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On the abominable properties of the almost Mathieu operator with well approximated frequencies

We show that some spectral properties of the almost Mathieu operator with frequency well approximated by rationals can be as poor as at all possible in the class of all one-dimensional discrete Schroedinger operators. For the class of critical coupling, we show that the Hausdorff measure of the spectrum may vanish (for appropriately chosen frequencies) whenever the gauge function tends to zero faster than logarithmically. For arbitrary coupling, we show that modulus of continuity of the integrated density of states can be arbitrary close to logarithmic; we also prove a similar result for the Lyapunov exponent as a function of the spectral parameter. Finally, we show that (for any coupling) there exist frequencies for which the spectrum is not homogeneous in the sense of Carleson, and, moreover, fails the Parreau-Widom condition. The frequencies for which these properties hold are explicitly described in terms of the growth of the denominators of the convergents.


Introduction
This paper concerns the almost Mathieu operator (1.1) (H α,λ,θ ψ)(n) = ψ(n + 1) + ψ(n − 1) + 2λ cos(2παn + θ)ψ(n), where α, λ, θ ∈ R and ψ ∈ 2 (Z → C). The almost Mathieu operator was first introduced by Peierls [59] as a Hamiltonian describing the motion of an electron on a two-dimensional lattice in the presence of a homogeneous magnetic field [41,61]. This model has been extensively studied not only because of its importance in physics [12,58], but also as a fascinating mathematical object. Indeed, by varying the parameters λ, α, and θ, one sees surprising spectral richness and it thus serves as a primary example for many interesting spectral phenomena, for example, the Cantor structure of the spectrum [8], sharp phase transitions between three spectral types [4,11], and also the universal hierarchical structure of quasiperiodic eigenfunctions [47]. It is known that the spectral properties of the almost Mathieu operator depend sensitively on the arithmetic properties of α. We illustrate this phenomenon with one archetypal example. It was found by Gordon [40] that Schrödinger operators which are well approximated by periodic ones can only have continuous spectrum. Using this result, Avron and Simon [13] showed that for well approximated α the spectrum of H α,λ,θ ,, is purely singular continuous for λ > 1. On the other hand, if α satisfies a Diophantine condition of the form then H α,λ,θ has Anderson localization for λ > 1 and a.e. θ [45].
This indicated the possibility of a phase transition between singular continuous spectrum and pure point spectrum. And indeed, in [44] Jitomirskaya introduced the arithmetic parameter where pn qn is the sequence of convergents of α, and conjectured that the spectrum is purely singular continuous if 1 ≤ λ < e β(α) , while it is pure point for (an arithmetically defined set of) a.e. θ if λ > e β(α) . This conjecture has been recently proved in [11,47].
The goal of the current paper is to show that some spectral properties of H α,λ,θ with very well approximated α can be as poor as at all possible in the class of all Schrödinger operators acting on 2 (Z) by (1.4) (Hψ)(n) = ψ(n + 1) + ψ(n − 1) + V (n)ψ(n) .
We focus on the following characteristics: the Hausdorff measure of the spectrum, regularity of the integrated density of states and of the Lyapunov exponent, homogeneity and the Parreau-Widom property of the spectrum. Now let us state the results precisely. ω t (s) = 1 log t 1 es (for t ∈ (0, ∞)) and ω t (s) = s t (for t ∈ (0, 1]) .
The ω-Hausdorff measure H ω (K) of a set K ⊂ R is by definition the quantity It is known that there exist Schrödinger operators with spectrum of zero Hausdorff dimension (i.e. zero ω t -Hausdorff measure for any t > 0), see the discussion below and also the recent work of Damanik and Fillman [30] and references therein. On the other hand, it is known (see Remark 4.2) that the spectrum of any one dimensional discrete Schrödinger operator H of the form (1.4) is of positive ω 1 -Hausdorff measure. Our first result bridges the gap by showing that the latter can not be improved, even for the almost Mathieu operator.
However, Wilkinson and Austin [75] argued that the fractal dimension of the spectrum should be sensitive to the arithmetic properties of α. The problem of determining the Hausdorff dimension of the spectrum of the critical almost Mathieu operator was also recently advertised by Simon in his lecture [64]. We are aware of the following rigorous results. Last [54] showed that dim H S(α, 1) ≤ 1 2 if lim sup q n+1 q −3 n > 0. In [55] two of the authors proved that there exists a G δ -dense set of α such that dim H S(α, 1) = 0. Recently, Helffer, Liu, Qu and Zhou [42] showed that there is also a positive Hausdorff-dimensional set of α with β(α) = 0 and dim H S(α, 1) > 0. Finally, Jitomirskaya and Krasovsky [48] showed that dim H S(α, 1) ≤ 1 2 for all α ∈ R \ Q. We also mention that Jitomirskaya and Zhang [49] proved that if β(α) > 0, then the box-counting (Minkowski) dimension of the spectrum is equal to 1. The dramatic difference between the Hausdorff and the Minkowski dimensions is a signature of the multifractal structure of the spectrum.
1.2. Regularity of the integrated density of states and of the Lyapunov exponent. Recall the following basic setup (see [36]). Let (Ω, F, P) be a probability space, let T : Ω → Ω be an invertible ergodic transformation, and let v : Ω → R be a measurable function. For each ∈ Ω construct a Schrödinger operator of the form (1.4) by taking (1.8) V (n) = v(T n ) .
Operators obtained by replacing this v by another continuous v : R/2πZ → R are called onefrequency operators.
For an operator H defined by (1.8), the integrated density of states is defined by where A (N ) E ( , ·) is the 2 × 2 matrix appearing in the second component of the N -th iterate of (1.10). In both (1.9) and (1.11) the expectation E can be replaced with an almost sure limit, and, if T is uniquely ergodic, (1.9) can even be replaced with a pointwise limit. The integrated density of states and the Lyapunov exponent are related by the formula (1.12) γ(E) = log |E − E |d N (E ) going back to the work of Thouless, Herbert and Jones (see [28]). We are interested in the uniform continuity of γ(E) and N (E). If ω is a gauge function, let be the class of uniformly continuous functions with modulus of continuity majorated by ω. Craig and Simon [29] showed using (1.12) that for any ergodic operator H (1.14) where ω 1 is as defined in (1.5) with t = 1. It is known from the work of Craig [27] and Gan-Krüger [52] that (1.14) is optimal in the class of operators (1.4) with almost periodic potentials. The operators constructed in [27] have pure point spectrum of arbitrary Lebesgue measure; those of [52] have spectrum of zero Lebesgue measure, which is consequently also singular. We also mention the recent work of Duarte-Klein-Santos [35], who constructed a class of random Schrödinger operators such that N / ∈ t>2 UC [ω t ]. In all these cases, the Lyapunov exponent γ is also non-regular, due to a lemma of Goldstein-Schlag that we reproduce in a generaliazed form as Proposition 2.2.
If ω is a gauge function satisfying (1.7), then for any λ = 0 there exists a G δ -dense set of This provides the first explicit examples of quasiperiodic Schrödinger operators with low regularity of N α,λ and γ α,λ . An interesting feature of our construction for λ < 1 is that the spectrum is purely absolutely continuous, and for all λ = 1 N α,λ is absolutely continuous [6].
1.3. Homogeneity and the Parreau-Widom property of the spectrum. Our last two results are motivated by the inverse spectral problem for quasiperiodic operators: given a compact set K ⊂ R, what are the spectral properties of the Jacobi operators whose spectrum coincides with K. In [67,68] Sodin and Yuditskii studied the inverse spectral problem in the class of Jacobi operators with almost periodic potentials. In particular, they showed that if the spectrum of a Jacobi operator is homogeneous (see Definition 1.1) and the diagonal elements of the corresponding resolvent operator are purely imaginary on the spectrum, then the operator is almost periodic.
To the best of our knowledge, all the works on the inverse spectral problem, starting from [67,68], require either homogeneity of the spectrum, meaning that it can not be too meager near any point, or the weaker Parreau-Widom property (Definition 1.1), which means that the space of analytic functions on its complement is sufficiently rich [73,74]. Therefore it is natural to ask whether the homogeneity or at least the Parreau-Widom property hold for simple quasiperiodic operators, namely to consider the above questions in the converse direction. Somewhat surprisingly, the answer is not always positive.
To state the results precisely, recall Definition 1.1. A set K ⊂ R is called homogeneous (in the sense of Carleson) if there exist 0 > 0 and 0 < τ ≤ 1 such that for any E ∈ K and for any 0 A set K ⊂ R is said to satisfy the Parreau-Widom condition if where g(z) is the Green function of the Dirichlet Laplacian in C \ K, and the sum is over the bounded connected components (a, b) of R \ K.
For continual quasiperiodic Schrödinger operators of the form H = − d 2 dx 2 + V , homogeneity of the spectrum was shown by Damanik-Goldstein-Lukic [31], provided that V is small enough and that α satisfies the Diophantine condition. For discrete quasiperiodic (one-frequency) Schrödinger operators Damanik-Goldstein-Schlag-Voda have obtained [32,Theorem H] that in the regime of positive Lyapunov exponent, for strong Diophantine α, each non-empty intersection of the spectrum with an open interval is homogeneous. Leguil-You-Zhao-Zhou [57] further proved that if α is strong Diophantine, then for a (measure-theoretically) typical real-analytic potential, the spectrum is homogeneous. Later, Liu-Shi [56] proved the spectrum is also homogeneous for weak Liouvillean frequency. Very recently, K. Tao proved [70] that the same is true for a class of Gevrey potentials when the frequency satisfies a weak Diophantine condition.
Recently, Simon [65] conjectured that the spectrum S(α, λ) of the operator H α,λ,θ is homogeneous for any λ = ±1. We disprove this conjecture in the following strong sense.
To the best of our knowledge, these are the first examples of ergodic Schrödinger operators with a spectrum of positive Lebesgue measure which is not homogeneous and does not even satisfy the Parreau-Widom condition. These examples show that a solution of the inverse spectral problem which would include the almost Mathieu operator with Liouville frequencies may require essentially new methods.
1.4. Structure of the paper, and an avalanche lemma. The proofs of the three theorems are based on the approximation of H α,λ,θ by periodic operators H p q ,λ,θ . The main arguments are developed for λ ≤ 1 and then extended to λ > 1 by Aubry-André duality.
For fixed θ, the spectrum of H p q ,λ,θ consists of q bands. Using Chambers' formula (Proposition 3.11), we show that for λ ≤ 1, the lengths of these bands are not exponentially small in q, whereas as we vary θ, the edges of a band vary by an amount of order (λ + o(1)) q . The small 1 intervals near the band edges which lie in the spectrum for some θ but not for other ones will be the main focus of our attention.
When we pass from p q to p q , the spectrum moves by a quantity which is bounded by C| p q − p q | 1 2 (see Proposition 2.8 due to Avron, van Mouche and Simon). Our assumptions on α ensure that, for an appropriately chosen sequence of approximants, this quantity is much smaller than the lengths of the intervals described above. Thus the intervals have to contain some spectrum of H p q ,λ,θ , and, eventually, of H α,λ,θ . In Proposition 2.9, possibly of independent interest, we prove a counterpart of Proposition 2.8 for the integrated density of states. It is used to show that the (θ-averaged) integrated density of states that H p q ,λ,θ and H α,λ,θ assign to these intervals is also not too small.
On the other hand, in Corollaries 3.2 and 3.4 we show that the spectrum of H p q ,λ,θ in these intervals is very meager: it is small when measured by any measure having a decent amount of regularity. This basic fact, combined with the lower bound on the amount of spectrum in the intervals as described above, eventually implies all the results of the paper. One ingredient used in the deductions and elsewhere in the paper is the Frostman lemma (Lemma 4.1).
Corollaries 3.2 and 3.4 are deduced from Propositions 3.1 and 3.3, which are the main technical results of the current paper. These propositions provide a lower bound on the Lyapunov exponent γ p q ,λ,θ (E + i ) when E lies in the intervals described above, and can be made very small. These estimates improve and generalize [55,Theorem 2], where only the case λ = 1 was considered, and much stronger approximability was assumed. These improvements are possible due to combination of several new ingredients. One of these is a new avalanche-type lemma, which we state here as it may be of independent interest: Proposition 3.5. For every 0 < c < 1 the following holds for sufficiently small 0 < b < b 0 (c). Let A j ∈ SL 2 (C) and let 0 < δ j < b, 1 ≤ j ≤ n, be such that This lemma falls into the category of avalanche principles, which deduce the global growth of a matrix product from conditions on pairs of adjacent matrices. The original avalanche principle was introduced by Goldstein-Schlag [38]; a version incorporating subsequent refinements can be found in the monograph of Duarte-Klein [34]. The current lemma, improving on [55,Theorem 3], is useful when the hyberbolicity of the terms in the matrix product is weak.
1.5. Dependency chart. The structure of the paper is illustrated by the chart in Figure 1.

Three auxiliary propositions
This section contains several auxiliary results which we will use in the sequel. In the first part, we prove two statements which we need in greater generality than we could find in the literature (their common feature is that estimates of singular integrals appear in the proof). The first one, Proposition 2.2, relates the regularity of the integrated density of states to that of the Lyapunov exponent. The argument closely follows Goldstein-Schlag [38,Lemma 10.3], who proved a similar but less general statement (e.g. their formulation applies to ω t but not to ω t ). The second one, Proposition 2.4, shows that a decently regular measure µ assigns little mass to the set of energies E for which the Lyapunov exponent evaluated at E differs significantly from its value at E + i . It is an extension of [69, Lemma 2] due to Surace, who considered the case when µ is the Lebesgue measure.
In the second part of the section, we recall a result of Avron, van Mouche and Simon [15] pertaining to the continuity of the spectrum of a one-frequency operator as a function of the frequency α (Proposition 2.8), and prove its counterpart for the integrated density of states (Proposition 2.9).
2.1. Two estimates on singular integrals. To state the results, we introduce some notation.
Let ω : (0, 1] → (0, ∞) (it will be convenient not to insist that ω is non-decreasing or continuous) and define UC [ω] as in (1.13). For any j ≥ 0 define , and extend W i ω, for i = 1, 2, 3, to (0, 1] by Remark 2.1. One can check by direct computation that for ω t and ω t be as in (1.5) (1) if t > 2, then (2) for any 0 < t ≤ 1, Let ρ be a compactly supported probability measure on R such that By the Thouless formula, if N (E) is the integrated density of states of an ergodic Schrödinger operator, then γ is the Lyapunov exponent. The regularity properties of N can be inferred from those of γ (and vice versa) with the help of Proposition 2.2. Assume ω is non-decreasing, and ρ is a compactly supported probability measure on R satisfying (2.1).
Proposition 2.2 and Remark 2.1 imply: We do not know whether this corollary is valid for t ∈ (1, 2]. For t = 1, one can construct ρ such that N ∈ UC[ω 1 ] but γ ρ | R is discontinuous [17,72]. The next proposition shows the set of E for which γ ρ (E + i ) is far from γ ρ (E) is very meager.

Proposition 2.4.
Assume ω is non-decreasing with lim →0 + ω( ) = 0. Suppose ρ is a compactly supported probability measure on R satisfying (2.1) and µ is a compactly supported probability measure such that N µ ∈ UC [ω]. For any ∈ (0, 1] and for any ξ > 0 where C µ > 0 depends only on µ. and for any x at most two terms in (2.2) are not equal to zero. Let so that for any integrable function u where * denotes convolution.
Lemma 2.5. Let u ∈ C 0 (R) be a continuous compactly supported function.
(1) If ω is non-decreasing and u ∈ UC [ω], then sup Proof. Using an appropriate smooth partition of unity, we can assume without loss of generality that diam supp u ≤ 1. Let us prove (1). Observe that φ is compactly supported, hence for sufficiently large j we have supp φ j ⊂ − 1 2 , 1 2 . Then Since φ is a Schwartz function, in particular |φ(y)| ≤ C |y| 3 for some constant C > 0. Therefore, for any j ≥ 0 we have |φ j (y)| ≤ C 2 2j |y| 3 . For k ≤ j we obtain Note that k<0φ k is compactly supported. By Bernstein's inequality (see, e.g [76, Proposition 5.2]) we obtain On the other hand, for 2 −j−1 ≤ |x − y| ≤ 2 −j we have Proof. It is sufficient to estimate (W 2 W 1 ω)(2 −j ). This is done as follows: and let us rewrite is (up to a constant) the Hilbert transform of u 1 . By (1) of Lemma 2.5, we have for any j ≥ 0 Let Φ be another Schwartz function satisfying (2.2) and Observe that where the first inequality follows from [62, Inequality after Exercise 6.7], the equality holds since the Hilbert transform is an isometry in L 2 (up to a constant), and the second inequality follows from [62, Lemma 6.12]. The last inequality holds since by the definition of Φ j (x) we get Therefore, by (2) of Lemma 2.5 and by Lemma 2.6 we conclude from (2.3) that The second statement (2) is proved by the same argument.
2.1.2. Proof of Proposition 2.4. We need the following lemma.
Lemma 2.7. Assume that ω is non-decreasing and let µ be a compactly supported probability measure such that N µ ∈ UC [ω]. Then, for any 0 < ≤ 1 where C µ > 0 is a constant that depends only on µ.
Therefore, it suffices to prove (2.4) for measures with diam supp µ ≤ 1. We can also without loss of generality let E = 0 (since the assumption is invariant under shifts) and that = 2 −j . Then we have Lemma 2.7 now implies that 2.2. Continuity estimates. Let H α,θ be a one-frequency operator: where φ : R/Z → R satisfies a uniform Lipschitz condition The following result due to Avron-van Mouche-Simon [15] shows that the set S(α) = θ σ(H α,θ ), where σ(H α,θ ) is the spectrum of H α,θ , depends continuously on α, with a quantitative estimate.
Our next proposition, possibly of independent interest, is a counterpart of this fact for the integrated density of states; it quantifies a result of Avron and Simon [65,Theorem 3.3]. We mention similar-looking results for random Schrödinger operators recently obtained in [43,66]. Let be the θ-averaged integrated density of states, (where N α,θ is the integrated density of states corresponding to H α,θ ), and let ρ α be the corresponding measure, then we have Remark 2.10. It would be interesting to know whether this result could be significantly improved in the regime of positive Lyapunov exponent (without any further assumptions on α).
Proof of Proposition 2.9. Let H L α,θ be the restriction of the operator H α,θ to a finite box {0, . . . , L− 1}. For any α , α ∈ R we have By definition we have is a sum of k − 1 matrices of rank 2, thus Therefore, using the interlacing property, we obtain Integrating over θ and taking k → ∞ gives (2.9).
For an L × L matrix A and an interval I ∈ R denote n(A; I) = Tr 1 I (A) .
since the eigenvalues of A can be shifted by (at most) distance κ. Thus if (2.7) holds, then we obtain Therefore, using (2.9) we get namely (2.8) holds.

The main estimate
The proofs of Theorems 1-3 rely on the following propositions, which are improvements and generalizations of [55,Theorem 2]. Recall that the spectrum of the periodic almost Mathieu operator H p q ,λ,θ depends on θ and denote These sets are also well defined in the case that α ∈ R \ Q, in which we have (consistently with the notation in the introduction). Propositions 3.1, 3.3 and their Corollaries 3.2 and 3.4 show that for p q sufficiently close to p q , H p q ,λ,θ has very little spectrum outside S − p q , λ . The difference between the two propositions is that the latter one has stronger assumptions (| p q − p q | = 1 q q ), and consequently a stronger conclusion.
To state the propositions precisely, we need a polynomial ∆ p q ,λ (E) (called the discriminant) which is defined in (3.22) below, and satisfies: As we recall below, S − p q , λ = ∅ for λ > 1, and which consist of energies that lie outside the spectrum σ p q , λ, θ for a sizeable fraction of θ.
Then, for any E ∈ J δ and for any ≥ exp − e rq 15000q 2 λ q 2 , the Lyapunov exponent, that corresponds to H p q ,λ,θ obeys for any θ ∈ [0, 2π) Proposition 3.1, Proposition 2.2, and the Thouless formula (1.12) will imply (see Section 3.6): Let ω : (0, 1] → R + be non-decreasing function such that lim →0 + ω( ) = 0, and let W 3 ω be as in Proposition 2.2. Let µ be a probability measure on R such that Then, in the setting of Proposition 3.1 we have If p q and p q are successive convergents of α, one can significantly relax the restriction on .
be the sequence of convergents of α. For any r > 0 and δ ∈ (0, 1) there exists n 0 = n 0 (r, α) such that the following holds. If, for some n ≥ n 0 , then for any E ∈ J δ , ≥ exp − c q n+1 δ qn , and θ ∈ [0, 2π) Similarly to Corollary 3.2, the combination of Proposition 3.3, Proposition 2.2, and the Thouless formula (1.12) implies the following: Corollary 3.4. In the setting of Proposition 3.3 we have where C 1 > 0 is a constant and | · | denotes the Lebesgue measure.
3.1. An avalanche estimate. One of the important new ingredients in the proofs of Propositon 3.1 and Propositon 3.3 is the following avalanche-type proposition (already stated in the introduction).
Proposition 3.5. For every 0 < c < 1 the following holds for sufficiently small Proof. Let u 0 ∈ C 2 be a unit vector such that By (3.5) we obtain (3.6) for j = 1. Assume that (3.6) holds for some 1 ≤ j ≤ n − 1. Then, we get Since det A j = 1 for every 1 ≤ j ≤ n, we obtain where the last inequality follows from conditions (2), (3) and from (3.7). Therefore, to show (3.6) for j + 1, it is sufficient to show that This is equivalent to showing that Since 0 < 1 − c < 1, we obtain for sufficiently small b > 0 where the last inequality holds since c > 0 is sufficiently small, therefore, cosh( c) < 1 and since by our assumption 0 < δ j < b. By assumption (1) we get where the last inequality holds since for sufficiently small ξ > 0 we have 3.2. Preliminaries. In this section, we introduce the notation and collect several (mostly) known facts which we use in the proof of Propositions 3.1 and 3.3.
3.2.1. Geometric resolvent expansions, and the Combes-Thomas estimate. Here we discuss several claims that hold true in the following general setting. Let H be one-dimensional discrete Schrödinger operator of the form (1.4). Define the restriction Define the corresponding Green functions via where δ j is the vector in 2 (Z), with entries δ j (n) that are 1 if n = j and 0 otherwise. We write  c] ) . Then, we have the following.
Finally, we recall the Combes-Thomas estimate (cf. [51, Theorem 11.2]) : There exists a numerical constant c > 0 such that for any z ∈ C \ σ(H) and any m, n ∈ Z, the Green function G of H satisfies 3.2.2. The periodic almost Mathieu operator. Consider the almost Mathieu operator H α,λ,θ , where α = p q , p, q ∈ N, and assume that p < q are relatively prime. For any j, n ≥ 1 let be the n-step and the one-step transfer matrices respectively. Denote by the trace of a one-period transfer matrix. It is a polynomial of degree q that has q real simple zeros. The θ dependence of D p q ,λ,θ (E) is described by the following formula, due to Chambers [24]. Proposition 3.11 (Chambers). If p, q are relatively prime, then: Moreover, one can see from (3.22) that if λ > 1 then S − p q , λ = ∅, and if λ ≤ 1 then as we claimed in (3.1). From the result of Choi, Elliott, and Yui [25] the set S p q , λ is the union of q closed intervals (bands), that may intersect only at the edges. In particular, ∆ p q ,λ (E) ≥ 2+2λ q at all its maxima points, and ∆ p q ,λ (E) ≤ −2 − 2λ q at all its minima points. Recall that the Lyapunov exponent is defined by By unique ergodicity, if α ∈ R \ Q, this expression does not depend on θ. In the periodic case, α = p q , it is determined by the one-period (q-step) transfer matrix as follows.

3.2.3.
Norms of transfer matrix products. To estimate the norm of transfer matrices we need the following lemma, similar to one used in the work [11]. For completeness, we recall the formulation and sketch the proof. Recall that Φ m (·, ·, θ) is the m-step transfer matrix defined by (3.20) corresponding to the operator H α,λ,θ .

3.3.
Overview, and preliminary reductions. The general strategy of the proof of Proposition 3.1 is as follows. Consider the matrix product corresponding to p q . Inside this product, we identify long stretches of indices j for which E / ∈ σ( p q , λ, θ + 2π p q j). On each such interval, we approximate the transfer matrices corresponding to p q by those corresponding to p q and use the avalanche Proposition 3.5 to ensure that the growth of the matrix product does not deteriorate too much. On the remaining intervals, we use a rough bound relying on the imaginary part of the spectral parameter. These pieces are glued together using an intermediate operator, which we define in (3.45) below. For the convenience of the reader, we record the main steps: To implement this strategy, we need some preliminary reductions. First, we observe that if , then for any θ ∈ T Indeed, the first inequality holds since by the Thouless formula (1.12) the Lyapunov exponent is an increasing function of , and the second one follows from the Combes-Thomas estimate (3.19).
Thus it suffices to prove (3.3) for E and satisfiying where c 0 is from assumption (3), which we used for the second inequality. Further, we decompose To prove (3.3), we can also assume that (3.37) = exp − e rq 15000 q 2 λ q 2 , since by the Thouless formula the Lyapunov exponent γ p q ,λ,θ (E + i ) is an increasing function of . By q-periodicity of H p q ,λ,θ we obtain Define two following sets: where the last inequality follows from the rough bound (3.39). The main step of the proof is to show that where we set Note that, the assumption (1) and the upper bound on δ in the assumption (3) guarantee that 1 ≤ l 0 ≤ q. Having (3.43) at hand, combining (3.38) and (3.42), we obtain for ≥ exp − l 0 √ δ 90 where the second inequality follows from (3.41). This concludes the proof of the proposition.

Construction of an intermediate operator.
The proof of (3.43) relies on a construction of an intermediate operator. For the rest of the argument we fix j = j k ∈ J . Consider the following operator (depending on j) where V per is a periodic potential of period q with the period determined by Therefore (3.43) is equivalent to and now we turn to the proof of this inequality. First, using (3.12) of Proposition 3.7, we have where the second inequality follows from the rough bound (3.39), and the last one holds since 3.4.1. Proof of (3.48). We may assume without loss of generality that j = 1. Indeed G [j,j+l 0 q−1] (j, j+ l 0 q − 1) corresponding to our θ is equal to G [1,l 0 q] (1, l 0 q) corresponding to θ = θ + 2π j p q , and if j lies in J corresponds to θ, then 1 lies in J corresponds to θ . For 1 ≤ m, k ≤ q, set The matrix Q −1 sq+m is the one-step transfer matrix corresponding to the intermediate problem, and Q −1 sq+k is the one-step transfer matrix that corresponds to the periodic almost Mathieu operator H p q ,λ,θ sq+1 of period q. Let

(3.49)
For any n ≥ l 0 q the intermediate problem H is periodic of period q, thus we obtain (3.50) If the assumptions of Proposition 3.5 hold, namely if for and for the vector u defined by (3.50) then, an application of Proposition 3.5 gives Using the rough bound (3.39), we obtain This concludes the proof of (3.48). It is clear that (3.51) holds for any b > 0 and δ s = δ 100 ≡ δ, thus it is left to verify (3.52), (3.53), and (3.54).
By Proposition 3.7, we have  This completes the verification of (3.54), thus we have completed the proof of Proposition 3.1.

Proof of Proposition 3.3.
Invoking the Combes-Thomas estimate as in the proof of Proposition 3.1 (3.36) we may assume that Furthermore, we only need to consider the case E ∈ J − δ , namely ∆ p q ,λ (E) < −2 + 2λ q − δ, since the case E ∈ J + δ can be dealt with similarly. By our assumption pn qn is the sequence of convergents of α, thus Unlike in the proof of Proposition 3.1 here we make use of only one intermediate problem, which is defined as follows. Let where 1 ≤ l 0 ≤ q n+1 is an integer chosen by and Let us show that the estimates (3.42) and (3.48) remain true with the current definitions if we set j = 1. We restate these estimates as follows.
(1) For any ≥ exp − l 0 √ δ 90 (2) With these estimates in hand, the proof is concluded in the same way as in Proposition 3.1.
3.7.1. Proof of (3.71). As we can see from the proof of (3.48), the key ingredients are Lemma 3.12 and Claim 3.14. Let us first verify the conditions of Lemma 3.12. Let ν = r 2 , then for any n ≥ n 0 c q 2 where η r 2 , α is given by Lemma 3.12. In the notation of Lemma 3.12, let α = p n+1 q n+1 , θ = θ (s+1)qn+j − θ sqn+j , 1 ≤ j ≤ q n . Now, assume that (3.4) holds, then for any n ≥ n 0 , we have On the other hand, by (3.69), we have By the continuity of the spectrum (Proposition 2.8) we get Thus Lemma 3.12 is applicable and we apply it in the same way as in the proof of Proposition 3.1.
Next, we need the following version of Claim 3.14.
Claim 3.15. For any E ∈ J − δ and any 1 ≤ k ≤ l 0 q n , we have Proof of Claim 3.15. As in the proof of Proposition 3.1 we may assume that θ = 0. Then, for Therefore, for any E ∈ J − δ and any 1 ≤ k ≤ l 0 q n D pn qn ,λ,θ k (E) = ∆ pn qn ,λ (E) − 2λ qn cos(θ k q n ) where the last inequality holds since 0 < λ ≤ 1.
Having this estimate at hand the proof of (3.71) is concluded in the same way as in Proposition 3.1. We omit the details. 3.7.2. Proof of (3.72). We closely follow [55, Proof of Theorem 2]. By (3.11) of Proposition 3.7 for any k < m < l To estimate the difference, first we use the second resolvent identity and obtain Using (3.73) we obtain for k ≥ l 0 q n + 1 where in the last step we used the Cauchy-Schwartz inequality. Recall that for any Therefore, (3.74), and the last inequality with the rough bound (3.39) we obtain , where the last inequality follows from the assumption that < 1 and λ ≤ 1. This concludes the proof of (3.72) and of Proposition 3.3. We omit the proof of Corollary 3.4, which is parallel to that of Corollary 3.2 (see Section 3.6).

Proof of Theorem 1
We repeatedly use Frostman's lemma, which we now state. Recall that the ω-Hausdorff content of a set K ⊂ R is defined via This is consistent with setting = ∞ in (1.6), and in particular H ω ∞ (K) ≤ H ω (K). Lemma 4.1 (Frostman, see [16]). Let ω be a gauge function. Let K ⊂ R be a compact set with positive ω-Hausdorff content, H ω ∞ (K) > 0. Then there exists a non-zero Borel measure µ ≥ 0 supported on K satisfying Vice versa, if there exists a non-zero Borel measure µ ≥ 0 supported on K satisfying (4.1), then H ω ∞ (K) > 0. Remark 4.2. Lemma 4.1 and the result of Craig and Simon [29] imply that H ω 1 (σ(H)) ≥ H ω ∞ (σ(H)) > 0 for any ergodic Schrödinger operator H. Using a result of Bourgain and Klein [21] in place of [29], we see that the same conclusion holds for any discrete Schrödinger operator.
From Proposition 2.8, we have where on the right hand side we have the Minkowski sum By (4.2) and (4.3), the set S(α, 1)\S P k Q k , 1 is contained in the union of (at most) 2Q k intervals of length ≤ Ce − (β(α)−r)Q k 2 , since S P k Q k , 1 has at most Q k bands. Therefore, by the assumption (2), for sufficiently large k we obtain since t > 2. By our assumptions (1)-(2) above, the measure µ restricted to the set S P k Q k , 1 , which we denote by µ k = µ S P k Q k , 1 , obeys the following: 100t , we obtain for sufficiently large k Thus we can apply Corollary 3.2 with p q = P k−1 Q k−1 , p q = P k Q k , δ = e −νQ k−1 and get using Remark 2.1 Since t > 2 and 0 < ν < (t − 2)r, we obtain

By [55, equation (5.3), Lemma 5.1], we obtain
therefore the set S P k Q k , 1 \ J δ is contained in the union of (at most) Q k−1 intervals of total length at most 2eδ where the last inequality holds by the choice of δ. Therefore, since t > 2. Combining the third condition (3) on µ k with the estimates (4.5) and (4.6), we obtain a contradiction.

4.2.
Proof of Theorem 1 (2). Let pn qn → α be the sequence of convergents of α. We show that H ω (S(α, 1)) = 0 if the denominators q n grow sufficiently fast. We construct such α via its convergents by induction: for any fixed n 0 the terms q 1 , . . . , q n 0 can be chosen in an arbitrary way. If q 1 , . . . , q n , n > n 0 , are given, we find x n > 0 such that for any 0 < x < x n (4.7) ω(x) log 1 x ≤ e −2qn , and we choose where C 1 > 0 is the constant from Corollary 3.4. It is clear that the set of α thus constructed is G δ -dense. For each n, we choose δ = e − qn 2 , then by the continuity of the spectrum(Proposition 2.8), we can construct a cover of S(α, 1) as follows: To estimate H ω (S(α, 1)), one can proceed as follows. First note that X n is a union of (at most) 2q n+1 intervals of length ≤ 12 √ 2 α − p n+1 q n+1 ≤ 1 √ q n+2 (for sufficiently large n). By (4.8) we conclude ω , thus the contribution of these intervals to (1.6) is On the other hand, by Corollary 3.4 we have This set can be covered by (at most) q n+1 + q n ≤ 2q n+1 intervals that contribute to (1.6) at most 2q n+1 ω exp − q n+1 2C 1 qn e −qn . Observe that by (4.8) we have q n+1 ≥ 2C 1 e qn log 1 xn for x n defined by (4.7), namely exp − q n+1 2C 1 qn e −qn ≤ x n . Therefore, (4.7) implies that Finally, by [55, equation (5.3), Lemma 5.1], we obtain J c δ is a union of q n intervals, thus its contribution to (1.6) is where the convergence follows by the assumption (1.7). This concludes the whole proof.
In this section, we construct a family of intervals F τ (0 < τ ≤ 1 2 ) contained in S p q , λ but disjoint from S − p q , λ , as illustrated in Figure 2. Clearly, these intervals will be found close to the edges of one of the bands of S p q , λ . We prove that these intervals are not too small, both in the sense of Lebesgue measure (see Proposition 5.1): and spectrally (see Lemma 5.3): where ρ p q ,λ is defined as in (2.6). Using Proposition 2.9, we shall deduce from (5.4) that also While (5.3) will play a rôle in the proof of Theorem 2-(3), Theorem 2-(1) and (2) will follow by comparing (5.5) with Corollaries 3.2 and 3.4, which tell us that regular measures can not assign large mass to F τ ⊂ J δ . A somewhat similar argument will be used in the proof of Theorem 3.
The construction goes as follows. First, for −1 ≤ τ ≤ 1 let These sets interpolate between S −1 Then F τ will eventually be defined as the "largest" among the 2q intervals , · · · , q} (a formal and slightly more careful definition is given before (5.8) below).
It is not hard to see that, for any q ≥ 1 and any x ≥ 0, T q (1 + x) ≤ exp(q 2 x).
Proof of Proposition 5.
where the second inequality follows from the Mean Value Theorem and the last inequality holds since 1 + τ λ q ≤ 2. This finishes the proof of the first statement of the proposition.
To prove the second part, we first note that since 1 1+ ≤ 1 − 2 for any 0 ≤ < 1, and then conclude from (5.7) Thus we get where the last inequality holds since for every τ ≥ τ we have I j τ,ξ ⊂ I j τ ,ξ . As proved in [15], for any 0 < λ < 1, therefore, there exist 1 ≤ j 0 ≤ q and ξ ∈ {+1, −1} such that I j 0 −1,ξ ⊂ S − p q , λ such that Thus we conclude that Let 1 ≤ j 0 ≤ q, ξ ∈ {+1, −1}, be as in the second part of Proposition 5.1. For 0 < τ ≤ 1 2 let Now we show that the mass assigned to F τ by the (θ-averaged) integrated density of states is not too small.
It remains to prove the claim. The inequality (6.3) ensures that the assumptions of Propositioin 3.1 hold for δ = λ q and p q = Pm Qm when m > k. Therefore, if N α,λ ∈ UC [ω t ], then Corollary 3.2 and part (i) of Remark 2.1 imply that where J λ q is the set J δ defined by (3.2) for δ = λ q . On the other hand, by the continuity of the spectrum (Proposition 2.8) Hence S(α, λ)\S Pm Qm , λ can be covered by (at most) 2Q m intervals of length ≤ 12 where the last inequality follows from (6.3). Thus, Letting m → ∞ and recalling that t > 2 we obtain (6.5).
7. Proof of Theorem 3 7.1. Non-homogeneity of the spectrum. We deduce Theorem 3-(1) from the following Proposition 7.2, which involves a relaxed notion of homogeneity which we now introduce.
Definition 7.1. Let ω(s) be a gauge function. A set S ⊂ R is called ω-homogeneous if there exist κ > 0 and 0 > 0 such that for any E ∈ S and for any 0 < ≤ 0 , we have the spectrum S(α, λ) is not ω 2 -homogeneous as well, thus not homogeneous.
Proof of Proposition 7.2. We start similarly to the proof of Proposition 6.1. Choose r > 0 sufficietly small to ensure that β(α) − r > 3 2 | log λ|. Then choose a sequence P k Q k → α satisfying (6.3), namely α − P k Q k ≤ e −(β(α)−r)Q k . For sufficiently large k, we have Q k ≥ q 0 (r, α), and the choice of r guarantees that p q = P k Q k satisfies We shall use the sets and note that (E − , E + ) ⊂ F 1 2 ⊂ J λ q . By Corollary 3.2 (and Remark 2.1) applied to the Lebesgue measure with ω 1 ( ) = so that (W 3 ω 1 )( ) ≤ C , we obtain By the continuity of the spectrum (Proposition 2.8), we have Since the set S(α, λ) \ S Pm Qm , λ is contained in the union of (at most) 2Q m intervals and since α − Pm Qm ≤ e −(β(α)−r)Qm , we conclude that The second term tends to zero as Q m → ∞, thus we conclude that for any κ > 0 and sufficiently large q. Thus the spectrum is not ω 2 -homogeneous.

7.2.
Failure of Parreau-Widom condition. By (5.2) we only need to prove Theorem 3-(2) for 0 < λ ≤ 1. Note that a Parreau-Widom set is always of positive Lebesgue measure [26, pp.6] thus in the case λ = 1, β(α) > 0 Theorem 3-(2) follows from results of Avila-Krikorian [10], Last [54] that the measure of the spectrum |S(α, λ)| = 0 for any irrational α. Therefore, from this point we assume that 0 < λ < 1. According to [63,Theorem 1.15] the Green function g(z) of the Dirichlet Laplacian in C\S(α, λ) is equal to the Lyapunov exponent γ α,λ (z) that corresponds to H α,λ,θ . To prove that the Parreau-Widom condition fails, we shall use Proposition 3.1 to show that at most of the points in the intervals F τ which we constructed above, the Lyapunov exponent is not too small (≥ const q −2 λ q 2 ). On the other hand, we show that the spectrum of H α,λ,θ in F τ has many (≥ const q −3 λ −(1+r)q 2 ) gaps. Therefore the contribution of F τ to the sum (1.16) is lower-bounded by a quantity which diverges as q → ∞.
To implement this plan, we first recall that β(α) + 3 log λ > 0, therefore there exists r > 0 and a sequence P k Q k → α such that Let F 1 2 be as defined in (5.8) for p q = P k Q k , then by (5.8) we have Using the continuity of the spectrum (Proposition 2.8) and (7.3) we obtain for m > k that for any E ∈ F 1 2 there exists E ∈ S Pm Qm , λ such that (3+r)q 2 . , k = 1, 2, · · · 9.
On the other hand, we can apply Proposition 3.1 to It yields that for any E ∈ J λ q , where J λ q is the set J δ defined by (3.2) for δ = λ q , and for any θ γ Pm Qm ,λ,θ (E + i ) ≥ δ 9600 q 2 λ q 2 = λ q 2 C 1 q 2 .
In particular, the same lower bound holds true for the θ-averaged Lyapunov exponentγ Pm Qm ,λ (E+i ). Applying Proposition 2.4 to the Lebesgue measure with ω 1 ( ) = so that (W 3 ω 1 )( ) ≤ C as in Remark 2.1, with ξ = λ q 2 C 1 q 2 , we obtain Since |j 5 | = 25λ (3+r)q 2 ≥ C 2 q 2 λ q 2 , (7.8) implies that there exists E = E Pm Qm , j ∈ j 5 such that Thus, we get using (7.6) (7.9) This holds for Pm Qm for any m > k. By compactness we can choose a subsequence Pm l Qm l → α, Q m l > q, such that for any interval j ∈ J q the sequence of points E Pm l Qm l , j converges to a number which we denote by E(α, j).
Since by (3.27) the Lyapunov exponent is equal to zero on the spectrum, we obtain that E(α, j) / ∈ S(α, λ). Since j 5 is closed we conclude that E(α, j) ∈ j 5 , and by (7.7) we obtain that different energies E(α, j) lie in different gaps of H α,λ,θ (in particular, all these energies are distinct). From (7.10) we obtain as in (7.9) j∈Jq γ α,λ (E (α, j)) ≥ λ In particular, the left-hand side of (1.16) is bounded from below by this expression. Since this holds for any q = Q k , we get that this sum is not bounded, therefore the Parreau-Widom condition fails.
We construct a sequence pn qn of convergents of α (arising from the continuous fraction expansion) as follows. For any fixed n 0 the terms q 1 , . . . , q n 0 can be chosen in an arbitrary way. If q 1 , . . . , q n , n > n 0 , are given, we find x n > 0 such that for any 0 < x < x n where we set δ = λ qn , and we choose (8.2) q n+1 ≥ max q 6 n λ −2qn , C q n λ qn log 5 x n , where C > 0 is the constant from Lemma 8.1 below. Denote the set of α ∈ R \ Q for which the sequence of convergents satisfies (8.2) by Ω(λ). Clearly, this set is G δ -dense in R.