# On the spectra of separable 2D almost Mathieu operators

Alberto Takase  Department of Mathematics, University of California, Irvine,
CA 92697, USA<EMAIL_ADDRESS>
###### Abstract.

We consider separable 2D discrete Schrödinger operators generated by 1D almost
Mathieu operators. For fixed Diophantine frequencies we prove that for
sufficiently small couplings the spectrum must be an interval. This
complements a result by J. Bourgain establishing that for fixed couplings the
spectrum has gaps for some (positive measure) Diophantine frequencies. Our
result generalizes to separable multidimensional discrete Schrödinger
operators generated by 1D quasiperiodic operators whose potential is analytic
and whose frequency is Diophantine. The proof is based on the study of the
thickness of the spectrum of the almost Mathieu operator, and utilizes the
Newhouse Gap Lemma on sums of Cantor sets.

The project was supported in part by the NSF grant DMS–1855541 (PI - A.
Gorodetski).

##  1\. Introduction

The almost Mathieu operator is the discrete Schrödinger operator
$H_{\lambda,\alpha,\omega}$ on $\ell^{2}(\mathbb{Z})$ defined by

$\textstyle[H_{\lambda,\alpha,\omega}\psi](n)=\psi(n+1)+\psi(n-1)+2\lambda\cos(2\pi(n\alpha+\omega))\psi(n)$

for every $\psi\in\ell^{2}(\mathbb{Z}),n\in\mathbb{Z}$. Here
$\lambda\in\mathbb{R}$ is the coupling,
$\alpha\in\mathbb{R}/\mathbb{Z}\eqqcolon\mathbb{T}$ is the frequency, and
$\omega\in\mathbb{T}$ is the phase. The operator has its origins in solid-
state physics and the study of electrons. The operator also has a connection
to the quantum Hall effect—a Nobel-prize-worthy discovery by K. von Klitzing,
G. Dorda, and M. Pepper in 1980 [27]. The connection was made by D. J.
Thouless, M. Kohmoto, M. P. Nightingale, and M. den Nijs in 1982 [37]. For a
discussion of discrete Schrödinger operators and the almost Mathieu operator
see the 2019 preprint by S. Jitomirskaya [24]. For more discussions see [23,
14, 30, 22, 28, 9, 36, 6].

P. G. Harper in 1955 [19, 20] under the tutelage of R. E. Peierls established
that electrons in a cubic lattice and under a magnetic field have nondiscrete
nonevenly-spaced broadened energy values. Harper used a tight-binding
approximation now called the Harper model. The Schrödinger operator governing
the spectrum of the Harper model is the almost Mathieu operator
$H_{\lambda,\alpha,\omega}$ with $\lambda=1$. Physicists sought to better
understand the topological structure of the spectrum
$\Sigma_{\lambda,\alpha,\omega}$ of $H_{\lambda,\alpha,\omega}$. M. Azbel in
1964 [5] conjectured and D. Hofstadter in 1976 [21] computationally supported
that $\Sigma_{\lambda,\alpha,\omega}$ has the characteristic of being either
band-like for rational $\alpha$ or fractal-like for irrational $\alpha$.
Decades later A. Avila and S. Jitomirskaya in 2009 [4] made the final
contribution to the resolution of the Ten Martini Problem which sought to
confirm the conjectured topological structure of
$\Sigma_{\lambda,\alpha,\omega}$. If
$\alpha=\frac{p}{q}\in\mathbb{T}\cap\mathbb{Q}$, then
$\Sigma_{\lambda,\alpha,\omega}$ is a disjoint union of at most $q$-many
compact intervals. If $\alpha\in\mathbb{T}\setminus\mathbb{Q}$, then
$\Sigma_{\lambda,\alpha,\omega}$ is a Cantor set. Also,
$\Sigma_{\lambda,\alpha,\omega}$ is independent of the phase $\omega$ when the
frequency $\alpha$ is irrational. This last fact can be found in the 1982
review paper by B. Simon [36] (where the name of the almost Mathieu operator
was introduced) and applies to operators with quasiperiodic potentials; the
definition of quasiperiodic potentials can be found within subsection 2.3. In
general, phase independence of the spectrum for operators with quasiperiodic
potentials follows from the fact that the underlying topological dynamical
system $(\mathbb{T}^{b},\mathbb{Z})$ is minimal [14].

Let $d\geq 2$ be a positive integer. We consider the $d$-dimensional analog
operator $\vphantom{H}\smash{\widehat{H}}$ generated by almost Mathieu
operators. Specifically, $\vphantom{H}\smash{\widehat{H}}$ is the discrete
Schrödinger operator on $\ell^{2}(\mathbb{Z}^{d})$ defined by

$\textstyle[\vphantom{H}\smash{\widehat{H}}\psi](n)=\big{(}\sum_{m\in\left\\{e_{1},\ldots,e_{d}\right\\}}\psi(n+m)+\psi(n-m)\big{)}+\big{(}\sum_{k\in\left\\{1,\ldots,d\right\\}}2\lambda_{k}\cos(2\pi(n_{k}\alpha_{k}+\omega_{k}))\big{)}\psi(n)$

for every $\psi\in\ell^{2}(\mathbb{Z}^{d}),n\in\mathbb{Z}^{d}$;
$\left\\{e_{1},\ldots,e_{d}\right\\}$ is the standard basis. The operator
$\vphantom{H}\smash{\widehat{H}}$ and the spectrum
$\vphantom{\Sigma}\smash{\widehat{\Sigma}}=\Sigma_{\lambda_{1},\alpha_{1},\omega_{1}}+\cdots+\Sigma_{\lambda_{d},\alpha_{d},\omega_{d}}$
are the main objects of study in this paper. The theory of the almost Mathieu
operator provides insight into the theory of
$\vphantom{H}\smash{\widehat{H}}$. Consider the following theorem established
by A. Avila and D. Damanik in 2008 [3]: If $\lambda\neq 1$ and
$\alpha\in\mathbb{T}\setminus\mathbb{Q}$, then the integrated density of
states of the almost Mathieu operator is absolutely continuous. Along with a
theorem by Steinhaus, which states that the sum of two sets with positive
Lebesgue-measure contains an open interval, we immediately obtain the
following proposition.

######  Proposition 1.1.

Assume at least two among $\lambda_{1},\ldots,\lambda_{d}$ are not equal to
$1$ and $\alpha_{1},\ldots,\alpha_{d}\in\mathbb{T}\setminus\mathbb{Q}$. Then
$\vphantom{\Sigma}\smash{\widehat{\Sigma}}$ has a dense interior.

More can be obtained in the small coupling and Diophantine frequency regime.
This is the context of the main theorem in this paper.

######  Theorem 1.2.

Assume $\alpha_{1},\ldots,\alpha_{d}$ are irrational and satisfy a Diophantine
condition. There exists
$\varepsilon=\varepsilon(\alpha_{1},\ldots,\alpha_{d})>0$ such that if
$0<|\lambda_{1}|,\ldots,|\lambda_{d}|<\varepsilon$, then
$\vphantom{\Sigma}\smash{\widehat{\Sigma}}$ is an interval.

We mention for comparison a theorem by J. Bourgain in 2002 [7].

######  Theorem 1.3 (J. Bourgain [7]).

Assume $\lambda_{1},\ldots,\lambda_{d}=\lambda>0$. There exist positive Haar-
measure sets
$\ooalign{$A_{\lambda}$\cr$\phantom{A}{}^{(1)}$\cr$\phantom{A^{(1)}_{\lambda}}$},\ldots,\ooalign{$A_{\lambda}$\cr$\phantom{A}{}^{(d)}$\cr$\phantom{A^{(d)}_{\lambda}}$}\subseteq\mathbb{T}$
such that if
$\alpha_{1}\in\ooalign{$A_{\lambda}$\cr$\phantom{A}{}^{(1)}$\cr$\phantom{A^{(1)}_{\lambda}}$},\ldots,\alpha_{d}\in\ooalign{$A_{\lambda}$\cr$\phantom{A}{}^{(d)}$\cr$\phantom{A^{(d)}_{\lambda}}$}$,
then $\vphantom{\Sigma}\smash{\widehat{\Sigma}}$ has gaps. Furthermore,
$A_{\lambda}$ $\phantom{A}{}^{(k)}$ can be chosen to be a subset of the
Diophantine frequencies and lie within $[0,\varepsilon]$ for any
$\varepsilon>0$.

We say a few words on the proof of Theorem 1.2. Observe
$\vphantom{\Sigma}\smash{\widehat{\Sigma}}$ is the sum of Cantor spectra
$\Sigma_{1}+\cdots+\Sigma_{d}$, where henceforth
$\Sigma_{k}\coloneqq\Sigma_{\lambda_{k},\alpha_{k},\omega_{k}}$. Indeed, the
potential of $\vphantom{H}\smash{\widehat{H}}$ is separable. In general, the
spectra of separable multidimensional operators are sums of the spectra of 1D
operators. This fact can be found within [12]. See also another proof
involving the convolution of density of states measures within [13]. The
Newhouse Gap Lemma may be utilized to guarantee that
$\Sigma_{1}+\cdots+\Sigma_{d}$ is an interval thereby establishing Theorem
1.2. The notion of thickness, which is a quantitative characterization of
nonempty compact subsets $K$ of $\mathbb{R}$ often denoted $\tau(K)$, was
utilized by S. Newhouse in the 1970s [31, 32, 33] to prove the namesake
Newhouse Gap Lemma. See also for a short proof section 4.2 on page 63 within
[34]. The definition of thickness can be found within subsection 2.6. We
abridge the Newhouse Gap Lemma: Let $K_{1}$ and $K_{2}$ be nonempty compact
subsets of $\mathbb{R}$. Assume the maximal-gap-lengths of $K_{1}$ and $K_{2}$
are sufficiently small relative to the diameters of $K_{1}$ and $K_{2}$, and
$1\leq\tau(K_{1})\cdot\tau(K_{2})$. Then $K_{1}+K_{2}$ is an interval. S.
Astels in 1999 [1, 2] generalized the Newhouse Gap Lemma to obtain the
following; see Theorem 2.7 for the unabridged version.

######  Theorem 1.4 (S. Astels [1, 2]).

Let $K_{1},\ldots,K_{d}$ ($d\geq 2$) be nonempty compact subsets of
$\mathbb{R}$. Assume the maximal-gap-lengths of $K_{1},\ldots,K_{d}$ are
sufficiently small relative to the diameters of $K_{1},\ldots,K_{d}$, and
$1\leq\frac{\tau(K_{1})}{\tau(K_{1})+1}+\cdots+\frac{\tau(K_{d})}{\tau(K_{d})+1}$.
Then $K_{1}+\cdots+K_{d}$ is an interval.

Because $\lambda_{k}$ is small, one can think of
$H_{\lambda_{k},\alpha_{k},\omega_{k}}$ as a perturbation of the discrete
Laplacian whose spectrum is the interval $[-2,2]$. Specifically, $\Sigma_{k}$
converges to $[-2,2]$ in the Hausdorff metric as $\lambda_{k}\to 0$. Also, the
diameter of $\Sigma_{k}$ converges to the diameter of $[-2,2]$ as
$\lambda_{k}\to 0$. Indeed,
$|\mathrm{diam}\,\sigma(A)-\mathrm{diam}\,\sigma(B)|\leq
2\,\mathrm{dist}_{\mathsf{Haus}}(\sigma(A),\sigma(B))\leq 2\left\lVert
A-B\right\rVert$ for bounded self-adjoint operators $A$ and $B$. Therefore
$\Sigma_{k}$ has a small maximal-gap-length relative to the diameters of
$\Sigma_{1},\ldots,\Sigma_{d}$. As a result, to utilize the Newhouse Gap Lemma
it is enough to establish that the thickness of the spectrum of the almost
Mathieu operator approaches infinity as the coupling approaches zero.

######  Theorem 1.5.

The spectrum of the almost Mathieu operator $H_{\lambda,\alpha,\omega}$ with
irrational frequency fixed and assumed to satisfy a Diophantine condition

$\textstyle\alpha\in\bigcup_{\begin{subarray}{c}c>0\\\
t>1\end{subarray}}\bigcap_{\frac{p}{q}\in\mathbb{Q}}\left\\{x\in\mathbb{R}:|qx-p|\geq\frac{c}{|q|^{t-1}}\right\\}\mathrlap{{}\eqqcolon\mathsf{DC}}$

has thickness approaching infinity as the coupling $\lambda$ approaches zero
i.e.

$\left(\forall\alpha\in\mathsf{DC}\right)[\textstyle\lim_{\lambda\to
0}\tau(\Sigma_{\lambda,\alpha,\omega})=+\infty].$

Theorem 1.5 extends to 1D quasiperiodic operators whose potential is analytic
and whose frequency is Diophantine; combine Theorem 2.4 and Theorem 2.5 and
Lemma 4.1. As a result, Theorem 1.2 extends to separable multidimensional
discrete Schrödinger operators generated by 1D quasiperiodic operators whose
potential is analytic and whose frequency is Diophantine.

######  Theorem 1.6.

For fixed Diophantine frequencies $\alpha_{1},\ldots,\alpha_{d}$, the spectrum
is an interval for separable $d$-dimensional discrete Schrödinger operators
generated by 1D $\alpha_{k}$-quasiperiodic analytic operators
$H_{k}=\Delta+V_{k}$ with $\alpha_{k}$-dependent sufficiently small norms
$\lVert V_{k}\rVert$.

Theorem 1.6 is necessarily perturbative due to Theorem 1.3. Specifically, the
smallness of the norm depends on the Diophantine frequency. We conclude this
section with a few words on related mathematical results and questions. The
definitions of limit-periodic and almost-periodic potential can be found
within [15] and [36], respectively. It suffices to say that the collection of
all almost-periodic potentials is a broad class of potentials which contains
the periodic and limit-periodic and quasiperiodic potentials. The definitions
of box-counting and Hausdorff dimension can be found within [16]. A Cantorval
is a nonempty compact subset $C$ of $\mathbb{R}$ such that $C$ has no isolated
connected components and $C$ has a dense interior; we mention for comparison
that a Cantor set is a nonempty compact subset $C$ of $\mathbb{R}$ such that
$C$ has no isolated points and $C$ has no interior points.

  1. (a)

We establish a single-interval-characterization for the spectra of separable
multidimensional discrete Schrödinger operators generated by 1D quasiperiodic
analytic operators $H_{k}=\Delta+V_{k}$ with Diophantine-frequency-dependent
sufficiently small norms $\lVert V_{k}\rVert$. The same characterization can
be said about periodic $V_{k}$ [18] and cannot be said about limit-periodic
$V_{k}$ [15].

  2. (b)

R. Han and S. Jitomirskaya [18] proved that periodic (not necessarily
separable) multidimensional discrete Schrödinger operators have interval
spectra when the norm of the potential is small and at least one period is
odd. This is the discrete analog of the L. Parnovski [35] resolution of the
continuous Bethe–Sommerfeld Conjecture. Moreover, the resolution was extended
to quasiperiodic (not necessarily separable) multidimensional continuous
Schrödinger operators for almost-all frequencies [26]. This paragraph is the
only place where continuous Schrödinger operators are mentioned, and it is
done so to motivate the following question. Can one extend the [18, 35, 26]
results to quasiperiodic (not necessarily separable) multidimensional discrete
Schrödinger operators for almost-all frequencies? Specifically, can one remove
the separability condition from Theorem 1.6? Moreover, can one remove the
Diophantine condition from Theorem 1.6? For example, can the frequencies
satisfy a Liouvillian condition? In a sense, this means that the irrational
frequencies are well approximated by rational numbers.

  3. (c)

D. Damanik, J. Fillman, and A. Gorodetski [15] proved that there exists a
dense subset $\mathcal{B}$ of 1D limit-periodic potentials such that
$\mathcal{B}$-type operators have Cantor spectra with zero (lower) box-
counting dimension. Furthermore, separable multidimensional discrete
Schrödinger operators generated by $\mathcal{B}$-type operators have Cantor
spectra with zero (lower) box-counting dimension. Is it true that the spectra
of separable multidimensional discrete Schrödinger operators generated by 1D
almost-periodic operators is either a finite union of disjoint intervals or a
Cantor set or a Cantorval? In view of Proposition 1.1, does there exist
$\lambda_{1},\ldots,\lambda_{d}$ with at least two not equal to $1$ and
$\alpha_{1},\ldots,\alpha_{d}\in\mathbb{T}\setminus\mathbb{Q}$ such that
$\vphantom{\Sigma}\smash{\widehat{\Sigma}}$ is a Cantorval?

  4. (d)

D. Damanik and A. Gorodetski [12] proved that 1D Fibonacci Hamiltonians have
Cantor spectra $\Sigma_{\lambda}$ with Hausdorff dimension strictly between
zero and one and with thickness strictly greater than zero, and
$\lim_{\lambda\to 0}\mathrm{dim}_{\mathsf{Haus}}(\Sigma_{\lambda})=1$ and
$\lim_{\lambda\to 0}\tau(\Sigma_{\lambda})=+\infty$. Consequently, they
establish a single-interval-characterization for the spectra of separable
multidimensional discrete Schrödinger operators generated by 1D Fibonacci
Hamiltonians with sufficiently small couplings.

  5. (e)

For fixed $s\in\mathbb{R}\setminus\left\\{-1,0,1\right\\}$ and for fixed
Diophantine frequency in the 2-torus M. Goldstein, W. Schlag, and M. Voda [17]
proved that for sufficiently large couplings the 1D operators with 2-frequency
quasiperiodic potential
$V_{\lambda,s,\alpha}:\mathbb{Z}\to\mathbb{R}:n\mapsto\lambda(\cos(2\pi
n\alpha_{1})+s\cos(2\pi n\alpha_{2}))$ must have an interval spectrum. This
complements a result by J. Bourgain [8] establishing that for fixed small
coupling the 1D operator with potential $V_{\lambda,1,\alpha}$ has gaps in its
spectrum for some (positive measure) Diophantine frequencies.

  6. (f)

The separable operator considered in Theorem 1.6 but with added background
potential (so the resulting operator is not necessarily separable) has been
studied by J. Bourgain and I. Kachkovskiy [10]. Similar but distinct
quasiperiodic (not necessarily separable) operators have been studied by S.
Jitomirskaya, W. Liu, and Y. Shi [25]. The [10, 25] results pertain to the
spectral type but not the topological structure of the spectrum as a set.

In section 2 we state the preliminaries. In section 3 we prove the main
theorem. In section 4 we prove a lemma used in the proof of the main theorem.

##  2\. Preliminaries

Let $A:\mathcal{H}\to\mathcal{H}$ be a bounded operator. The spectrum of $A$
is denoted $\sigma(A)$. Note $\sigma(A)$ is a nonempty compact subset of
$\mathbb{C}$. Also note, if $A$ is self-adjoint, then
$\sigma(A)\subseteq\mathbb{R}$.

###  2.1. Potentials and Schrödinger Operators

A ($d$-dimensional lattice) potential is a bounded real-valued function $V$ on
$\mathbb{Z}^{d}$. Moreover, $V$ also denotes the bounded self-adjoint
multiplication operator on $\ell^{2}(\mathbb{Z}^{d})$ defined by
$[V\psi](n)=V(n)\psi(n)$ for every
$\psi\in\ell^{2}(\mathbb{Z}^{d}),n\in\mathbb{Z}^{d}$. The ($d$-dimensional
lattice) Schrödinger operator with respect to a potential
$V:\mathbb{Z}^{d}\to\mathbb{R}$ is the bounded self-adjoint operator $H$ on
$\ell^{2}(\mathbb{Z}^{d})$ defined by $H=\Delta+V$, where
$\Delta=\Delta^{(d)}$ is the ($d$-dimensional lattice) Laplacian defined by
$[\Delta\psi](n)=\sum_{m\in\left\\{e_{1},\ldots,e_{d}\right\\}}\psi(n+m)+\psi(n-m)$
for every $\psi\in\ell^{2}(\mathbb{Z}^{d}),n\in\mathbb{Z}^{d}$;
$\left\\{e_{1},\ldots,e_{d}\right\\}$ is the standard basis.

###  2.2. Separable Potentials and The Laplacian

Let $V:\mathbb{Z}^{d}\to\mathbb{R}$ be a potential. Say $V$ is separable if
there exist (sub)potentials $V_{1},\ldots,V_{d}:\mathbb{Z}\to\mathbb{R}$ such
that $V(n)=V_{1}(n_{1})+\cdots+V_{d}(n_{d})$ for every $n\in\mathbb{Z}^{d}$.
The proof of the following theorem can be found within [12].

######  Theorem 2.1.

Let $V:\mathbb{Z}^{d}\to\mathbb{R}:n\mapsto V_{1}(n_{1})+\cdots+V_{d}(n_{d})$
be a separable potential. Define $H\coloneqq\Delta^{(d)}+V$. For each $k$,
define $H_{k}\coloneqq\Delta^{(1)}+V_{k}$. Then

$\sigma(H)=\sigma(H_{1})+\cdots+\sigma(H_{d}).$

By the spectral mapping theorem,

$\textstyle\sigma(\Delta^{(1)})=\sigma(\ooalign{$U_{1}$\cr$\phantom{U}{}^{*}$\cr$\phantom{U^{*}_{1}}$}+U_{1})=\sigma(\Phi_{U_{1}}(z^{*}+z))=\\{z^{*}+z:z\in\sigma(U_{1})\\}=\\{z^{*}+z:|z|=1\\}=[-2,2].$

By Theorem 2.1,

$\textstyle\sigma(\Delta^{(d)})=\sigma(\Delta^{(1)})+\cdots+\sigma(\Delta^{(1)})\textnormal{
($d$ terms)}=[-2d,2d].$

Here $U_{m}$ is the unitary operator from $\ell^{2}(\mathbb{Z}^{d})$ to
$\ell^{2}(\mathbb{Z}^{d})$ defined by $[U_{m}\psi](n)=\psi(n-m)$ for every
$\psi\in\ell^{2}(\mathbb{Z}^{d}),n\in\mathbb{Z}^{d}$. Also here, $\Phi_{A}$ is
the Borel functional calculus with respect to a bounded operator $A$.

###  2.3. Quasiperiodic Potentials and The Almost Mathieu Operator

Let $V:\mathbb{Z}\to\mathbb{R}$ be a potential. Let $b$ be a positive integer.
Say $V$ is ($b$-frequency) quasiperiodic if there exists $v\in
C(\mathbb{T}^{b},\mathbb{R})$ and there exist $\alpha,\omega\in\mathbb{T}^{b}$
such that $v$ is nonconstant and
$\left\\{1,\alpha_{1},\ldots,\alpha_{b}\right\\}$ is independent over the
rationals and $V(n)=v(n\alpha+\omega)$ for every $n\in\mathbb{Z}$. The proof
of the following theorem can be found within [36].

######  Theorem 2.2.

Let $V_{\omega}:\mathbb{Z}\to\mathbb{R}:n\mapsto
v\big{|}_{\mathbb{T}^{b}}(n\alpha+\omega)$ be a quasiperiodic potential with
parameter $\omega$. Let $H_{\omega}$ be the Schrödinger operator. Then
$\sigma(H_{\omega})=\Sigma_{\omega}\eqqcolon\Sigma$ is independent of
$\omega$. Furthermore, $\Sigma$ is a nonempty compact subset of $\mathbb{R}$
and $\Sigma$ has no isolated points.

The potential of the almost Mathieu operator
$V_{\lambda,\alpha,\omega}:\mathbb{Z}\to\mathbb{R}:n\mapsto
2\lambda\cos(2\pi(n\alpha+\omega))$ is $1$-frequency quasiperiodic when
$\alpha$ is irrational. By the resolution of the Ten Martini Problem within
[4], the spectrum $\Sigma_{\lambda,\alpha,\omega}$ of the almost Mathieu
operator is a Cantor set when $\alpha$ is irrational.

###  2.4. Integrated Density of States

Let $V_{\omega}:\mathbb{Z}\to\mathbb{R}:n\mapsto
v\big{|}_{\mathbb{T}^{b}}(n\alpha+\omega)$ be a quasiperiodic potential with
parameter $\omega$. Let $H_{\omega}$ be the Schrödinger operator. The
integrated density of states (IDS) is

$\displaystyle\mathfrak{N}:\mathbb{R}\to[0,1]:x$
$\displaystyle\mapsto\textstyle\int\langle\delta_{0},\mathbbm{1}_{(-\infty,x]}(H_{\theta})\delta_{0}\rangle
d\mu(\theta).$

Here $\mu$ is the normalized Haar measure on $\mathbb{T}^{b}$. Note
$\mathfrak{N}\upharpoonright(-\infty,\inf\Sigma]\equiv 0$ and
$\mathfrak{N}\upharpoonright[\sup\Sigma,+\infty)\equiv 1$. The proof of
Theorem 2.3 can be found within [36] and of Theorem 2.4 can be found within
[11].

######  Theorem 2.3 (Gap Labeling).

Let $V:\mathbb{Z}\to\mathbb{R}:n\mapsto
v\big{|}_{\mathbb{T}^{b}}(n\alpha+\omega)$ be a quasiperiodic potential. Let
$H$ be the Schrödinger operator. Let $\mathfrak{N}$ be the IDS. Define
$\Sigma\coloneqq\sigma(H)$. Let $\mathrm{Gap}_{\mathsf{b}}(\Sigma)$ be the
collection of all bounded gaps of $\Sigma$.

  1. (i)

$\mathfrak{N}$ is monotone and continuous.

  2. (ii)

$\left\\{x\in\mathbb{R}:\left(\forall\varepsilon>0\right)[\mathfrak{N}(x+\varepsilon)-\mathfrak{N}(x-\varepsilon)>0]\right\\}=\Sigma$.

  3. (iii)

$\left\\{\mathfrak{N}(x):x\in
U\in\mathrm{Gap}_{\mathsf{b}}(\Sigma)\right\\}\subseteq\\{\mathbf{n}\alpha-\left\lfloor\mathbf{n}\alpha\right\rfloor:\mathbf{n}\in\mathbb{Z}^{b}\\}\setminus\left\\{0\right\\}$.

  4. (iv)

$\left\\{\mathfrak{N}(x):x\in
U\in\mathrm{Gap}_{\mathsf{b}}(\Sigma)\right\\}\supseteq\\{\mathbf{n}\alpha-\left\lfloor\mathbf{n}\alpha\right\rfloor:\mathbf{n}\in\mathbb{Z}^{b}\\}\setminus\left\\{0\right\\}$
$\mspace{5.0mu}\Longrightarrow\mspace{5.0mu}$ $\Sigma$ is a Cantor set.

######  Theorem 2.4 (Hölder Continuity).

Let $V_{\lambda,v,\alpha,\omega}:\mathbb{Z}\to\mathbb{R}:n\mapsto\lambda
v\big{|}_{\mathbb{T}^{b}}(n\alpha+\omega)$ be a quasiperiodic potential with
parameters $\lambda,v,\alpha,\omega$. Let $H_{\lambda,v,\alpha,\omega}$ be the
Schrödinger operator. Fix $v,\alpha$. Let $\mathfrak{N}_{\lambda}$ be the IDS.
Assume the following.

  1. (i)

$\alpha$ satisfies a Diophantine condition:

$\textstyle\smash{\alpha\in\bigcap_{\begin{subarray}{l}p\in\mathbb{Z}\\\
\smash{\mathbf{q}\in\mathbb{Z}^{b}\setminus\left\\{\mathbf{0}\right\\}}\end{subarray}}\left\\{\mathbf{x}\in\mathbb{R}^{b}:|\mathbf{q}\mathbf{x}-2\pi
p|\geq\frac{c}{\lVert\mathbf{q}\rVert^{t-1}}\right\\}\mathrlap{{}\eqqcolon\vphantom{\mathsf{DC}}\smash{\widetilde{\mathsf{DC}}}{}^{b}(c,t)}}$

for some $c>0,t>b$.

  2. (ii)

$v$ satisfies a regularity condition: $v$ is $r$-differentiable on
$\mathbb{T}^{b}$ for some $r\geq 550t$.

There exist
$\lambda_{0}=\lambda_{0}(v,b,c,t,r)>0,C_{\mathsf{H}}=C_{\mathsf{H}}(b,c,t)>0$
such that

$\smash{|\mathfrak{N}_{\lambda}(x)-\mathfrak{N}_{\lambda}(y)|\leq
C_{\mathsf{H}}|x-y|^{{1}/{2}}}$

for every $\lambda_{0}\geq|\lambda|>0,x,y$.

###  2.5. Spectral Gaps

Let $V:\mathbb{Z}\to\mathbb{R}:n\mapsto
v\big{|}_{\mathbb{T}^{b}}(n\alpha+\omega)$ be a quasiperiodic potential. Let
$H$ be the Schrödinger operator. Let $\mathfrak{N}$ be the IDS. Define
$\Sigma\coloneqq\sigma(H)$. Let $\mathrm{Gap}_{\mathsf{b}}(\Sigma)$ be the
collection of all bounded gaps of $\Sigma$. Fix
$\mathbf{n}\in\mathbb{Z}^{b}\setminus\left\\{\mathbf{0}\right\\}$. The
$\mathbf{n}$-th spectral gap of $\Sigma$ is either the element
$G^{(\mathbf{n})}$ in $\mathrm{Gap}_{\mathsf{b}}(\Sigma)$ such that
$\mathfrak{N}\upharpoonright\smash{\overline{G^{(\mathbf{n})}}}\equiv\mathbf{n}\alpha-\lfloor\mathbf{n}\alpha\rfloor$
or the empty set when such an element $G^{(\mathbf{n})}$ does not exist.
Moreover, $(-\infty,\inf\Sigma)\cup(\sup\Sigma,+\infty)$ is called the
$\mathbf{0}$-th spectral gap of $\Sigma$. The proof of the following theorem
can be found within [29].

######  Theorem 2.5 (Gap Estimate).

Let $V_{v,\alpha,\omega}:\mathbb{Z}\to\mathbb{R}:n\mapsto
v\big{|}_{\mathbb{T}^{b}}(n\alpha+\omega)$ be a quasiperiodic potential with
parameters $v,\alpha,\omega$. Let $H_{v,\alpha,\omega}$ be the Schrödinger
operator. Fix $\alpha$. Fix $r_{0}>r>0$. Define
$\Sigma_{v}\coloneqq\sigma(H_{v,\alpha,\omega})$. For each
$\mathbf{n}\neq\mathbf{0}$, let
$\ooalign{$G_{v}$\cr$\phantom{G}{}^{(\mathbf{n})}$\cr$\phantom{G^{(\mathbf{n})}_{v}}$}=(\ooalign{$E_{v}$\cr$\phantom{E}{}^{(\mathbf{n}){\scriptscriptstyle-}}$\cr$\phantom{E^{(\mathbf{n}){\scriptscriptstyle-}}_{v}}$},\ooalign{$E_{v}$\cr$\phantom{E}{}^{(\mathbf{n}){\scriptscriptstyle+}}$\cr$\phantom{E^{(\mathbf{n}){\scriptscriptstyle+}}_{v}}$})$
be the $\mathbf{n}$-th spectral gap of $\Sigma_{v}$. Assume the following.

  1. (i)

$\alpha$ satisfies a Diophantine condition:

$\textstyle\smash{\alpha\in\bigcap_{\begin{subarray}{l}p\in\mathbb{Z}\\\
\smash{\mathbf{q}\in\mathbb{Z}^{b}\setminus\left\\{\mathbf{0}\right\\}}\end{subarray}}\left\\{\mathbf{x}\in\mathbb{R}^{b}:|\mathbf{q}\mathbf{x}-p|\geq\frac{c}{\lVert\mathbf{q}\rVert^{t-1}}\right\\}\mathrlap{{}\eqqcolon\mathsf{DC}{}^{b}(c,t)}}$

for some $c>0,t>b$.

  2. (ii)

$v$ satisfies a regularity condition: $v$ is analytic on
$(\mathbb{T}+i(-r_{0},r_{0}))^{b}$.

There exists $\varepsilon=\varepsilon(b,c,t,r_{0},r)>0$ such that

$\textstyle\smash{|\ooalign{$E_{v}$\cr$\phantom{E}{}^{(\mathbf{n}){\scriptscriptstyle+}}$\cr$\phantom{E^{(\mathbf{n}){\scriptscriptstyle+}}_{v}}$}-\ooalign{$E_{v}$\cr$\phantom{E}{}^{(\mathbf{n}){\scriptscriptstyle-}}$\cr$\phantom{E^{(\mathbf{n}){\scriptscriptstyle-}}_{v}}$}|\leq(|v|_{r_{0}})^{2/3}e^{-2\pi
r\lVert\mathbf{n}\rVert}}$

for every $\varepsilon\geq|v|_{r_{0}}>0,\mathbf{n}\neq\mathbf{0}$.

###  2.6. Thickness and Gap Lemmas

Let $K$ be a nonempty compact subset of $\mathbb{R}$. Henceforth
$K^{\scriptscriptstyle-}\coloneqq\inf K$ and
$K^{\scriptscriptstyle+}\coloneqq\sup K$. Let $\mathrm{Gap}_{\mathsf{b}}(K)$
be the collection of all bounded gaps of $K$. Let $U$ be a bounded gap of $K$.
The left-plank of $U$, denoted $\pi_{\scriptscriptstyle-}(K,U)$, is the
length-maximal interval $[a,b]$ contained in
$[K^{\scriptscriptstyle-},K^{\scriptscriptstyle+}]$ such that $b\in\partial U$
and for each $V\in\mathrm{Gap}_{\mathsf{b}}(K)$, if
$V\cap[a,b]\neq\varnothing$, then $\mathrm{length}(V)<\mathrm{length}(U)$. The
right-plank of $U$, denoted $\pi_{\scriptscriptstyle+}(K,U)$, is the length-
maximal interval $[a,b]$ contained in
$[K^{\scriptscriptstyle-},K^{\scriptscriptstyle+}]$ such that $a\in\partial U$
and for each $V\in\mathrm{Gap}_{\mathsf{b}}(K)$, if
$V\cap[a,b]\neq\varnothing$, then $\mathrm{length}(V)<\mathrm{length}(U)$. The
(local) thickness of $U$ is

$\inf\nolimits_{{\bullet}\in\left\\{-,+\right\\}}\frac{\mathrm{length}(\pi_{\scriptscriptstyle\bullet}(K,U))}{\mathrm{length}(U)}\mathrlap{{}\eqqcolon\tau(K,U).}$

The thickness of $K$ is

$\smash{\inf\nolimits_{U\in\mathrm{Gap}_{\mathsf{b}}(K)}\tau(K,U)\mathrlap{{}\eqqcolon\tau(K).}}$

Note $\tau(K)=0$ when $K$ has an isolated point, $\tau(K)=1$ when $K$ is the
middle-thirds Cantor set, and $\tau(K)=+\infty$ when $K$ is an interval. The
proof of Theorem 2.6 can be found within [12] and of Theorem 2.7 can be found
within [1, 2]. The former theorem is the classical Gap Lemma, but we require
the latter theorem which is a generalized version.

######  Theorem 2.6 (Gap Lemma).

Let $K_{1}$ and $K_{2}$ be nonempty compact subsets of $\mathbb{R}$. For each
nonempty compact subset $K$ of $\mathbb{R}$, let
$\mathrm{Gap}_{\mathsf{b}}(K)$ be the collection of all bounded gaps of $K$.
For each nonempty compact subset $K$ of $\mathbb{R}$, define
$\Gamma(K)\coloneqq\sup\\{\mathrm{length}(U):U\in\mathrm{Gap}_{\mathsf{b}}(K)\\}$.
Assume
$[K_{1}^{\scriptscriptstyle-},K_{1}^{\scriptscriptstyle+}]\cap[K_{2}^{\scriptscriptstyle-},K_{2}^{\scriptscriptstyle+}]\neq\varnothing$,
$\Gamma(K_{2})\leq\mathrm{diam}(K_{1})$,
$\Gamma(K_{1})\leq\mathrm{diam}(K_{2})$, and
$1\leq\tau(K_{1})\cdot\tau(K_{2})$. Then $K_{1}\cap K_{2}\neq\varnothing$.
Furthermore,
$K_{1}+K_{2}=[K_{1}^{\scriptscriptstyle-}+K_{2}^{\scriptscriptstyle-},K_{1}^{\scriptscriptstyle+}+K_{2}^{\scriptscriptstyle+}]$.

######  Theorem 2.7 (Gap Lemma).

Let $K_{1},\ldots,K_{d}$ ($d\geq 2$) be nonempty compact subsets of
$\mathbb{R}$. For each nonempty compact subset $K$ of $\mathbb{R}$, let
$\mathrm{Gap}_{\mathsf{b}}(K)$ be the collection of all bounded gaps of $K$.
For each nonempty compact subset $K$ of $\mathbb{R}$, define
$\Gamma(K)\coloneqq\sup\\{\mathrm{length}(U):U\in\mathrm{Gap}_{\mathsf{b}}(K)\\}$.
Assume

$\begin{cases}\left(\forall i:2\leq i\leq d\right)\left(\forall j:1\leq j\leq
i-1\right)[\Gamma(K_{j})\leq\mathrm{diam}(K_{i})],\\\ \left(\forall i:2\leq
i\leq
d\right)[\Gamma(K_{i})\leq\mathrm{diam}(K_{1})+\cdots+\mathrm{diam}(K_{i-1})].\end{cases}$

If
$1\leq\frac{\tau(K_{1})}{\tau(K_{1})+1}+\cdots+\frac{\tau(K_{d})}{\tau(K_{d})+1}$,
then $\tau(K_{1}+\cdots+K_{d})=+\infty$ and

$K_{1}+\cdots+K_{d}=[K_{1}^{\scriptscriptstyle-}+\cdots+K_{d}^{\scriptscriptstyle-},K_{1}^{\scriptscriptstyle+}+\cdots+K_{d}^{\scriptscriptstyle+}].$

If
$\frac{\tau(K_{1})}{\tau(K_{1})+1}+\cdots+\frac{\tau(K_{d})}{\tau(K_{d})+1}<1$,
then
$\tau(K_{1}+\cdots+K_{d})\geq\frac{\frac{\tau(K_{1})}{\tau(K_{1})+1}+\cdots+\frac{\tau(K_{d})}{\tau(K_{d})+1}}{1-(\frac{\tau(K_{1})}{\tau(K_{1})+1}+\cdots+\frac{\tau(K_{d})}{\tau(K_{d})+1})}$.

##  3\. Proof of Main Theorem

######  Lemma 3.1.

Let $A:\mathcal{H}\to\mathcal{H}$ and $B:\mathcal{H}\to\mathcal{H}$ be bounded
self-adjoint operators. Then

$|\mathrm{diam}\,\sigma(A)-\mathrm{diam}\,\sigma(B)|\leq
2\,\mathrm{dist}_{\mathsf{Haus}}(\sigma(A),\sigma(B))\leq 2\left\lVert
A-B\right\rVert.$

###### Proof..

Define $\xi\coloneqq\min\sigma(A)-\min\sigma(B)$,
$\zeta\coloneqq\max\sigma(A)-\max\sigma(B)$. Define $s\curlyvee
t\coloneqq\sup\left\\{s,t\right\\}$. Because

$\displaystyle|\xi|\curlyvee|\zeta|$
$\displaystyle\textstyle\leq\sup_{x\in\sigma(A)}\mathrm{dist}(x,\sigma(B))\curlyvee\sup_{x\in\sigma(B)}\mathrm{dist}(x,\sigma(A))\mathrlap{{}\eqqcolon\mathrm{dist}_{\mathsf{Haus}}(\sigma(A),\sigma(B))}$
$\displaystyle\textstyle=\sup_{x\in\sigma(A)}\inf_{\left\lVert\psi\right\rVert=1}\left\lVert(B-xI)\psi\right\rVert\curlyvee\sup_{x\in\sigma(B)}\inf_{\left\lVert\psi\right\rVert=1}\left\lVert(A-xI)\psi\right\rVert$
$\displaystyle\textstyle\leq\sup_{x\in\sigma(A)}\left\lVert\smash{B-A}\right\rVert\curlyvee\sup_{x\in\sigma(B)}\left\lVert\smash{A-B}\right\rVert$
$\displaystyle\textstyle=\left\lVert A-B\right\rVert,$

$|\mathrm{diam}\,\sigma(A)-\mathrm{diam}\,\sigma(B)|=|\zeta-\xi|\leq|\zeta|+|\xi|\leq
2\,\mathrm{dist}_{\mathsf{Haus}}(\sigma(A),\sigma(B))\leq 2\left\lVert
A-B\right\rVert$. ∎

######  Lemma 3.2.

Let $H_{\lambda,\alpha,\omega}=\Delta+V_{\lambda,\alpha,\omega}$ be the almost
Mathieu operator. Fix $\alpha\in\mathbb{T}\setminus\mathbb{Q}$. Define
$\Sigma_{\lambda}\coloneqq\sigma(H_{\lambda,\alpha,\omega})$. Assume $\alpha$
satisfies a Diophantine condition:

$\textstyle\alpha\in\bigcup_{\begin{subarray}{c}c>0\\\
t>1\end{subarray}}\bigcap_{\frac{p}{q}\in\mathbb{Q}}\left\\{x\in\mathbb{R}:|qx-p|\geq\frac{c}{|q|^{t-1}}\right\\}.$

Then $\lim_{\lambda\to 0}\tau(\Sigma_{\lambda})=+\infty$.

The proof of Lemma 3.2 can be found within section 4.

######  Theorem 3.3 (Main Theorem).

Fix
$\alpha_{1},\ldots,\alpha_{d},\omega_{1},\ldots,\omega_{d}\in[0,1]\eqqcolon\mathbb{T}$.
Let $\vphantom{H}\smash{\widehat{H}}$ be the bounded self-adjoint operator
$\vphantom{H}\smash{\widehat{H}}:\ell^{2}(\mathbb{Z}^{d})\to\ell^{2}(\mathbb{Z}^{d})$
defined by

$\textstyle[\vphantom{H}\smash{\widehat{H}}\psi](n)=\big{(}\sum_{m\in\left\\{e_{1},\ldots,e_{d}\right\\}}\psi(n+m)+\psi(n-m)\big{)}+\big{(}\sum_{k\in\left\\{1,\ldots,d\right\\}}2\lambda_{k}\cos(2\pi(n_{k}\alpha_{k}+\omega_{k}))\big{)}\psi(n)$

for every $\psi\in\ell^{2}(\mathbb{Z}^{d}),n\in\mathbb{Z}^{d}$;
$\left\\{e_{1},\ldots,e_{d}\right\\}$ is the standard basis.

Let $H_{\lambda,\alpha,\omega}=\Delta+V_{\lambda,\alpha,\omega}$ be the almost
Mathieu operator. For each $k$, define
$\Sigma_{k}\coloneqq\sigma(H_{\lambda_{k},\alpha_{k},\omega_{k}})$. Assume
$\alpha_{1},\ldots,\alpha_{d}$ ($d\geq 2$) are irrational and satisfy a
Diophantine condition:

$\textstyle\alpha\in\bigcup_{\begin{subarray}{c}c>0\\\
t>1\end{subarray}}\bigcap_{\frac{p}{q}\in\mathbb{Q}}\left\\{x\in\mathbb{R}:|qx-p|\geq\frac{c}{|q|^{t-1}}\right\\}.$

There exists $\varepsilon=\varepsilon(\alpha_{1},\ldots,\alpha_{d})>0$ such
that if $0<|\lambda_{1}|,\ldots,|\lambda_{d}|<\varepsilon$, then
$\sigma(\vphantom{H}\smash{\widehat{H}})$, which is a sum of Cantor spectra
$\Sigma_{1}+\cdots+\Sigma_{d}$, is an interval.

###### Proof..

For each nonempty compact subset $K$ of $\mathbb{R}$, let
$\mathrm{Gap}_{\mathsf{b}}(K)$ be the collection of all bounded gaps of $K$.
For each nonempty compact subset $K$ of $\mathbb{R}$, define
$\Gamma(K)\coloneqq\sup\\{\mathrm{length}(U):U\in\mathrm{Gap}_{\mathsf{b}}(K)\\}$.
By Lemma 3.1, $\lim_{\lambda\to
0}\mathrm{dist}_{\mathsf{Haus}}(\sigma(H_{\lambda,\alpha,\omega}),[-2,2])=0$
and $\lim_{\lambda\to
0}|\mathrm{diam}\,\sigma(H_{\lambda,\alpha,\omega})-4|=0$. By Lemma 3.2,
$\lim_{\lambda\to 0}\tau(\sigma(H_{\lambda,\alpha_{k},\omega}))=+\infty$. As a
result, there exists $\varepsilon=\varepsilon(\alpha_{1},\ldots,\alpha_{d})>0$
such that if $0<|\lambda_{1}|,\ldots,|\lambda_{d}|<\varepsilon$, then

$\begin{cases}\left(\forall i:2\leq i\leq d\right)\left(\forall j:1\leq j\leq
i-1\right)[\Gamma(\Sigma_{j})\leq\mathrm{diam}(\Sigma_{i})],\\\ \left(\forall
i:2\leq i\leq
d\right)[\Gamma(\Sigma_{i})\leq\mathrm{diam}(\Sigma_{1})+\cdots+\mathrm{diam}(\Sigma_{i-1})],\\\
1\leq\frac{\tau(\Sigma_{1})}{\tau(\Sigma_{1})+1}+\cdots+\frac{\tau(\Sigma_{d})}{\tau(\Sigma_{d})+1}.\end{cases}$

By Theorem 2.7, $\Sigma_{1}+\cdots+\Sigma_{d}$ is an interval. By Theorem 2.1,
$\sigma(\vphantom{H}\smash{\widehat{H}})=\Sigma_{1}+\cdots+\Sigma_{d}$. ∎

##  4\. Proof of Lemma 3.2

######  Lemma 4.1.

Let $V_{\lambda,\alpha,\omega}:\mathbb{Z}\to\mathbb{R}:n\mapsto\lambda
v\big{|}_{\mathbb{T}^{b}}(n\alpha+\omega)$ be a quasiperiodic potential with
parameters $\lambda,\alpha,\omega$. Let $H_{\lambda,\alpha,\omega}$ be the
Schrödinger operator. Fix $\alpha$. Let $\mathfrak{N}_{\lambda}$ be the IDS.
Define $\Sigma_{\lambda}\coloneqq\sigma(H_{\lambda,\alpha,\omega})$. For each
$\mathbf{n}\neq\mathbf{0}$, let
$\ooalign{$G_{\lambda}$\cr$\phantom{G}{}^{(\mathbf{n})}$\cr$\phantom{G^{(\mathbf{n})}_{\lambda}}$}=(\ooalign{$E_{\lambda}$\cr$\phantom{E}{}^{(\mathbf{n}){\scriptscriptstyle-}}$\cr$\phantom{E^{(\mathbf{n}){\scriptscriptstyle-}}_{\lambda}}$},\ooalign{$E_{\lambda}$\cr$\phantom{E}{}^{(\mathbf{n}){\scriptscriptstyle+}}$\cr$\phantom{E^{(\mathbf{n}){\scriptscriptstyle+}}_{\lambda}}$})$
be the $\mathbf{n}$-th spectral gap of $\Sigma_{\lambda}$. Assume the
following.

  1. (i)

$\alpha$ satisfies a Diophantine condition:

$\textstyle\alpha\in\bigcap_{\begin{subarray}{l}p\in\mathbb{Z}\\\
\smash{\mathbf{q}\in\mathbb{Z}^{b}\setminus\left\\{\mathbf{0}\right\\}}\end{subarray}}\left\\{\mathbf{x}\in\mathbb{R}^{b}:|\mathbf{q}\mathbf{x}-p|\geq\frac{c}{\lVert\mathbf{q}\rVert^{t-1}}\right\\}\mathrlap{{}\eqqcolon\mathsf{DC}{}^{b}(c,t)}$

for some $c>0,t>b$.

  2. (ii)

There exist
$\lambda_{0}=\lambda_{0}(b,c,t)>0,C_{\mathsf{H}}=C_{\mathsf{H}}(b,c,t)>0,1\geq
h=h(b,c,t)>0$ such that

$|\mathfrak{N}_{\lambda}(x)-\mathfrak{N}_{\lambda}(y)|\leq
C_{\mathsf{H}}|x-y|^{h}$

for every $\lambda_{0}\geq|\lambda|>0,x,y$.

  3. (iii)

There exist
$\lambda_{1}=\lambda_{1}(b,c,t)>0,C(\lambda)=C(\lambda,b,c,t)>0,C_{\mathsf{E}}=C_{\mathsf{E}}(b,c,t)>0$
such that

$|\ooalign{$E_{\lambda}$\cr$\phantom{E}{}^{(\mathbf{n}){\scriptscriptstyle+}}$\cr$\phantom{E^{(\mathbf{n}){\scriptscriptstyle+}}_{\lambda}}$}-\ooalign{$E_{\lambda}$\cr$\phantom{E}{}^{(\mathbf{n}){\scriptscriptstyle-}}$\cr$\phantom{E^{(\mathbf{n}){\scriptscriptstyle-}}_{\lambda}}$}|\leq
C(\lambda)e^{-C_{\mathsf{E}}\lVert\mathbf{n}\rVert}$

for every $\lambda_{1}\geq|\lambda|>0,\mathbf{n}\neq\mathbf{0}$ and
$\lim_{\lambda\to 0}C(\lambda)=0$.

Then $\textstyle\lim_{\lambda\to 0}\tau(\Sigma_{\lambda})=+\infty$.

###### Proof..

For each $\mathbf{n}\neq\mathbf{0}$, let
$\pi(\Sigma_{\lambda},\ooalign{$G_{\lambda}$\cr$\phantom{G}{}^{(\mathbf{n})}$\cr$\phantom{G^{(\mathbf{n})}_{\lambda}}$})$
be a length-minimal plank of
$\ooalign{$G_{\lambda}$\cr$\phantom{G}{}^{(\mathbf{n})}$\cr$\phantom{G^{(\mathbf{n})}_{\lambda}}$}\neq\varnothing$.
Without loss of generality, assume

$\pi(\Sigma_{\lambda},\ooalign{$G_{\lambda}$\cr$\phantom{G}{}^{(\mathbf{n})}$\cr$\phantom{G^{(\mathbf{n})}_{\lambda}}$})=\pi_{\scriptscriptstyle+}(\Sigma_{\lambda},\ooalign{$G_{\lambda}$\cr$\phantom{G}{}^{(\mathbf{n})}$\cr$\phantom{G^{(\mathbf{n})}_{\lambda}}$})=[\ooalign{$E_{\lambda}$\cr$\phantom{E}{}^{(\mathbf{n}){\scriptscriptstyle+}}$\cr$\phantom{E^{(\mathbf{n}){\scriptscriptstyle+}}_{\lambda}}$},\ooalign{$F_{\lambda}$\cr$\phantom{F}{}^{(\mathbf{n})}$\cr$\phantom{F^{(\mathbf{n})}_{\lambda}}$}]$

for some
$\ooalign{$F_{\lambda}$\cr$\phantom{F}{}^{(\mathbf{n})}$\cr$\phantom{F^{(\mathbf{n})}_{\lambda}}$}\in\Sigma_{\lambda}$.
Observe

$\displaystyle\tau(\Sigma_{\lambda},\ooalign{$G_{\lambda}$\cr$\phantom{G}{}^{(\mathbf{n})}$\cr$\phantom{G^{(\mathbf{n})}_{\lambda}}$})=\frac{\mathrm{length}(\pi(\Sigma_{\lambda},\ooalign{$G_{\lambda}$\cr$\phantom{G}{}^{(\mathbf{n})}$\cr$\phantom{G^{(\mathbf{n})}_{\lambda}}$}))}{\mathrm{length}(\ooalign{$G_{\lambda}$\cr$\phantom{G}{}^{(\mathbf{n})}$\cr$\phantom{G^{(\mathbf{n})}_{\lambda}}$})}=\frac{|\ooalign{$F_{\lambda}$\cr$\phantom{F}{}^{(\mathbf{n})}$\cr$\phantom{F^{(\mathbf{n})}_{\lambda}}$}-\ooalign{$E_{\lambda}$\cr$\phantom{E}{}^{(\mathbf{n}){\scriptscriptstyle+}}$\cr$\phantom{E^{(\mathbf{n}){\scriptscriptstyle+}}_{\lambda}}$}|}{|\ooalign{$E_{\lambda}$\cr$\phantom{E}{}^{(\mathbf{n}){\scriptscriptstyle+}}$\cr$\phantom{E^{(\mathbf{n}){\scriptscriptstyle+}}_{\lambda}}$}-\ooalign{$E_{\lambda}$\cr$\phantom{E}{}^{(\mathbf{n}){\scriptscriptstyle-}}$\cr$\phantom{E^{(\mathbf{n}){\scriptscriptstyle-}}_{\lambda}}$}|}.$

Because $\Sigma_{\lambda}$ has no isolated points,

$\ooalign{$E_{\lambda}$\cr$\phantom{E}{}^{(\mathbf{n}){\scriptscriptstyle+}}$\cr$\phantom{E^{(\mathbf{n}){\scriptscriptstyle+}}_{\lambda}}$}<\ooalign{$F_{\lambda}$\cr$\phantom{F}{}^{(\mathbf{n})}$\cr$\phantom{F^{(\mathbf{n})}_{\lambda}}$}\leq\sup\Sigma_{\lambda}.$

Temporarily, assume
$\ooalign{$F_{\lambda}$\cr$\phantom{F}{}^{(\mathbf{n})}$\cr$\phantom{F^{(\mathbf{n})}_{\lambda}}$}=\sup\Sigma_{\lambda}$.
Therefore

$|\mathfrak{N}_{\lambda}(\ooalign{$F_{\lambda}$\cr$\phantom{F}{}^{(\mathbf{n})}$\cr$\phantom{F^{(\mathbf{n})}_{\lambda}}$})-\mathfrak{N}_{\lambda}(\ooalign{$E_{\lambda}$\cr$\phantom{E}{}^{(\mathbf{n}){\scriptscriptstyle+}}$\cr$\phantom{E^{(\mathbf{n}){\scriptscriptstyle+}}_{\lambda}}$})|=|1-(\mathbf{n}\alpha-\lfloor\mathbf{n}\alpha\rfloor)|=|\mathbf{n}\alpha-\lfloor\mathbf{n}\alpha\rfloor-1|.$

By (i–iii), for sufficiently small $\lambda$,

$\displaystyle\frac{|\ooalign{$F_{\lambda}$\cr$\phantom{F}{}^{(\mathbf{n})}$\cr$\phantom{F^{(\mathbf{n})}_{\lambda}}$}-\ooalign{$E_{\lambda}$\cr$\phantom{E}{}^{(\mathbf{n}){\scriptscriptstyle+}}$\cr$\phantom{E^{(\mathbf{n}){\scriptscriptstyle+}}_{\lambda}}$}|}{|\ooalign{$E_{\lambda}$\cr$\phantom{E}{}^{(\mathbf{n}){\scriptscriptstyle+}}$\cr$\phantom{E^{(\mathbf{n}){\scriptscriptstyle+}}_{\lambda}}$}-\ooalign{$E_{\lambda}$\cr$\phantom{E}{}^{(\mathbf{n}){\scriptscriptstyle-}}$\cr$\phantom{E^{(\mathbf{n}){\scriptscriptstyle-}}_{\lambda}}$}|}$
$\displaystyle\geq\frac{(\frac{1}{C_{\mathsf{H}}})^{\frac{1}{h}}|\mathfrak{N}_{\lambda}(\ooalign{$F_{\lambda}$\cr$\phantom{F}{}^{(\mathbf{n})}$\cr$\phantom{F^{(\mathbf{n})}_{\lambda}}$})-\mathfrak{N}_{\lambda}(\ooalign{$E_{\lambda}$\cr$\phantom{E}{}^{(\mathbf{n}){\scriptscriptstyle+}}$\cr$\phantom{E^{(\mathbf{n}){\scriptscriptstyle+}}_{\lambda}}$})|^{\frac{1}{h}}}{|\ooalign{$E_{\lambda}$\cr$\phantom{E}{}^{(\mathbf{n}){\scriptscriptstyle+}}$\cr$\phantom{E^{(\mathbf{n}){\scriptscriptstyle+}}_{\lambda}}$}-\ooalign{$E_{\lambda}$\cr$\phantom{E}{}^{(\mathbf{n}){\scriptscriptstyle-}}$\cr$\phantom{E^{(\mathbf{n}){\scriptscriptstyle-}}_{\lambda}}$}|}\geq\frac{(\frac{1}{C_{\mathsf{H}}})^{\frac{1}{h}}|\frac{c}{\lVert\mathbf{n}\rVert^{t-1}}|^{\frac{1}{h}}}{|\ooalign{$E_{\lambda}$\cr$\phantom{E}{}^{(\mathbf{n}){\scriptscriptstyle+}}$\cr$\phantom{E^{(\mathbf{n}){\scriptscriptstyle+}}_{\lambda}}$}-\ooalign{$E_{\lambda}$\cr$\phantom{E}{}^{(\mathbf{n}){\scriptscriptstyle-}}$\cr$\phantom{E^{(\mathbf{n}){\scriptscriptstyle-}}_{\lambda}}$}|}\geq\frac{(\frac{c}{C_{\mathsf{H}}})^{{1}/{h}}}{C(\lambda)e^{-C_{\mathsf{E}}\lVert\mathbf{n}\rVert}\lVert\mathbf{n}\rVert^{{(t-1)}/{h}}}.$

Temporarily, assume
$\ooalign{$F_{\lambda}$\cr$\phantom{F}{}^{(\mathbf{n})}$\cr$\phantom{F^{(\mathbf{n})}_{\lambda}}$}<\sup\Sigma_{\lambda}$.
Therefore

$|\mathfrak{N}_{\lambda}(\ooalign{$F_{\lambda}$\cr$\phantom{F}{}^{(\mathbf{n})}$\cr$\phantom{F^{(\mathbf{n})}_{\lambda}}$})-\mathfrak{N}_{\lambda}(\ooalign{$E_{\lambda}$\cr$\phantom{E}{}^{(\mathbf{n}){\scriptscriptstyle+}}$\cr$\phantom{E^{(\mathbf{n}){\scriptscriptstyle+}}_{\lambda}}$})|=|(\mathbf{m}_{\mathbf{n}}\alpha-\lfloor\mathbf{m}_{\mathbf{n}}\alpha\rfloor)-(\mathbf{n}\alpha-\lfloor\mathbf{n}\alpha\rfloor)|=|(\mathbf{m}_{\mathbf{n}}-\mathbf{n})\alpha-(\lfloor\mathbf{m}_{\mathbf{n}}\alpha\rfloor-\lfloor\mathbf{n}\alpha\rfloor)|$

for some $\mathbf{m}_{\mathbf{n}}$ such that
$\ooalign{$G_{\lambda}$\cr$\phantom{G}{}^{(\mathbf{m}_{\mathbf{n}})}$\cr$\phantom{G^{(\mathbf{m}_{\mathbf{n}})}_{\lambda}}$}\neq\varnothing$
and
$\mathrm{length}(\ooalign{$G_{\lambda}$\cr$\phantom{G}{}^{(\mathbf{n})}$\cr$\phantom{G^{(\mathbf{n})}_{\lambda}}$})<\mathrm{length}(\ooalign{$G_{\lambda}$\cr$\phantom{G}{}^{(\mathbf{m}_{\mathbf{n}})}$\cr$\phantom{G^{(\mathbf{m}_{\mathbf{n}})}_{\lambda}}$})$
and
$\ooalign{$F_{\lambda}$\cr$\phantom{F}{}^{(\mathbf{n})}$\cr$\phantom{F^{(\mathbf{n})}_{\lambda}}$}=\ooalign{$E_{\lambda}$\cr$\phantom{E}{}^{(\mathbf{m}_{\mathbf{n}}){\scriptscriptstyle-}}$\cr$\phantom{E^{(\mathbf{m}_{\mathbf{n}}){\scriptscriptstyle-}}_{\lambda}}$}$
and

$\textstyle\lVert\mathbf{m}_{\mathbf{n}}\rVert\leq\frac{1}{-C_{\mathsf{E}}}\log(\frac{1}{C(\lambda)}\mathrm{length}(\ooalign{$G_{\lambda}$\cr$\phantom{G}{}^{(\mathbf{n})}$\cr$\phantom{G^{(\mathbf{n})}_{\lambda}}$}))\mathrlap{{}\eqqcolon\kappa_{\lambda}(\mathbf{n}),}$

where
$\mathrm{length}(\ooalign{$G_{\lambda}$\cr$\phantom{G}{}^{(\mathbf{n})}$\cr$\phantom{G^{(\mathbf{n})}_{\lambda}}$})=C(\lambda)e^{-C_{\mathsf{E}}\kappa_{\lambda}(\mathbf{n})}$.
Because $\lVert\mathbf{n}\rVert\leq\kappa_{\lambda}(\mathbf{n})$, for
sufficiently small $\lambda$,

$\displaystyle\frac{|\ooalign{$F_{\lambda}$\cr$\phantom{F}{}^{(\mathbf{n})}$\cr$\phantom{F^{(\mathbf{n})}_{\lambda}}$}-\ooalign{$E_{\lambda}$\cr$\phantom{E}{}^{(\mathbf{n}){\scriptscriptstyle+}}$\cr$\phantom{E^{(\mathbf{n}){\scriptscriptstyle+}}_{\lambda}}$}|}{|\ooalign{$E_{\lambda}$\cr$\phantom{E}{}^{(\mathbf{n}){\scriptscriptstyle+}}$\cr$\phantom{E^{(\mathbf{n}){\scriptscriptstyle+}}_{\lambda}}$}-\ooalign{$E_{\lambda}$\cr$\phantom{E}{}^{(\mathbf{n}){\scriptscriptstyle-}}$\cr$\phantom{E^{(\mathbf{n}){\scriptscriptstyle-}}_{\lambda}}$}|}\geq\frac{(\frac{c}{C_{\mathsf{H}}})^{1/h}}{C(\lambda)e^{-C_{\mathsf{E}}\kappa_{\lambda}(\mathbf{n})}\lVert\mathbf{m}_{\mathbf{n}}-\mathbf{n}\rVert^{{(t-1)}/{h}}}\geq\frac{(\frac{c}{C_{\mathsf{H}}})^{1/h}}{C(\lambda)e^{-C_{\mathsf{E}}\kappa_{\lambda}(\mathbf{n})}(2\kappa_{\lambda}(\mathbf{n}))^{{(t-1)}/{h}}}.$

As a result, $\textstyle\lim_{\lambda\to
0}\tau(\Sigma_{\lambda})=\lim_{\lambda\to
0}\inf_{\begin{subarray}{l}\mathbf{n}\neq\mathbf{0}\\\
\hphantom{\mathbf{n}}\mathllap{\ooalign{$\scriptstyle
G_{\scriptscriptstyle\lambda}$\cr$\phantom{\scriptstyle
G}{}^{\scriptscriptstyle(\mathbf{n})}$\cr$\phantom{\scriptstyle
G^{\scriptscriptstyle(\mathbf{n})}_{\scriptscriptstyle\lambda}}$}}\neq\varnothing\end{subarray}}\tau(\Sigma_{\lambda},\ooalign{$G_{\lambda}$\cr$\phantom{G}{}^{(\mathbf{n})}$\cr$\phantom{G^{(\mathbf{n})}_{\lambda}}$})=+\infty$.
∎

The proof of Lemma 4.1 is a combination of three components. For simplicity,
we restrict this discussion to the case $b=1$. The first component is that the
frequency $\alpha$ satisfies a Diophantine condition. Specifically, there
exists $c>0$ and there exists $t>1$ such that

$|q\alpha-p|\geq\frac{c}{|q|^{t-1}}$

for every $\frac{p}{q}\in\mathbb{Q}$. The second component is that the
integrated density of states $\mathfrak{N}_{\lambda}$ is Hölder continuous for
small couplings. Specifically, there exists $C_{\mathsf{H}}>0$ and there
exists $h>0$ such that

$|\mathfrak{N}_{\lambda}(x)-\mathfrak{N}(y)|\leq C_{\mathsf{H}}|x-y|^{h}$

for sufficiently small $\lambda$ and for every $x,y$ and the constants
$C_{\mathsf{H}}$ and $h$ are independent of the sufficiently small $\lambda$.
The existence of $C_{\mathsf{H}}$ and $h$ is established by A. Cai, C.
Chavaudret, J. You, and Q. Zhou in 2019 [11]; see Theorem 2.4 for the
unabridged statement. The third component is that the lengths of spectral gaps
exponentially decay for large gap-labels. Specifically, there exists
$C(\lambda)>0$ and there exists $C_{\mathsf{E}}>0$ such that

$\mathrm{length}(\ooalign{$G_{\lambda}$\cr$\phantom{G}{}^{(n)}$\cr$\phantom{G^{(n)}_{\lambda}}$})\leq
C(\lambda)e^{-C_{\mathsf{E}}|n|}$

for sufficiently small $\lambda$ and for every $n\neq 0$ and the constant
$C_{\mathsf{E}}$ is independent of the sufficiently small $\lambda$ and the
constant $C(\lambda)$ satisfies $\lim_{\lambda\to 0}C(\lambda)=0$. Here
$G_{\lambda}$ $\phantom{G}{}^{(n)}$ is a spectral gap satisfying
$\mathfrak{N}_{\lambda}\upharpoonright\overline{\ooalign{$G_{\lambda}$\cr$\phantom{G}{}^{(n)}$\cr$\phantom{G^{(n)}_{\lambda}}$}\vphantom{G^{(n)}}}\equiv
n\alpha-\lfloor n\alpha\rfloor$ for every $n\neq 0$. The existence of
$C_{\mathsf{E}}$ and $C(\lambda)$ is established by M. Leguil, J. You, Z.
Zhao, and Q. Zhou in 2017 [29]; see Theorem 2.5 for the unabridged statement.
Then we combine the three components to deduce

$\frac{\mathrm{length}([E_{\lambda},F_{\lambda}])}{\mathrm{length}(\ooalign{$G_{\lambda}$\cr$\phantom{G}{}^{(n)}$\cr$\phantom{G^{(n)}_{\lambda}}$})}\geq\frac{|F_{\lambda}-E_{\lambda}|}{C(\lambda)e^{-C_{\mathsf{E}}|n|}}\geq\frac{(\frac{1}{C_{\mathsf{H}}})^{\frac{1}{h}}|\mathfrak{N}_{\lambda}(F_{\lambda})-\mathfrak{N}_{\lambda}(E_{\lambda})|{}^{\frac{1}{h}}}{C(\lambda)e^{-C_{\mathsf{E}}|n|}}$

for every $n\neq 0,E_{\lambda},F_{\lambda}\in\Sigma_{\lambda}$ with
$E_{\lambda}<F_{\lambda}$. If  $G_{\lambda}$ $\phantom{G}{}^{(n_{1})}$ is
left-adjacent to $[E_{\lambda},F_{\lambda}]$ and  $G_{\lambda}$
$\phantom{G}{}^{(n_{2})}$ is right-adjacent to $[E_{\lambda},F_{\lambda}]$,
then $\mathfrak{N}_{\lambda}(E_{\lambda})=n_{1}\alpha-\lfloor
n_{1}\alpha\rfloor$ and
$\mathfrak{N}_{\lambda}(F_{\lambda})=n_{2}\alpha-\lfloor n_{2}\alpha\rfloor$
and

$\textstyle|\mathfrak{N}_{\lambda}(F_{\lambda})-\mathfrak{N}_{\lambda}(E_{\lambda})|\geq\frac{c}{|n_{2}-n_{1}|^{t-1}}$

and

$\frac{\mathrm{length}([E_{\lambda},F_{\lambda}])}{\mathrm{length}(\ooalign{$G_{\lambda}$\cr$\phantom{G}{}^{(n)}$\cr$\phantom{G^{(n)}_{\lambda}}$})}\geq\frac{(\frac{c}{C_{\mathsf{H}}})^{{1}/{h}}}{C(\lambda)e^{-C_{\mathsf{E}}|n|}|n_{2}-n_{1}|^{{(t-1)}/{h}}}.$

Because the frequency is fixed and because the spectrum is the same for any
phase, we define $\Sigma_{\lambda}\coloneqq\Sigma_{\lambda,\alpha,\omega}$.
The spectral-thickness $\tau(\Sigma_{\lambda})$ is the infimum-length-ratio of
planks over gaps; more details can be found within subsection 2.6. Therefore
it can be similarly shown that

$\tau(\Sigma_{\lambda})\geq\inf_{\begin{subarray}{l}n\neq 0\\\
\hphantom{n}\mathllap{\ooalign{$\scriptstyle
G_{\scriptscriptstyle\lambda}$\cr$\phantom{\scriptstyle
G}{}^{\scriptscriptstyle(n)}$\cr$\phantom{\scriptstyle
G^{\scriptscriptstyle(n)}_{\scriptscriptstyle\lambda}}$}}\neq\varnothing\end{subarray}}\frac{(\frac{c}{C_{\mathsf{H}}})^{1/h}}{C(\lambda)e^{-C_{\mathsf{E}}\kappa_{\lambda}(n)}(2\kappa_{\lambda}(n))^{{(t-1)}/{h}}},$

where $\kappa_{\lambda}(n)$ is a positive real-valued function of $n$. Taking
the limit as $\lambda$ approaches zero concludes the proof of Lemma 4.1.

######  Theorem 4.2 (Lemma 3.2).

Let $H_{\lambda,\alpha,\omega}=\Delta+V_{\lambda,\alpha,\omega}$ be the almost
Mathieu operator. Fix $\alpha\in\mathbb{T}\setminus\mathbb{Q}$. Assume
$\alpha$ satisfies a Diophantine condition:

$\textstyle\alpha\in\bigcup_{\begin{subarray}{c}c>0\\\
t>1\end{subarray}}\bigcap_{\frac{p}{q}\in\mathbb{Q}}\left\\{x\in\mathbb{R}:|qx-p|\geq\frac{c}{|q|^{t-1}}\right\\}\mathrlap{{}\eqqcolon\mathsf{DC}.}$

Then $\lim_{\lambda\to 0}\tau(\sigma(H_{\lambda,\alpha,\omega}))=+\infty$.

###### Proof..

Observe $\cos(z)$ is holomorphic on $\mathbb{C}$. By Theorem 2.4 for $b=1$ and
Theorem 2.5 for $b=1$ and Lemma 4.1 for $b=1$,
$\left(\forall\alpha\in\mathsf{DC}\right)[\textstyle\lim_{\lambda\to
0}\tau(\sigma(H_{\lambda,\alpha,\omega}))=+\infty]$. ∎

### Acknowledgements

I want to thank Anton Gorodetski for his guidance and direction; this paper
would not exist in its current form without the support. I want to thank David
Damanik and Svetlana Jitormirskaya for their feedback which resulted in a more
polished version of this paper. Along with the previous people mentioned, I
want to thank Siegfried Beckus, Lingrui Ge, Ilya Kachovskiy, Abel Klein, and
Wencai Liu for the mathematics-related discussions that in some form or
another positively influenced me and positively influenced this paper.

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