1.1 Cantor spectrum for 1D Schrödinger operators

When $ d=1 $ , the most prominent example of the operator defined in (1.1) with a Cantor spectrum is the Almost Mathieu Operator (AMO) $H_{\lambda ,\alpha ,\theta }$ , defined on $\ell ^2(\mathbb {Z})$ as follows:

(1.2) $$ \begin{align} (H_{\lambda,\alpha,\theta} u)_n = u_{n+1} + u_{n-1} + 2\lambda \cos 2\pi (n\alpha + \theta) u_n. \end{align} $$

The AMO was originally introduced by Peierls [Reference Peierls62] as a model for an electron on a two-dimensional lattice subjected to a homogeneous magnetic field [Reference Harper40, Reference Rauh65]. The famous “Ten Martini Problem” [Reference Kac52, Reference Simon66] asserts that $ H_{\lambda ,\alpha ,\theta } $ has a Cantor spectrum for any $ \lambda \neq 0 $ and $ \alpha \in \mathbb {R} \setminus \mathbb {Q} $ . This was ultimately proved by Avila and Jitomirskaya [Reference Avila and Jitomirskaya8], building on earlier contributions [Reference Avila and Krikorian10, Reference Bellissard and Simon16, Reference Choi, Elliott and Yui21, Reference Helffer and Sjöstrand45, Reference Last57, Reference Puig63]. Building on methods developed for the quantitative aspects of Avila’s global theory [Reference Avila5, Reference Ge, Jitomirskaya, You and Zhou33], it was recently shown that any Type-1 operator with a triorthogonal potential has Cantor spectrum [Reference Ge, Jitomirskaya and You31, Reference Ge, Jitomirskaya and You32]. Meanwhile, Argentieri and Avila [Reference Argentieri and Avila3] proved that there exist analytic perturbations of subcritical AMO that exhibit interval spectrum. For the critical almost Mathieu operator, the “Dry Ten Martini Problem”—whether all spectral gaps are open—remains open [Reference Avila and Jitomirskaya8, Reference Avila, You and Zhou12, Reference Kac52, Reference Simon66]. A recent result by Li-Xu-Zhou [Reference Li, Xu and Zhou59] shows that any Type-1 operator with a supercritical triorthogonal potential has all gaps open, thereby partially confirming the conjecture in [Reference Ge, Jitomirskaya and You31, Reference You76].

For more general potentials $ V $ , Simon [Reference Simon66] conjectured that the operator defined in (1.1) generically exhibits a Cantor spectrum. The first foundational result establishing the existence of a Cantor spectrum was obtained by Sinai [Reference Sinai68] for large $C^2$ cosine-type potentials; this result was later reproved using different methods in [Reference Forman and VandenBoom29, Reference Wang and Zhang74]. Eliasson [Reference Eliasson25] proved that for any fixed Diophantine frequency $ \alpha \in \mathbb {R} \backslash \mathbb {Q} $ and for generic small analytic potentials, (1.1) possesses a Cantor spectrum. This result was subsequently extended to the entire subcritical regime and any irrational $ \alpha \in \mathbb {R} \backslash \mathbb {Q} $ [Reference Li and Zhou60, Reference Puig64], via global-to-local reduction [Reference Avila4, Reference Avila6, Reference Avila and Jitomirskaya9, Reference Ge30, Reference Han and Schlag39]. Goldstein and Schlag [Reference Goldstein and Schlag35] established that for generic $ \alpha \in \mathbb {R} \setminus \mathbb {Q} $ in the region of positive Lyapunov exponents, the spectrum forms a Cantor set. Avila-Bochi-Damanik [Reference Avila, Bochi and Damanik7] showed that for any fixed $ \alpha \in \mathbb {R} \setminus \mathbb {Q} $ and any generic $ V \in C^0(\mathbb {T},\mathbb {R}) $ in the $ C^0 $ -topology, the spectrum is also a Cantor set [Reference Avila, Bochi and Damanik7]. In the $ C^k $ -topology (for $ 1 \leq k \leq \infty $ or even in the analytic category), it has been proven that for generic $ \alpha \in \mathbb {R} \setminus \mathbb {Q} $ and generic $ V \in C^k(\mathbb {T},\mathbb {R}) $ , the spectrum of (1.1) is a Cantor set. For an outline of the proof, we refer to footnote 1 in [Reference Wang, Zhou and Jäger73].

1.2 Cantor spectrum for multidimensional Schrödinger operators

Historically, it was widely anticipated that the Cantor spectrum is predominantly a one-dimensional phenomenon, rarely appearing when $d \ge 2$ . A related conjecture, known as the Bethe-Sommerfeld conjecture, posits that for $ d \geq 2 $ and any periodic function $ V: \mathbb {R}^d \rightarrow \mathbb {R} $ , the spectrum of the continuous Schrödinger operator $ -\Delta + V $ contains only finitely many gaps; that is, there are no gaps at high energy levels. This conjecture has received extensive studies over the years and was proved by Parnovski [Reference Parnovski61]. For a more detailed discussion, please refer to [Reference Kuchment56, Reference Parnovski61] and their references. Additionally, the discrete version of the Bethe-Sommerfeld conjecture has also been resolved recently [Reference Embree and Fillman26, Reference Han and Jitomirskaya38]. More recently, the resolution of the Bethe-Sommerfeld conjecture has been extended to quasiperiodic (not necessarily separable) multidimensional continuous Schrödinger operators for almost all frequencies [Reference Karpeshina, Parnovski and Shterenberg53, Reference Karpeshina and Shterenberg54].

In high-energy regimes, continuous Schrödinger operators can be treated as perturbations of the free Laplacian. Thus, in the discrete framework, it is plausible to expect that when $ \lambda $ is small, the spectrum of (1.1) forms an interval. This was recently proved by Takase [Reference Takase70] under the assumption that $ V(\vec \theta ) $ is separable (i.e., $ V(\vec \theta ) = V_1(\theta _1) + \cdots + V_d(\theta _d) $ ). These findings elucidate the behavior under conditions of small potential.

However, if $ \lambda $ is large (or if the potential is large), one would still expect that the spectrum contains an interval. The first result concerning the interval spectrum was obtained by Chulaevsky and Sinai [Reference Chulaevsky and Sinai20], later reinforced by Goldstein-Schlag-Voda [Reference Goldstein, Schlag and Voda36], who constructed a class of real-analytic functions $ W $ on $ \mathbb {T}^d $ (considered to represent a generic condition) in which, if the frequency is Diophantine and $ \lambda $ is sufficiently large, the spectrum of the one-dimensional operator

(1.3) $$ \begin{align} (H_W u)_n = u_{n+1} + u_{n-1} + \lambda W(\vec\theta+n\vec\alpha) u_n, \end{align} $$

comprises a single interval. This indicates that in high-dimensional cases, if $ V(\vec \theta ) $ is degenerate or separable and $ \lambda $ is large, then the spectrum of (1.1) is likely an interval.

Based on these discussions, one might conclude that the Cantor spectrum is predominantly a one-dimensional phenomenon. However, Damanik-Fillman-Gorodetski [Reference Damanik, Fillman and Gorodetski24] constructed a class of multidimensional limit-periodic Schrödinger operators whose spectrum is a Cantor set, spurring interest in identifying a quasiperiodic example. This task is not trivial; the spectral behavior of limit-periodic Schrödinger operators diverges significantly from that of quasiperiodic operators [Reference Damanik and Fillman23], making the direct application of [Reference Damanik, Fillman and Gorodetski24]’s methods to quasiperiodic settings challenging. Nonetheless, through Aubry duality [Reference Aubry and André2] and Eliasson’s results [Reference Eliasson25], one can readily construct the long-range operator

(1.4) $$ \begin{align} (Hu)_{\boldsymbol{n}} = (Lu)_{\boldsymbol{n}} + (Vu)_{\boldsymbol{n}} = \sum_{\boldsymbol{k} \in \mathbb{Z}^d} V_{\boldsymbol{k}} u_{\boldsymbol{n} - \boldsymbol{k}} + 2\lambda \cos 2\pi(\theta + \langle \boldsymbol{n}, \vec\alpha \rangle) u_{\boldsymbol{n}}, \end{align} $$

which possesses a Cantor spectrum when $ V_{\boldsymbol {k}} $ decays exponentially fast. However, the challenge remains in the Schrödinger case. Indeed, as expressed by Damanik, Fillman, and Gorodetski [Reference Damanik, Fillman and Gorodetski24]:

In this paper, we aim to address this question by analyzing the multidimensional Almost Mathieu Operator (AMO):

(1.5) $$ \begin{align} (M_{\lambda \cos, \vec{\alpha}}\, u)({\boldsymbol{n}}) = \sum_{|\boldsymbol{m}-\boldsymbol{n}|=1} u({\boldsymbol{m}}) + 2\lambda \left( \sum_{j=1}^{d} \cos 2\pi (\theta_j + n_j \alpha_j) \right) u({\boldsymbol{n}}), \end{align} $$

as studied by Bourgain [Reference Bourgain18]. This operator is also referred to as the multidimensional Aubry-André model in the physical literature, and it has attracted significant interest due to its localization properties, which are found to be substantially more complex compared to the one-dimensional case [Reference Bordia, Lüschen, Hodgman, Schreiber, Bloch and Schneider17, Reference Johnstone, Öhberg and Duncan51, Reference Szabó and Schneider69]. Below, we present our main result:

1.3 Fractal dimensions of the spectrum of the critical AMO

Now we consider the critical AMO:

(1.6) $$ \begin{align} (H_{1,\alpha,\theta} u)_n = u_{n+1} + u_{n-1} + 2 \cos 2\pi(n\alpha + \theta) u_n, \end{align} $$

which is also known as Harper or Azbel-Hofstadter model in physical literature [Reference Avron, Osadchy and Seiler13, Reference Harper40]. When $\alpha $ is irrational, the spectrum does not depend on $\theta $ and we denote the spectrum as $\Sigma _{\alpha }$ . The spectrum is beautifully described via the Hofstadter butterfly [Reference Hofstadter48].

Regarding the finer fractal properties of $\Sigma _{\alpha }$ , it was widely believed until the mid-1990s that the box dimension $\dim _B\Sigma _{\alpha }$ is equal to $\frac {1}{2}$ for almost all $\alpha $ . Numerical and heuristic evidence supporting this conjecture can be found in [Reference Bell and Stinchcombe15, Reference Geisel, Ketzmerick and Petshel34, Reference Tang and Kohmoto71]. The upper (resp. lower) box dimension, denoted $\overline {\dim }_{B}S$ (resp. $\underline {\dim }_ BS$ ), of a bounded set $S \subseteq \mathbb {R}$ is defined as follows:

$$ \begin{align*}\overline{\dim}_{B}S = \limsup_{r \downarrow 0} \frac{\log N_r( S)}{\log(r^{-1})}, \qquad \underline{\dim}_{ B} S = \liminf_{r \downarrow 0} \frac{\log N_r(S)}{\log(r^{-1})}, \end{align*} $$

where $N_r(S)$ represents the minimum number of intervals of length r required to cover S. When $\overline {\dim }_{ B}S=\underline {\dim }_{ B}S$ , we call the common value the box dimension of S and denote by $\dim _B S.$ In 1994, Wilkinson and Austin [Reference Wilkinson and Austin75] provided numerical evidence that $\dim _B\Sigma _{\alpha } = 0.498$ for $\alpha = \frac {\sqrt {5}-1}{2}$ , thereby conjecturing that $\overline {\dim }_B\Sigma _{\alpha } < \frac {1}{2}$ for every irrational $\alpha $ . However, Jitomirskaya and Zhang [Reference Jitomirskaya and Zhang50] disproved this conjecture by demonstrating that if $\beta (\alpha )> 0$ , then $\overline {\dim }_B\Sigma _{\alpha } = 1$ . Here, $\beta (\alpha )$ measures the Liouvillean property of $\alpha $ and is defined as follows:

(1.7) $$ \begin{align} \beta(\alpha) := \limsup_{n \rightarrow \infty} \frac{\log q_{n+1}(\alpha)}{q_n(\alpha)}, \end{align} $$

where $q_n(\alpha )$ denotes the denominator of the n-th convergent of $\alpha $ . Note that $\beta (\alpha ) = 0$ for every Diophantine $\alpha $ , so in particular, $\beta =0$ for a.e. $\alpha $ .

Considering these results, a natural question arises:

If $\beta (\alpha ) = 0$ , what is the box dimension of the spectrum of the critical AMO?

In this paper, we answer this question as follows:

Theorem 1.3. For any $\delta \in (0,1)$ , there exists a dense and positive Hausdorff dimension set of $\alpha \in {\mathbb R}\backslash {\mathbb Q}$ with $\beta (\alpha )=0$ , such that the spectrum $\Sigma _\alpha $ of $H_{1,\alpha ,\theta }$ satisfies

$$ \begin{align*}0< \dim_H\Sigma_{\alpha}\leq \overline \dim_B \, \Sigma_{\alpha} \le\delta.\end{align*} $$

Here the Hausdorff dimension of a set $S \subset \mathbb {R}$ is defined as follows:

$$ \begin{align*}\dim_H S = \inf \left\{ t \in \mathbb{R^{+}} \, \big{|} \, \lim_{\epsilon \rightarrow 0} \inf_{\epsilon \text{-covers}} \sum_{n} |U_n|^t < \infty \right\}, \end{align*} $$

where an $\epsilon $ -cover of S is a family $(U_n)_n$ such that $S \subset \cup _{n=1}^{\infty } U_n$ , and each $U_n$ is an interval of length smaller than $\epsilon $ ; $|A|$ denotes the Lebesgue measure of $A\subset {\mathbb R}$ . Indeed, in recent years, there has been an increasing interest in determining the Hausdorff dimension of $\Sigma _{\alpha }$ . Around 1995, J. BellissardFootnote ¹ conjectured that there exists some $\kappa \in (0, 1/2]$ such that $\dim _H \Sigma _{\alpha } = \kappa $ for almost every $\alpha $ . Recently, B. Simon included the problem of determining the Hausdorff dimension of the spectrum of the critical almost Mathieu operator in his new list of significant unsolved problems [Reference Simon67].

Let us review the advances related to these conjectures and provide comments on Theorem 1.3. We introduce the set

$$ \begin{align*}\mathcal{L} := \{\alpha \in \mathbb{R} \setminus \mathbb{Q} : \beta(\alpha)> 0\}. \end{align*} $$

In 1994, Last proved that for a dense subset containing $\mathcal {L}$ , the Hausdorff dimension of $\Sigma _{\alpha }$ satisfies $\dim _H \Sigma _{\alpha } \leq \frac {1}{2}$ . This conclusion was recently strengthened by Jitomirskaya and Krasovsky [Reference Jitomirskaya and Krasovsky49] to include all $\alpha $ . Last and Shamis [Reference Last and Shamis58] showed that there exists a dense subset of $\mathcal {L}$ for which $\dim _H \Sigma _{\alpha } = 0$ . The results in [Reference Last and Shamis58] were further enhanced by Avila, Last, Shamis, and Zhou [Reference Avila, Last, Shamis and Zhou11], who established that for any $\alpha \in \mathcal {L}$ , $\dim _H \Sigma _{\alpha } = 0$ . However, the box (resp. Hausdorff) dimension may depend sensitively on the arithmetic properties of $\alpha $ . To illustrate this, let us fix any $\delta \in (0,1)$ and define two sets of frequencies as follows:

$$ \begin{align*} \mathscr{F}_H(\delta) := \{\alpha \in \mathbb{R} \setminus \mathbb{Q}: \dim_H \Sigma_{\alpha} \leq \delta\}; \quad \mathscr{F}_B(\delta) := \{\alpha \in \mathbb{R} \setminus \mathbb{Q} : \overline{\dim}_B \Sigma_{\alpha} \leq \delta\}. \end{align*} $$

It is evident that $\mathscr {F}_B(\delta ) \subset \mathscr {F}_H(\delta )$ , since for any set $A \subset [0,1]$ , it holds that $\dim _H A \leq \overline {\dim }_B A$ . Moreover, we have $\mathscr {F}_B(\delta ) \subsetneq \mathscr {F}_H(\delta )$ , as [Reference Avila, Last, Shamis and Zhou11] implies that $\mathcal {L} \subset \mathscr {F}_H(\delta )$ , while [Reference Jitomirskaya and Zhang50] indicates that $\mathscr {F}_B(\delta ) \cap \mathcal {L} = \emptyset $ . On the other hand, Helffer-Liu-Qu-Zhou [Reference Helffer, Liu, Qu and Zhou42] illustrated the existence of dense frequencies (with positive Hausdorff dimension) in the set $\mathbb {R} \setminus \mathcal {L}$ , such that $\dim _H \Sigma _{\alpha }> 0$ . Theorem 1.3 is a consequence of the following:

1.4 Ideas of the proof

The semiclassical analysis of $M_{\cos ,\vec {\alpha }}$ was established in [Reference Assel1]; however, the estimates provided there are too crude to yield precise spectral information. For multidimensional Schrödinger operators with separable potentials as defined in (1.5), their spectra can be expressed as Minkowski sums of one-dimensional spectra. In the limit-periodic case, Damanik, Fillman, and Gorodetski [Reference Damanik, Fillman and Gorodetski24] constructed a Cantor spectrum by analyzing a one-dimensional limit-periodic Schrödinger operator with a spectrum of lower box dimension zero; then the result follows from standard arguments about Minkowski sums of fractal sets. Their methodology, however, fails in our context due to the positivity of the Hausdorff dimension. To address this, we explicitly prove that the upper box dimension can be made arbitrarily small (Theorem 1.4), which forces the Minkowski sum to have Lebesgue measure zero (Theorem 1.1). The crux of the proof lies in Theorem 1.4, whose analysis relies on the semiclassical analysis of the almost Mathieu operator (AMO) developed by Helffer and Sjöstrand [Reference Helffer and Sjöstrand43, Reference Helffer and Sjöstrand44, Reference Helffer and Sjöstrand45], which will be shortly recalled in the appendices.

1.4.1 Structure of the spectrum

In our previous work [Reference Helffer, Liu, Qu and Zhou42], we established a lower bound for the Hausdorff dimension of the spectrum for frequencies $\alpha = [a_1, a_2, \ldots ]$ where $a_n$ is eventually sufficiently large. Our work is based on the fine covering structure of the spectrum established in [Reference Helffer and Sjöstrand43, Reference Helffer and Sjöstrand44]. In [Reference Helffer and Sjöstrand43], Helffer and Sjöstrand inductively constructed a family of nested coverings $\{\mathcal {B}_n: n \in \mathbb {N}\}$ of the spectrum. Let us fix a small $\epsilon> 0$ . The covering $\mathcal {B}_0$ consists of a single interval (or band) $B_\emptyset $ , which is approximately $[-4, 4]$ . The covering $\mathcal {B}_1$ is a disjoint family of subintervals of $B_\emptyset $ defined as follows:

$$ \begin{align*}\mathcal{B}_1 = \{B_{-p}, \ldots, B_{-1}, B_0, B_1, \ldots, B_q\},\end{align*} $$

where $B_0 = [-\epsilon , \epsilon ]$ and

(1.8) $$ \begin{align} p, q \sim a_1; \quad \frac{|B_i|}{|B_\emptyset|} \sim e^{-a_1}, \quad i \neq 0. \end{align} $$

(Here and later in the introduction, $a \sim b$ roughly means that a and b have the same order. However we warn that it is not so precise and we make it vague on purpose for the moment; the exact meaning will be made precise, see especially Definition 3.2.) See Figure 1 for an illustration. Here, $B_0$ is considered a “black box” in the sense that complete information about its internal structure is unavailable. The construction of $\mathcal {B}_2$ proceeds as follows: for each $i \neq 0$ , there exists a disjoint family of subintervals of $B_i$ given by

$$ \begin{align*}\mathcal{J}_i = \{B_{i(-p_i)}, \ldots, B_{i(-1)}, B_{i0}, B_{i1}, \ldots, B_{iq_i}\},\end{align*} $$

with $B_{i0}$ positioned in the “middle” of $B_i$ and satisfying $|B_{i0}|/|B_i| \sim \epsilon $ , such that

(1.9) $$ \begin{align} p_i, q_i \sim a_2; \quad \frac{|B_{i\ell}|}{|B_i|} \sim e^{-a_2}, \quad \ell \neq 0. \end{align} $$

Again, $B_{i0}$ is treated as a black box. The second-order covering is then defined as

$$ \begin{align*}\mathcal{B}_2 = \{B_0\} \cup \bigcup_{i \neq 0} \mathcal{J}_i.\end{align*} $$

For each interval in $\mathcal {B}_2$ that is not a black box, the above construction can be repeated. By aggregating all resulting intervals, we obtain $\mathcal {B}_3$ , and so forth.

Figure 1

The covering ${\mathcal B}_1$ and zoom-in of $B_i$ .

Figure 1 Long description

The diagram consists of two parallel horizontal sequences. The upper sequence, labeled B sub empty set, starts at B sub negative p on the far left, followed by B sub negative 2, B sub negative 1, minus epsilon, B sub 0, epsilon, B sub 1, B sub 2, B sub i, and ends at B sub q on the far right. Red squares mark each node, and two thick black rectangles are positioned at minus epsilon and epsilon. Dashed red arrows connect B sub i on the upper sequence to B sub i q i on the lower sequence, and from B sub negative p to B sub i negative p i. The lower sequence, starting at B sub i negative p i, continues with B sub i negative 2, B sub i negative 1, B sub i 0, B sub i 1, B sub i 2, and ends at B sub i q i. Each node in the lower sequence is also marked by a red square. Dashed red lines indicate correspondence between specific nodes in the upper and lower sequences.

For each n, if we disregard all black boxes in $\mathcal {B}_n$ and denote the new family as $\mathcal {B}_n^*$ , we establish a nested covering structure $\{\mathcal {B}_n^*: n \geq 0\}$ . The limit set X of this covering structure satisfies

$$ \begin{align*}X := \bigcap_n \bigcup_{B \in \mathcal{B}_n^*} B \subset \Sigma_\alpha.\end{align*} $$

Using the precise information given by (1.8) and (1.9), we can estimate the lower bound of the Hausdorff dimension of the spectrum.

However, to derive an upper bound for the upper box dimension of the spectrum, partial information about the structure of the spectrum $\Sigma _\alpha $ is not sufficient; one must understand how the black boxes evolve during each semiclassical approximation procedure. This is precisely what Helffer and Sjöstrand accomplished in [Reference Helffer and Sjöstrand45]. We will elucidate the situation concerning the first black box $B_0 = [-\epsilon , \epsilon ]$ , see Figure 2 for an illustration. In [Reference Helffer and Sjöstrand45], Helffer and Sjöstrand replaced the black box $B_0$ with a family of subintervals $\{B_0^{(i)}: i \in \mathscr {I}\}$ , for which they maintained exact control over $\# \mathscr {I}$ and the ratios $|B_0^{(i)}|/|B_\emptyset |$ , similar to (1.8) and (1.9). More specifically, we introduce $ h \sim 1/a_1$ and fix a constant $M> 1$ . Define

$$ \begin{align*}\mathscr I_{in}:=\{i\in \mathscr I: B_0^{(i)} \cap [-Mh, Mh]\ne \emptyset\} \ \text{ and }\ \ \mathscr I_{mid}:=\mathscr I\setminus \mathscr I_{in}.\end{align*} $$

Then $0\in \mathscr I_{in}$ and $B_0^{(0)}$ is at the center of $B_0$ , of size h, and is the largest subinterval. For $i\in \mathscr I_{in}\setminus \{0\}$ ,

(1.10) $$ \begin{align} |B_0^{(i)}|\sim -h/\log h \ \ \text{ and }\ \ \#\mathscr I_{in}\sim -\log h. \end{align} $$

For the bands in $B_0 \setminus [-Mh, Mh]$ , the lengths decrease progressively, ranging roughly from $-h/\log h$ to $e^{-1/h}$ . The exact description is a bit complex, we refer the reader to Definition 3.2 (vi). It is in this part that the bands exhibit typical multiscale nature. Consequently, we obtain an upgraded covering

$$ \begin{align*}\tilde{\mathcal{B}}_1 := \{B_i: i \neq 0\} \cup \{B_0^{(i)}: i\in \mathscr I_{in}\} \cup \{B_0^{(i)}: i\in \mathscr I_{mid}\}\end{align*} $$

such that for the pair $(B_\emptyset , \tilde {\mathcal {B}}_1)$ , we possess complete information regarding the cardinality of $\tilde {\mathcal {B}}_1$ and the ratios $|B|/|B_\emptyset |$ for all $B \in \tilde {\mathcal {B}}_1$ . This motivates our definition of a special type of configuration, which will be explicitly detailed in Section 3.

Figure 2

The black box $B_0$ .

Figure 2 Long description

From left to right, the horizontal axis is bounded by minus epsilon and epsilon. Green blocks represent j in I sub mid, grouped on both sides of the central region. The central region, labeled I sub in, contains yellow blocks between minus M h and M h, with boundaries B zero super i and B zero super zero. The entire interval is denoted B zero, spanning all regions. Dashed lines above the axis indicate corresponding intervals for each block. Labels B zero super j and B zero super j prime mark the boundaries of the green regions. The arrangement visually separates the mid and in sets, with color and bracket annotations clarifying group membership.

The strength of [Reference Helffer and Sjöstrand45] lies in the ability to iterate this process. The authors introduce two types. Specifically, starting from any band B in $\tilde {\mathcal {B}}_1$ with a certain type, one can repeat the aforementioned construction to obtain another pair $(B, \tilde {\mathcal {B}}_B)$ , which either behaves essentially the same as $(B_\emptyset , \tilde {\mathcal {B}}_1)$ if B has type $\mathbf {1}$ , or can be decomposed into finitely many (with uniform upper bound) pairs $\{(\hat {B}_j, \tilde {\mathcal {B}}_B^{(j)})\}$ , each of which behaves similarly to $(B_\emptyset , \tilde {\mathcal {B}}_1)$ if B has type $\mathbf {2}$ (see also Definitions B.2 and B.4). In this manner, we can construct $\tilde {{\mathcal B}}_2$ , ensuring a precise control over the cardinality of $\tilde {{\mathcal B}}_2$ and the ratios of lengths between any band and its sub-bands. This process can be continued to construct the entire family $\{\tilde {\mathcal {B}}_n: n \geq 1\}$ . With this family established, we can ultimately estimate the upper bound of the upper box dimension of the spectrum.

1.4.2 Estimate the fractal dimensions

In [Reference Helffer and Sjöstrand44, Reference Helffer and Sjöstrand45], it was established that the spectrum $\Sigma _\alpha $ is a Cantor set. Furthermore, [Reference Helffer and Sjöstrand45] suggested that this result might enable a detailed analysis of the Hausdorff dimension of $\Sigma _\alpha $ . The present work realizes this suggestion by providing a concrete study of the Hausdorff dimension, even its box dimension.

To estimate the fractal dimensions of $\Sigma _\alpha $ , we analyze the nested covering structure $\{\tilde {\mathcal {B}}_n : n \geq 1\}$ . While this structure shares similarities with the Moran sets studied in [Reference Feng, Wen and Wu28], it exhibits greater complexity. Specifically, the spectrum can be interpreted as a nonlinear, graph-directed, and highly nonhomogeneous variant of a Moran set.

Estimating the upper bound for the Hausdorff dimension is relatively straightforward due to the natural covering of the spectrum provided by each $\tilde {\mathcal {B}}_n$ . Given the precise metrical information encoded in these coverings, deriving this bound requires only standard arguments. In contrast, the upper box dimension poses significant challenges. For fixed $r> 0$ , one has to estimate $N_r(\Sigma _\alpha )$ , the minimal number of intervals of length r needed to cover $\Sigma _\alpha $ . The difficulty arises from the multiscale structure of the bands in $\tilde {\mathcal {B}}_B$ : the configuration $(B, \tilde {\mathcal {B}}_B)$ exhibits pronounced nonhomogeneity in band lengths.

A critical insight resolves this issue: by imposing an upper bound on the coefficients $a_n$ , we ensure a lower bound on the length ratio between the shortest band in $\tilde {\mathcal B}_B$ and $B.$ This constraint allows us to effectively control $N_r(\Sigma _\alpha )$ , thereby bounding the upper box dimension.

Finally, let us explain why the upper box dimension of the spectrum can be made arbitrarily small. Recall that for a self-similar set S with open set condition, the Hausdorff and box dimensions are equal and the common value s is determined through $\sum _{i=1}^k c_i^s=1$ , where $c_i$ is the i-th contraction ratio. By analogy with this, if for any small $\delta \in (0,1)$ and a typical configuration like $(B_\emptyset , \tilde {\mathcal B}_1)$ , we have

$$ \begin{align*}\sum_{B\in \tilde {\mathcal B}_1} r_B^\delta\le 1, \ \ \text{ where }\ \ r_B=\frac{|B|}{|B_\emptyset|},\end{align*} $$

then we can use a similar argument to show that the Hausdorff dimension of the spectrum is less than $\delta $ . If in addition, $\{a_n:n\in {\mathbb N}\}$ is bounded from above, then the upper box dimension of the spectrum is also less than $\delta $ . By the definition of $\tilde {\mathcal B}_1$ , we can write

$$ \begin{align*}\sum_{B\in \tilde {\mathcal B}_1} r_B^\delta=\sum_{i\ne 0}\left(\frac{|B_i|}{|B_\emptyset|}\right)^\delta+\sum_{i\in \mathscr I_{in}}\left(\frac{|B_0^{(i)}|}{|B_\emptyset|}\right)^\delta+\sum_{i\in \mathscr I_{mid}}\left(\frac{|B_0^{(i)}|}{|B_\emptyset|}\right)^\delta=: S_{out}(\delta)+S_{in}(\delta)+S_{mid}(\delta).\end{align*} $$

By (1.8) and (1.10), it is seen that for $a_1$ large enough (equivalent, for h small enough), we have $S_{out}(\delta )<1/3$ and $S_{in }(\delta )<1/3$ . But for $S_{mid}(\delta )$ , there is no easy path to get the estimate. We need to decompose the sum according to the different scales and estimate them separately. Although quite tricky, it turns out that one can choose suitable h such that $S_{mid}(\delta )<1/3.$ Consequently, we obtain the desired estimates.

The above argument assumes $a_n$ is large for all $n \geq 1$ . To extend these fractal dimension estimates to a dense set of frequencies $\alpha $ , we relax this condition: it suffices for them that there exists $m \in \mathbb {N}$ such that $a_n$ is sufficiently large for all $n \geq m$ . To achieve this, we need the full strength of [Reference Helffer and Sjöstrand44] and to extend slightly [Reference Helffer and Sjöstrand45]. For details, we refer the reader to Appendix C.

1.5 Outline of the paper

The rest of the paper is organized as follows. In Section 2, we prove Theorem 1.1 by assuming Theorem 1.4. In Section 3, we define certain abstract configurations and study their basic properties. In Section 4, we define an abstract nested covering structure and obtain the upper bound estimates of the Hausdorff and upper box dimensions of the limit set. In Section 5, we apply the result established in Section 4 to the critical AMO and prove Theorem 1.4 and Theorem 1.3. In Appendices A, B, and C we extract from [Reference Helffer and Sjöstrand43, Reference Helffer and Sjöstrand44, Reference Helffer and Sjöstrand45] what we need for the description of the nested covering structure.