arXiv:2110.07974v1 [math-ph] 15 Oct 2021

ON THE ABOMINABLE PROPERTIES OF THE ALMOST MATHIEUOPERATOR WITH WELL APPROXIMATED FREQUENCIES

ARTUR AVILA, YORAM LAST, MIRA SHAMIS, AND QI ZHOU

Abstract. We show that some spectral properties of the almost Mathieu operator with frequencywell approximated by rationals can be as poor as at all possible in the class of all one-dimensionaldiscrete Schrodinger operators. For the case of critical coupling, we show that the Hausdorffmeasure of the spectrum may vanish (for appropriately chosen frequencies) whenever the gaugefunction tends to zero faster than logarithmically. For arbitrary coupling, we show that modulusof continuity of the integrated density of states can be arbitrary close to logarithmic; we also provea similar result for the Lyapunov exponent as a function of the spectral parameter. Finally, weshow that (for any coupling) there exist frequencies for which the spectrum is not homogeneousin the sense of Carleson, and, moreover, fails the Parreau–Widom condition. The frequencies forwhich these properties hold are explicitly described in terms of the growth of the denominators ofthe convergents.

1. Introduction

This paper concerns the almost Mathieu operator

(1.1) (Hα,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 byPeierls [59] as a Hamiltonian describing the motion of an electron on a two-dimensional lattice inthe presence of a homogeneous magnetic field [41, 61]. This model has been extensively studied notonly 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 thusserves as a primary example for many interesting spectral phenomena, for example, the Cantorstructure of the spectrum [8], sharp phase transitions between three spectral types [4, 11], and alsothe universal hierarchical structure of quasiperiodic eigenfunctions [47].

It is known that the spectral properties of the almost Mathieu operator depend sensitively onthe arithmetic properties of α. We illustrate this phenomenon with one archetypal example. Itwas found by Gordon [40] that Schrodinger operators which are well approximated by periodicones can only have continuous spectrum. Using this result, Avron and Simon [13] showed that forwell approximated α the spectrum of Hα,λ,θ,, is purely singular continuous for λ > 1. On the otherhand, if α satisfies a Diophantine condition of the form

(1.2) infp∈Z|nα− p| ≥ γ

|n|τ, n ∈ Z \ 0 ,

then Hα,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

(1.3) β(α) = lim supq→∞

1

qlog min

p

∣∣∣∣α− p

q

∣∣∣∣ = lim supn→∞

log qn+1

qn,

Date: October 18, 2021.1

arX

iv:2

110.

0797

4v1

[m

ath-

ph]

15

Oct

202

1

2 ARTUR AVILA, YORAM LAST, MIRA SHAMIS, AND QI ZHOU

wherepnqn

is the sequence of convergents of α, and conjectured that the spectrum is purely

singular continuous if 1 ≤ λ < eβ(α), while it is pure point for (an arithmetically defined set of)a.e. θ if λ > eβ(α). This conjecture has been recently proved in [11, 47].

The goal of the current paper is to show that some spectral properties of Hα,λ,θ with very wellapproximated α can be as poor as at all possible in the class of all Schrodinger operators actingon `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, regularityof 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.

1.1. Hausdorff measure of the spectrum. Let ω : [0, 1] → [0,∞) be a gauge function, i.e. anon-decreasing function with ω(0) = 0. The main examples are

(1.5) ωt(s) =1

logt 1es

(for t ∈ (0,∞)) and ωt(s) = st (for t ∈ (0, 1]) .

The ω-Hausdorff measure Hω(K) of a set K ⊂ R is by definition the quantity

Hω(K) = limε→0Hωε (K) ,

where

(1.6) Hωε (K) = inf

∞∑j=1

ω(bj − aj)∣∣ K ⊂

∞⋃j=1

(aj, bj), bj − aj ≤ ε

.

It is known that there exist Schrodinger operators with spectrum of zero Hausdorff dimension (i.e.zero ωt-Hausdorff measure for any t > 0), see the discussion below and also the recent work ofDamanik 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 Schrodinger operator H of the form (1.4) is ofpositive ω1-Hausdorff measure. Our first result bridges the gap by showing that the latter can notbe improved, even for the almost Mathieu operator.

Theorem 1. For any α ∈ R \Q, let S(α, λ) be the spectrum of Hα,λ,θ. The following holds:

(1) If α ∈ R\Q is such that β(α) > 0, then Hωt(S(α, 1)) = 0 for any t > 2,with ωt as in (1.5).(2) If ω is any gauge function decaying faster than ω1, i.e. such that

(1.7) lims→+0

ω(s) log1

s= 0 ,

then there exists a Gδ-dense set of α ∈ R \Q for which Hω(S(α, 1)) = 0.

To put this result in context, recall that [10, 54] for any irrational α the spectrum S(α, 1) isa Cantor set of zero Lebesgue measure. A conjecture attributed to Thouless [75] asserts that itsHausdorff dimension equals 1

2:

dimH S(α, 1)def= sup

t > 0 | Hωt(S(α, 1)) > 0

=

1

2.

However, Wilkinson and Austin [75] argued that the fractal dimension of the spectrum should besensitive to the arithmetic properties of α. The problem of determining the Hausdorff dimensionof the spectrum of the critical almost Mathieu operator was also recently advertised by Simon inhis lecture [64].

We are aware of the following rigorous results. Last [54] showed that dimH S(α, 1) ≤ 12

iflim sup qn+1q

−3n > 0. In [55] two of the authors proved that there exists a Gδ-dense set of α

3

such that dimH S(α, 1) = 0. Recently, Helffer, Liu, Qu and Zhou [42] showed that there is also apositive Hausdorff-dimensional set of α with β(α) = 0 and dimH S(α, 1) > 0. Finally, Jitomirskayaand Krasovsky [48] showed that dimH S(α, 1) ≤ 1

2for all α ∈ R \ Q. We also mention that

Jitomirskaya and Zhang [49] proved that if β(α) > 0, then the box-counting (Minkowski) dimensionof the spectrum is equal to 1. The dramatic difference between the Hausdorff and the Minkowskidimensions 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 bean invertible ergodic transformation, and let v : Ω→ R be a measurable function. For each $ ∈ Ωconstruct a Schrodinger operator of the form (1.4) by taking

(1.8) V$(n) = v(T n$) .

The almost Mathieu operator is obtained by takingΩ = R/2πZ , α ∈ R \QΩ = θ′ ∈ R/2πZ : θ′ − θ ∈ 2π

qZ , α = p

q∈ Q

T (θ) = θ + 2πα , and v(θ) = 2λ cos θ .

Operators obtained by replacing this v by another continuous v : R/2πZ → R are called one-frequency operators.

For an operator H$ defined by (1.8), the integrated density of states is defined by

(1.9) N (E) = limN→∞

1

NE$number of eigenvalues ofH [0,...,N−1]

$ in (−∞, E] ,

where H[0,...,N−1]$ denotes the restriction of the operator H$ to the interval [0, . . . , N − 1] ⊂ Z.

For each E ∈ R, we associate with H$ the Schrodinger cocycle

(1.10) AE : Ω× R2 → Ω× R2, AE($, x) =

(T$,

(E − v($) −1

1 0

)x

),

and define the Lyapunov exponent

(1.11) γ(E) = limN→∞

1

NE$ log ‖A(N)

E ($, ·)‖ ,

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 integrateddensity of states and the Lyapunov exponent are related by the formula

(1.12) γ(E) =

∫log |E − E ′|dN (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

(1.13) UC[ω] =

f : R→ C | sup

0<|E−E′|≤1

|f(E)− f(E ′)|ω(|E − E ′|)

<∞

be the class of uniformly continuous functions with modulus of continuity majorated by ω. Craigand Simon [29] showed using (1.12) that for any ergodic operator H$

(1.14) N ∈ UC[ω1] ,


where ω1 is as defined in (1.5) with t = 1. It is known from the work of Craig [27] and Gan–Kruger[52] that (1.14) is optimal in the class of operators (1.4) with almost periodic potentials. Theoperators 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 therecent work of Duarte-Klein-Santos [35], who constructed a class of random Schrodinger operatorssuch that N /∈

⋃t>2 UC [ωt]. 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.In the context of the almost Mathieu operator, the following was known. In the critical case

(λ = 1), it follows from [19, Section 8] and [55] that there exist irrational α for which the integrateddensity of states Nα,1 /∈

⋃t>0 UC [ωt]. For arbitrary λ 6= 0, it was pointed out by Avila and

Jitomirskaya[7], Nα,λ /∈⋃t>0 UC [ωt] for Baire-generic α. These results stand in contrast with what

is known for Diophantine frequencies: it was shown in [4], following earlier works of Goldstein–Schlag, Avila–Jitomirskaya and Bourgain [7, 18, 38, 39] (in which stronger conditions on α wereimposed), that if β(α) = 0, then Nα,λ ∈ UC[ω 1

2] for any λ 6= 1. The cited works, as well as [1, 22],

are applicable to more general quasiperiodic operators. We also mention UC[ω 12] continuity is sharp

for general quasi-periodic operators [60], the density of optimal (pointwise) t−Holder continuitywith 1

2< t < 1 was recently proved in the subcritical regime [53].

Theorem 1 (combined with Frostman’s lemma stated as Lemma 4.1 below) implies that (1.14)is optimal for the critical case λ = 1. The next result shows that (1.14) is optimal for arbitraryλ 6= 0.

Theorem 2. Consider the almost Mathieu operator Hα,λ,θ with α ∈ R \Q.

(1) If e−2β(α)

3 < λ < e2β(α)

3 , then (a) Nα,λ /∈⋃t>3 UC [ωt] and (b) γα,λ /∈

⋃t>4 UC [ωt] .

(2) If e−β(α)2 < λ < e

β(α)2 , then (a) Nα,λ /∈

⋃t>2 UC [ωt] and (b) γα,λ /∈

⋃t>3 UC [ωt] .

(3) If ω is a gauge function satisfying (1.7), then for any λ 6= 0 there exists a Gδ-dense set ofα ∈ R \Q such that (a) Nα,λ /∈ UC [ω], and (b) γα,λ /∈

⋃t>2 UC [ωt] .

This provides the first explicit examples of quasiperiodic Schrodinger operators with low regu-larity of Nα,λ and γα,λ. An interesting feature of our construction for λ < 1 is that the spectrumis purely absolutely continuous, and for all λ 6= 1 Nα,λ is absolutely continuous [6].

1.3. Homogeneity and the Parreau-Widom property of the spectrum. Our last two re-sults are motivated by the inverse spectral problem for quasiperiodic operators: given a compactset K ⊂ R, what are the spectral properties of the Jacobi operators whose spectrum coincideswith K. In [67, 68] Sodin and Yuditskii studied the inverse spectral problem in the class of Jacobioperators with almost periodic potentials. In particular, they showed that if the spectrum of aJacobi operator is homogeneous (see Definition 1.1) and the diagonal elements of the correspondingresolvent 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 analyticfunctions on its complement is sufficiently rich [73, 74]. Therefore it is natural to ask whetherthe 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, theanswer 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 ε0 > 0and 0 < τ ≤ 1 such that for any E ∈ K and for any 0 < ε ≤ ε0

(1.15) |(E − ε, E + ε) ∩K| ≥ τε .

5

A set K ⊂ R is said to satisfy the Parreau-Widom condition if

(1.16)∑(a,b)

maxE∈(a,b)

g(E) <∞,

where g(z) is the Green function of the Dirichlet Laplacian in C \ K, and the sum is over thebounded connected components (a, b) of R \K.

Remark 1.2. Every homogeneous set satisfies the Parreau–Widom property, while the converseimplication is not true – see Jones and Marshall [50, pp. 297––298].

For continual quasiperiodic Schrodinger operators of the form H = − d2

dx2+ V , homogeneity of

the spectrum was shown by Damanik-Goldstein-Lukic [31], provided that V is small enough andthat α satisfies the Diophantine condition. For discrete quasiperiodic (one-frequency) Schrodingeroperators Damanik-Goldstein-Schlag-Voda have obtained [32, Theorem H] that in the regime ofpositive Lyapunov exponent, for strong Diophantine α, each non-empty intersection of the spec-trum with an open interval is homogeneous. Leguil-You-Zhao-Zhou [57] further proved that if α isstrong Diophantine, then for a (measure-theoretically) typical real-analytic potential, the spectrumis homogeneous. Later, Liu-Shi [56] proved the spectrum is also homogeneous for weak Liouvilleanfrequency. Very recently, K. Tao proved [70] that the same is true for a class of Gevrey potentialswhen the frequency satisfies a weak Diophantine condition.

Recently, Simon [65] conjectured that the spectrum S(α, λ) of the operator Hα,λ,θ is homogeneousfor any λ 6= ±1. We disprove this conjecture in the following strong sense.

Theorem 3. Assume α ∈ R \Q. Then we have the following:

(1) If e−2β(α)

3 < λ < e2β(α)

3 , then S(α, λ) is not homogeneous.

(2) If e−β(α)3 < λ < e

β(α)3 , then S(α, λ) does not satisfy the Parreau-Widom condition.

To the best of our knowledge, these are the first examples of ergodic Schrodinger operators witha spectrum of positive Lebesgue measure which is not homogeneous and does not even satisfy theParreau–Widom condition. These examples show that a solution of the inverse spectral problemwhich would include the almost Mathieu operator with Liouville frequencies may require essentiallynew methods.

1.4. Structure of the paper, and an avalanche lemma. The proofs of the three theoremsare based on the approximation of Hα,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–Andre duality.For fixed θ, the spectrum of H p

q,λ,θ consists of q bands. Using Chambers’ formula (Proposi-

tion 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 small1

intervals near the band edges which lie in the spectrum for some θ but not for other ones will bethe main focus of our attention.

When we pass from pq

to pq, the spectrum moves by a quantity which is bounded by C|p

q− p

q| 12

(see Proposition 2.8 due to Avron, van Mouche and Simon). Our assumptions on α ensure that, foran appropriately chosen sequence of approximants, this quantity is much smaller than the lengthsof the intervals described above. Thus the intervals have to contain some spectrum of H p

q,λ,θ, and,

eventually, of Hα,λ,θ. In Proposition 2.9, possibly of independent interest, we prove a counterpartof 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 Hα,λ,θ assign to these intervals is also not too small.

1note that the intervals grow as λ→ 1− 0, and for λ = 1 they cover most of the spectrum


On the other hand, in Corollaries 3.2 and 3.4 we show that the spectrum of H pq,λ,θ in these

intervals is very meager: it is small when measured by any measure having a decent amount ofregularity. This basic fact, combined with the lower bound on the amount of spectrum in theintervals as described above, eventually implies all the results of the paper. One ingredient usedin 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 technicalresults of the current paper. These propositions provide a lower bound on the Lyapunov exponentγ pq,λ,θ(E + iε) when E lies in the intervals described above, and ε can be made very small. These

estimates improve and generalize [55, Theorem 2], where only the case λ = 1 was considered, andmuch stronger approximability was assumed. These improvements are possible due to combinationof several new ingredients. One of these is a new avalanche-type lemma, which we state here as itmay be of independent interest:

Proposition 3.5. For every 0 < c < 1 the following holds for sufficiently small 0 < b < b0(c).Let Aj ∈ SL2(C) and let 0 < δj < b, 1 ≤ j ≤ n, be such that

(1) δj+1 ≤ δj + bδ3/2j , 1 ≤ j ≤ n− 1,

(2) ‖Aj+1 − Aj‖ ≤ bδj, 1 ≤ j ≤ n− 1,(3) |TrAj| ≥ 2 + (1− b)δj, 1 ≤ j ≤ n,

Then, for any vector u0 ∈ C2 such that ‖A1u0‖ ≥ exp((1− c)√δ1)‖u0‖,

(1.17) ‖An · · ·A1u0‖ ≥ exp

((1− c)

n∑j=1

√δj

)‖u0‖.

This lemma falls into the category of avalanche principles, which deduce the global growth ofa matrix product from conditions on pairs of adjacent matrices. The original avalanche principlewas introduced by Goldstein–Schlag [38]; a version incorporating subsequent refinements can befound 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.

2. 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 Lyapunovexponent. The argument closely follows Goldstein–Schlag [38, Lemma 10.3], who proved a similarbut less general statement (e.g. their formulation applies to ωt but not to ωt). The second one,Proposition 2.4, shows that a decently regular measure µ assigns little mass to the set of energiesE for which the Lyapunov exponent evaluated at E differs significantly from its value at E+ iε. Itis an extension of [69, Lemma 2] due to Surace, who considered the case when µ is the Lebesguemeasure.

In the second part of the section, we recall a result of Avron, van Mouche and Simon [15] pertain-ing 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

7

Figure 1. Dependency Chart

(1) (W1ω)(2−j) =∑

0≤k≤j 2−2(j−k)ω(2−k),

(2) (W2ω)(2−j) =∑

0≤k≤j 2−(j−k)ω(2−k) +∑

k>j ω(2−k),

(3) (W3ω)(2−j) =∑

0≤k≤j 2−2(j−k)ω(2−k) + j∑∞

l=1 lω(2−jl),

and extend Wiω, for i = 1, 2, 3, to (0, 1] by

(Wiω)(ε) = (Wiω)(2−j), 2−j−1 < ε ≤ 2−j.

Remark 2.1. One can check by direct computation that for ωt and ωt be as in (1.5)

(1) if t > 2, then

W1 ωt ≤ Ct ωt, W2 ωt ≤ Ct ωt−1, W3 ωt ≤ Ct ωt−1 .

(2) for any 0 < t ≤ 1,

Wi ωt ≤ Ct ωt, i = 1, 2, 3.

Let ρ be a compactly supported probability measure on R such that

(2.1) infE∈R

∫log− |E − E ′|dρ(E ′) > −∞.

Set

(1) N (E) = Nρ(E) = ρ(−∞, E] for E ∈ R,(2) γ(z) = γρ(z) =

∫log |z − E|dρ(E) for z ∈ C.


By the Thouless formula, if N (E) is the integrated density of states of an ergodic Schrodingeroperator, then γ is the Lyapunov exponent. The regularity properties of N can be inferred fromthose of γ (and vice versa) with the help of

Proposition 2.2. Assume ω is non-decreasing, and ρ is a compactly supported probability measureon R satisfying (2.1).

(1) If Nρ ∈ UC [ω], then the restriction of γρ to R satisfies γρ|R ∈ UC [W2ω].(2) If γρ|R ∈ UC [ω], then Nρ ∈ UC [W2ω].

Proposition 2.2 and Remark 2.1 imply:

Corollary 2.3. For any t > 2, we have

(1) if Nρ ∈ UC[ωt] , then γρ|R ∈ UC[ωt−1];(2) if γρ|R ∈ UC[ωt], then Nρ ∈ UC[ωt−1].

We do not know whether this corollary is valid for t ∈ (1, 2]. For t = 1, one can construct ρsuch that N ∈ UC[ω1] but γρ|R is discontinuous [17, 72].

The next proposition shows the set of E for which γρ(E + iε) is far from γρ(E) is very meager.

Proposition 2.4. Assume ω is non-decreasing with limε→0+ ω(ε) = 0. Suppose ρ is a compactlysupported probability measure on R satisfying (2.1) and µ is a compactly supported probabilitymeasure such that Nµ ∈ UC [ω]. For any ε ∈ (0, 1] and for any ξ > 0

µE ∈ R| |γρ(E + iε)− γρ(E)| ≥ ξ ≤ Cµ(W3ω)(ε)

ξ,

where Cµ > 0 depends only on µ.

2.1.1. Proof of Proposition 2.2. The proof of Proposition 2.2 relies on the Littlewood-Paley de-composition (see e.g. [62, Lemma 8.3]). We fix a Schwartz function φ the Fourier transform φ ofwhich is compactly supported in R \ 0, and such that

(2.2)∞∑

j=−∞

φ(2−jx) ≡ 1,

and for any x at most two terms in (2.2) are not equal to zero. Let

φj(x) = 2jφ(2jx),

so that for any integrable function u

u =∑j

u ∗ φj ,

where ∗ denotes convolution.

Lemma 2.5. Let u ∈ C0(R) be a continuous compactly supported function.

(1) If ω is non-decreasing and u ∈ UC [ω], then supj≥0

‖u∗φj‖∞(W1ω)(2−j)

<∞.

(2) If supj≥0

‖u∗φj‖∞ω(2−j)

<∞, then u ∈ UC [W2ω].

9

Proof. Using an appropriate smooth partition of unity, we can assume without loss of generalitythat diam suppu ≤ 1. Let us prove (1). Observe that φ is compactly supported, hence forsufficiently large j we have suppφj ⊂

[−1

2, 1

2

]. Then

|(u ∗ φj)(x)| =∣∣∣∣∫ (u(x− y)− u(x))φj(y)dy

∣∣∣∣≤∑k≤j

∫2−k−1<|y|≤2−k

|u(x− y)− u(x)||φj(y)|dy +

∫|y|≤2−j−1

|u(x− y)− u(x)||φj(y)|dy

≤∑k≤j

ω(2−k)

∫2−k−1<|y|≤2−k

|φj(y)|dy + ω(2−j−1)

∫|y|≤2−j−1

|φj(y)|dy.

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 |φj(y)| ≤ C22j |y|3 . For k ≤ j we obtain∫

2−k−1<|y|≤2−k|φj(y)|dy ≤ 2−k

C

22j2−3(k+1)= 8C2−2(j−k),

where 2−k is the length of the interval 2−k−1 < |y| ≤ 2−k. Also,∫|y|≤2−j−1

|φj(y)|dy =

∫|y|≤ 1

2

|φ(y)|dy ≤ C.

Now we prove (2). Let us write

u =∞∑

k=−∞

u ∗ φk =

(∑k<0

+∑k≥0

)u ∗ φk ≡ u− + u+, so that u− =

(u ∗∑k<0

φk

)∧= u

∑k<0

φk.

Note that∑

k<0 φk is compactly supported. By Bernstein’s inequality (see, e.g [76, Proposition5.2]) we obtain

‖(u−)′‖∞ =

∥∥∥∥∥(∑k≥0

u ∗ φk

)′∥∥∥∥∥∞

≤ C

∥∥∥∥∥u ∗∑k≥0

φk

∥∥∥∥∥∞

<∞.

On the other hand, for 2−j−1 ≤ |x− y| ≤ 2−j we have

|u+(x)− u+(y)| ≤∑k≥0

|(u ∗ φk)(x)− (u ∗ φk)(y)|.

For k ≥ j we obtain

|(u ∗ φk)(x)− (u ∗ φk)(y)| ≤ 2‖u ∗ φk‖∞ ≤ Cuω(2−k).

For 0 ≤ k ≤ j we get

|(u ∗ φk)(x)− (u ∗ φk)(y)| ≤ |x− y|‖(u ∗ φk)′‖∞ ≤ C|x− y|‖u ∗ φk‖∞2k

≤ C2−j+k‖u ∗ φk‖∞ ≤ Cu2−(j−k)ω(2−k).

Lemma 2.6. Assume that ω is non-decreasing. Then W2W1ω ≤ 4W2ω.


Proof. It is sufficient to estimate (W2W1ω)(2−j). This is done as follows:

(W2W1ω)(2−j) =∑k≤j

2−(j−k)(W1ω)(2−k) +∑k>j

(W1ω)(2−k)

=∑m≤j

ω(2−m)2−2(j−m)

( ∑m≤k≤j

2j−k +∑k>j

2−2(k−j)

)+∑m>j

ω(2−m)∑k≥m

2−2(k−m)

≤∑m≤j

ω(2−m)2−2(j−m)(2j−m+1 + 1) + 2∑m>j

ω(2−m)

≤ 4

(∑m≤j

ω(2−m)2−(j−m) +∑m>j

ω(2−m)

).

Proof of Proposition 2.2. Suppose that supp ρ ⊂ [−A,A]. For E /∈ [−A− 1, A+ 1]

|γ′ρ(E)| =∣∣∣∣∫ 1

E − E ′d ρ(E ′)

∣∣∣∣ ≤ 1.

Thus it suffices to estimate |γρ(E1) − γρ(E2)| for E1, E2 ∈ [−A − 1, A + 1]. Choose a smoothcompactly supported function χ ∈ C∞0 so that χ ≥ 0 and the restriction of χ to the interval[−A− 2, A+ 2] χ|[−A−2,A+1] ≡ 1. Denote by

u1 = Nρχ, u2 = Nρ(1− χ),

and let us rewrite

γρ(E) =

∫log |E − E ′|d ρ(E ′)

=

∫log |E − E ′|du1 (E ′) +

∫log |E − E ′|du2 (E ′) ≡ v1(E) + v2(E).

Then the derivative v′2 is bounded in [−A− 1, A+ 1], whereas u1 ∈ C0(R) and

v′1(E) =

∫1

E − E ′du1(E ′) = u1(E)

is (up to a constant) the Hilbert transform of u1. By (1) of Lemma 2.5, we have for any j ≥ 0

‖u1 ∗ φj‖∞ ≤ C(W1ω)(2−j).

Let Φ be another Schwartz function satisfying (2.2) and Φ|supp φ ≡ 1. Define Φj(x) = 2jΦ(2jx)satisfying Φj ∗ φj = φj for all j. Then we have

(2.3) ‖u1 ∗ φj‖∞ = ‖u1 ∗ Φj ∗ φj‖∞ = ‖Φj ∗ u1 ∗ φj‖∞ ≤ ‖Φj‖1‖u ∗ φj‖∞.

Observe that

suppΦj ⊂ supp Φj ⊂ [−C2j, C2j] \ [−c2j, c2j] .

Then

‖Φj‖1 ≤ C2−j2‖Φj‖2 = C ′2−

j2‖Φj‖2 ≤ C ′′|Φj‖1 ≤ C ′′′,

where the first inequality follows from [62, Inequality after Exercise 6.7], the equality holds sincethe Hilbert transform is an isometry in L2 (up to a constant), and the second inequality followsfrom [62, Lemma 6.12]. The last inequality holds since by the definition of Φj(x) we get∫

|Φj|dx =

∫|Φ|dx ≤ Const .

11

Therefore, by (2) of Lemma 2.5 and by Lemma 2.6 we conclude from (2.3) that

u1 ∈ UC[W2W1ω] ⊂ UC[W2ω] .

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 probabilitymeasure such that Nµ ∈ UC [ω]. Then, for any 0 < ε ≤ 1

(2.4) supE′∈R

∫R

log

(1 +

ε2

(E − E ′)2

)dµ(E) ≤ Cµ(W3ω)(ε),

where Cµ > 0 is a constant that depends only on µ.

Proof. Let µ = c1µ1 + c2µ2 + · · ·+ ckµk, 1 ≤ k ≤ ∞, where diam suppµk ≤ 1 and ck ≥ 0 for everyk, with

∑k ck ≡ 1. Then,∫

log

(1 +

ε2

(E − E ′)2

)dµ(E) =

∑k≥1

ck

∫log

(1 +

ε2

(E − E ′)2

)dµk(E).

Therefore, it suffices to prove (2.4) for measures with diam suppµ ≤ 1. We can also without lossof generality let E ′ = 0 (since the assumption is invariant under shifts) and that ε = 2−j. Thenwe have∫

log

(1 +

ε2

E2

)dµ(E) ≤ 2

∫|E|≤2−j

log2−j

|E|dµ(E) +

∫2−j≤|E|≤1

2−2j

E2dµ(E)

≤ C∞∑l=1

∫2−j(l+1)≤|E|≤2−jl

jldµ(E) + C

j−1∑k=0

∫2−k−1≤|E|≤2−k

2−2(j−k)dµ(E)

≤ Cµ

(∞∑l=1

jl ω(2−jl) +

j−1∑k=0

2−2(j−k)ω(2−k)

)≤ Cµ(W3ω)(2−j).

Proof of Proposition 2.4. Denote Λ = E ∈ R | |γρ(E + iε)− γρ(E)| ≥ ξ, then by Chebyshev’sinequality and Fubini’s theorem, we obtain

µ(Λ) ≤ 1

ξ

∫Λ

|γρ(E + iε)− γρ(E)|dµ(E)

≤ 1

2ξ

∫Λ

dµ(E)

∫R

log

(1 +

ε2

(E − E ′)2

)dρ(E ′)

=1

2ξ

∫R

dρ(E ′)

∫Λ

log

(1 +

ε2

(E − E ′)2

)dµ(E)

≤ 1

2ξsupE′

∫R

log

(1 +

ε2

(E − E ′)2

)dµ(E).

Lemma 2.7 now implies that

µ E ∈ R | |γρ(E + iε)− γρ(E)| ≥ ξ ≤ 1

2ξCµ(W3ω)(ε).


2.2. Continuity estimates. Let Hα,θ be a one-frequency operator:

(2.5) (Hα,θψ)(n) = ψ(n+ 1) + ψ(n− 1) + φ(2παn+ θ)ψ(n) ,

where φ : R/Z→ R satisfies a uniform Lipschitz condition

‖φ‖Lip = max|φ(x)− φ(y)||x− y|

<∞ .

The following result due to Avron–van Mouche–Simon [15] shows that the set S(α) =⋃θ σ(Hα,θ),

where σ(Hα,θ) is the spectrum of Hα,θ, depends continuously on α, with a quantitative estimate.

Proposition 2.8 (Avron–van Mouche–Simon [15]). Let Hα,θ be as in (2.5). If |α−α′| is sufficientlysmall, then

distH(S(α′), S(α)) ≤ 6(‖φ‖Lip|α− α′|)12 ,

where the Hausdorff distance between two sets A,B ⊂ R is defined as

distH(A,B) = max(supE∈A

dist(E,B), supE∈B

dist(E,A)) = max(supE∈A

infE′∈B|E − E ′|, sup

E∈BinfE′∈A|E − E ′|) .

Our next proposition, possibly of independent interest, is a counterpart of this fact for theintegrated density of states; it quantifies a result of Avron and Simon [65, Theorem 3.3]. Wemention similar-looking results for random Schrodinger operators recently obtained in [43, 66].

Let

N α(E) =

∫dθNα,θ(E)

be the θ-averaged integrated density of states, (where Nα,θ is the integrated density of statescorresponding to Hα,θ), and let ρα be the corresponding measure,

(2.6) ρα(E1, E2] = N α(E2)−N α(E1) .

Proposition 2.9. Let Hα,θ be as in (2.5). For any [r−, r+] ⊂ R, α, α′ ∈ R, L ≥ 1, if

(2.7) κ ≥ 2π‖φ‖Lip|α′ − α|L,then we have

(2.8) ρα′ [r−, r+] ≥ ρα[r− + κ, r+ − κ]− 4

L.

Remark 2.10. It would be interesting to know whether this result could be significantly improvedin the regime of positive Lyapunov exponent (without any further assumptions on α).

Proof of Proposition 2.9. Let HLα,θ be the restriction of the operator Hα,θ to a finite box 0, . . . , L−

1. For any α′, α ∈ R we have

‖HLα′,θ −HL

α,θ‖op ≤ 2π‖φ‖Lip|α′ − α|L ≤ κ .

First, let us show that for any interval I ∈ R

(2.9)

∣∣∣∣ρα(I)−∫dθ

1

LTr 1I(H

Lα,θ)

∣∣∣∣ ≤ 2

L.

By definition we have

ρα(I) = limk→∞

1

kL

∫dθ

1

LTr 1I(H

kLα,θ) .

The matrix

HkLα,θ −

k−1⊕j=0

HLα,θ+jLα

13

is a sum of k − 1 matrices of rank 2, thus

rank

[HkLα,θ −

k−1⊕j=0

HLα,θ+jLα

]≤ 2(k − 1) ≤ 2k .

Therefore, using the interlacing property, we obtain∣∣∣∣∣ 1

kLTr 1I(H

kLα,θ)−

1

k

k−1∑j=0

1

LTr 1I(H

Lα,θ+jLα)

∣∣∣∣∣ ≤ 2

L.

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 1I(A) .

Then, for any L× L matrix A with ‖A− A‖op ≤ κ for some κ > 0

n(A; [r−, r+]) ≥ n(A; [r− + κ, r+ − κ]) ,

since the eigenvalues of A can be shifted by (at most) distance κ. Thus if (2.7) holds, then weobtain ∫

dθ1

L

[n(HL

α′,θ; [r−, r+])− n(HLα,θ; [r− + κ, r+ − κ])

]≥ 0 .

Therefore, using (2.9) we get

ρα′ [r−, r+]− ρα[r− + κ, r+ − κ] = ρα′ [r−, r+]−∫dθ

1

Ln(HL

α′,θ; [r−, r+])

+

∫dθ

1

L

[n(HL

α′,θ; [r−, r+])− n(HLα,θ; [r− + κ, r+ − κ])

]+

∫dθ

1

Ln(HL

α,θ; [r− + κ, r+ − κ])− ρα[r− + κ, r+ − κ] ≥ − 2

L+ 0− 2

L,

namely (2.8) holds.

3. The main estimate

The proofs of Theorems 1-3 rely on the following propositions, which are improvements andgeneralizations of [55, Theorem 2]. Recall that the spectrum of the periodic almost Mathieuoperator H p

q,λ,θ depends on θ and denote

S

(p

q, λ

)=⋃θ

σ

(p

q, λ, θ

), S−

(p

q, λ

)=⋂θ

σ

(p

q, λ, θ

).

These sets are also well defined in the case that α ∈ R \Q, in which we have

S(α, λ) = S−(α, λ) = σ(α, λ, θ)

(consistently with the notation in the introduction). Propositions 3.1, 3.3 and their Corollaries 3.2

and 3.4 show that for pq

sufficiently close to pq, H p

q,λ,θ has very little spectrum outside S−

(pq, λ)

. The

difference between the two propositions is that the latter one has stronger assumptions (|pq− p

q| =

1qq

), and consequently a stronger conclusion.


To state the propositions precisely, we need a polynomial ∆ pq,λ(E) (called the discriminant)

which is defined in (3.22) below, and satisfies:

σ

(p

q, λ, θ

)=E ∈ R

∣∣ ∣∣∣∆ pq,λ(E)− 2λq cos(θ q)

∣∣∣ ≤ 2.

As we recall below, S−

(pq, λ)

= ∅ for λ > 1, and

(3.1) S−

(p

q, λ

)=E : |∆ p

q,λ(E)| ≤ 2− 2λq

if λ ≤ 1. This leads us to define, for λ ≤ 1, a family of sets Jδ ⊂ R \ S−

(pq, λ)

:

(3.2) Jδ =E ∈ R

∣∣ |∆ pq,λ(E)| > 2− 2λq + δ

, δ > 0 ,

which consist of energies that lie outside the spectrum σ(pq, λ, θ

)for a sizeable fraction of θ.

Proposition 3.1. Let 0 < λ ≤ 1 and α ∈ R \Q. For any r > 0 there exist q0 = q0(r, α) ∈ N and

c0 = c0(r, α) > 0 such that the following holds. Assume that pq, pq∈ Q, δ ∈ (0, 1) are such that

(1)∣∣∣pq − α∣∣∣ , ∣∣∣ pq − α∣∣∣ < δe−rq ,

(2) q > q ≥ q0 ,(3) exp(−e rq2 ) < δ ≤ c0(r, α) q2λq .

Then, for any E ∈ Jδ and for any ε ≥ exp(− erq

15000q2λq2

), the Lyapunov exponent, that corresponds

to H pq,λ,θ obeys for any θ ∈ [0, 2π)

(3.3) γ pq,λ,θ(E + iε) ≥ δ

9600q2λq2

.

Proposition 3.1, Proposition 2.2, and the Thouless formula (1.12) will imply (see Section 3.6):

Corollary 3.2. Let ω : (0, 1] → R+ be non-decreasing function such that limε→0+ ω(ε) = 0, andlet W3 ω be as in Proposition 2.2. Let µ be a probability measure on R such that

µ(a, b) ≤ C ω(b− a), b− a ≤ 1 .

Then, in the setting of Proposition 3.1 we have

µ

(S

(p

q, λ

)⋂Jδ

)≤ Cµ q

2 λq2

δ(W3 ω)

[exp

(− erq

15000q2λq2

)].

If pq

and pq

are successive convergents of α, one can significantly relax the restriction on ε.

Proposition 3.3. Let 0 < λ ≤ 1, α ∈ R \ Q and letpnqn

be the sequence of convergents of α.

For any r > 0 and δ ∈ (0, 1) there exists n0 = n0(r, α) such that the following holds. If, for somen ≥ n0,

(3.4) qn+1δ > erqn ,

then for any E ∈ Jδ, ε ≥ exp(− c qn+1 δ

qn

), and θ ∈ [0, 2π)

γ pn+1qn+1

,λ,θ(E + iε) ≥ c δ

q2n

.

15

Similarly to Corollary 3.2, the combination of Proposition 3.3, Proposition 2.2, and the Thoulessformula (1.12) implies the following:

Corollary 3.4. In the setting of Proposition 3.3 we have∣∣∣∣S (pn+1

qn+1

, λ

)⋂Jδ

∣∣∣∣ ≤ C q2n

δexp

(−qn+1δ

C1 qn

),

where C1 > 0 is a constant and | · | denotes the Lebesgue measure.

3.1. An avalanche estimate. One of the important new ingredients in the proofs of Proposi-ton 3.1 and Propositon 3.3 is the following avalanche-type proposition (already stated in theintroduction).

Proposition 3.5. For every 0 < c < 1 the following holds for sufficiently small 0 < b < b0(c). LetAj ∈ SL2(C) and let 0 < δj < b, 1 ≤ j ≤ n, be such that

(1) δj+1 ≤ δj + bδ32j , 1 ≤ j ≤ n− 1,

(2) ‖Aj+1 − Aj‖ ≤ bδj, 1 ≤ j ≤ n− 1,(3) |TrAj| ≥ 2 + (1− b)δj, 1 ≤ j ≤ n,

Then, for any vector u0 ∈ C2 such that ‖A1u0‖ ≥ exp((1− c)√δ1)‖u0‖,

‖An · · ·A1u0‖ ≥ exp

((1− c)

n∑j=1

√δj

)‖u0‖.

Remark 3.6. One can check that b0(1/2) ≥ 1/10.

Proof. Let u0 ∈ C2 be a unit vector such that

(3.5) ‖A1u0‖ ≥ exp((1− c)√δ1).

Denote Bj = Aj · · ·A1, 1 ≤ j ≤ n. It is sufficient to show that for every 1 ≤ j ≤ n

(3.6) ‖Bju0‖ ≥ exp((1− c)√δj)‖Bj−1u0‖.

By (3.5) we obtain (3.6) for j = 1. Assume that (3.6) holds for some 1 ≤ j ≤ n− 1. Then, we get

exp(−(1− c)√δj)‖Bju0‖ ≥ ‖Bj−1u0‖ = ‖A−1

j AjAj−1 · · ·A1u0‖= ‖A−1

j Bju0‖.(3.7)

Since detAj = 1 for every 1 ≤ j ≤ n, we obtain

Aj + A−1j = (TrAj)Id2×2.

Therefore, we get

‖Bj+1u0‖ = ‖Aj+1Bju0‖ = ‖[(Aj+1 − Aj) + (Aj + A−1j )− A−1

j ]Bju0‖≥ ‖(Aj + A−1

j )Bju0‖ − ‖Aj+1 − Aj‖‖Bju0‖ − ‖A−1j Bju0‖

≥ (2 + (1− b)δj − bδj − exp(−(1− c)√δj))‖Bju0‖,

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

2 + (1− 2b)δj − exp(−(1− c)√δj) ≥ exp((1− c)

√δj+1).

This is equivalent to showing that

(1− 2b)δj ≥ (e(1−c)√δj + e−(1−c)

√δj − 2) + (e(1−c)

√δj+1 − e(1−c)

√δj).


Since 0 < 1− c < 1, we obtain for sufficiently small b > 0

e(1−c)√δj + e−(1−c)

√δj − 2 ≤ 2

(1 +

(1− c)2

2δj +

(1− c)4

4!δ2j cosh(c)− 1

)< ((1− c)2 + b)δj,

where the last inequality holds since c > 0 is sufficiently small, therefore, cosh(c) < 1 and since byour assumption 0 < δj < b. By assumption (1) we get√

δj+1 ≤√δj(1 + b

√δj) ≤

√δj

(1 +

b

2

√δj

),

namely,

(3.8)√δj+1 −

√δj ≤

b

2δj.

Using (3.8) we obtain for δj ≤ ξ ≤ δj+1

e(1−c)√δj+1 − e(1−c)

√δj ≤ e(1−c)

√ξ(√

δj+1 −√δj

)≤ e(1−c)

√ξ b

2δj ≤ bδj,

where the last inequality holds since for sufficiently small ξ > 0 we have e(1−c)√ξ ≤ 2. Therefore,

since (1 − 2b)δj ≥ ((1 − c)2 + b)δj, namely, (1 − c)2 + 4b < 1 for sufficiently small b, we concludethe proof.

3.2. Preliminaries. In this section, we introduce the notation and collect several (mostly) knownfacts 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 sev-eral claims that hold true in the following general setting. Let H be one-dimensional discreteSchrodinger operator of the form (1.4). Define the restriction H [a,b] of H to an interval [a, b] ⊂ Zas follows. (

H [a,b]ψ)

(n) = φ(n) ,

where

φ(n) =

ψ(n+ 1) + ψ(n− 1) + V (n) if a < n < b,ψ(a+ 1) + V (a) if n = a > −∞,ψ(b− 1) + V (b) if n = b < +∞ .

Define the corresponding Green functions via

(3.9) G[a,b](n,m; z) = 〈δm, (H [a,b] − z)−1δn〉,

where δj is the vector in `2(Z), with entries δj(n) that are 1 if n = j and 0 otherwise. We writeG[a,b](n,m; z) ≡ G[a,b](n,m). Then we have (see, e.g. [51, Theorem 5.20]).

Proposition 3.7. Let a, b, c ∈ Z, a ≤ c ≤ b ≤ ∞, z 6∈(σ(H [a,b])

⋃σ(H [a,c])

). Then, we have the

following.

(1) If a ≤ m ≤ c < n ≤ b, then

G[a,b](m,n) = G[a,c](m, c)G[a,b](c+ 1, n) ,(3.10)

G[a,b](n,m) = G[a,b](n, c)G[c+1,b](c+ 1,m) .(3.11)

(2) If a ≤ m, n ≤ c, then

G[a,b](m,n)−G[a,c](m,n) = G[a,b](m, c+ 1)G[a,c](c, n) .(3.12)

17

Claim 3.8. Let

−∞ ≤ a ≤ i < c1 < c2 < · · · cM < j ≤ b ≤ ∞ .

Then

G[a,b](i, j) = G[a,c1](i, c1)G[a,b](c1 + 1, c2)

G[c2+1,c3](c2 + 1, c3)G[c2+1,b](c3 + 1, c4) · · ·G[cM−1+1,cM ](cM−1 + 1, cM)G[cM−1+1,b](cM + 1, j),

where G[·,·](·, ·) is the corresponding Green’s function defined in (3.9).

Proof. By (3.10) of Proposition 3.7, we have

(3.13) G[a,b](i, j) = G[a,c](i, c)G[a,b](c+ 1, j) .

Now deduce from (3.11) of Proposition 3.7

(3.14) G[a,b](i, j) = G[a,b](i, c)G[c+1,b](c+ 1, j).

Using (3.13), we obtain

(3.15) G[a,b](i, j) = G[a,c1](i, c1)G[a,b](c1 + 1, j),

and using (3.14), we obtain

(3.16) G[a,b](c1 + 1, j) = G[a,b](c1 + 1, c2)G[c2+1,b](c2 + 1, j).

An additional use of (3.13) gives

(3.17) G[c2+1,b](c2 + 1, j) = G[c2+1,c3](c2 + 1, c3)G[c2+1,b](c3 + 1, j).

Plugging (3.17) into (3.16) and the resulting expression into (3.15), we obtain

G[a,b](i, j) = G[a,c1](i, c1)G[a,b](c1 + 1, c2)

G[c2+1,c3](c2 + 1, c3)G[c2+1,b](c3 + 1, j) .

Continuing this way (now expand G[c2+1,b](c3 + 1, j)) and so on, we obtain the desired claim.

Finally, we recall the Combes–Thomas estimate (cf. [51, Theorem 11.2]) :

Proposition 3.9 (Combes–Thomas). There exists a numerical constant c > 0 such that for anyz ∈ C \ σ(H) and any m,n ∈ Z, the Green function G of H satisfies

(3.18) |G(n,m; z)| ≤ 1

κexp(−cκ|m− n|) , where κ = min(1, dist(z, σ(H))) .

Remark 3.10. For the case of ergodic Schrodinger operators, (3.18) and the general relation

γ(z) = − limn→∞

E$1

nlog |G(1, n; z)| , z /∈ σ(H)

(or (3.38) below) imply:

(3.19) γ(z) ≥ cmin(1, dist(z, σ(H))) .


3.2.2. The periodic almost Mathieu operator. Consider the almost Mathieu operator Hα,λ,θ, whereα = p

q, p, q ∈ N, and assume that p < q are relatively prime. For any j, n ≥ 1 let

(3.20)

Φn(E,α, θ) = Tn(E) · · ·T2(E)T1(E) , Tj(E) = Tj(E,α, θ) =

(E − 2λ cos(2παj + θ) −1

1 0

)be the n-step and the one-step transfer matrices respectively. Denote by

(3.21) D pq,λ,θ(E) = Tr

(Φq

(E,

p

q, θ

)),

the trace of a one-period transfer matrix. It is a polynomial of degree q that has q real simplezeros. 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:

(3.22) D pq,λ,θ(E) = ∆ p

q,λ(E)− 2λq cos θq,

where ∆ pq,λ(E) ≡ D p

q,λ, π

2q(E).

From this formula

(3.23) S

(p

q, λ

)=E : |∆ p

q,λ(E)| ≤ 2 + 2λq

.

Moreover, one can see from (3.22) that if λ > 1 then S−

(pq, λ)

= ∅, and if λ ≤ 1 then

(3.24) S−

(p

q, λ

)=E : |∆ p

q,λ(E)| ≤ 2− 2λq

as we claimed in (3.1). From the result of Choi, Elliott, and Yui [25] the set S

(pq, λ)

is the union

of q closed intervals (bands), that may intersect only at the edges. In particular, ∆ pq,λ(E) ≥ 2+2λq

at all its maxima points, and ∆ pq,λ(E) ≤ −2− 2λq at all its minima points.

Recall that the Lyapunov exponent is defined by

(3.25) γα,λ,θ(E) = limn→∞

1

nln ‖Φn(E,α, θ)‖ .

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.

(3.26) γ pq,λ,θ(E) = lim

n→∞

1

nln

∥∥∥∥Φn

(E,

p

q, θ

)∥∥∥∥ =1

qln Spr

(Φq

(E,

p

q, θ

)),

where Spr (·) denotes the spectral radius.If α ∈ R \ Q, E ∈ S(α, λ), the Lyapunov exponent that corresponds to the almost Mathieu

operator Hα,λ,θ is given by [20, Corollary 2]

(3.27) γ(E) = max0, ln |λ| .

19

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

Lemma 3.12. Let α ∈ R \ Q, 0 < λ ≤ 1, ν > 0. Then, there exist η = η(ν, α) > 0 and

q0 = q0(ν, α) > 0 such that for every |α′ − α|, |pq− α| < η, |ε| < η, |θ| < η, and for every E ∈ R

with dist (E, S(α, λ)) < η, we have for any θ ∈ T and for any q ≥ q0,∥∥∥∥Φq

(E + iε,

p

q, θ

)− Φq

(E,

p

q, θ

)∥∥∥∥ ≤ |ε|eνq ,(3.28) ∥∥∥∥Φq (E + iε, α′, θ)− Φq

(E + iε,

p

q, θ

)∥∥∥∥ ≤ ∣∣∣∣α′ − p

q

∣∣∣∣ eνq ,(3.29) ∥∥∥Φq (E + iε, α′, θ)− Φq

(E + iε, α′, θ + θ

)∥∥∥ ≤ |θ|eνq.(3.30)

The main ingredient in the proof of Lemma 3.12 is the following lemma.

Lemma 3.13. Let α ∈ R \ Q, 0 < λ ≤ 1, ν > 0. Then, there exist η = η(ν, α) > 0 and m0 =m0(ν, α) > 0 such that for any |α′ − α| < η, |ε| < η and for every E ′ with dist (E ′, S(α, λ)) < η,we have for any |m| ≥ m0

(3.31) supθ∈T‖Φm(E ′ + iε, α′, θ)‖ ≤ eνm.

Proof. Since 0 < λ ≤ 1, by (3.27), we have γα,λ(E) = 0 for any E ∈ S(α, λ). By continuity of theLyapunov exponent [20, 46], for any ν > 0 there exists ε0 = ε0(ν) > 0, such that if |ε| ≤ ε0, thenγα,λ(E + iε) ≤ ν

3for any E ∈ S(α, λ). The rotation of the circle by α ∈ R \Q is uniquely ergodic

and the sequence ln ‖Φm(θ, α, E)‖m is subadditive, therefore by a result of Furman [37] we obtainthe following. For any E ∈ S(α, λ) there exists m0(E, ν, α) such that for any m ≥ m0(E, ν, α)

supθ∈T

1

mln ‖Φm(E,α, θ)‖ < 2ν

3.

This implies that there exists η = η(E, ν, α) > 0 such that if |α′ − α| < η(E, ν, α), |E ′ − E| <η(E, ν, α), then for any m0(E, ν, α) ≤ m ≤ 2m0(E, ν, α) + 1

(3.32) supθ∈T

1

mln ‖Φm(E ′ + iε, α′, θ)‖ < ν .

By subadditivity, (3.32) holds for every m > m0(E, ν, α). By compactness of S(α, λ), there existη = η(ν, α) > 0, m0 = m0(ν, α) > 0, such that if |α′ − α| < η(ν, α), |ε| < η, and E ′ is such thatdistH (E ′, S(α, λ)) < η(ν, α), then (3.31) holds true for any |m| ≥ m0.

Proof of Lemma 3.12. Let us prove (3.28), the proofs of other inequalities are similar. Using thecontinuity of the spectra (Proposition 2.8), we conclude that if E ∈ S(p

q, λ), then

(3.33) dist (E, S(α, λ)) <

∣∣∣∣α− p

q

∣∣∣∣ 12 .In the notations of Lemma 3.13, let α′ = p

q. Having (3.33) Lemma 3.13 implies that there exists

q0 = q0(ν, α) > 2m0(ν, α), such that for any q > q0(ν, α) and m ≥ m0(ν, α)

supθ

∥∥∥∥Φm

(E + iε,

p

q, θ

)∥∥∥∥ ≤ e2νm3 ,(3.34)

supθ

∥∥∥∥Φm

(E,

p

q, θ

)∥∥∥∥ ≤ e2νm3 .(3.35)


Denote by Φq

(E + iε, p

q, θ)

= T εq · · ·T ε1 and Φq

(E, p

q, θ)

= Tq · · ·T1, where T εj = Tj(E + iε), Tj =

Tj(E) are one-step transfer matrices defined by (3.20). Then,

Φq

(E + iε,

p

q, θ

)− Φq

(E,

p

q, θ

)=

q∑i=1

T εq · · ·T εi+1

(T εi − Ti

)Ti−1 · · ·T1

=( m0∑i=1

+

q−m0∑i=m0+1

+

q∑i=q−m0+1

)T εq · · ·T εi+1

(T εi − Ti

)Ti−1 · · ·T1

= I1 + I2 + I3 .

By definition (3.20) of Tj we have ‖T εi − Ti‖ = |ε|, thus for any 1 ≤ j ≤ q, combining (3.34) and(3.35), we obtain

‖I1‖ ≤ |ε|m0∑i=1

(4λ+ 3)i−1e(q−i) 2ν3 ,

‖I2‖ ≤ |ε|q−m0∑i=m0+1

e(q−1) 2ν3 ,

‖I3‖ ≤ |ε|q∑

i=q−m0+1

(4λ+ 3)q−ie(i−1) 2ν3 ,

thus, (3.28) holds for sufficiently large q.

3.3. Overview, and preliminary reductions. The general strategy of the proof of Proposi-tion 3.1 is as follows. Consider the matrix product

Φn

(E + iε,

p

q, θ

)= Tn

(E + iε,

p

q, θ

)Tn−1

(E + iε,

p

q, θ

)· · ·T1

(E + iε,

p

q, θ

)corresponding to p

q. Inside this product, we identify long stretches of indices j for which E /∈

σ(pq, λ, θ+ 2π p

qj). On each such interval, we approximate the transfer matrices corresponding to p

q

by those corresponding to pq

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 boundrelying on the imaginary part ε of the spectral parameter. These pieces are glued together usingan intermediate operator, which we define in (3.45) below. For the convenience of the reader, werecord the main steps:

Proposition 3.1⇐= (3.43)⇐= (3.48)⇐= Proposition 3.5 +

(3.52)

(3.53)

(3.54)

To implement this strategy, we need some preliminary reductions. First, we observe that if

ε ≥ C1δ

q2λq2

or E ∈ Jδ and dist(E, S

(pq, λ))≥ C1δ

q2λq2

, then for any θ ∈ T

γ pq,λ,θ(E + iε) ≥ γ p

q,λ,θ(E) ≥ cδ

q2λq2

.

Indeed, the first inequality holds since by the Thouless formula (1.12) the Lyapunov exponent isan increasing function of ε, and the second one follows from the Combes-Thomas estimate (3.19).

21

Thus it suffices to prove (3.3) for E and ε satisfiying

(3.36) max

(ε, dist

(E, S

(p

q, λ

)))<

C1δ

q2λq2

≤ C1c0(r, α)λq2 ≤ C1c0(r, α) ,

where c0 is from assumption (3), which we used for the second inequality.Further, we decompose

Jδ = J+δ

⋃J−δ =

∆ p

q,λ(E) > 2− 2λq + δ

⋃∆ p

q,λ(E) < −2 + 2λq − δ

.

To prove (3.3), we can also assume that

(3.37) ε = exp

(− erq

15000 q2 λq2

),

since by the Thouless formula the Lyapunov exponent γ pq,λ,θ(E + iε) is an increasing function of ε.

By q-periodicity of H pq,λ,θ we obtain

(3.38) γ pq,λ,θ(E + iε) = −1

qln∣∣G[1,∞)(1, q, E + iε)

∣∣ = − 1

q qln∣∣G[1,∞)(1, q q, E + iε)

∣∣ ,where G[1,∞)(·, ·, E + iε) is the restricted Green function corresponding to the restricted operator

H[1,∞)pq,λ,θ

. Thus our main goal is to obtain an upper bound on∣∣G[1,∞)(1, q q, E + iε)

∣∣ ≡ ∣∣G[1,∞)(1, q q)∣∣.

Recall the following rough bound. For any interval I ⊂ Z

(3.39) ‖GI‖ = ‖(HI − E − iε)‖ ≤ 1

ε.

Define

(3.40) θk = θ + 2π

(p

q− p

q

)k, θ ∈ [0, 2π) .

Define two following sets:

J− =

j ∈ 1, . . . , qq

∣∣ |(qθj)mod2π| ≤√δ

10λq2

, and |(qθj−1)mod2π| >√δ

10λq2

= j−1 , . . . , j−M− ,

and

J+ =

j ∈ 1, . . . , qq

∣∣ |(qθj + π)mod2π| ≤√δ

10λq2

, and |(qθj−1 + π)mod2π| >√δ

10λq2

= j+

1 , . . . , j+M+ .

Denote J = J+ ∪ J−. Observe that by assumption (1)

1

qq≤∣∣∣∣pq − p

q

∣∣∣∣ < 2δe−rq ≡ 2η,

hence

(3.41) #J± = M± ≥ [q2qδe−rq] ≡ [q2q η] .


Applying Claim 3.8 to G[1,∞)(1, q q) with j+k ∈ J+ if E ∈ J+

δ , 1 ≤ k ≤ M+, and with j−k ∈ J− ifE ∈ J−δ , 1 ≤ k ≤M−, we obtain∣∣G[1,∞)(1, q q)

∣∣=

M−1∏k=1

[|G[j±k ,j

±k +l0q−1](j±k , j

±k + l0q − 1)| × |G[j±k ,∞)(j±k + l0q, j

±k+1 − 1)|

]|G[j±M ,j

±M+l0q−1](j±M , j

±M + l0q − 1)| × |G[j±M ,∞)(j±M + l0q, qq)|

≤M∏k=1

[|G[j±k ,j

±k +l0q−1](j±k , j

±k + l0q − 1)|1

ε

],

(3.42)

where the last inequality follows from the rough bound (3.39). The main step of the proof is toshow that

(3.43) |G[j±k ,j±k +l0q−1](j±k , j

±k + l0q − 1)| ≤ ε exp

(− l0√δ

60

),

where we set

(3.44) l0 =

[ √δ

160q3λq2 η

], η ≡ δe−rq .

Note that, the assumption (1) and the upper bound on δ in the assumption (3) guarantee that

1 ≤ l0 ≤ q. Having (3.43) at hand, combining (3.38) and (3.42), we obtain for ε ≥ exp(− l0

√δ

90

)γ pq,λ,θ(E + iε) ≥ l0

√δM±

60 q q≥ l0√δ q2 q η

60q q=l0√δ

60η q ≥ δ

9600 q2 λq2

,

where the second inequality follows from (3.41). This concludes the proof of the proposition.

3.4. Construction of an intermediate operator. The proof of (3.43) relies on a constructionof an intermediate operator. For the rest of the argument we fix j = jk ∈ J . Consider thefollowing operator (depending on j)

(3.45) (Hψ)(n) =

ψ(n+ 1) + ψ(n− 1) + V (n)ψ(n), if n > j

ψ(j + 1) + V (j)ψ(j), if n = j

that acts on `2([j,∞) ⊂ Z), where

(3.46) V (n) =

2λ cos

(2π p

qn+ θn

)if j ≤ n ≤ j + l0q − 1

V per(n) if n > j + l0q − 1,

where V per is a periodic potential of period q with the period determined by

V per(m) = 2λ cos

(2πp

qm+ θm

), if j + (l0 − 1)q ≤ m < j + l0q − 1 ,

namely, we repeat the last q values periodically. Denote by G[a,b](·, ·, E+ iε) ≡ G[a,b](·, ·) the Green

function corresponding to the restriction of the operator H to an interval [a, b] ⊂ [j,∞) ⊂ Z. Note

that the restricted operator H[j,j+l0q−1]pq,λ,θ

coincides with the intermediate problem restricted to that

interval H [j,j+l0q−1], thus

G[j,j+l0q−1](j, j + l0q − 1) = G[j,j+l0q−1](j, j + l0q − 1) .

23

Therefore (3.43) is equivalent to

(3.47) |G[j,j+l0q−1](j, j + l0q − 1)| ≤ ε exp

(− l0√δ

60

),

and now we turn to the proof of this inequality.First, using (3.12) of Proposition 3.7, we have

(G[j,∞) − G[j,j+l0q−1])(j, j + l0q − 1) = G[j,∞)(j, j + l0q)G[j,j+l0q−1](j + l0q − 1, j + l0q − 1) ,

thus

|G[j,j+l0q−1](j, j + l0q − 1)|

≤ |G[j,∞)(j, j + l0q − 1)|+ |G[j,∞)(j, j + l0q)||G[j,j+l0q−1](j + l0q − 1, j + l0q − 1)|

≤ |G[j,∞)(j, j + l0q − 1)|+ |G[j,∞)(j, j + l0q)|1

ε

≤ 2

ε

[|G[j,∞)(j, j + l0q − 1)|+ |G[j,∞)(j, j + l0q)|

],

where the second inequality follows from the rough bound (3.39), and the last one holds since

0 < ε < 1. Observe that ε ≥ exp(− l0

√δ

90

), and note that l0

√δ ≥

[erq

160 q3λq2

]≥ 240, therefore (3.47)

is implied by

(3.48) |G[j,∞)(j, j + l0q − 1)|, |G[j,∞)(j, j + l0q)| ≤2

εexp

(− l0√δ

20

).

3.4.1. Proof of (3.48). We may assume without loss of generality that j = 1. IndeedG[j,j+l0q−1](j, j+

l0q − 1) corresponding to our θ is equal to G[1,l0q](1, l0q) corresponding to θ′ = θ + 2π j pq, and if j

lies in J corresponds to θ, then 1 lies in J corresponds to θ′.For 1 ≤ m, k ≤ q, set

Q−1sq+m(E + iε) =

(0 1

−1 E + iε− 2λ cos(

2π pq(sq +m) + θsq+m

) ),

Q−1sq+k(E + iε) =

(0 1

−1 E + iε− 2λ cos(

2π pq(sq + k) + θsq+1

) ).

The matrix Q−1sq+m is the one-step transfer matrix corresponding to the intermediate problem, and

Q−1sq+k is the one-step transfer matrix that corresponds to the periodic almost Mathieu operator

H pq,λ,θsq+1

of period q. Let

T−1s (E + iε) = (Q−1

sq+1 . . . Q−1(s+1)q)(E + iε), 0 ≤ s ≤ l0 − 1,

T−1s (E + iε) = (Q−1

sq+1 . . . Q−1(s+1)q)(E + iε), 0 ≤ s ≤ l0 − 1,

Φ−1l0q

(E + iε,

p

q, θ

)= (T−1

0 . . . T−1l0−1)(E + iε) .

(3.49)


For any n ≥ l0q the intermediate problem H is periodic of period q, thus we obtain(G[1,∞)(1, 1)

1

)= Φ−1

l0q

(E + iε,

p

q, θ

)(G[1,∞)(1, l0q + 1)

G[1,∞)(1, l0q)

)≡ Φ−1

l0q

(E + iε,

p

q, θ

)u .

(3.50)

If the assumptions of Proposition 3.5 hold, namely if for

δs = δ =δ

100, 0 ≤ s ≤ l0 − 1, b =

1

10, c =

1

2,

δs ≤ δs + bδ32s ,(3.51)

‖T−1s (E + iε)− T−1

s+1(E + iε)‖ ≤ δ

10,(3.52)

|TrT−1s (E + iε)| ≥ 2 + (1− b)δ,(3.53)

and for the vector u defined by (3.50)

(3.54) ‖T−1l0−1u‖ ≥ exp

(√δ

2

)‖u‖,

then, an application of Proposition 3.5 gives

1

‖u‖

∥∥∥∥Φ−1l0q

(E + iε,

p

q, θ

)u

∥∥∥∥ ≥ exp

(1

2

l0−1∑s=0

√δ

)= exp

(l0√δ

2

)= exp

(l0√δ

20

).

Using the rough bound (3.39), we obtain

|G[1,∞)(1, l0q + 1)|, |G[1,∞)(1, l0q)| ≤ ‖u‖ ≤ exp

(− l0√δ

20

)∥∥∥∥Φ−1l0q

(E + iε,

p

q, θ

)u

∥∥∥∥= exp

(− l0√δ

20

)√|G[1,∞)(1, 1)|2 + 1 ≤ exp

(− l0√δ

20

)2

ε.

This concludes the proof of (3.48).

It is clear that (3.51) holds for any b > 0 and δs = δ100≡ δ, thus it is left to verify (3.52), (3.53),

and (3.54).

3.5. Verification of (3.52), (3.53), and (3.54). In the notation of Proposition 3.5 let Aj =T−1s , 0 ≤ s ≤ l0 − 1, and An · · ·A1 u0 = Φ−1

l0qu.

3.5.1. Verification of (3.52). We apply Lemma 3.12 with ν = r2, α′ = p

q, ε = exp

(− erq

15000 q2 λq2

),

and θ = θ(s+1)q+j − θsq+j for 1 ≤ j ≤ q. Then, by assumption (1) for sufficiently large q∣∣∣∣α− p

q

∣∣∣∣ , ∣∣∣∣α− p

q

∣∣∣∣ < 1

100η(r

2, α)2

,

and

(3.55) |θ(s+1)q+j − θsq+j| = 2π q

∣∣∣∣pq − p

q

∣∣∣∣ = 2π q

∣∣∣∣ pq − p

q

∣∣∣∣ < η(r

2, α).

25

Further, our reduction (3.36) yields that

max

(ε, dist

(E, S

(p

q, λ

)))<

1

2η(r

2, α)

provided that we make sure that c0(r, α) ≤ 12C1

η(r2, α). By the continuity of the spectrum (Propo-

sition 2.8) for sufficiently large q ≥ q0

dist (E, S(α, λ)) ≤ dist

(E, S

(p

q, λ

))+ distH

(S

(p

q, λ

), S(α, λ)

)≤ 1

2η(r

2, α)

+1

2η(r

2, α)

= η(r

2, α).

Thus Lemma 3.12 is applicable. Also observe that the lower bound (3) on δ and the assumption(3.37) imply that ε ≤ δe−rq. Therefore using (3.28)

(3.56) ‖T−1s (E)− T−1

s (E + iε)‖ ≤ δe−rqerq2 <

δ

1000=

δ

10.

The inequality (3.29) yields that

(3.57) ‖T−1s (E + iε)− T−1

s (E + iε)‖ ≤ δe−rqerq2 <

δ

1000=

δ

10,

and, lastly, (3.55) and (3.30) imply that

(3.58) ‖T−1s (E + iε)− T−1

s+1(E + iε)‖ ≤ 4πqδe−rqerq2 <

δ

1000=

δ

10.

Thus (3.52) holds true.

3.5.2. Verification of (3.53). We need to show that for any 0 ≤ s ≤ l0 − 1, E ∈ Jδ, and ε =

exp(− erq

15000 q2 λq2

)|TrT−1

s (E + iε)| ≥ 2 + (1− b)δ .For any 0 ≤ s ≤ l0 − 1 we have

(3.59) |TrT−1s (E + iε)| ≥ |Tr T−1

s (E)| − 2‖T−1s (E)− T−1

s (E + iε)‖ .Combining (3.56) and (3.57) we obtain

2‖T−1s (E)− T−1

s (E + iε)‖ ≤ 2‖T−1s (E)− T−1

s (E + iε)‖+ 2‖T−1s (E + iε)− T−1

s (E + iε)‖

≤ 4δe−rqerq2 ≤ δ

5.

(3.60)

Therefore, we need to show that

(3.61) |Tr T−1s (E)| ≥ 2 +

3

4δ .

By definition (3.49) the matrices T−1s are the one-period (q-step) transfer matrices that correspond

to the periodic almost Mathieu operator H pq,λ,θsq+1

. Let us show that our choice (3.44) of l0guarantees that for all k ∈ [j, j + l0q − 1] the energy E is sufficiently far from the spectrum of theoperator H p

q,λ,θk with θk defined by (3.40).

Claim 3.14. If E ∈ J+δ and j ∈ J+ or E ∈ J−δ and j ∈ J−, then for all k ∈ [j, j + l0q − 1]∣∣∣D p

q,λ,θk(E)

∣∣∣ ≥ 2 +3

4δ .


Proof of Claim 3.14. Assume that E ∈ J+δ and j ∈ J+. Then |(qθj + π)mod 2π| <

√δ

10λq2

, therefore

by our choice (3.44) of l0 =[ √

δ

160 q3 λq2 η

], η ≡ δe−rq we get for any k ∈ [j, j + l0q − 1]

|(qθk + π)mod 2π| ≤ |(qθj + π)mod 2π|+ |q(θk − θj)|

≤√δ

10λq2

+ 4π q η |k − j| ≤√δ

10λq2

+ 4π q η l0q

≤√δ

10λq2

+4π

160

√δ

λq2

≤√δ

5λq2

.

(3.62)

Since E ∈ J+δ we have ∆ p

q,λ(E) > 2 − 2λq + δ, thus by Chambers’ formula (3.22) for any k ∈

[j, j + l0q − 1]

D pq,λ,θk(E) = ∆ p

q,λ(E)− 2λq cos(qθk)

≥ 2− 2λq + δ − 2λq(−1 +

δ

12π2λq

)= 2− 2λq + δ + 2λq − δ

6π2≥ 2 +

3

4δ ,

where the first inequality holds since for sufficiently large q, using (3.62) we obtain

cos(qθk) ≤ −1 +2

π2(qθk + π)2 ≤ −1 +

2

π2

δ

25λq≤ −1 +

δ

12π2λq.

Now let E ∈ J−δ and j ∈ J−. In the same way we obtain that for any k ∈ [j, j + l0q − 1]

(3.63) |qθk mod 2π| ≤√δ

5λq2

.

Since E ∈ J−δ we have ∆ pq,λ(E) < −2 + 2λq − δ, thus by Chambers’ formula (3.22) for any

k ∈ [j, j + l0q − 1]

D pq,λ,θk(E) ≤ −2 + 2λq − δ − 2λq

(1− δ

50λq

)< −2− 3

4δ,

where the first inequality holds since (3.63) implies for sufficiently large q

cos(qθk) ≥ 1− (qθk)2

2≥ 1− δ

50λq.

Thus combining (3.59), (3.60), and (3.61) we conclude

(3.64) |TrT−1s (E + iε)| > 2 +

3

4δ − δ

5≥ 2 +

9

10δ = 2 + (1− b)δ,

where the equality follows from our choice δ = δ100, b = 1

10.

3.5.3. Verification of (3.54). We need to verify that for the vector u defined by (3.50)

‖T−1l0−1u‖ ≥ exp

(√δ

2

)‖u‖ .

By (3.64) we have for s = l0 − 1 ∣∣TrT−1l0−1(E + iε)

∣∣ > 2 +9

10δ .

27

Denote by e±(γs+iζs)q the eigenvalues of T−1s (E + iε). Then,

|TrT−1l0−1(E + iε)| = |e(γl0−1+iζl0−1)q + e−(γl0−1+iζl0−1)q| ,

thus we have

(3.65) γl0−1q > cosh−1

(1 +

9

20δ

)≥ 1

2

√δ .

By Proposition 3.7, we have

G[1,∞)(1, l0q + 1) = G[1,∞)(1, (l0 − 1)q + 1)G[(l0−1)q+2,∞)((l0 − 1)q + 2, l0q + 1),

G[1,∞)(1, l0q) = G[1,∞)(1, (l0 − 1)q)G[(l0−1)q+1,∞)((l0 − 1)q + 1, l0q) .(3.66)

By definition (3.46) of the intermediate operator H, using the periodicity and the relation (3.38)between the Lyapunov exponent and Green’s function, we get

|G[(l0−1)q+2,∞)((l0 − 1)q + 2, l0q + 1)| = |G[(l0−1)q+1,∞)((l0 − 1)q + 1, l0q)|= e−γl0−1q .

(3.67)

By definition of the intermediate operator H(G[1,∞)(1, (l0 − 1)q + 1)

G[1,∞)(1, (l0 − 1)q)

)= T−1

l0−1(E + iε)u ,

thus, combining the definition (3.50) of u, and identities (3.66) and (3.67), we obtain

‖u‖ = ‖T−1l0−1u‖e

−γl0−1q .

Using the lower bound on γl0−1q given by (3.65), we conclude

‖T−1l0−1u‖ = eγl0−1q‖u‖ ≥ exp

(1

2

√δ

)‖u‖ .

This completes the verification of (3.54), thus we have completed the proof of Proposition 3.1.

3.6. Proof of Corollary 3.2. Fix an arbitrary θ ∈ [0, 2π), e.g. θ = 0. First, observe that for any

for any E ∈ S(pq, λ)

, 0 < λ ≤ 1, by Proposition 3.11 we have |∆ pq,λ(E)| ≤ 2 + 2λq, thus for E

such that γ pq,λ,0(E) 6= 0, we obtain

6 ≥ 2 + 4λq ≥ |D pq,λ,0(E)| = e

γ pq,λ,0

(E)q

+ e−γ p

q,λ,0

(E)q

≥ eγ pq,λ,0

(E)q

,

namely

γ pq,λ,0(E) ≤ ln(2 + 4λq)

q≤ ln 6

q.

This inequality obviously holds also for γ pq,λ,0(E) = 0. Let ε = exp

(− erq

15000 q2 λq2

). For any measure

µ for which

µ(a, b) ≤ Cω(b− a), b− a ≤ 1 ,

we obtain using Proposition 2.4

µ

E ∈ R

∣∣ ∣∣∣γ pq,λ,0(E + iε)− γ p

q,λ,0(E)

∣∣∣ ≥ δ

19200q2λq2

≤ Cµ 19200 q2λ

q2

δ(W3 ω)

[exp

(− erq

15000 q2 λq2

)].

(3.68)


By the assumption (1) of Proposition 3.1 we have

1

q q≤∣∣∣∣pq − p

q

∣∣∣∣ < 2δe−rq ,

namely q > erq

2δ q, thus for sufficiently large q

ln 6

q≤ δ

19200 q2 λq2

.

Therefore, using (3.68) and the estimate (3.3) given by Proposition 3.1, we obtain

µ

(E ∈ R

∣∣ γ pq,λ,0(E) ≤ ln 6

q

⋂Jδ

)≤ µ

(E∣∣ γ p

q,λ,0(E) ≤ δ

19200q2λq2

⋂E∣∣ γ p

q,λ,0(E + iε) ≥ δ

9600q2λq2

)≤ µ

E ∈ R

∣∣ ∣∣∣γ pq,λ,0(E + iε)− γ p

q,λ,0(E)

∣∣∣ ≥ δ

19200q2λq2

≤ Cµ q

2λq2

δ(W3 ω)

[exp

(− erq

15000 q2 λq2

)].

This concludes the proof of Corollary 3.2.

3.7. Proof of Proposition 3.3. Invoking the Combes–Thomas estimate as in the proof of Propo-sition 3.1 (3.36) we may assume that

(3.69) max

(ε, dist

(E, S

(pn+1

qn+1

, λ

)))≤ cδ

q2n

.

Furthermore, we only need to consider the case E ∈ J−δ , namely ∆ pq,λ(E) < −2+2λq−δ, since the

case E ∈ J+δ can be dealt with similarly. By our assumption

pnqn

is the sequence of convergents

of α, thus

|pnqn+1 − pn+1qn| = 1,

∣∣∣∣pnqn − pn+1

qn+1

∣∣∣∣ =1

qn qn+1

.

Unlike in the proof of Proposition 3.1 here we make use of only one intermediate problem, whichis defined as follows. Let

V (m) =

2λ cos(

2π pnqnm+ θm

)if 1 ≤ m < l0q

2λ cos(

2π pnqnm+ θl0q

)if m ≥ l0q,

where 1 ≤ l0 ≤ qn+1 is an integer chosen by

(3.70) l0 =

[qn+1

√δ

100 qn

],

and

θm = θ + 2π

(pn+1

qn+1

− pnqn

)m, θ ∈ [0, 2π) .

Note that l0 ≥ 1 by (3.4). Then we define the operator H as in (3.45) acting on `2([1,∞) ⊂ Z).By definition

H[1,l0qn]pn+1qn+1

,λ,θ= H [1,l0qn],

29

thus we have an equality between the corresponding Green functions

G[1,l0qn](1, l0qn) = G[1,l0qn](1, l0qn) .

Let us show that the estimates (3.42) and (3.48) remain true with the current definitions if we setj = 1. We restate these estimates as follows.

(1) For any ε ≥ exp(− l0

√δ

90

)(3.71) |G[1,∞)(1, l0qn)| ≤ 2

εexp

(− l0√δ

20

).

(2)

(3.72) |G[1,∞)(1, qn qn+1)| ≤ 5

ε3|G[1,∞)(1, l0qn)| ≤ exp

(− l0√δ

60

).

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 3.12and Claim 3.14. Let us first verify the conditions of Lemma 3.12. Let ν = r

2, then for any n ≥ n0

c

q2n

<1

2η(r

2, α),

where η(r2, α)

is given by Lemma 3.12. In the notation of Lemma 3.12, let α′ = pn+1

qn+1, θ =

θ(s+1)qn+j − θsqn+j, 1 ≤ j ≤ qn. Now, assume that (3.4) holds, then for any n ≥ n0, we have∣∣∣∣α− pnqn

∣∣∣∣ , ∣∣∣∣α− pn+1

qn+1

∣∣∣∣ ≤ 1

qn qn+1

< 2δe−rqn <1

100η(r

2, α)2

,

|θ| = 2π qn


qn+1

∣∣∣∣ =2π

qn+1

<1

100η(r

2, α)2

.

On the other hand, by (3.69), we have

max

(ε, dist

(E, S

(pn+1

qn+1

, λ

)))≤ cδ

q2n

<1

2η(r

2, α).

By the continuity of the spectrum (Proposition 2.8) we get

dist (E, S(α, λ)) ≤ dist

(E, S

(pn+1

qn+1

, λ

))+

∣∣∣∣α− pn+1

qn+1

∣∣∣∣ 12 ≤ η(r

2, α).

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 ≤ l0qn, we have∣∣∣D pnqn,λ,θk(E)

∣∣∣ > 2 +3

4δ .

Proof of Claim 3.15. As in the proof of Proposition 3.1 we may assume that θ = 0. Then, for

l0 =[qn+1

√δ

100qn

]we get

|θk| ≤ |θl0qn| = 2πl0qn


qn+1

∣∣∣∣ =2πl0qn+1

=2π√δ

100qn,


thus

cos(qnθl0qn) ≥ 1− (qnθl0qn)2

2≥ 1− 2π2 δ

10000.

Therefore, for any E ∈ J−δ and any 1 ≤ k ≤ l0qn

D pnqn,λ,θk(E) = ∆ pn

qn,λ(E)− 2λqn cos(θkqn)

< −2 + 2λqn − δ − 2λqn +4λqnπ2δ

10000< −2− 3

4δ ,

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 Proposi-tion 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 3.7for any k < m < l

(3.73) G[k,∞)(k, l) = G[k,∞)(k,m)G[m+1,∞)(m+ 1, l) ,

therefore

|G[1,∞)(1, qnqn+1)| ≤ |G[1,∞)(1, l0qn)G[l0qn+1,∞)(l0qn + 1, qnqn+1)|≤ |G[l0qn+1,∞)(l0qn + 1, qnqn+1)|×(|G[1,∞)(1, l0qn)|+ |G[1,∞)(1, l0qn)− G[1,∞)(1, l0qn)|

).

(3.74)

To estimate the difference, first we use the second resolvent identity and obtain∣∣∣G[1,∞)(1, l0qn)− G[1,∞)(1, l0qn)∣∣∣

≤∞∑

k=l0qn+1

∣∣∣G[1,∞)(1, k)(V (k)− V (k))G[1,∞)(k, l0qn)∣∣∣

≤ 4λ∞∑

k=l0qn+1

∣∣∣G[1,∞)(1, k)G[1,∞)(k, l0qn)∣∣∣ .

Using (3.73) we obtain for k ≥ l0qn + 1

G[1,∞)(1, k) = G[1,∞)(1, l0qn)G[l0qn+1,∞)(l0qn + 1, k),

thus ∣∣∣G[1,∞)(1, l0qn)− G[1,∞)(1, l0qn)∣∣∣

≤ 4λ∣∣∣G[1,∞)(1, l0qn)

∣∣∣ ∞∑k=l0qn+1

∣∣∣G[l0qn+1,∞)(l0qn + 1, k)G[1,∞)(k, l0qn)∣∣∣

≤ 4λ∣∣∣G[1,∞)(1, l0qn)

∣∣∣( ∞∑k=1

∣∣∣G[1,∞)(1, k)∣∣∣2) 1

2(∞∑k=1

∣∣G[1,∞)(k, l0qn)∣∣2) 1

2

,

where in the last step we used the Cauchy-Schwartz inequality. Recall that for any k0 ∈ [1,∞)

∞∑k=1

∣∣G[1,∞)(k0, k)∣∣2 =

1

εIm G[1,∞)(k0, k0) ≤ 1

ε2,

31

thus we conclude that∞∑k=1

∣∣∣G[1,∞)(1, k)∣∣∣2 , ∞∑

k=1

∣∣G[1,∞)(k, l0qn)∣∣2 ≤ 1

ε2.

Therefore,

|G[1,∞)(1, l0qn)− G[1,∞)(1, l0qn)| ≤ 4λ

ε2|G[1,∞)(1, l0qn)| .

Combining (3.74), and the last inequality with the rough bound (3.39) we obtain∣∣G[1,∞)(1, qnqn+1)∣∣ ≤ ∣∣∣G[1,∞)(1, l0qn)

∣∣∣ (1 +4λ

ε2

) ∣∣G[l0qn+1,∞)(l0qn + 1, qnqn+1)∣∣

≤ 5

ε3

∣∣∣G[1,∞)(1, l0qn)∣∣∣ ,

where the last inequality follows from the assumption that ε < 1 and λ ≤ 1. This concludes theproof of (3.72) and of Proposition 3.3. We omit the proof of Corollary 3.4, which is parallel tothat of Corollary 3.2 (see Section 3.6).

4. Proof of Theorem 1

We repeatedly use Frostman’s lemma, which we now state. Recall that the ω-Hausdorff contentof a set K ⊂ R is defined via

Hω∞(K) = inf

∞∑j=1

ω(bj − aj)

.

This is consistent with setting ε =∞ in (1.6), and in particular Hω∞(K) ≤ Hω(K).

Lemma 4.1 (Frostman, see [16]). Let ω be a gauge function. Let K ⊂ R be a compact set withpositive ω-Hausdorff content, Hω

∞(K) > 0. Then there exists a non-zero Borel measure µ ≥ 0supported on K satisfying

(4.1) µ(a, b) ≤ Cω(b− a) .

Vice versa, if there exists a non-zero Borel measure µ ≥ 0 supported on K satisfying (4.1), thenHω∞(K) > 0.

Remark 4.2. Lemma 4.1 and the result of Craig and Simon [29] imply that Hω1(σ(H)) ≥Hω∞(σ(H)) > 0 for any ergodic Schrodinger operator H. Using a result of Bourgain and Klein [21]

in place of [29], we see that the same conclusion holds for any discrete Schrodinger operator.

4.1. Proof of Theorem 1 (1). The proof is by contradiction. Assume that Hωt(S(α, 1)) > 0.Then, by Frostman’s Lemma (Lemma 4.1) there exists a non-zero Borel measure µ (we can assumewithout loss of generality that µ is supported on an interval of length 1

e), such that

(1) suppµ ⊂ S(α, 1),(2) µ[a, b] ≤ C

lnt 1|b−a|

for any a < b, t > 2 .

Denote µ(S(α, 1)) = m > 0. Assume that β(α) > 0. Then, for any 0 < r < β(α), there exists asequence Pk

Qk→ α (a subsequence of convergents of α) such that∣∣∣∣α− Pk

Qk

∣∣∣∣ ≤ e−(β(α)−r)Qk ,(4.2)

In the current proof, we choose r = β(α)100t

.


From Proposition 2.8, we have

(4.3) S(α, 1) ⊂ S

(PkQk

, 1

)+

(−6

√2

∣∣∣∣α− PkQk

∣∣∣∣, 6

√2


∣∣∣∣)

,

where on the right hand side we have the Minkowski sum

A+B =a+ b

∣∣ a ∈ A, b ∈ B , A,B ⊂ R .

By (4.2) and (4.3), the set S(α, 1)\S(PkQk, 1)

is contained in the union of (at most) 2Qk intervals

of length ≤ Ce−(β(α)−r)Qk

2 , since S(PkQk, 1)

has at most Qk bands. Therefore, by the assumption

(2), for sufficiently large k we obtain

(4.4) µ

(S(α, 1) \ S

(PkQk

, 1

))≤ 2CtQk

(β(α)− r)tQtk

≤ Ct

Qt−1k

≤ m

2,

since t > 2. By our assumptions (1)–(2) above, the measure µ restricted to the set S(PkQk, 1)

,

which we denote by µk = µ S(PkQk, 1)

, obeys the following:

(1) suppµk ⊂ S(PkQk, 1)

,

(2) µk[a, b] ≤ Clnt 1|b−a|

,

(3) µk

(S(PkQk, 1))≥ m

2.

Let 0 < ν < (t− 2)r. From (4.2) and since r = β(α)100t

, we obtain for sufficiently large k∣∣∣∣α− Pk−1

Qk−1

∣∣∣∣ , ∣∣∣∣α− PkQk

∣∣∣∣ ≤ e−(β(α)−r)Qk−1 ≤ e−rQk−1e−νQk−1 .

Thus we can apply Corollary 3.2 with pq

= Pk−1

Qk−1, pq

= PkQk, δ = e−νQk−1 and get using Remark 2.1

µk

(S

(PkQk

, 1

)⋂Jδ

)≤ CtQ

2tk−1e

−(t−1)rQk−1eνQk−1 ,

where

Jδ =

E∣∣ ∣∣∣∣∆ Pk−1

Qk−1,1

(E)

∣∣∣∣ > δ

.

Since t > 2 and 0 < ν < (t− 2)r, we obtain

(4.5) limk→∞

µk

(S

(PkQk

, 1

)⋂Jδ

)= 0 .

By [55, equation (5.3), Lemma 5.1], we obtain

|J cδ | ≤2eδ

Qk−1

,

therefore the set S(PkQk, 1)\Jδ is contained in the union of (at most) Qk−1 intervals of total length

at most2eδ

Qk−1

+ Ce−(β(α)−r)Qk−1

2 ≤ Ce−ν2Qk−1 ,

where the last inequality holds by the choice of δ. Therefore,

(4.6) µk

(S

(PkQk

, 1

)\ Jδ

)≤ Qk−1

Ct

lnt(eν2Qk−1)

−→k→∞

0 ,

33

since t > 2. Combining the third condition (3) on µk with the estimates (4.5) and (4.6), we obtaina contradiction.

4.2. Proof of Theorem 1 (2). Let pnqn→ α be the sequence of convergents of α. We show

that Hω(S(α, 1)) = 0 if the denominators qn grow sufficiently fast. We construct such α via itsconvergents by induction: for any fixed n0 the terms q1, . . . , qn0 can be chosen in an arbitrary way.If q1, . . . , qn, n > n0, are given, we find xn > 0 such that for any 0 < x < xn

(4.7) ω(x) log1

x≤ e−2qn ,

and we choose

(4.8) qn+1 ≥ max

2C1eqn log

1

xn,

1

ω−1(

1q2n

)2

,

where C1 > 0 is the constant from Corollary 3.4. It is clear that the set of α thus constructed isGδ-dense.

For each n, we choose δ = e−qn2 , then by the continuity of the spectrum(Proposition 2.8), we

can construct a cover of S(α, 1) as follows:

S(α, 1) ⊂ S

(pn+1

qn+1

, 1

)+

(−6√

2

√∣∣∣∣α− pn+1

qn+1

∣∣∣∣,+6√

2

√∣∣∣∣α− pn+1

qn+1

∣∣∣∣)

⊂ S

(pn+1

qn+1

, 1

)⋃Xn ,

⊂(S

(pn+1

qn+1

, 1

)⋂Jδ

)⋃J cδ⋃

Xn.

To estimate Hω(S(α, 1)), one can proceed as follows. First note that Xn is a union of (at most)

2qn+1 intervals of length ≤ 12√

2

√∣∣∣α− pn+1

qn+1

∣∣∣ ≤ 1√qn+2

(for sufficiently large n). By (4.8) we

conclude ω(

1√qn+2

)≤ 1

q2n+1, thus the contribution of these intervals to (1.6) is

≤ ω

(1

√qn+2

)× 2qn+1 ≤

1

q2n+1

× 2qn+1 −→n→∞

0 .

On the other hand, by Corollary 3.4 we have∣∣∣∣S (pn+1

qn+1

, 1

)⋂Jδ

∣∣∣∣ ≤ Cq2ne

qn2 exp

(− qn+1

C1qne−

qn2

)≤ exp

(− qn+1

2C1qne−qn

).

This set can be covered by (at most) qn+1 + qn ≤ 2qn+1 intervals that contribute to (1.6) at most

2qn+1ω(

exp(− qn+1

2C1qne−qn

)). Observe that by (4.8) we have qn+1 ≥ 2C1e

qn log 1xn

for xn defined by

(4.7), namely exp(− qn+1

2C1qne−qn

)≤ xn. Therefore, (4.7) implies that

ω

(exp

(− qn+1

2C1qne−qn

))log

1

exp(− qn+1

2C1qne−qn

) ≤ e−2qn ,


Figure 2. Construction of Fτ

thus we obtain

2qn+1ω

(exp

(− qn+1

2C1qne−qn

))≤ 4C1qne

−qn −→n→∞

0.

Finally, by [55, equation (5.3), Lemma 5.1], we obtain

|J cδ | ≤2eδ

qn=

2e

qne−

qn2 .

J cδ is a union of qn intervals, thus its contribution to (1.6) is

≤ qnω

(2e

qne−

qn2

)≤ 4ω

(2e

qne−

qn2

)log

12eqne−

qn2

−→n→∞

0,

where the convergence follows by the assumption (1.7). This concludes the whole proof.

5. Construction of auxiliary intervals

Let Nα,λ,θ(E) be the density of states measure corresponding to the almost Mathieu operatorHα,λ,θ. For α ∈ Q, Nα,λ,θ depends on θ, therefore, similarly to Section 2.2, we define

(5.1) N α,λ(E) =1

2π

∫ 2π

0

Nα,λ,θ(E)dθ .

For irrational α, N α,λ(E) = Nα,λ(E) = Nα,λ,θ(E).Due to Aubry–Andre duality [2, 3], the following holds true for any λ 6= 0

N α,λ(E) = N α, 2λ

(E

λ

),

S(α, λ) = λS

(α,

2

λ

).

(5.2)

Therefore, it is enough to prove the results for 0 < λ ≤ 1. For the time being, we assume that0 < λ < 1; a separate argument will be given for λ = 1.

In this section, we construct a family of intervals Fτ (0 < τ ≤ 12) contained in S

(pq, λ)

but

disjoint from S−

(pq, λ)

, as illustrated in Figure 2. Clearly, these intervals will be found close to

the edges of one of the bands of S(pq, λ)

. We prove that these intervals are not too small, both in

the sense of Lebesgue measure (see Proposition 5.1):

(5.3) |Fτ | ≥(1− λ)τ

2q3λq

35

and spectrally (see Lemma 5.3):

(5.4) ρ pq,λ(Fτ ) ≥

τ arccos τ

π

λq2

q

where ρ pq,λ is defined as in (2.6). Using Proposition 2.9, we shall deduce from (5.4) that also

(5.5) ρα,λ(Fτ ) ≥τ arccos τ

2π

λq2

q.

While (5.3) will play a role in the proof of Theorem 2-(3), Theorem 2-(1) and (2) will follow bycomparing (5.5) with Corollaries 3.2 and 3.4, which tell us that regular measures can not assignlarge mass to Fτ ⊂ Jδ. A somewhat similar argument will be used in the proof of Theorem 3.

The construction goes as follows. First, for −1 ≤ τ ≤ 1 let

Sτ

(p

q, λ

)=E ∈ R : |∆ p

q,λ(E)| ≤ 2 + 2τλq

=

q⋃j=1

Ijτ .

These sets interpolate between S−1

(pq, λ)

= S−

(pq, λ)

and S+1

(pq, λ)

= S(pq, λ)

. For ξ ∈−1,+1 let

(5.6) Ijτ,ξ = Ijτ⋂

E ∈ R : sign ∆ pq,λ(E) = ξ

.

Then Fτ will eventually be defined as the “largest” among the 2q intervals

Ijτ,ξ \ Ij−τ,ξ , ξ ∈ −1,+1, j ∈ 1, · · · , q

(a formal and slightly more careful definition is given before (5.8) below).

Proposition 5.1. Let 0 < λ < 1. We have the following:

(1) For any 1 ≤ j ≤ q and any ξ ∈ −1,+1, −1 ≤ τ ≤ τ ′ ≤ 1,

|Ijτ ′,ξ||Ijτ,ξ|

≥ 1 +λq

4q2(τ ′ − τ) .

(2) There exist 1 ≤ j0 ≤ q and ξ ∈ −1,+1, such that for any −1 ≤ τ ≤ τ ′ ≤ 1

|Ij0τ ′,ξ| − |Ij0τ,ξ| ≥

1− λ4q3

λq(τ ′ − τ) .

In the proof of the proposition we need the following inequality due to Chebyshev. Recall thatthe Chebyshev polynomial (of the first kind) of degree q is defined by

Tq(cos θ) = cos(qθ), Tq(cosh θ) = cosh(qθ).

It is not hard to see that, for any q ≥ 1 and any x ≥ 0, Tq(1 + x) ≤ exp(q2x).

Lemma 5.2 (Chebyshev, see e.g. pp. 67–68 [71]). Let [c, d] ⊂ [a, b] be two intervals and let p(x)be a polynomial of degree q. Then

supx∈[a,b]

|p(x)| ≤ Tq

(2

t− 1

)supx∈[c,d]

|p(x)| ≤ exp

(q2 2(1− t)

t

)supx∈[c,d]

|p(x)| , t =d− cb− a

.

Proof of Proposition 5.1. Observe that

maxE∈Ij

τ ′,ξ

|∆ pq,λ(E)| = 2 + 2τ ′λq, max

E∈Ijτ,ξ|∆ p

q,λ(E)| = 2 + 2τλq.


Invoking Lemma 5.2 we get

1 + τ ′λq

1 + τλq=

maxE∈Ijτ ′,ξ|∆ p

q,λ(E)|

maxE∈Ijτ,ξ|∆ p

q,λ(E)|

≤ exp

(2q2

(|Ijτ ′,ξ||Ijτ,ξ|

− 1

)).

Therefore

|Ijτ ′,ξ||Ijτ,ξ|

≥ 1 +1

2q2(ln (1 + τ ′λq)− ln (1 + τλq))

≥ 1 +λq(τ ′ − τ)

2q2(1 + τ ′λq)≥ 1 +

1

4q2(τ ′ − τ)λq,

(5.7)

where the second inequality follows from the Mean Value Theorem and the last inequality holdssince 1 + τ ′λq ≤ 2. This finishes the proof of the first statement of the proposition.

To prove the second part, we first note that since 11+ε≤ 1 − ε

2for any 0 ≤ ε < 1, and then

conclude from (5.7)

|Ijτ,ξ||Ijτ ′,ξ|

≤ 1− 1

8q2λq(τ ′ − τ).

Thus we get

|Ijτ ′,ξ| − |Ijτ,ξ| = |I

jτ ′,ξ|

(1−|Ijτ,ξ||Ijτ ′,ξ|

)≥ |Ijτ ′,ξ|

λq(τ ′ − τ)

8q2≥ |Ij−1,ξ|

λq(τ ′ − τ)

8q2,

where the last inequality holds since for every τ ′ ≥ τ we have Ijτ,ξ ⊂ Ijτ ′,ξ. As proved in [15], forany 0 < λ < 1, ∣∣∣∣S−(pq , λ

)∣∣∣∣ = 4− 4λ,

therefore, there exist 1 ≤ j0 ≤ q and ξ ∈ +1,−1 such that Ij0−1,ξ ⊂ S−

(pq, λ)

such that

|Ij0−1,ξ| ≥4− 4λ

2q.

Thus we conclude that

|Ij0τ ′,ξ| − |Ij0τ,ξ| = |I

j0τ ′,ξ|

(1−|Ij0τ,ξ||Ij0τ ′,ξ|

)≥ 1− λ

4q3λq(τ ′ − τ) .

Let 1 ≤ j0 ≤ q, ξ ∈ +1,−1, be as in the second part of Proposition 5.1. For 0 < τ ≤ 12

let

(5.8) Fτ = Ij0τ,ξ \ Ij0−τ,ξ = (aτ , bτ ), |Fτ | ≥

(1− λ)τ

2q3λq .

Now we show that the mass assigned to Fτ by the (θ-averaged) integrated density of states is nottoo small.

Lemma 5.3. For any 0 < λ < 1, we have

ρ pq,λ(Fτ ) ≥ cτ

λq2

q, cτ =

τ arccos τ

π.

37

Proof. Without loss of generality assume that ξ = +1, namely

Fτ = Ij0τ,+1 \ Ij0−τ,+1 = (aτ , bτ ).

Then,ρ pq,λ,θ(Fτ ) = N p

q,λ,θ(bτ )−N p

q,λ,θ(aτ ).

Consider the setA = θ ∈ [−π, π) : cos θq > τ , |A| = 2 arccos τ.

Using a result of Delyon-Souillard [33] one obtains that

(5.9) D pq,λ,θ(E) = 2 cos

(2πqN p

q,λ,θ(E)

),

where D pq,λ,θ(E) is defined by (3.21). Thus, for any θ ∈ A combining (5.9), the definition (5.6) of

Ijτ , and Chambers’ formula (Proposition 3.11), we get

2 cos(

2πqN pq,λ,θ(bτ )

)= 2 + 2τλq − 2λq cos θq,

2 cos(

2πqN pq,λ,θ(aτ )

)= 2− 2τλq − 2λq cos θq ,

namely

cos(


)− cos

(2πqN p

q,λ,θ(aτ )

)= 2τλq.

On the other hand∣∣∣cos(


)− cos

(2πqN p

q,λ,θ(aτ )

)∣∣∣=∣∣∣2 sin

(πq(N p

q,λ,θ(bτ ) +N p

q,λ,θ(aτ )

))sin(πq(N p

q,λ,θ(bτ )−N p

q,λ,θ(aτ )

))∣∣∣ .(5.10)

Since | sinx| ≤ |x| for any x, we obtain∣∣∣sin(πq (N pq,λ,θ(bτ )−N p

q,λ,θ(aτ )

))∣∣∣ ≤ ∣∣∣πq (N pq,λ,θ(bτ )−N p

q,λ,θ(aτ )

)∣∣∣ .We also note by our selection

j0 − 12

q≤ N p

q,λ,θ(aτ ) < N p

q,λ,θ(bτ ) ≤

j0

q,

thence we have ∣∣∣2 sin(πq(N p

q,λ,θ(bτ ) +N p

q,λ,θ(aτ )

))∣∣∣≤ 2

∣∣∣sin(2πqN pq,λ,θ(aτ )

)∣∣∣ = 2

√1− cos2

(2πqN p

q,λ,θ(aτ )

)≤ 2√

2λq(1 + τ)− λ2q(1 + τ)2 ≤ 4λq2 ,

where the last inequality holds since 0 < τ ≤ 12, and in the third inequality we used that

cos2(

2πqN pq,λ,θ(aτ )

)= (1− τλq − λq cos(θq))2 ≥ (1− λq(1 + τ))2.

Consequently, we have using (5.10):

2τλq ≤ 4λq2πq

(N p

q,λ,θ(bτ )−N p

q,λ,θ(aτ )

),

namely

N pq,λ,θ(bτ )−N p

q,λ,θ(aτ ) = ρ p

q,λ,θ(Fτ ) ≥

τ

2πqλq2 .


Thus we conclude that

ρ pq,λ(Fτ ) ≥

τ

2πqλq2 |A| = τ arccos τ

πqλq2 .

Next, we show that if α is sufficiently close to pq, a similar estimate holds for ρα,λ(Fτ ) = ρα,λ(Fτ )

in place of ρ pq,λ(Fτ ).

Lemma 5.4. Let 0 < λ < 1 and assume that α ∈ R is such that

(5.11)

∣∣∣∣α− p

q

∣∣∣∣ ≤ τ 2 arccos τ2

256π2 q4λ

3q2 .

Then, the intervals Fτ corresponding to pq

satisfy

ρα,λ(Fτ ) ≥τ arccos τ

4π

λq2

q= cτ

λq2

q.

Proof. Let τ ′ = τ2, then by Lemma 5.3, we have

(5.12) ρ pq,λ(Fτ ′) ≥

τ ′ arccos τ ′

π

λq2

q=τ arccos τ

2

2π

λq2

q.

Denote Fτ ′ = (aτ ′ , bτ ′) ⊂ Fτ = (aτ , bτ ). Applying Proposition 5.1 once with − τ2,−τ and once with

τ, τ2

in place of τ ′, τ , we obtain

|aτ ′ − aτ |, |bτ − bτ ′ | ≥1− λ4q3

λq(τ − τ ′) =(1− λ)τ

8q3λq.

In the notations of Proposition 2.9, let

κ =(1− λ)τ

8q3λq , L =

κ

4πλ∣∣∣α− p

q

∣∣∣ =(1− λ)τ

32π q3λ∣∣∣α− p

q

∣∣∣λq .Then Proposition 2.9 implies that

ρα,λ(Fτ ) ≥ ρ pq,λ(Fτ ′)−

4

L≥ ρ p

q,λ(Fτ ′)−

128πλ

(1− λ)τ

q3

λq

∣∣∣∣α− p

q

∣∣∣∣≥τ arccos τ

2

2π

λq2

q

(1− λ

1− λλq

q3

)≥ τ arccos τ

4π

λq2

q,

where the third inequality follows from (5.12) and the assumption (5.11) on∣∣∣α− p

q

∣∣∣.

6. Proof of Theorem 2: (1) and (2)

The items (1) and (2) of Theorem 2 follow from the following proposition.

Proposition 6.1. If α ∈ R \Q, 0 < λ < 1, and t > 2 are such that

(6.1) β(α) > max

(3

2,

t

t− 1

)| log λ|,

then Nα,λ /∈ UC[ωt].

39

Proof of Theorem 2: (1) and (2). The case λ = 1 follows from Theorem 1 and the second half ofthe Frostman’s Lemma (Lemma 4.1), since supp ρα,λ = S(α, λ). By (5.2) the case λ > 1 followsfrom the case λ < 1, therefore we focus on the latter.

If β(α) > 32| log λ|, then (6.1) holds with any t > 3, hence Nα,λ /∈

⋃t>3 UC [ωt], thus by Corol-

lary 2.3 we have γα,λ /∈⋃t>4 UC [ωt]. If β(α) > 2| log λ|, then (6.1) holds with any t > 2, hence

Nα,λ /∈⋃t>2 UC [ωt], thus by Corollary 2.3 we have γα,λ /∈

⋃t>3 UC [ωt].

Proof of Proposition 6.1. Fix t > 2 satisfying (6.1) and let

(6.2) r =1

2(t− 1)max(0, 3− t)| log λ|+ ν,

where ν > 0 is sufficiently small to ensure that β(α)− r > 32| log λ|. Then we can find a sequence

of rationals PkQk→ α such that

(6.3)


∣∣∣∣ ≤ e−(β(α)−r)Qk .

For sufficiently large k, we have Qk ≥ q0(r, α), and the choice of r guarantees that pq

= PkQk

satisfies∣∣∣∣α− p

q

∣∣∣∣ ≤ τ 2 arccos τ

256π2 q4λ

3q2 , with τ =

1

2.

Define F 12

corresponding to one such pq

using (5.8). Then, Lemma 5.4 gives

(6.4) ρα,λ(F 12) ≥ λ

q2

20 q.

Claim. If Nα,λ ∈ UC [ωt] for some t > 2, then

(6.5) ρα,λ(F 12) ≤ Ct q

2t λq2

(t−2) e−rq(t−1).

For sufficiently large q, the lower bound (6.4) with our choice (6.2) of r contradicts the upperbound (6.5), therefore the assumption Nα,λ ∈ UC [ωt] for some t > 2 can not hold.

It remains to prove the claim.The inequality (6.3) ensures that the assumptions of Propositioin 3.1 hold for δ = λq and p

q= Pm

Qm

when m > k. Therefore, if Nα,λ ∈ UC [ωt], then Corollary 3.2 and part (i) of Remark 2.1 implythat

ρα,λ

(S

(PmQm

, λ

)⋂F 1

2

)≤ ρα,λ

(S

(PmQm

, λ

)⋂Jλq

)≤ Ct q

2t λq2

(t−2) e−rq(t−1),

where Jλq is the set Jδ defined by (3.2) for δ = λq.On the other hand, by the continuity of the spectrum (Proposition 2.8)

S(α, λ) ⊂ S

(PmQm

, λ

)+ 6√

2λ

(−

√∣∣∣∣α− PmQm

∣∣∣∣,+√∣∣∣∣α− Pm

Qm

∣∣∣∣).

Hence S(α, λ)\S(PmQm, λ)

can be covered by (at most) 2Qm intervals of length≤ 12√

2λ

√∣∣∣α− PmQm

∣∣∣.Therefore, if Nα,λ ∈ UC [ωt], then

ρα,λ

(S(α, λ) \ S

(PmQm

, λ

))≤ C

Qm

logt 1

|α− PmQm|≤ CQ1−t

m

(β(α)− r)t,


where the last inequality follows from (6.3). Thus,

ρα,λ(F 12) = ρα,λ(S(α, λ) ∩ F 1

2) ≤ Ct q

2t λq2

(t−2) e−rq(t−1) +CQ1−t

m

(β(α)− r)t.

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 Propo-sition 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 existκ > 0 and ε0 > 0 such that for any E ∈ S and for any 0 < ε ≤ ε0, we have

(7.1) ω(|(E − ε, E + ε) ∩ S|) ≥ κ ε .

Proposition 7.2. For any α ∈ R\Q, e−2β(α)

3 < λ < 1, the spectrum S(α, λ) is not ω2-homogeneous.

Proof of Theorem 3-(1) . If λ = 1, then by the results of Avila–Krikorian [10] and Last [54] themeasure of the spectrum |S(α, λ)| = 0 for any irrational α. Thus S(α, λ) is not ω2-homogeneous

and not homogeneous. If e−2β(α)

3 < λ < 1, then by Proposition 7.2 the spectrum S(α, λ) is not

ω2-homogeneous, thus not homogeneous. By (5.2), if 1 < λ < e2β(α)

3 the spectrum S(α, λ) is notω2-homogeneous as well, thus not homogeneous.

Proof of Proposition 7.2. We start similarly to the proof of Proposition 6.1. Choose r > 0 suffici-etly small to ensure that β(α) − r > 3

2| log λ|. Then choose a sequence Pk

Qk→ α satisfying (6.3),

namely∣∣∣α− Pk

Qk

∣∣∣ ≤ e−(β(α)−r)Qk . For sufficiently large k, we have Qk ≥ q0(r, α), and the choice of

r guarantees that pq

= PkQk

satisfies

(7.2)

∣∣∣∣α− p

q

∣∣∣∣ ≤ τ 2 arccos τ

256 π2 q4λ

3q2 , with τ =

1

4.

We shall use the sets F 12

= (a, b) and F 14

= (a, b) from (5.8). Applying the second part of

Proposition 5.1 once with τ ′ = −14, τ = −1

2, and once with τ ′ = 1

2, τ = 1

4, we obtain

|b− b|, |a− a| ≥ 1− λ16 q3

λq .

By (7.2) the condition of Lemma 5.4 holds true, therefore applying Lemma 5.4 with τ = 14, we get

ρα,λ(F 14) ≥ λ

q2

10 q> 0 ,

and in particular F 14∩ S(α, λ) 6= ∅. Let

E ∈ F 14∩ S(α, λ), ε =

1− λ16 q3

λq,p

q=PmQm

, m > k ,

and note that (E − ε, E + ε) ⊂ F 12⊂ Jλq . By Corollary 3.2 (and Remark 2.1) applied to the

Lebesgue measure with ω1(ε) = ε so that (W3 ω1)(ε) ≤ Cε, we obtain∣∣∣∣(E − ε, E + ε)⋂

S

(PmQm

, λ

)∣∣∣∣ ≤ ∣∣∣∣F 12

⋂S

(PmQm

, λ

)∣∣∣∣ ≤ ∣∣∣∣Jδ⋂S

(PmQm

)∣∣∣∣≤ C q2

λq2

exp

(− erq

15000q2λq2

).

41

By the continuity of the spectrum (Proposition 2.8), we have

S(α, λ) ⊂ S

(PmQm

, λ

)+

(−6√

2


∣∣∣∣,+6√

2


∣∣∣∣)

.

Since the set S(α, λ) \ S(PmQm, λ)

is contained in the union of (at most) 2Qm intervals and since∣∣∣α− PmQm

∣∣∣ ≤ e−(β(α)−r)Qm , we conclude that

|(E − ε, E + ε) ∩ S(α, λ)| ≤∣∣∣∣(E − ε, E + ε)

⋂S

(PmQm

, λ

)∣∣∣∣+ 2Qm 6√

2


∣∣∣∣≤ C q2

λq2

exp

(− erq

15000q2λq2

)+ 2Qm e

−(β(α)−r)Qm/2 .

The second term tends to zero as Qm →∞, thus we conclude that

|(E − ε, E + ε) ∩ S(α, λ)| ≤ C q2

λq2

exp

(− erq

15000q2λq2

)≤ exp

(−e

rq2 λ−

q2

),

hence1

log2 1|(E−ε,E+ε)∩S(α,λ)|

≤ e−rq λq < κ1− λ16 q3

λq = κ ε ,

for any κ > 0 and sufficiently large q. Thus the spectrum is not ω2-homogeneous.

7.2. Failure of Parreau–Widom condition. By (5.2) we only need to prove Theorem 3-(2) for0 < λ ≤ 1. Note that a Parreau–Widom set is always of positive Lebesgue measure [26, pp.6] thusin 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 weassume 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 γα,λ(z) that corresponds to Hα,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 theintervals Fτ which we constructed above, the Lyapunov exponent is not too small (≥ const q−2λ

q2 ).

On the other hand, we show that the spectrum of Hα,λ,θ in Fτ has many (≥ const q−3λ−(1+r)q

2 )gaps. Therefore the contribution of Fτ to the sum (1.16) is lower-bounded by a quantity whichdiverges as q →∞.

To implement this plan, we first recall that β(α) + 3 log λ > 0, therefore there exists r > 0 anda sequence Pk

Qk→ α such that

(7.3)


∣∣∣∣ ≤ e−(β(α)−r)Qk ≤ λ(3+r)Qk .

Let F 12

be as defined in (5.8) for pq

= PkQk

, then by (5.8) we have

(7.4) |F 12| ≥ 1− λ

4 q3λq.

Using the continuity of the spectrum (Proposition 2.8) and (7.3) we obtain for m > k that for any

E ∈ F 12

there exists E ′ ∈ S(PmQm, λ)

such that

(7.5) |E − E ′| < 6

√2λ

∣∣∣∣pq − PmQm

∣∣∣∣ ≤ 12λ(3+r)q

2 .


Therefore, by (7.4) we can find a collection Jq of disjoint closed intervals j ⊂ F 12

of length |j| =

225λ(3+r)q

2 , so that

(7.6) #Jq ≥

⌊1−λ4 q3

λq

225λ(3+r)q

2

⌋≥ (1− λ)

1000q3λ−

(1+r)q2 .

Let j ∈ Jq be an interval and divide it into nine equal closed subintervals j1, . . . , j9 of length

|jk| =225

9λ

(3+r)q2 = 25λ

(3+r)q2 , k = 1, 2, · · · 9.

Then, by (7.5) applied to the centers of j2, j8 there exist s2 ∈ S(PmQm, λ)⋂

j2 and s8 ∈ S(PmQm, λ)⋂

j8.

Another use of Proposition 2.8 and (7.3) imply the existence of s′2, s′8 ∈ S(α, λ) such that

|s′2 − s2|, |s′8 − s8| ≤ 6

(2λ

∣∣∣∣α− PmQm

∣∣∣∣) 12

≤ 12λ(3+r)Qm

2 <|jk|2

,

and in particular

(7.7) S(α, λ) ∩ (j1 ∪ j2 ∪ j3), S(α, λ) ∩ (j7 ∪ j8 ∪ j9) 6= ∅ .

On the other hand, we can apply Proposition 3.1 to

p

q=PkQk

,p

q=PmQm

, m > k, δ = λq, ε = exp

(− erq

15000q2λq2

).

It yields that for any E ∈ Jλq , where Jλq is the set Jδ defined by (3.2) for δ = λq, and for any θ

γ PmQm

,λ,θ(E + iε) ≥ δ

9600 q2 λq2

=λq2

C1 q2.

In particular, the same lower bound holds true for the θ-averaged Lyapunov exponent γ PmQm

,λ(E+iε).

Applying Proposition 2.4 to the Lebesgue measure with ω1(ε) = ε so that (W3 ω1)(ε) ≤ Cε as in

Remark 2.1, with ξ = λq2

C1 q2, we obtain

(7.8)

∣∣∣∣E | ∣∣∣γ PmQm ,λ (E + iε)− γ PmQm

,λ (E)∣∣∣ ≥ λ

q2

2C1 q2

∣∣∣∣ ≤ C2 q2

λq2

ε .

Since |j5| = 25λ(3+r)q

2 ≥ C2 q2

λq2ε, (7.8) implies that there exists E = E

(PmQm, j)∈ j5 such that

γ PmQm

,λ

(E

(PmQm

, j

))≥ λ

q2

4C1q2.

Thus, we get using (7.6)∑j∈Jq

γ PmQm

,λ

(E

(PmQm

, j

))≥ λ

q2

4C1q2#Jq ≥

λq2

4C1q2

1− λ1000 q3

λ−(1+r)q

2

≥ 1− λ4000C1 q5

λ−rq2 .

(7.9)

This holds for PmQm

for any m > k. By compactness we can choose a subsequencePmlQml→ α, Qml > q,

such that for any interval j ∈ Jq the sequence of points E(PmlQml

, j)

converges to a number which

we denote by E(α, j).

43

By a result of Bourgain-Jitomirskaya [20], the θ-averaged Lyapunov exponent is jointly contin-uous in (E,α) for every E and every irrational α, thence

limQml→∞

γ PmlQml

,λ

(E

(PmlQml

, j

))= γα,λ(E(α, j)) = γα,λ(E(α, j)),

in particular,

(7.10) γα,λ(E(α, j)) ≥ λq2

4C1 q2> 0.

Since by (3.27) the Lyapunov exponent is equal to zero on the spectrum, we obtain that E(α, j) /∈S(α, λ). Since j5 is closed we conclude that E(α, j) ∈ j5, and by (7.7) we obtain that differentenergies E(α, j) lie in different gaps of Hα,λ,θ (in particular, all these energies are distinct). From(7.10) we obtain as in (7.9)∑

j∈Jq

γα,λ (E (α, j)) ≥ λq2

4C1q2#Jq ≥

1− λ4000C1 q5

λ−rq2 .

In particular, the left-hand side of (1.16) is bounded from below by this expression. Since thisholds for any q = Qk, we get that this sum is not bounded, therefore the Parreau–Widom conditionfails.

8. Proof of Theorem 2-(3)

If Nα,λ /∈ UC [ω], then by Corollary 2.3 we have γα,λ /∈⋃t>2 UC [ωt]. Thus we only need to

prove that Nα,λ /∈ UC [ω]. The case λ = 1 follows from Theorem 1, (2) and the second half of theFrostman’s Lemma (Lemma 4.1), since supp ρα,λ = S(α, λ). By (5.2) the case λ > 1 follows fromthe case λ < 1, therefore we focus on the latter.

We construct a sequencepnqn

of convergents of α (arising from the continuous fraction ex-

pansion) as follows. For any fixed n0 the terms q1, . . . , qn0 can be chosen in an arbitrary way. Ifq1, . . . , qn, n > n0, are given, we find xn > 0 such that for any 0 < x < xn

(8.1) ω(x) log1

x≤ δ

q2n

=λqn

q2n

,

where we set δ = λqn , and we choose

(8.2) qn+1 ≥ max

(q6nλ−2qn ,

C ′qnλqn

log5

xn

),

where C ′ > 0 is the constant from Lemma 8.1 below. Denote the set of α ∈ R \ Q for which thesequence of convergents satisfies (8.2) by Ω(λ). Clearly, this set is Gδ-dense in R.

Lemma 8.1. If α ∈ Ω(λ), then the sets F 12

= (a, b) defined for pq

= pnqn

as in (5.8) satisfy:

(1)∣∣∣S (pn+1

qn+1, λ)⋂

F 12

∣∣∣ ≤ exp(− qn+1λqn

C′qn

)≤ 1

100|F 1

2| .

(2) S(pn+1

qn+1, λ)⋂

F 12

is 1100|F 1

2|-dense in

(a+

|F 12|

100, b−

|F 12|

100

), i.e.

∀E ∈

(a+|F 1

2|

100, b−

|F 12|

100

)∃E ′ ∈ S

(pn+1

qn+1

, λ

)⋂F 1

2: |E − E ′| ≤ 1

100|F 1

2| .


Proof. First, by construction

(8.3) F 12⊂ S

(pnqn, λ

), |F 1

2| ≥ 1− λ

4q3n

λqn .

In the notation of Proposition 3.3 let δ = λqn , r = | lnλ|. Then from (8.2) we obtain

qn+1δ ≥ λqnq6nλ−2qn ≥ λ−qn = erqn ,

thus Corollary 3.4 is applicable and we obtain∣∣∣∣S (pn+1

qn+1

, λ

)⋂F 1

2

∣∣∣∣ ≤ ∣∣∣∣S (pn+1

qn+1

, λ

)⋂Jλqn

∣∣∣∣ ≤ Cq2n

λqnexp

(−qn+1λ

qn

C1qn

)≤ exp

(−qn+1λ

qn

C ′qn

)≤ 1

100|F 1

2| ,

where C ′ > 0 is a constant, and on the last step we have used (8.2) and (8.3). This proves the firststatement.

By continuity of the spectrum (Proposition 2.8) for any

E ∈

(a+|F 1

2|

100, b−

|F 12|

100

)⊂ S

(pnqn, λ

)there exists E ′ ∈ S

(pn+1

qn+1, λ)

such that

|E − E ′| < 6

(2λ

∣∣∣∣pn+1

qn+1

− pnqn

∣∣∣∣) 12

=6√

2λ√qn+1qn

≤|F 1

2|

100,(8.4)

where the last inequality follows from (8.2) and (8.3) . Thus, (8.4) implies that E ′ ∈ S(pn+1

qn+1, λ)⋂

F 12,

therefore we conclude that S(pn+1

qn+1, λ)⋂

F 12

is|F 1

2|

100-dense in the interval

(a+

|F 12|

100, b−

|F 12|

100

).

We also need the following elementary claim.

Claim 8.2. Let A ⊂ (a, b) be a finite union of intervals such that

(1) |A| ≤ τ |b− a|, for some 0 ≤ τ ≤ 13.

(2) A is τ |b− a|-dense in (a+ τ |b− a|, b− τ |b− a|).

Then the number of intervals comprising A is ≥ 1−2τ3τ

.

Proof. Let A =⊎c∈C(c− ac, c+ ac). Then, by (2)

(a, b) ⊂⋃c∈C

(c− ac − τ |b− a|, c+ ac + τ |b− a|) .

Therefore, (1) yields

(1− 2τ)|b− a| ≤∑c∈C

(2ac + 2τ |b− a|) ≤ 3τ |b− a|#c,

namely, #C = #( intervals in A) ≥ 1−2τ3τ

.

By Lemma 8.1 the conditions of Claim 8.2 hold with τ = 1100

and we obtain

#

(bands of S

(pn+1

qn+1

, λ

)lying entirely in F 1

2

)≥ 1− 2τ

3τ− 2 ≥ 30.

45

Let I ⊂ F 12

be one band of S(pn+1

qn+1, λ)

. From the first part of Lemma 8.1 we obtain

|I| ≤ exp

(−qn+1λ

qn

C ′qn

).

Let us define an extended interval I+ as the Minkowski sum

I+ = I +

(−2 exp

(−qn+1λ

qn

C ′qn

),+2 exp

(−qn+1λ

qn

C ′qn

)).

Then,

|I+| ≤ 5 exp

(−qn+1λ

qn

C ′qn

).

Let ρ pn+1qn+1

,λ be the θ-averaged density of states defined as in (2.6), corresponding to H pn+1qn+1

,λ,θθ.

Since I is a band of S(pn+1

qn+1, λ)

we have

ρ pn+1qn+1

,λ(J) =1

qn+1

.

Now apply Propsition 2.9 with κ = 2 exp(− qn+1λqn

C′qn

)and L = 10qn+1. The condition (8.2) implies

that |α− pn+1

qn+1| ≤ λ2qn+1 , hence (for sufficiently large n)

κ = 2 exp

(−qn+1λ

qn

C ′qn

)≥ 4πλ

∣∣∣∣α− pn+1

qn+1

∣∣∣∣L .

Using the condition (8.2) on qn+1, the definition of L, and applying Proposition 2.9 we obtain

ρα,λ(I+) ≥ ρ pn+1qn+1

,λ(I)− 4L≥ 1

qn+1− 2

5qn+1≥ 1

2qn+1.

By the condition (8.2) on qn+1 we obtain

5 exp

(−qn+1λ

qn

C ′qn

)≤ xn,

whence

ω

(5 exp

(−qn+1λ

qn

C ′qn

))log

1

5 exp(− qn+1λqn

C′qn

) ≤ λqn

q2n

,

therefore by (8.1), we have

1

2qn+1

≥ q2n

2qn+1λqnω

(5 exp

(−qn+1λ

qn

C ′qn

))log

1

5 exp(− qn+1λqn

C′qn

) .

Consequently, we have

ρα,λ(I+) ≥ q2n

2qn+1λqnω

(5 exp

(−qn+1λ

qn

C ′qn

))qn+1λ

qn

C ′′qn

≥ qnC ′′′

ω

(5 exp

(−qn+1λ

qn

C ′qn

))≥ qnC ′′′

ω(|I+|) ,

and in particular

limn→∞

ρα,λ(I+)

ω(|I+|)=∞ .

This concludes the proof of Theorem 2-(3).


Acknowledgements

The authors thank S. Jitomirskaya for useful discussions on the homogeneity of the spectrum.A. Avila was supported by the ERC Starting Grant “Quasiperiodic”. This work was initiatedwhile M. Shamis was a postdoc at Weizmann Institute, supported by the ISF grant 147/15,and while Q. Zhou was a postdoc at Universite Paris Diderot supported by ERC Starting Grant“Quasiperiodic”. Q. Zhou was also partially supported by National Key R&D Program of China(2020YFA0713300), NSFC grant 12071232 and Nankai Zhide Foundation.

References

[1] S. Amor, Holder continuity of the rotation number for the quasiperiodic cocycles in SL(2,R),Commun. Math. Phys. 287 565–588, (2009)

[2] S. Aubry and G. Andre, Analyticity breaking and Anderson localization in incommensuratelatices, Ann. Israel. Phys. Soc. 3, 133–164 (1980)

[3] S. Aubry, The new concept of transitions by breaking of analyticity in a crystallographic model,Solitons and Condensed Matter Physics. Springer Berlin Heidelberg, 264–277 (1978)

[4] A. Avila, The absolutely continuous spectrum of the almost Mathieu operator, arXiv preprintarXiv:0810.2965 (2008)

[5] A. Avila. Global theory of one-frequency Schrodinger operators. Acta Math. 215 (2015), 1-54.[6] A. Avila, D. Damanik, Absolute continuity of the integrated density of states for the almostMathieu operator with non-critical coupling. Inventiones Mathematicae 172 (2008), 439-453.

[7] A. Avila, S. Jitomirskaya, Almost localization and almost reducibility, J. Eur. Math. Soc. 12,93–131 (2010)

[8] A. Avila, S. Jitomirskaya, The Ten Martini Problem, Ann. of Math. 170, 303–342 (2009)[9] A. Avila, S. Jitomirskaya, C. Sadel, Complex one-frequency cocycles, J. Eur. Math. Soc. 161915–1935, (2013)

[10] A. Avila, R. Krikorian, Reducibility or nonuniform hyperbolicity for quasiperiodic Schrodingercocycles, Ann. of Math. No. 1 911–940 (2006)

[11] A. Avila, J. You, and Z. Zhou, Sharp Phase transitions for the almost Mathieu operator, DukeMath. J. 166, 2697–2718 (2017)

[12] J. E. Avron, D. Osadchy, R. Seiler, A topological look at the quantum Hall effect, Physicstoday, 38–42 (2003)

[13] J. Avron, B. Simon, Singular continuous spectrum for a class of almost periodic Jacobi ma-trices, Bulletin of the American Mathematical Society, 6(1), 81–85 (1982)

[14] J. Avron, B. Simon, Almost periodic Schrodinger operators. II. The integrated density of states.Duke Math. J. 50, 369–391 (1983)

[15] J. Avron, P. van Mouche and B. Simon, On the measure of the spectrum for the almostMathieu operator, Commun. Math. Phys. 132, 103–118 (1990)

[16] C. J. Bishop, Y. Peres, Fractals in probability and analysis, Vol. 162 Cambridge UniversityPress 2017

[17] J. Bochi, Genericity of zero Lyapunov exponents, Ergodic Theory Dynam. Systems 22(6)1667–1696 (2002)

[18] J. Bourgain, Holder regularity of integrated density of states for the almost Mathieu operatorin a perturbative regime, Lett. Math. Phys. 51− 2 83–118 (2000)

[19] J. Bourgain, Green’s function estimates for lattice Schrodinger operators and applications.Annals of Mathematics Studies, 158. Princeton University Press, Princeton, NJ, 2005. x+173 pp.

[20] J. Bourgain, S. Jitomirskaya, Continuity of the Lyapunov exponent for quasiperiodic operatorswith analytic potential, Dedicated to David Ruelle and Yasha Sinai on the occasion of their 65thbirthdays, J. Statist. Phys. 108 no. 5-6, 1203–1218 (2002)

47

[21] J. Bourgain, A. Klein, Bounds on the density of states for Schrodinger operators, Inventionesmathematicae, 194(1), 41–72 (2013)

[22] A. Cai, C. Chavaudret, J. You, Q. Zhou, Sharp Holder continuity of the Lyapunov exponentof finitely differentiable quasiperiodic cocycles, Math. Z. 291, no. 3-4, 931–958 (2019).

[23] L. Carleson, On H∞ in multiply connected domains, Conference on harmonic analysis in honorof Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981), 349–372, Wadsworth Math. Ser., Wadsworth,Belmont, CA, (1983)

[24] W. Chambers, Linear network model for magnetic breakdown in two dimensions. Phys. Rev.A 140, 135–143 (1965)

[25] M-D. Choi, G. A. Elliott, and N. Yui, Gauss polynomials and the rotation algebra. Invent.Math. 99.1, 225–246 (1990)

[26] J. S. Christiansen, Dynamics in the Szego class and polynomial asymptotics, Journal d’AnalyseMathematique, 137, 723–749 (2019)

[27] W. Craig, Pure point spectrum for discrete almost periodic Schrodinger operators, Comm.Math. Phys. 88 (1983), no. 1, 113–131.

[28] W. Craig, B. Simon, Subharmonicity of the Lyaponov index, Duke Math. J. 50, no. 2, 551–560(1983).

[29] W. Craig, B. Simon, Log Holder Continuity of the Integrated Density of States for StochasticJacobi Matrices. Commun. Math. Phys. 90, 207–218 (1983)

[30] D. Damanik, J. Fillman, Schrodinger operators with thin spectra, arXiv:2007.01402[31] D. Damanik, M. Goldstein, and M. Lukic, The spectrum of a Schrodinger operator with smallquasiperiodic potential is homogeneous, J. Spec. Theory, 6, 415–427 (2016).

[32] D. Damanik, M. Goldstein, W. Schlag, and M. Voda, Homogeneity of the spectrum forquasiperiodic Schrodinger operators, J. Eur. Math. Soc. 20, no. 12, 3073–3111 (2018).

[33] F. Delyon, B. Souillard. The rotation number for finite difference operators and its properties,Commun. Math. Phys. 89.3 415–426 (1983)

[34] P. Duarte, S. Klein, Lyapunov exponents of linear cocycles, Continuity via large deviations.Atlantis Studies in Dynamical Systems, 3. Atlantis Press, Paris, 2016. xiii+263 pp.

[35] P. Duarte, S. Klein and M. Santos, A random cocycle with non Holder Lyapunov exponent.Discrete Contin. Dyn. Syst. 39 (2019), no. 8, 4841–4861.

[36] A. Figotin, L. Pastur, Spectra of random and almost-periodic operators, Spectra of randomand almost-periodic operators. Grundlehren der Mathematischen Wissenschaften [FundamentalPrinciples of Mathematical Sciences], 297. Springer-Verlag, Berlin, 1992. viii+587 pp.

[37] A. Furman, On the multiplicative ergodic theorem for uniquely ergodic systems. Annales del’Institut Henri Poincare (B) Probability and Statistics. Vol. 33. No. 6 (1997)

[38] M. Goldstein, W. Schlag, Holder continuity of the integrated density of states for quasiperiodicSchrodinger equations and averages of shifts of subharmonic functions. Ann. of Math. 155–203(2001)

[39] M. Goldstein, W. Schlag, Fine properties of the integrated density of states and a quantitativeseparation property of the Dirichlet eigenvalues, Geom. Funct. Anal. 18, 755–869 (2008)

[40] A. Ya. Gordon, The point spectrum of the one-dimensional Schrodinger operator, UspekhiMatematicheskikh Nauk 31 no. 4, 257–258 (1976)

[41] P. G. Harper, Single band motion of conduction electrons in a uniform magnetic field, Proc.Phys. Soc. London A., 68 874–892 (1955)

[42] B. Helffer, Q. Liu, Y. Qu, Q. Zhou, Positive Hausdorff dimensional spectrum for the criticalalmost Mathieu operator, Commun. Math. Phys. 368 369–382 (2019)

[43] P. D. Hislop, C. A. Marx, Dependence of the density of states on the probability distributionfor discrete random Schrodinger operators, Int. Math. Res. Not. IMRN 17, 5279—5341 (2020),


[44] S. Jitomirskaya, Almost everything about the almost Mathieu operator. II, In XIth InternationalCongress of Mathematical Physics (Paris, 1994), 373–382 (1994)

[45] S. Jitomirskaya, Metal-insulator transition for the almost Mathieu operator, Ann. of Math.150(3), 1159–1175 (1999)

[46] S. Jitomirskaya, D. A. Koslover, M. S. Schulteis, Continuity of the Lyapunov exponent foranalytic quasiperiodic cocycles, Ergodic Theory Dynam. Systems 29 1881–1905 (2009)

[47] S. Jitormiskya, W. Liu, Universal hierarchical structure of quasiperiodic eigenfuctions, Ann.of Math. 187 no.3 (2018)

[48] S. Jitomirskaya, I. Krasovsky, Critical almost Mathieu operator: hidden singularity, gap con-tinuity, and the Hausdorff dimension of the spectrum, arXiv:1909.04429 (2019)

[49] S. Jitomirskaya, S. Zhang, Quantitative continuity of singular continuous spectral measuresand arithmetic criteria for quasiperiodic Schrodinger operators, to appear in J. Eur. Math. Soc.

[50] P. W. Jones, D. E. Marshall, Critical points of Green’s function, harmonic measure, and thecorona problem, Ark. Mat. 23, no. 2, 281—314 (1985)

[51] W. Kirsch, An invitation to random Schrodinger operators, With an appendix by FredericKlopp. Panor. Syntheses, 25, Random Schrodinger operators, 1–119, Soc. Math. France, Paris,2008.

[52] H. Kruger and Z. Gan, Optimality of log Holder continuity of the integrated density of states,Math. Nachr. 284 (2011), no. 14-15, 1919–1923.

[53] N. Karaliolios, X. Xu and Q. Zhou, Anosov-Katok constructions for quasi-periodic SL(2, R)cocycles, arXiv:2012.11069

[54] Y. Last, Zero measure spectrum for the almost Mathieu Operator, Commun. Math. Phys. 164,421–432 (1994)

[55] Y. Last, M. Shamis, Zero Hausdorff Dimension Spectrum for the Almost Mathieu Operator.Commun. Math. Phys. 348. No. 3, 729–750 (2016)

[56] W. Liu and Y. Shi, Upper bounds on the spectral gaps of quasi-periodic Schrodinger operatorswith Liouville frequencies. J. Spectr. Theory 9 (2019), no. 4, 1223–1248.

[57] M. Leguil, J. You, Z. Zhao, Q. Zhou, Asymptotics of spectral gaps of quasiperiodic Schrodingeroperators, Https://arxiv.org/pdf/1712.04700.pdf

[58] D. Osadchy, J. E. Avron, Hofstadter butterfly as quantum phase diagram, J. Math. Phys. 42,5665–5671 (2001)

[59] R. Peierls, Zur Theorie des Diamagnetismus von Leitungselektronen, Z. Phys. 80, 763–791(1933)

[60] J. Puig, A nonperturbative Eliasson’s reducibility theorem, Nonlinearity, 19(2), 355–376(2006).

[61] A. Rauh, Degeneracy of Landau levels in chrystals, Phys. Status Solidi B 65, 131–135 (1974)[62] W. Schlag, Harmonic analysis notes, https://gauss.math.yale.edu/~ws442/

harmonicnotes_old.pdf

[63] B. Simon, Equilibrium measures and capacities in spectral theory, Inverse Probl. Imaging 1(2007), no. 4, 713–772.

[64] B. Simon. Fifty Years of the Spectral Theory of Schrodinger Operators, Linde Hall InauguralMath Symposium, Caltech, Feb. 2019, https://www.youtube.com/watch?v=HLWoJYbdBUA

[65] B. Simon. Public communication at ”A Young Researcher Symposium on the Occasion ofthe 70th Birthday of Barry Simon”. http://www.fields.utoronto.ca/video-archive/2016/08/1962-15727

[66] M. Shamis, On the continuity of the integrated density of states in the disorder, Int. Math.Res. Not. IMRN 12 (2019), rnz321

https://gauss.math.yale.edu/~ws442/harmonicnotes_old.pdf

https://gauss.math.yale.edu/~ws442/harmonicnotes_old.pdf

https://www.youtube.com/watch?v=HLWoJYbdBUA

http://www.fields.utoronto.ca/video-archive/2016/08/1962-15727

http://www.fields.utoronto.ca/video-archive/2016/08/1962-15727

49

[67] M. Sodin, P. Yuditskii Almost periodic Sturm-Liouville operators with Cantor homogeneousspectrum. Commentarii Mathematici Helvetici 70.1, 639–658 (1995)

[68] M. Sodin, P. Yuditskii, Almost periodic Jacobi matrices with homogeneous spectrum, infinitedimensional Jacobi inversion, and Hardy spaces of character-automorphic functions. J. of Geom.Anal. 7(3), 387–435 (1997)

[69] S. Surace, Positive Lyapunov exponent for a class of ergodic Schrodinger operators. Commun.Math. Phys. 162, 529–537 (1994)

[70] K. Tao, Non-perturbative homogeneous spectrum of some quasi-periodic Gevrey Jacobi opera-tors with weak Diophantine frequencies, arXiv:2102.12851

[71] A. F. Timan, Theory of approximation of functions of a real variable, New York: PergamonPress (1963)

[72] Y. Wang, J. You, Examples of discontinuity of Lyapunov exponent in smooth quasiperiodiccocycles, Duke Math. 162 2363–2412, (2013)

[73] H. Widom, The maximum principle for multiple-valued analytic functions, Acta Math. 126,63–82 (1971)

[74] H. Widom,Hp-sections of vector bundles over Riemann surfaces, Ann. of Math. (2) 94 304–324(1971)

[75] M. Wilkinson, E. J. Austin, Spectral dimension and dynamics for Harper’s equation, Phys.Rev. B 50, 1420–1430 (1994)

[76] T. H. Wolff, Lectures on harmonic analysis, University Lecture Series, Vol. 29, AmericanMathematical Soc. (2003)

Institut fur Mathematik, Universitat Zurich Winterthurerstrasse 190, CH-8057 Zurich, Switzer-land & IMPA, Estrada Dona Castorina 110, 22460-320, Rio de Janeiro, Brazil

Email address: [email protected]

Institute of Mathematics, Hebrew University, 91904 Jerusalem, Israel.Email address: [email protected]

School of Mathematical Sciences, Queen Mary University of London, Mile End Road, LondonE1 4NS, England

Email address: [email protected]

Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, ChinaEmail address: [email protected]

arXiv:2110.07974v1 [math-ph] 15 Oct 2021

Documents

Transcript of arXiv:2110.07974v1 [math-ph] 15 Oct 2021