsequence of $n$, the most
expanding $d$-planes of $M_{n,E}(\theta)$ and $M_{n,E}^{*}(\theta)$ are not
entirely orthogonal.

######  Lemma 5.3.

Assume that $\beta(\alpha)=0$. Then for any small $\varepsilon>0$, there
exists $N_{0}=N_{0}(\varepsilon,\alpha,E,d)$ and
$\kappa_{0}=\kappa_{0}(\varepsilon,\alpha,E,d,N_{0})>0$ such that for any
$\kappa_{0}$-admissible $N_{1}$ (those satisfying
$\|N_{1}d\alpha\|_{\mathbb{T}}\leq\kappa_{0}$), and $N_{1}\geq 8N_{0}$, we
have

(5.477)
$\displaystyle\mathrm{mes}(\tilde{\mathcal{B}}^{d}_{N_{1},E,\varepsilon}):=\mathrm{mes}\Big{(}\left\\{\theta:\,\left|\frac{1}{2N_{1}}\log\|\wedge^{d}\left(M_{N_{1},E}^{2}(\theta)\right)\|-\hat{L}^{d}\right|>\varepsilon\cdot\hat{L}^{d}\right\\}\Big{)}\leq\frac{1}{20}$

###### Proof.

By Lemma 2.2, we can choose $N_{0}:=N_{0}(\varepsilon,\alpha,E,d)$ so that for
$n\geq N_{0}$, one has $\mathrm{mes}(\mathcal{B}_{n,E,\varepsilon}^{d})\leq
e^{-n\varepsilon^{2}\cdot\hat{L}^{d}(d\alpha,M_{E})}$ and, furthermore,

(5.478)
$\displaystyle\sum_{j=0}^{\infty}e^{-2^{j}c_{1}\varepsilon^{2}N_{0}\cdot\hat{L}^{d}(d\alpha,M_{E})}<1/100,$

where $c_{1}$ is the constant in Lemma 2.2, as well as

(5.479)
$\displaystyle\sum_{j=0}^{\infty}e^{-2^{j+1}N_{0}(1-\varepsilon)\hat{L}^{d}(d\alpha,M_{E})}\cdot
e^{2^{j+5}N_{0}\varepsilon\hat{L}^{d}(d\alpha,M_{E})}\cdot
e^{8\tilde{C}_{0}c_{0}}<c_{0},$

where $\tilde{C}_{0},c_{0}>0$ are the constants in Theorem 2.4. For the
purposes of this proof, we let
$\mathcal{G}_{n}:=(\mathcal{B}_{n,E,\varepsilon}^{d})^{c}$, omitting the fixed
parameters $E$ and $\varepsilon$ for simplicity. We also denote
$\hat{L}^{k}(d\alpha,M_{E})$ by $\hat{L}^{k}$. First, we need the following
estimate at the fixed scale $N_{0}$.

######  Lemma 5.4.

There exists $\kappa_{0}=\kappa_{0}(\varepsilon,\alpha,E,d,N_{0})$ such that
for any $\|\kappa\|_{\mathbb{T}}\leq\kappa_{0}$, and for any $\theta_{0}$
satisfying

(5.480) $\displaystyle\|\wedge^{d}M_{2N_{0},E}(\theta_{0})\|\geq
e^{2N_{0}(1-\varepsilon)\hat{L}^{d}}$

one has the bound

(5.481)
$\displaystyle\|\wedge^{d}\left(M_{N_{0},E}(\theta_{0}+N_{0}d\alpha+\kappa)M_{N_{0},E}(\theta_{0})\right)\|\geq
e^{2N_{0}(1-2\varepsilon)\hat{L}^{d}}.$

###### Proof.

We use the following estimate from [DK]:

######  Lemma 5.5.

[DK]*Lemma 2.8 For $k\times k$ complex matrices $h_{1},h_{2}$, for any $1\leq
j\leq k$,

(5.482) $\displaystyle\|\wedge^{j}h_{1}-\wedge^{j}h_{2}\|\leq
j\max\\{1,\|h_{1}\|,\|h_{2}\|\\}^{j-1}\,\|h_{1}-h_{2}\|.$

We apply Lemma 5.5 to $k=2d$, $j=d$, and

(5.483) $\displaystyle h_{1}(\theta)=$ $\displaystyle
M_{N_{0},E}(\theta+N_{0}d\alpha+\kappa)M_{N_{0},E}(\theta)$ (5.484)
$\displaystyle h_{2}(\theta)=$ $\displaystyle
M_{N_{0},E}(\theta+N_{0}d\alpha)M_{N_{0},E}(\theta)$

By Lemma 2.1, uniformly in $\theta$,

(5.485) $\displaystyle\max(\|h_{1}(\theta)\|,\|h_{2}(\theta)\|)\leq
e^{2N_{0}(1+\varepsilon)\hat{L}^{d}},\text{ and }\|M_{N_{0},E}(\theta)\|\leq
e^{N_{0}(1+\varepsilon)\hat{L}^{d}}.$

Hence, Lemma 5.5 implies furthermore that uniformly in $\theta$,

(5.486) $\displaystyle\|\wedge^{d}h_{1}(\theta)-\wedge^{d}h_{2}(\theta)\|\leq$
$\displaystyle
de^{2dN_{0}(1+\varepsilon)\hat{L}^{d}}\cdot\|h_{1}(\theta)-h_{2}(\theta)\|$
(5.487) $\displaystyle\leq$ $\displaystyle
de^{3dN_{0}(1+\varepsilon)\hat{L}^{d}}\cdot\|M_{N_{0},E}(\theta+N_{0}d\alpha+\kappa)-M_{N_{0},E}(\theta+N_{0}d\alpha)\|$
(5.488) $\displaystyle\leq$ $\displaystyle
C_{\alpha,d,E,N_{0}}e^{4dN_{0}(1+\varepsilon)\hat{L}^{d}}\cdot\|\kappa\|_{\mathbb{T}},$

where in the last step we used a telescoping argument. Hence for
$\|\kappa\|_{\mathbb{T}}\leq\kappa_{0}(\varepsilon,\alpha,E,d,N_{0})$, one has
uniformly in $\theta$,

(5.489) $\displaystyle\|\wedge^{d}h_{1}(\theta)-\wedge^{d}h_{2}(\theta)\|\leq
e^{2N_{0}(1-2\varepsilon)\hat{L}^{d}}.$

This, combined with (5.480) at $\theta=\theta_{0}$, implies that

(5.490)
$\displaystyle\|\wedge^{d}(M_{N_{0},E}(\theta_{0}+N_{0}d\alpha+\kappa)M_{N_{0},E}(\theta_{0}))\|\geq
e^{2N_{0}(1-\varepsilon)\hat{L}^{d}}-e^{2N_{0}(1-2\varepsilon)\hat{L}^{d}}\geq
e^{2N_{0}(1-2\varepsilon)\hat{L}^{d}},$

which is the claimed result. ∎

Next, we pick an arbitrary $\kappa_{0}$-admissible $N_{1}$ (i.e.,
$\|N_{1}d\alpha\|_{\mathbb{T}}\leq\kappa_{0}$), satisfying $N_{1}\geq 8N_{0}$.
Let $k\geq 2$ be such that $2^{k+1}N_{0}\leq N_{1}<2^{k+2}N_{0}$, and
$n_{0}=N_{1}-2^{k}N_{0}=N_{1}-N_{0}-\sum_{\ell=0}^{k-1}2^{\ell}N_{0}$ be the
remainder. Clearly

(5.491) $\displaystyle 2^{k}N_{0}\leq n_{0}\leq(3\cdot 2^{k}-1)N_{0}.$

Our goal is to estimate

(5.492)
$\displaystyle\log\|\wedge^{d}(M_{N_{1},E}^{2}(\theta))\|=\log\|\wedge^{d}(M_{N_{1},E}(\theta+N_{1}d\alpha+\kappa)M_{N_{1},E}(\theta))\|,$

where $\kappa=-N_{1}d\alpha$ (modulo $\mathbb{Z}$) satisfies
$\|\kappa\|_{\mathbb{T}}\leq\kappa_{0}$. In order to accomplish this, we
decompose

(5.493) $\displaystyle
M_{N_{1},E}(\theta+N_{1}d\alpha+\kappa)M_{N_{1},E}(\theta)=M_{N_{1},E}(\theta)^{2}$
(5.494) $\displaystyle=$ $\displaystyle
M_{n_{0},E}(\theta+(2N_{1}-n_{0})d\alpha+\kappa)M_{2^{k-1}N_{0},E}(\theta+(N_{1}+2^{k-1}N_{0})d\alpha+\kappa)\cdots$
(5.495) $\displaystyle\cdots
M_{4N_{0},E}(\theta+(N_{1}+2^{2}N_{0})d\alpha+\kappa)M_{2N_{0},E}(\theta+(N_{1}+2N_{0})d\alpha+\kappa)$
(5.496) $\displaystyle\,\,\,\,\,\cdot
M_{N_{0},E}(\theta+(N_{1}+N_{0})d\alpha+\kappa)M_{N_{0},E}(\theta+N_{1}d\alpha+\kappa)\cdot$
(5.497) $\displaystyle\,\,\,\,\,\cdot
M_{N_{0},E}(\theta+(N_{1}-N_{0})d\alpha)M_{N_{0},E}(\theta+(N_{1}-2N_{0})d\alpha)$
(5.498) $\displaystyle\,\,\,\,\,\cdot
M_{2N_{0},E}(\theta+(N_{1}-2^{2}N_{0})d\alpha)M_{4N_{0},E}(\theta+(N_{1}-2^{3}N_{0})d\alpha)$
(5.499) $\displaystyle\cdots
M_{2^{k-1}N_{0},E}(\theta+n_{0}d\alpha)M_{n_{0},E}(\theta)$

Step 1. Let

(5.500) $\displaystyle\mathcal{G}_{N_{0}}^{(1)}:=$
$\displaystyle(\mathcal{G}_{N_{0}}-(N_{1}-2N_{0})d\alpha)\cap(\mathcal{G}_{N_{0}}-(N_{1}-N_{0})d\alpha)\cap(\mathcal{G}_{N_{0}}-N_{1}d\alpha-\kappa)\cap$
(5.501)
$\displaystyle\cap(\mathcal{G}_{N_{0}}-(N_{1}+N_{0})d\alpha-\kappa)\cap(\mathcal{G}_{2N_{0}}-(N_{1}-2N_{0})d\alpha)\cap(\mathcal{G}_{2N_{0}}-N_{1}d\alpha-\kappa)\cap$
(5.502) $\displaystyle\cap(\mathcal{G}_{2N_{0}}-(N_{1}-N_{0})d\alpha)$

We restrict to $\theta\in\mathcal{G}_{N_{0}}^{(1)}$ for the remainder of the
proof. Let

(5.503) $\displaystyle g_{N_{0},1}(\theta)=$
$\displaystyle\wedge^{d}M_{N_{0},E}(\theta+(N_{1}-2N_{0})d\alpha)$ (5.504)
$\displaystyle g_{N_{0},2}(\theta)=$
$\displaystyle\wedge^{d}M_{N_{0},E}(\theta+(N_{1}-N_{0})d\alpha)$ (5.505)
$\displaystyle g_{N_{0},3}(\theta)=$
$\displaystyle\wedge^{d}M_{N_{0},E}(\theta+N_{1}d\alpha+\kappa)$ (5.506)
$\displaystyle g_{N_{0},4}(\theta)=$
$\displaystyle\wedge^{d}M_{N_{0},E}(\theta+(N_{1}+N_{0})d\alpha+\kappa)$

The goal of Step 1 is to show the following.

######  Lemma 5.6.

For any $\theta\in\mathcal{G}_{N_{0}}^{(1)}$, one has

(5.507) $\displaystyle\log\|g_{N_{0},4}(\theta)\cdots
g_{N_{0},1}(\theta)\|\geq
4N_{0}(1-3\varepsilon)\hat{L}^{d}-4\tilde{C}_{0}\tilde{\kappa}_{N_{0}}\cdot\tilde{\varepsilon}_{N_{0}}^{-2},$

for the constant $\tilde{C}_{0}>0$ in Theorem 2.4 and with

(5.508)
$\displaystyle\tilde{\kappa}_{N_{0}}=e^{-2N_{0}(1-\varepsilon)\hat{L}_{d}+2^{2}N_{0}\varepsilon\hat{L}^{d-1}},\text{
and }\tilde{\varepsilon}_{N_{0}}=e^{-2^{3}N_{0}\varepsilon\hat{L}^{d}}.$

Furthermore,

(5.509) $\displaystyle\mathrm{mes}(\mathcal{G}_{N_{0}}^{(1)})\geq
1-4e^{-c_{1}\varepsilon^{2}N_{0}\hat{L}^{d}}-3e^{-2c_{1}\varepsilon^{2}N_{0}\hat{L}^{d}},$

where $c_{1}>0$ is the constant in Lemma 2.2.

###### Proof.

By Lemma 2.2, for all $\theta\in\mathcal{G}_{N_{0}}^{(1)}$, we have

(5.510) $\displaystyle
e^{N_{0}(1-\varepsilon)\hat{L}^{d}}\leq\sigma_{1}(g_{N_{0},j}(\theta))\leq
e^{N_{0}(1+\varepsilon)\hat{L}^{d}},\text{ for }j=1,2,3,4,$

and

(5.511) $\displaystyle\|g_{N_{0},4}(\theta)g_{N_{0},3}(\theta)\|\geq
e^{2N_{0}(1-\varepsilon)\hat{L}^{d}},\text{ and
}\|g_{N_{0},2}(\theta)g_{N_{0},1}(\theta)\|\geq
e^{2N_{0}(1-\varepsilon)\hat{L}^{d}}.$

Note that by definition
$\theta\in\mathcal{G}_{N_{0}}^{(1)}\subset\mathcal{G}_{2N_{0}}-(N_{1}-N_{0})d\alpha$,
hence

(5.512)
$\displaystyle\|\wedge^{d}(M_{N_{0},E}(\theta+N_{1}d\alpha)M_{N_{0},E}(\theta+(N_{1}-N_{0})d\alpha))\|\geq
e^{2N_{0}(1-\varepsilon)\hat{L}^{d}}.$

Combining the above (which verifies the assumptions of Lemma 5.4 for
$\theta_{0}=\theta+(N_{1}-N_{0})d\alpha$) with Lemma 5.4 and
$\|\kappa\|_{\mathbb{T}}\leq\kappa_{0}$, yields

(5.513) $\displaystyle\|g_{N_{0},3}(\theta)g_{N_{0},2}(\theta)\|=$
$\displaystyle\|\wedge^{d}(M_{N_{0},E}(\theta+N_{1}d\alpha+\kappa)M_{N_{0},E}(\theta+(N_{1}-N_{0})d\alpha))\|$
(5.514) $\displaystyle\geq$ $\displaystyle
e^{2N_{0}(1-2\varepsilon)\hat{L}^{d}}.$

By Lemma 2.1 applied to the matrix $\wedge^{2}(\wedge^{d}M_{N_{0},E})$ and

(5.515) $\displaystyle{L}^{1}(d\alpha,\wedge^{2}(\wedge^{d}M_{E}))=$
$\displaystyle{L}^{1}(d\alpha,\wedge^{d}M_{E})+{L}^{2}(d\alpha,\wedge^{d}M_{E})$
(5.516) $\displaystyle=$
$\displaystyle{L}^{d}+{L}^{d-1}+{L}_{d+1}=2{L}^{d-1},$

one has uniformly in $\theta\in\mathbb{T}$,

(5.517)
$\displaystyle\sigma_{1}(\wedge^{d}M_{N_{0},E}(\theta))\cdot\sigma_{2}(\wedge^{d}M_{N_{0},E}(\theta))=\sigma_{1}(\wedge^{2}(\wedge^{d}M_{N_{0},E}(\theta)))\leq
e^{2N_{0}(1+\varepsilon)\hat{L}^{d-1}}.$

Combining the above with the lower bounds in (5.510), we conclude that

(5.518) $\displaystyle\sigma_{2}(g_{N_{0},j}(\theta))\leq
e^{N_{0}(1-\varepsilon)(\hat{L}^{d-1}+\hat{L}_{d+1})}\cdot
e^{4N_{0}\varepsilon\hat{L}^{d-1}}.$

Note that (5.510), (5.518), (5.511) and (5.513) verify the assumptions of
Theorem 2.4, the Avalanche Principle. In fact, for $j=1,2,3,4$,

(5.519)
$\displaystyle\frac{\sigma_{1}(g_{N_{0},j}(\theta))}{\sigma_{2}(g_{N_{0},j}(\theta))}\geq
e^{2N_{0}(1-\varepsilon)\hat{L}_{d}}e^{-4N_{0}\varepsilon\hat{L}^{d-1}}=:\tilde{\kappa}_{N_{0}}^{-1},$

and for $j=1,2,3$,

(5.520)
$\displaystyle\frac{\|g_{N_{0},j+1}(\theta)g_{N_{0},j}(\theta)\|}{\sigma_{1}(g_{N_{0},j+1}(\theta))\cdot\sigma_{1}(g_{N_{0},j}(\theta))}\geq
e^{-2^{3}N_{0}\varepsilon\hat{L}^{d}}=:\tilde{\varepsilon}_{N_{0}}.$

Clearly,

(5.521)
$\displaystyle\tilde{\kappa}_{N_{0}}\cdot\tilde{\varepsilon}_{N_{0}}^{-2}\leq
e^{-2N_{0}(1-\varepsilon)\hat{L}_{d}}e^{(2^{4}+2^{2})N_{0}\varepsilon\hat{L}^{d}}<c_{0},$

due to (5.479). Therefore, by applying Theorem 2.4 to
$g_{N_{0},4}(\theta)\cdots g_{N_{0},1}(\theta)$, we obtain with the constant
$\tilde{C}_{0}>0$ in Theorem 2.4,

(5.522) $\displaystyle\log\|g_{N_{0},4}(\theta)\cdots
g_{N_{0},1}(\theta)\|\geq$
$\displaystyle\sum_{j=1}^{3}\log\|g_{N_{0},j+1}(\theta)g_{N_{0},j}(\theta)\|-\sum_{j=2}^{3}\log\|g_{N_{0},j}(\theta)\|-4\tilde{C}_{0}\tilde{\kappa}_{N_{0}}\cdot\tilde{\varepsilon}_{N_{0}}^{-2}$
(5.523) $\displaystyle\geq$ $\displaystyle
4N_{0}(1-3\varepsilon)\hat{L}^{d}-4\tilde{C}_{0}\tilde{\kappa}_{N_{0}}\cdot\tilde{\varepsilon}_{N_{0}}^{-2}.$

Regarding the measure estimate, we note that

(5.524) $\displaystyle\mathrm{mes}(\mathcal{G}_{N_{0}}^{(1)})\geq
1-4e^{-c_{1}\varepsilon^{2}N_{0}\hat{L}^{d}}-3e^{-2c_{1}\varepsilon^{2}N_{0}\hat{L}^{d}},$

where we used that for any $n\geq N_{0}$,
$\mathrm{mes}(\mathcal{G}_{n})=1-\mathrm{mes}(\mathcal{B}_{n,E,\varepsilon}^{d})\geq
1-e^{-c_{1}\varepsilon^{2}n\hat{L}^{d}}$, due to Lemma 2.2. ∎

To better illustrate the inductive procedure, we present the second step in
detail as well.

Step 2. Let

(5.525) $\displaystyle g_{2N_{0},1}(\theta)=$
$\displaystyle\wedge^{d}M_{2N_{0},E}(\theta+(N_{1}-2^{2}N_{0})d\alpha)$
(5.526) $\displaystyle g_{2N_{0},2}(\theta)=$
$\displaystyle\wedge^{d}M_{2N_{0},E}(\theta+(N_{1}-2N_{0})d\alpha)$ (5.527)
$\displaystyle g_{2N_{0},3}(\theta)=$
$\displaystyle\wedge^{d}M_{2N_{0},E}(\theta+N_{1}d\alpha+\kappa)$ (5.528)
$\displaystyle g_{2N_{0},4}(\theta)=$
$\displaystyle\wedge^{d}M_{2N_{0},E}(\theta+(N_{1}+2N_{0})d\alpha+\kappa)$

Note that

$g_{2N_{0},3}(\theta)g_{2N_{0},2}(\theta)=g_{N_{0},4}(\theta)\cdots
g_{N_{0},1}(\theta).$

Hence by Lemma 5.6, for $\theta\in\mathcal{G}_{N_{0}}^{(1)}$,

(5.529) $\displaystyle\log\|g_{2N_{0},3}(\theta)g_{2N_{0},2}(\theta)\|\geq
4N_{0}(1-3\varepsilon)\hat{L}^{d}-4\tilde{C}_{0}\tilde{\kappa}_{N_{0}}\cdot\tilde{\varepsilon}_{N_{0}}^{-2}.$

We further restrict $\theta$ to the following subset of
$\mathcal{G}_{N_{0}}^{(1)}$:

(5.530) $\displaystyle\theta\in\mathcal{G}_{2N_{0}}^{(1)}=$
$\displaystyle\mathcal{G}_{N_{0}}^{(1)}\cap(\mathcal{G}_{2N_{0}}-(N_{1}-2^{2}N_{0})d\alpha)\cap(\mathcal{G}_{2N_{0}}-(N_{1}+2N_{0})d\alpha-\kappa)\cap$
(5.531)
$\displaystyle\qquad\cap(\mathcal{G}_{4N_{0}}-(N_{1}-2^{2}N_{0})d\alpha)\cap(\mathcal{G}_{4N_{0}}-N_{1}d\alpha-\kappa).$

Our goal of Step 2 is to prove the following:

######  Lemma 5.7.

For $\theta\in\mathcal{G}_{2N_{0}}^{(1)}$,

(5.532) $\displaystyle\log\|g_{2N_{0},4}(\theta)\cdots
g_{2N_{0},1}(\theta)\|\geq
2^{3}N_{0}(1-3\varepsilon)\hat{L}^{d}-4\tilde{C}_{0}\sum_{\ell=0}^{1}\tilde{\kappa}_{2^{\ell}N_{0}}\cdot\tilde{\varepsilon}_{2^{\ell}N_{0}}^{-2},$

where

(5.533)
$\displaystyle\tilde{\kappa}_{2N_{0}}=e^{-2^{2}N_{0}(1-\varepsilon)\hat{L}_{d}}\cdot
e^{2^{3}N_{0}\varepsilon\hat{L}^{d-1}}\,\text{ and
}\tilde{\varepsilon}_{2N_{0}}=e^{-2^{4}N_{0}\varepsilon\hat{L}^{d}}\cdot
e^{-4\tilde{C}_{0}\tilde{\kappa}_{N_{0}}\cdot\tilde{\varepsilon}_{N_{0}}^{-2}}.$

Furthermore,

(5.534) $\displaystyle\mathrm{mes}(G_{2N_{0}}^{(1)})\geq
1-4e^{-c_{1}\varepsilon^{2}N_{0}\hat{L}^{d}}-5e^{-2c_{1}\varepsilon^{2}N_{0}\hat{L}^{d}}-2e^{-4c_{1}\varepsilon^{2}N_{0}\hat{L}^{d}}.$

###### Proof.

Since $\theta\in\mathcal{G}_{2N_{0}}^{(1)}$, by Lemma 2.2 one has for
$j=1,2,3,4$ that

(5.535) $\displaystyle
e^{2N_{0}(1-\varepsilon)\hat{L}^{d}}\leq\sigma_{1}(g_{2N_{0},j}(\theta))\leq
e^{2N_{0}(1-\varepsilon)\hat{L}^{d}},$

and for $j=1,3$ that

(5.536) $\displaystyle\|g_{2N_{0},j+1}(\theta)g_{2N_{0},j}(\theta)\|\geq
e^{4N_{0}(1-\varepsilon)\hat{L}^{d}}.$

In analogy to (5.518), we have for $j=1,2,3,4$ that

(5.537) $\displaystyle\sigma_{2}(g_{2N_{0},j}(\theta))\leq
e^{2N_{0}(1-\varepsilon)(\hat{L}^{d-1}+\hat{L}_{d+1})}\cdot
e^{8N_{0}\varepsilon\hat{L}^{d-1}}.$

We apply the avalanche principle (AP) Theorem 2.4 to
$g_{2N_{0},4}(\theta),...,g_{2N_{0},1}(\theta)$. The hypotheses of the AP hold
since: for $j=1,2,3,4$, by (5.535) and (5.537),

(5.538)
$\displaystyle\frac{\sigma_{1}(g_{2N_{0},j}(\theta))}{\sigma_{2}(g_{2N_{0},j}(\theta))}\geq
e^{2^{2}N_{0}(1-\varepsilon)\hat{L}_{d}}e^{-2^{3}N_{0}\varepsilon\hat{L}^{d-1}}=:\tilde{\kappa}_{2N_{0}}^{-1},$

and for $j=1,2,3$, by further taking (5.536) and (5.529) into account,

(5.539)
$\displaystyle\frac{\|g_{2N_{0},j+1}(\theta)g_{2N_{0},j}(\theta)\|}{\sigma_{1}(g_{2N_{0},j+1}(\theta))\cdot\sigma_{1}(g_{2N_{0},j}(\theta))}\geq
e^{-2^{4}N_{0}\varepsilon\hat{L}^{d}}e^{-4\tilde{C}_{0}\tilde{\kappa}_{N_{0}}\cdot\tilde{\varepsilon}_{N_{0}}^{-2}}=:\tilde{\varepsilon}_{2N_{0}}.$

Clearly,

(5.540)
$\displaystyle\tilde{\kappa}_{2N_{0}}\cdot\tilde{\varepsilon}_{2N_{0}}^{-2}\leq
e^{-2^{2}N_{0}(1-\varepsilon)\hat{L}_{d}}e^{(2^{5}+2^{3})N_{0}\varepsilon\hat{L}^{d}}e^{8\tilde{C}_{0}c_{0}}<c_{0},$

where we used (5.521) to bound
$\tilde{\kappa}_{N_{0}}\cdot\tilde{\varepsilon}_{N_{0}}^{-2}<c_{0}$.
Therefore, Theorem 2.4 implies that

(5.541) $\displaystyle\log\|g_{2N_{0},4}(\theta)\cdots
g_{2N_{0},1}(\theta)\|\geq$
$\displaystyle\sum_{j=1}^{3}\log\|g_{2N_{0},j+1}(\theta)g_{2N_{0},j}(\theta)\|-\sum_{j=2}^{3}\log\|g_{2N_{0},j}(\theta)\|-4\tilde{C}_{0}\tilde{\kappa}_{2N_{0}}\cdot\tilde{\varepsilon}_{2N_{0}}^{-2}$
(5.542) $\displaystyle\geq$ $\displaystyle
2^{3}N_{0}(1-3\varepsilon)\hat{L}^{d}-4\tilde{C}_{0}\sum_{\ell=0}^{1}\tilde{\kappa}_{2^{\ell}N_{0}}\cdot\tilde{\varepsilon}_{2^{\ell}N_{0}}^{-2}.$

Regarding the measure estimate, Lemma 2.2 implies that

(5.543) $\displaystyle\mathrm{mes}(\mathcal{G}_{2N_{0}}^{(1)})\geq$
$\displaystyle\mathrm{mes}(\mathcal{G}_{N_{0}}^{(1)})-2e^{-2c_{1}\varepsilon^{2}N_{0}}-2e^{-2c_{2}\varepsilon^{2}N_{0}}$
(5.544) $\displaystyle\geq$ $\displaystyle
1-4e^{-c_{1}\varepsilon^{2}N_{0}\hat{L}^{d}}-5e^{-2c_{1}\varepsilon^{2}N_{0}\hat{L}^{d}}-2e^{-4c_{1}\varepsilon^{2}N_{0}\hat{L}^{d}}.$

This is the claimed result. ∎

Next, we carry out the following inductive procedure. For any $\ell\geq 0$,
let

(5.545) $\displaystyle g_{2^{\ell}N_{0},1}(\theta)=$
$\displaystyle\wedge^{d}M_{2^{\ell}N_{0},E}(\theta+(N_{1}-2^{\ell+1}N_{0})d\alpha)$
(5.546) $\displaystyle g_{2^{\ell}N_{0},2}(\theta)=$
$\displaystyle\wedge^{d}M_{2^{\ell}N_{0},E}(\theta+(N_{1}-2^{\ell}N_{0})d\alpha)$
(5.547) $\displaystyle g_{2^{\ell}N_{0},3}(\theta)=$
$\displaystyle\wedge^{d}M_{2^{\ell}N_{0},E}(\theta+N_{1}d\alpha+\kappa)$
(5.548) $\displaystyle g_{2^{\ell}N_{0},4}(\theta)=$
$\displaystyle\wedge^{d}M_{2^{\ell}N_{0},E}(\theta+(N_{1}+2^{\ell}N_{0})d\alpha+\kappa)$

Note this is consistent with the definitions of $g_{N_{0},j}(\theta)$ and
$g_{2N_{0},j}(\theta)$ in (5.503) and (5.525). For any $\ell\geq 0$, let

(5.549) $\displaystyle\tilde{\kappa}_{2^{\ell}N_{0}}=$ $\displaystyle
e^{-2^{\ell+1}N_{0}(1-\varepsilon)\hat{L}_{d}}\cdot
e^{2^{\ell+2}N_{0}\varepsilon\hat{L}^{d-1}},$

and define $\tilde{\varepsilon}_{2^{\ell}N_{0}}$ inductively;
$\tilde{\varepsilon}_{N_{0}}$ is defined in (5.508) and for $\ell\geq 1$,

(5.550) $\displaystyle\tilde{\varepsilon}_{2^{\ell}N_{0}}=$ $\displaystyle
e^{-2^{\ell+3}N_{0}\varepsilon\hat{L}^{d}}\cdot
e^{-4\tilde{C}_{0}\sum_{r=0}^{\ell-1}\tilde{\kappa}_{2^{r}N_{0}}\cdot\tilde{\varepsilon}_{2^{r}N_{0}}^{-2}}.$

Note that these definitions are consistent with those in (5.508) and (5.533)
for $\ell=0,1$. Clearly, for any $n\geq 1$,

(5.551)
$\displaystyle\sum_{\ell=0}^{n}\tilde{\kappa}_{2^{\ell}N_{0}}\cdot\tilde{\varepsilon}_{2^{\ell}N_{0}}^{-2}\leq$
$\displaystyle\sum_{\ell=0}^{n}e^{-2^{\ell+1}N_{0}(1-\varepsilon)\hat{L}_{d}}\cdot
e^{(2^{\ell+4}+2^{\ell+2})N_{0}\varepsilon\hat{L}^{d}}\cdot
e^{8\tilde{C}_{0}\sum_{r=0}^{n-1}\tilde{\kappa}_{2^{r}N_{0}}\cdot\tilde{\varepsilon}_{2^{r}N_{0}}^{-2}}$
(5.552) $\displaystyle\leq$
$\displaystyle\sum_{\ell=0}^{n}e^{-2^{\ell+1}N_{0}(1-\varepsilon)\hat{L}_{d}}\cdot
e^{(2^{\ell+4}+2^{\ell+2})N_{0}\varepsilon\hat{L}^{d}}\cdot
e^{8\tilde{C}_{0}c_{0}}<c_{0},$

by (5.479), (5.521), and induction.

Step $m$ for $m=3,...,k$.

######  Lemma 5.8.

Suppose $\mathcal{G}_{2^{m-2}N_{0}}^{(1)}$ is defined and satisfies the
following two conditions:

  1. (1)

(5.553)
$\displaystyle\mathcal{G}_{2^{m-2}N_{0}}^{(1)}\subset(\mathcal{G}_{2^{m-1}N_{0}}-(N_{1}-2^{m-1}N_{0})d\alpha)\cap(\mathcal{G}_{2^{m-1}N_{0}}-N_{1}d\alpha-\kappa)$

  2. (2)

(5.554) $\displaystyle\mathrm{mes}(G_{2^{m-2}N_{0}}^{(1)})\geq
1-5\sum_{\ell=0}^{m-2}e^{-2^{\ell}c_{1}\varepsilon^{2}N_{0}\hat{L}^{d}}-2e^{-2^{m-1}c_{1}\varepsilon^{2}N_{0}\hat{L}^{d}}.$

Assume further that for any $\theta\in\mathcal{G}_{2^{m-2}N_{0}}^{(1)}$ the
following holds:

(5.555) $\displaystyle\log\|g_{2^{m-2}N_{0},4}(\theta)\cdots
g_{2^{m-2}N_{0},1}(\theta)\|\geq
2^{m}N_{0}(1-3\varepsilon)\hat{L}^{d}-4\tilde{C}_{0}\sum_{\ell=0}^{m-2}\tilde{\kappa}_{2^{\ell}N_{0}}\cdot\tilde{\varepsilon}_{2^{\ell}N_{0}}^{-2}.$

Let

(5.556) $\displaystyle\mathcal{G}_{2^{m-1}N_{0}}^{(1)}:=$
$\displaystyle\mathcal{G}_{2^{m-2}N_{0}}^{(1)}\cap(\mathcal{G}_{2^{m-1}N_{0}}-(N_{1}-2^{m}N_{0})d\alpha)\cap(\mathcal{G}_{2^{m-1}N_{0}}-(N_{1}+2^{m-1}N_{0})d\alpha-\kappa)\cap$
(5.557)
$\displaystyle\qquad\cap(\mathcal{G}_{2^{m}N_{0}}-(N_{1}-2^{m}N_{0})d\alpha)\cap(\mathcal{G}_{2^{m}N_{0}}-N_{1}d\alpha-\kappa).$

Then for any $\theta\in\mathcal{G}_{2^{m-1}N_{0}}^{(1)}$, one has

(5.558) $\displaystyle\log\|g_{2^{m-1}N_{0},4}(\theta)\cdots
g_{2^{m-1}N_{0},1}(\theta)\|\geq
2^{m+1}N_{0}(1-3\varepsilon)\hat{L}^{d}-4\tilde{C}_{0}\sum_{\ell=0}^{m-1}\tilde{\kappa}_{2^{\ell}N_{0}}\cdot\tilde{\varepsilon}_{2^{\ell}N_{0}}^{-2}.$

Furthermore, $\mathcal{G}_{2^{m-1}N_{0}}^{(1)}$ satisfies the following two
conditions:

  1. (1)

(5.559)
$\displaystyle\mathcal{G}_{2^{m-1}N_{0}}^{(1)}\subset(\mathcal{G}_{2^{m}N_{0}}-(N_{1}-2^{m}N_{0})d\alpha)\cap(\mathcal{G}_{2^{m}N_{0}}-N_{1}d\alpha-\kappa)$

  2. (2)

(5.560) $\displaystyle\mathrm{mes}(G_{2^{m-1}N_{0}}^{(1)})\geq
1-5\sum_{\ell=0}^{m-1}e^{-2^{\ell}c_{1}\varepsilon^{2}N_{0}\hat{L}^{d}}-2e^{-2^{m}c_{1}\varepsilon^{2}N_{0}\hat{L}^{d}}.$

###### Proof.

For any $\theta\in\mathcal{G}_{2^{m-1}N_{0}}^{(1)}$, Lemma 2.2 implies for
$j=1,2,3,4$ that

(5.561) $\displaystyle
e^{2^{m-1}N_{0}(1-\varepsilon)\hat{L}^{d}}\leq\sigma_{1}(g_{2^{m-1}N_{0},j}(\theta))\leq
e^{2^{m-1}N_{0}(1+\varepsilon)\hat{L}^{d}},$

and for $j=1,3$ that

(5.562)
$\displaystyle\|g_{2^{m-1}N_{0},j+1}(\theta)g_{2^{m-1}N_{0},j}(\theta)\|\geq
e^{2^{m}N_{0}(1-\varepsilon)\hat{L}^{d}}.$

By assumption (5.555), one has

(5.563)
$\displaystyle\log\|g_{2^{m-1}N_{0},3}(\theta)g_{2^{m-1}N_{0},2}(\theta)\|=$
$\displaystyle\log\|g_{2^{m-2}N_{0},4}(\theta)\cdots
g_{2^{m-2}N_{0},1}(\theta)\|$ (5.564) $\displaystyle\geq$ $\displaystyle
2^{m}N_{0}(1-3\varepsilon)\hat{L}^{d}-4\tilde{C}_{0}\sum_{\ell=0}^{m-2}\tilde{\kappa}_{2^{\ell}N_{0}}\cdot\tilde{\varepsilon}_{2^{\ell}N_{0}}^{-2}.$

In analogy to (5.518), we infer for $j=1,2,3,4$ that

(5.565) $\displaystyle\sigma_{2}(g_{2^{m-1}N_{0},j}(\theta))\leq
e^{2^{m-1}N_{0}(1-\varepsilon)(\hat{L}^{d-1}+\hat{L}_{d+1})}\cdot
e^{2^{m+1}N_{0}\varepsilon\hat{L}^{d-1}}.$

In order to apply Theorem 2.4 to
$g_{2^{m-1}N_{0},4}(\theta),...,g_{2^{m-1}N_{0},1}(\theta)$, we first check
that the assumptions are satisfied. By (5.561) and (5.565), one has for
$j=1,2,3,4$ that

(5.566)
$\displaystyle\frac{\sigma_{1}(g_{2^{m-1}N_{0},j}(\theta))}{\sigma_{2}(g_{2^{m-1}N_{0},j}(\theta))}\geq
e^{2^{m}N_{0}(1-\varepsilon)\hat{L}_{d}}\cdot
e^{-2^{m+1}N_{0}\varepsilon\hat{L}^{d-1}}=\tilde{\kappa}_{2^{m-1}N_{0}}^{-1}.$

By (5.561), (5.562) and (5.563) we have that for $j=1,2,3$,

(5.567)
$\displaystyle\frac{\|g_{2^{m-1}N_{0},j+1}(\theta)g_{2^{m-1}N_{0},j}(\theta)\|}{\sigma_{1}(g_{2^{m-1}N_{0},j+1}(\theta))\cdot\sigma_{1}(g_{2^{m-1}N_{0},j}(\theta))}\geq
e^{-2^{m+2}N_{0}\varepsilon\hat{L}^{d}}\cdot
e^{-4\tilde{C}_{0}\sum_{\ell=0}^{m-2}\tilde{\kappa}_{2^{\ell}N_{0}}\cdot\tilde{\varepsilon}_{2^{\ell}N_{0}}^{-2}}=\tilde{\varepsilon}_{2^{m-1}N_{0}}.$

Clearly,

(5.568)
$\displaystyle\tilde{\kappa}_{2^{m-1}N_{0}}\cdot\tilde{\varepsilon}_{2^{m-1}N_{0}}^{-2}\leq$
$\displaystyle e^{-2^{m}N_{0}(1-\varepsilon)\hat{L}_{d}}\cdot
e^{(2^{m+3}+2^{m+1})N_{0}\varepsilon\hat{L}^{d}}\cdot
e^{8\tilde{C}_{0}\sum_{\ell=0}^{m-2}\tilde{\kappa}_{2^{\ell}N_{0}}\cdot\tilde{\varepsilon}_{2^{\ell}N_{0}}^{-2}}$
(5.569) $\displaystyle\leq$ $\displaystyle
e^{-2^{m}N_{0}(1-\varepsilon)\hat{L}_{d}}\cdot
e^{(2^{m+3}+2^{m+1})N_{0}\varepsilon\hat{L}^{d}}\cdot
e^{8\tilde{C}_{0}c_{0}}<c_{0},$

where we used (5.551). Therefore, Theorem 2.4 implies that

(5.570) $\displaystyle\log\|g_{2^{m-1}N_{0},4}(\theta)\cdots
g_{2^{m-1}N_{0},1}(\theta)\|\geq$
$\displaystyle\sum_{j=1}^{3}\log\|g_{2^{m-1}N_{0},j+1}(\theta)g_{2^{m-1}N_{0},j}(\theta)\|-\sum_{j=2}^{3}\log\|g_{2^{m-1}N_{0},j}(\theta)\|$
(5.571)
$\displaystyle\qquad\qquad-4\tilde{C}_{0}\tilde{\kappa}_{2^{m-1}N_{0}}\cdot\tilde{\varepsilon}_{2^{m-1}N_{0}}^{-2}$
(5.572) $\displaystyle\geq$ $\displaystyle
2^{m+1}N_{0}(1-3\varepsilon)\hat{L}^{d}-4\tilde{C}_{0}\sum_{\ell=0}^{m-1}\tilde{\kappa}_{2^{\ell}N_{0}}\cdot\tilde{\varepsilon}_{2^{\ell}N_{0}}^{-2}.$

It suffices to prove the measure estimate for
$\mathcal{G}_{2^{m-1}N_{0}}^{(1)}$. Clearly, Lemma 2.2 implies that

(5.573)
$\displaystyle\mathrm{mes}(\mathcal{G}_{2^{m-1}N_{0}}^{(1)})\geq\mathrm{mes}(\mathcal{G}_{2^{m-2}N_{0}}^{(1)})-2e^{-2^{m-1}c_{1}\varepsilon^{2}N_{0}\hat{L}^{d}}-2e^{-2^{m}c_{1}\varepsilon^{2}N_{0}\hat{L}^{d}}$
(5.574) $\displaystyle\geq