Large Deviation Theorems for Dirichlet Determinants of Analytic Quasi-Periodic Jacobi Operators with Brjuno-Russmann¨ Frequency

COMMUNICATIONS ON doi:10.3934/cpaa.2020240 PURE AND APPLIED ANALYSIS Volume 19, Number 12, December 2020 pp. 5305–5335

LARGE DEVIATION THEOREMS FOR DIRICHLET DETERMINANTS OF ANALYTIC QUASI-PERIODIC JACOBI OPERATORS WITH BRJUNO-RUSSMANN¨ FREQUENCY

Wenmeng Geng and Kai Tao∗ College of Sciences, Hohai University, No.1 Xikang Road Nanjing, Jiangsu, 210098, China

(Communicated by Enrico Valdinoci)

Abstract. In this paper, we first study the strong Birkhoff Ergodic Theorem for subharmonic functions with the Brjuno-Rüssmannshift on the Torus. Then, we apply it to prove the large deviation theorems for the finite scale Dirichlet determinants of quasi-periodic analytic Jacobi operators with this frequency. It shows that the Brjuno-Rüssmannfunction, which reflects the irrationality of the frequency, plays the key role in these theorems via the smallest deviation. At last, as an application, we obtain a distribution of the eigenvalues of the Jacobi operators with Dirichlet boundary conditions, which also depends on the smallest deviation, essentially on the irrationality of the frequency.

1. Introduction. We study the following quasi-periodic analytic Jacobi operators H(x, ω) on l2(Z): [H(x, ω)φ](n) = − a(x + (n + 1)ω)φ(n + 1) − a(x + nω)φ(n − 1) + v(x + nω)φ(n), n ∈ Z, (1.1) where v : T → R is a real analytic function called potential, a : T → C is a complex analytic function and not identically zero. The characteristic equations H(x, ω)φ = Eφ can be expressed as φ(n + 1) φ(n) 1 v(x + nω) − E −a(x + nω) φ(n) = . (1.2) a(x + (n + 1)ω) a(x + (n + 1)ω) 0 φ(n − 1) Deﬁne 1 v(x) − E −a(x) M(x, E, ω) := (1.3) a(x + ω) a(x + ω) 0 and call a map (ω, M):(x,~v) 7→ (x + ω, M(x)~v)

2020 Mathematics Subject Classification. 37C55, 37F10, 37C40. Key words and phrases. Large deviation theorems; Jacobi operators; finite scale Dirichlet determinants; Brjuno-Rüssmannfrequency; strong Birkhoff ergodic theorem. The second author was supported by the Fundamental Research Funds for the Central Univer- sities(Grant B200202004) and China Postdoctoral Science Foundation (Grant 2019M650094). ∗ Corresponding author.

5305 5306 WENMENG GENG AND KAI TAO a Jacobi cocycle. Due to the fact that an analytic function only has ﬁnite zeros, M(x, E, ω) and the n-step transfer matrix 1 Y Mn(x, E, ω) := M(x + kω, E) k=n make sense almost everywhere. By the Kingman’s subadditive ergodic theorem, the Lyapunov exponent

L(E, ω) = lim Ln(E, ω) = inf Ln(E, ω) ≥ 0 (1.4) n→∞ n→∞ always exists, where 1 Z L (E, ω) = log kM (x, E, ω)kdx. n n n T Let H[m,n](x, ω) be the Jaocbi operator deﬁned by (1.1) on a ﬁnite interval [m, n] a with Dirichlet boundary conditions, φ(m−1)=0 and φ(n+1)=0. Let f[m,n](x, E, ω) = det(H[m,n](x, ω) − E) be its characteristic polynomial. One has a a f[m,n](x, E, ω) = fn−m+1 x + (m − 1)ω, E, ω , (1.5) where a fn (x, E, ω) = det Hn(x, ω) − E v x + ω − E −a x + 2ω 0 ··· 0

−ax + 2ω vx + 2ω − E −ax + 3ω ··· 0

= ...... 0 0 · · · −a x + nω v x + nω − E

a In this paper, the aim is to study the properties of fn (x, E, ω). To state our conclusions, we first make some introductions to the background of our topic. The operator (1.1) has the following important special case, which is called the Schrödingeroperator and has been studied extensively: s [H (x, ω)φ](n) = φ(n + 1) + φ(n − 1) + v(x + nω)φ(n), n ∈ Z. (1.6) s s s s Then, Mn(x, E, ω), L (E, ω), Ln(E, ω) and fn(x, E, ω) have the similar definitions. In [6], Bourgain and Goldstein proved that if Ls(E, ω) > 0, then for almost all ω, the operator Hs(0, ω) has Anderson Localization, which means that it has pure- point spectrum with exponentially decaying eigenfunction. In [11], Goldstein and Schlag obtained the Höldercontinuity of Ls(E, ω) in E with the strong Diophantine ω, i.e. for some α > 1 and any integer n, C knωk > ω , (1.7) |n| (log |n| + 1)α where kxk = min |x + n|. n∈Z It is well known that for a fixed α > 1, almost every irrational ω satisfies (1.7). Obviously, if we define the Diophantine number as C knωk > ω , (1.8) |n|α LDT DIRICHLET DETERMINANTS WITH BRJUNO-RUSSMANN¨ FREQUENCY 5307 then it also has a full measure. In these two references, the key lemmas are the s following so-called large deviation theorems (LDTs for short) for matrix Mn(x, E, ω) with these two types of frequencies: for the Diophantine ω, it was proved in [6] that there exists 0 < σ < 1 such that 1 s s −σ σ mes x : log kM (x, E, ω)k − L (E, ω) > n < exp (−n ) ; (1.9) n n n

(n) (log n)A for the strong Diophantine ω, it was proved in [11] that there exists δ0 = n (n) such that for any δ > δ0 1 s s 2 mes x : log kM (x, E, ω)k − L (E, ω) > δ < exp −cδ n . (1.10) n n n (n) Here δ0 is called the smallest deviation in the LDT and very important in our paper. Compared with the Schrödingercocycle, one of the distinguishing features of the Jacobi cocycle is that it is not SL(2, C). Then Jitomirskaya, Koslover and Schulteis [17], and Jitomirskaya and Marx [18] proved that the LDT (1.9) for Mn(x, E, ω) and the weak Höldercontinuity of the Lyapunov exponent of the analytic GL(2, C) cocycles hold with the Diophantine frequency. In [20], we showed that (1.10) can hold for Mn(x, E, ω) with the strong Diophantine ω and the continuity of the Lyapunov exponent of the Jacobi cocycles L(E, ω) can be Hölderin E. For any irrational ω, there exist its continued fraction approximates { ps }∞ , qs s=1 satisfying 1 p 1 < |ω − s | < . (1.11) qs(qs+1 + qs) qs qsqs+1 Define β as the exponential growth exponent of { ps }∞ as follows: qs s=1 log q β(ω) := lim sup s+1 ∈ [0, ∞]. s qs Obviously, both the sets of the strong Diophantine frequency and the Diophantine one are the subsets of {ω : β(ω) = 0}. We say ω is the Liouville number, if β(ω) ≥ 0. Recently, more and more attentions are paid to the question that what will happen to these operators with more generic ω, such as the one satisfying β(ω) = 0, the finite Liouville one satisfying 0 ≤ β(ω) < ∞ and the irrational one. So far, the most striking answers are mainly for the almost Mathieu operators (AMO for short), which is also a special case of the Jacobi ones m [H (x, ω, λ)φ](n) = φ(n + 1) + φ(n − 1) + 2λ cos (2π(x + nω)) φ(n). n ∈ Z. (1.12) The most famous one, the Ten Martini Problem, which was dubbed by Barry Simon and conjectures that for any irrational ω, the spectrum of AMO is a Cantor set, was completely solved by Avila and Jitomirskaya [1]. In that reference, they also proved that Hm(x, ω, λ) has Anderson Localization for almost every x ∈ T with 16 β β λ > e 9 . In [4], Avila, You and Zhou improved it to λ > e . While, the answers for the Schrödingeror Jacobi operators in the positive Lya- punov exponent regimes are mainly in the study of the continuity of the Lyapunov exponent. In [5], they proved that the Lyapunov exponent is continuous in E for any irrational ω. The first result that the Höldercontinuity holds for some weak Liouville frequency, which means that β(ω) < c, where c is a small constant depending only on the analytic potential v(x), is [22]. Recently, Han and Zhang [15] 5308 WENMENG GENG AND KAI TAO ameliorated it to λ > eCβ in the large coupling regimes, where the potential v is of the form λv0 with a general analytic v0 and C is a positive constant also depending only on v0. Our second author also proved the corresponding conclusion for the Ja- cobi operators in [21]. These two results are optimal, because Avila, Last, Shamis and Zhou [3] showed that the continuity of the Lyapunov exponent of the almost Mathieu operators can’t be Hölderif β > 0 and e−β < λ < eβ. Until now, we do not know much about the spectrum of the Schrödingeror Jacobi operators in the positive Lyapunov exponent regimes when the frequency is not strong Diophantine. The main reason is that we do not know much about the finite- a volume determinant fn (x, E, ω). While, for the almost Mathieu operators, it can be handled explicitly via the Lagrange interpolation for the trigonometric polynomial. This method can be applied for the following extend Harper’s operators, which also have the cosine potential, to obtain many spectral conclusions with the generic frequency, such as [2] and [14]: ω ω a(x) =λ exp[−2πi(x + )] + λ + λ exp[2πi(x + )], 0 ≤ λ , 0 ≤ λ + λ , 3 2 2 1 2 2 1 3 v(x) =2 cos(2πx).

However, the Lagrange interpolation can not work for the more general Schr¨odinger operators, since their potentials both are generic analytic functions. Therefore, in [12], Goldstein and Schlag applied the LDT (1.10) and the relationship that

s s s fn(x, E, ω) fn−1(x + ω, E, ω) Mn(x, E, ω) = s s (1.13) fn−1(x, E, ω) fn−2(x + ω, E, ω)

s to estimate the BMO norm of fn(x, E, ω). Then they obtained the following LDT s for fn(x, E, ω) with the strong Diophantine ω by the John-Nirenberg inequality:

−1 s s (n) mes {x ∈ T : |log |fn (x)| − hlog |fn|i| > nδ} ≤ C exp −cδn δ0 . (1.14)

This LDT was applied to get the Hölderexponent of the Höldercontinuity of s s L (E, ω) in E and the upper bound on the number of eigenvalues of Hn(x, ω) contained in an interval of size n−C . What’s more, with its help, the estimation on s the separation of the eigenvalues of Hn(x, ω) and the property that the spectrum of s H (x, ω), denoted by Sω, is a Cantor set were obtained in [13], and the homogeneity of Sω was proved in [10]. In [8] and [9], Binder and Voda applied this method to our analytic Jacobi operators (1.1). It must be noted that the above conclusions all hold s a only for the strong Diophantine ω and the LDTs for fn(x, E, ω) and fn (x, E, ω) are the key lemma in the method. Now, we can declare that the concrete content of our main aim is to obtain a the LDT for the finite-volume determinant fn (x, E, ω) with more generic ω. It is important in our field and is the key preparation for the study of the spectrum problem for discrete quasiperiodic operators of second order in the future. In this paper, we assume that the frequency ω is the Brjuno-Rüssmannnumber, which is a famous extension of the strong Diophantine number. It says that there exists a monotone increasing and continuous function ∆(t) : [1, ∞) → [1, ∞) such that ∆(1) = 1 and for any positive integer k > 0, C kkωk > ω , (1.15) ∆(k) LDT DIRICHLET DETERMINANTS WITH BRJUNO-RUSSMANN¨ FREQUENCY 5309 and Z ∞ log ∆(t) 2 < +∞. (1.16) 1 t α 1 For example, this Brjuno-Rüssmannfunction ∆(t) can be t(log t + 1) , exp(t α ), α α t −1 t , exp((log t) ) and exp( (log t)α ) with α > 1. Define Γω(n) = knωk . Due to (1.11), we have that

qs+1 < Γω(qs) < qs + qs+1, and Γω(n) < Γω(qs), ∀n ∈ (qs, qs+1). (1.17) Therefore, there exists another deﬁnition of the Brjuno-R¨ussmannnumber as follow: −1 There exists a function Ψω(t) = max{kkωk , ∀0 < k ≤ t, k ∈ Z} satisfying (1.16). ∞ Note that the denominator series {qs}s=1 and the function Ψω depend on ω. Thus, to make almost every irrational number satisfy (1.15), we assume that ∆(t) > t(log t + 1). (1.18)

∆0(t) 0 0 1 It implies that log ∆(t) > log t but it is false that ∆(t) = (log ∆(t)) > (log t) = t . However, it holds for all examples which we have given. So we make the following hypothesis: Hypothesis H.1. ∆(t) > t(log t + 1) and t∆0(t) ≥ ∆(t) for any t ≥ 1. a Then, our first LDT for fn (x, E, ω) is Theorem 1.1. Let ω be the Brjuno-Rüssmannnumber satisfying Hypothesis H.1 and L(E, ω) > 0. There exist constants c = c(a, v, E, ω) and C˘ = C˘(a, v, ω), and absolute constant C such that for any integer n ≥ 1 and δ > δH.1(n) := C˘ log ∆(n) −1 −1 1− , where ∆ (·) means the inverse function of ∆(·), then (∆ (Cω n)) a a −1 mes {x ∈ T : |log |fn (x)| − hlog |fn |i| > nδ} ≤ C exp −cδ(δH.1(n)) . Remark 1. In this paper, the notation A1− means A1− for any small absolute > 0. And A1+ has the similar definition.

Remark 2. If we assume the potential v is of the form λv0 with a general analytic v0, then the second author [21] proved that there exists λ0 = λ0(v0, a) such that the Lyapunov exponent L(E, ω) is always positive for any E and any irrational ω under the condition λ > λ0.

α C˘ log n If ∆(t) = t(log t + 1) , then δH.1(n) = n1− which is very close to the smallest (log n)A 1 α deviation for the strong Diophantine number n . But if ∆(t) = exp(t ), then 1 − δH.1(n) = Cn˘ α which is too large for us to apply Theorem 1.1 to the research of the spectrum of the analytic quasi-periodic operators (1.1) in our future work. Thus, we need to make some hypothesis to improve this smallest deviation when ∆(t) grows fast: Hypothesis H.2. ω satisﬁes Hypothesis H.1 and for any t ≥ 1, t ∆(t) < exp . (1.19) log t t From (1.16), ∆(t) has an upper bound of exp( log t ) generally. But it is possible that it grows very fast and exceeds this upper bound in some intervals, and in the rest it grows very slowly and makes the integral converge. Therefore, the aim of this hypothesis is to avoid this extreme possibility. Then, our second LDT for a fn (x, E, ω) is 5310 WENMENG GENG AND KAI TAO

Theorem 1.2. Let ω be the Brjuno-R¨ussmannnumber satisfying Hypothesis H.2 and L(E, ω) > 0. There exist constants c = c(a, v, E, ω) and C˘ = C˘(a, v, ω), and absolute constant C such that for any integer n ≥ 1 and δ > δH.2(n) := C˘ −1 1− , [log(∆ (Cω n))] a a −1 mes {x ∈ T : |log |fn (x)| − hlog |fn |i| > nδ} ≤ C exp −cδ(δH.2(n)) .

1 α t Remark 3. With this hypothesis, no matter ∆(t) equals to exp(t ) or exp( (log t)α ), C˘ the smallest deviation δH.2(n) = [log log n]1− 1 which satisfies what we need for the study of the spectrum, such as Theorem 1.5. As mentioned above, what we want to avoid is the case that ∆(t) grows faster than exp(t) in some intervals. But Hypothesis H.2 only requires that ∆(t) has an upper bound, but has no restriction on its derivative. Thus, we make the following hypothesis, which gives the mutual restriction between ∆(t) and ∆0(t) and looks also very reasonable: log ∆(t) Hypothesis H.3. ω satisfies Hypothesis H.1 and t is non-increasing for any t ≥ 1. Easy computation shows that this hypothesis is equivalent to the inequality ∆(t) log ∆(t) ∆0(t) ≤ . t Combined it with Hypothesis H.1, it shows that the bound of t∆0(t) is determined by ∆(t). Obviously, all examples of functions mentioned above satisfy this hypothesis. With its help, we improve Theorem 1.1 and 1.2 as follows: Theorem 1.3. Let ω be the Brjuno-Rüssmannnumber satisfying Hypothesis H.3 and L(E, ω) > 0. There exist constants c = c(a, v, E, ω) and C˘ = C˘(a, v, ω), and absolute constant C such that for any integer n ≥ 1 and δ > δH.3(n) := C˘ log(Cω n) −1 1− , [∆ (Cω n)] a a −1 mes {x ∈ T : |log |fn (x)| − hlog |fn |i| > nδ} ≤ C exp −cδ(δH.3(n)) . Remark 4. Since t(log t + 1) < ∆(t), it is obvious that

log ∆(n) log(Cωn) δH.1(n) = δH.3(n) = . −1 1− −1 1− ∆ (Cωn) ∆ (Cωn) On the other hand, due to the fact that 0 < ∆−1(t) < t < ∆(t), n ∆(log2 ∆−1(n)), we have that

1 log(Cωn) δH.2(n) = δH.3(n) = . −1 1− −1 1− log(∆ (Cωn)) ∆ (Cωn) a The key to prove these three LDTs for fn (x, E, ω) is an ergodic theorem for the subharmonic function shifting on T. Specifically, we know that if T : X → X is an ergodic transformation on a measurable space (X, Σ, m) and f is an m−integrable function, then the Birkhoff Ergodic Theorem tells that the time average functions 1 Pn−1 k 1 R fn(x) = n k=0 f(T x) converge to the space average hfi = m(X) X fdm for almost every x ∈ X. But it doesn’t tell us how fast do they converge. So, we call a theorem the strong Birkhoff Ergodic Theorem, if it gives the convergence rate. The following strong Birkhoff Ergodic Theorem for the subharmonic function shifting on T is the key which we just mentioned above: LDT DIRICHLET DETERMINANTS WITH BRJUNO-RUSSMANN¨ FREQUENCY 5311

Theorem 1.4. Let u :Ω → R be a subharmonic function on a domain Ω ⊂ C and ω be the Brjuno-Rüssmannumber satisfying Hypothesis H.1, or H.2, or H.3. Suppose that ∂Ω consists of finitely many piece-wise C1 curves and T is contained 0 0 in Ω b Ω(i.e., Ω is a compactly contained subregion of Ω). There exist constants ˘ ˘ (n) c = c(ω, u) and C = C(Ω, u) such that for any positive n and δ > δ0 , ( n )! X mes x ∈ T : | u(x + kω) − nhui| > δn ≤ exp (−cδn) , (1.20) k=1 where  C˘ log ∆(n) δH.1(n) := −1 1− , if ω satisfies H.1,  (∆ (Cω n)) (n)  C˘ δ = δH.2(n) := −1 1− , if ω satisfies H.2, (1.21) 0 [log(∆ (Cω n))] ˘  C log(Cω n) δH.3(n) := −1 1− , if ω satisfies H.3. [∆ (Cω n)] A very interesting thing we find is that no matter the irrational frequency is, the convergence rate of the exceptional measure is always exp (−cδn). The only (n) difference is the smallest deviation δ0 . If β(ω) > 0, then our second author (n) obtained in [21] that δ0 = cβ which is proved to be optimal in [3]; if β(ω) = 0, we obtain (1.21) which includes the result for the strong Diophantine number by Goldstein and Schlag. Correspondingly, the three LDTs we obtain in this paper can be unified into the following form: a a (n)−1 (n) mes x∈T:|log |fn (x)|−hlog |fn |i|>nδ ≤C exp −cδ δ0 , ∀δ >δ0 . (1.22)

(n) While, the exceptional measure in (1.22) will not converge when δ0 = cβ! The method created by Goldstein and Schlag and applied in this paper should be improved for the Liouville frequency. We think it is a good question for our further research in the future. Here we need to emphasize that our paper is not weaker version of [21]. That (n) shows that the smallest deviation is δ0 = cβ, and then the strong Birkhoff ergodic theorem and the LDTs for matrices hold when the deviation is larger than (n) δ0 . Letting the positive Lyapunov exponent be this deviation, our second author obtained the Höldercontinuity of the Lyapunov exponent. However, if we applied these results in our condition that β = 0, then the smallest deviation is 0! It is absurd! So, compared to [21], the main aim of our second section is to find the smallest deviation when β = 0. What’s more, we will find that in technology the P2m−1 qs−j+1 key is to estimate log qs−j+1. It is easy when β > 0: j=1 qs−j 2m−1 2m−1 X qs−j+1 X log q ≤ 2β q ≤ 8βn. q s−j+1 s−j+1 j=1 s−j j=1 While, when β = 0, the fact that { log qs+1 }∞ has different speeds, which depend qs s=1 on ∆(t), to converge to 0 makes this estimation much harder. On the other hand, a the aims of our Section 3 and 4 are to obtain the LDT for fn and its applications, which are nonexistent in [21]. In summary, the focus point of our paper is to show the importance of the smallest deviation of the strong Birkhoff ergodic theorem and calculate it when β = 0. Of course, when we need the LDTs for matrices and the Höldercontinuity of the Lyapunov exponent, such as Lemma 3.1 and 4.1, we can use the results from [21] directly. 5312 WENMENG GENG AND KAI TAO

At last, we have an application of our LDTs, which estimates the upper bound (n) 1 on the number of eigenvalues of Hn(x, ω) contained in an interval of size (δ0 ) h , where h is the Hölderexponent of the Höldercontinuity of L(E, ω), see Lemma 4.1. The distribution of the eigenvalues is very important in the further study of the spectrum problem for discrete quasiperiodic operators of second order. With fixed x and ω, the matrix Hn(x, ω) has n eigenvalues. So we have an intuition that these eigenvalues have a more uniform distribution when the frequency ω is “more irrational”. For the Brjuno-Rüssmannumber, it means that ∆(t) grows more slowly (n) and then δ0 is smaller. The following theorem verifies our intuition: Theorem 1.5. Let ω be the Brjuno-Rüssmannnumber satisfying Hypothesis H.1, or H.2, or H.3 and L(E, ω) > 0. Then, for any x0 ∈ T and E0 ∈ R, 1 a (n) h (n) # E ∈ R : fn (x0, E, ω) = 0, |E − E0| < δ0 ≤ 13nδ0 .

We organize this paper as follows. In Section 2, we prove Theorem 1.4, the strong Birkhoff Ergodic theorem for the subharmonic function shifting on T with our Brjuno-Rüssmannfrequency. We apply it to the analytic quasi-periodic Jacobi operator and obtain Theorem 1.1, Theorem 1.2 and Theorem 1.3 in Section 3, a which are all the LDTs for fn (x, E, ω) with different hypothesises. Then, we prove Theorem 1.5, an application of them, in the last section.

2. Strong birkhoff ergodic theorem for subharmonic functions with the Brjuno-Rüssmannshift. Let {x} = x − [x]. For any positive integer q, complex number ζ = ξ + iη and 0 ≤ x < 1, define

X Z 1 Fq,ζ (x) = log |{x + kω} − ζ| and I(ζ) = log |y − ζ|dy. 0≤k

qs Let |{x + k0ω} − ξ| = mink=1 |{x + kω} − ξ|, where qs is the denominator of the continued fraction approximants. In [11], Goldstein and Schlag proved Lemma 3.1 that for any irrational ω, there exists an absolute constant C such that

|Fqs,ζ (x) − qsI(ζ)| ≤ C log qs + |log |{x + k0ω} − ζ|| . (2.1) Then Lemma 2.1. For any irrational ω and integer l < qs+1 , qs

|Flsqs,ζ (x) − lsqsI(ζ)| < Cls log qs + | log D(x − ξ, ω, lsqs)| + 2ls log qs+1, n−1 where D(x, ω, n) := mink=0 {x + kω}.

qs−1 Proof. Deﬁne xh = x + hqsω and |{xh + khω} − ξ| = mink=0 |{xh + kω} − ξ|. Due to (2.1), we have

l −1 Xs |Flsqs,ζ (x) − lsqsI(ζ)| ≤ |Fqs,ζ (xh) − lsqsI(ζ)| h=0 l −1 Xs ≤ |log |{xh + khω} − ζ|| + Cls log qs. (2.2) h=0 LDT DIRICHLET DETERMINANTS WITH BRJUNO-RUSSMANN¨ FREQUENCY 5313

1 1 We declare that if there exists 0 ≤ j < qs such that |{x + jω} − ξ| ≤ − , 2qs qs+1 1 1 then j = k0. Actually, if |{x+jω}−ξ| ≤ − and j 6= k0, then |{x+k0ω}−ξ| ≤ 2qs qs+1 |{x + jω} − ξ| ≤ 1 − 1 , which implies 2qs qs+1 1 2 {x + k0ω} − {x + jω} ≤ − . qs qs+1 Due to (1.11), it has

k kps k 1 < |kω − | < ≤ , 0 < k < qs. (2.3) qs(qs+1 + qs) qs qsqs+1 qs+1 Then,

ps ps 1 {x + j } − {x + k0 } < . qs qs qs It is a contraction. Thus, there is at most one integer 0 ≤ k0 < qs such that 1 1 |{x + k0ω} − ξ| < − and 2qs qs+1 1 1 1 |{x + kω} − ξ| > − > , k 6= k0. (2.4) 2qs qs+1 4qs Due to (1.11) again, it has 1 1 1 < < |qsω − ps| < . (2.5) 2qs+1 qs + qs+1 qs+1

qs+1 1 Deﬁne Q = [ ] and let j be the number such that |{xj + kjω} − ξ| < . qs 4qs+1 Then by (2.5) and the above declaration, we have for any j − 2Q + 1 ≤ h < j and j < h ≤ j + 2Q − 1, 1 |{xh + khω} − ξ| > . 4qs+1 Thus there are at most one point which is small than 1 . Combining it with 4qs+1 (2.2), we have

1 |Flsqs,ζ (x) − lsqsI(ζ)| ≤| log D(x − ξ, ω, lsqs)| + Cls log qs + ls log 4qs+1

≤| log D(x − ξ, ω, lsqs)| + Cls log qs + 2ls log qs+1.

qs+1 Lemma 2.2. For any 0 < σ < 1, irrational ω and integer ls < , qs Z

exp (σ|Flsqs,ζ (x) − lsqsI(ζ)|) dx < exp (5σls log qs+1) . T Proof. We ﬁrst apply Lemma 3.2 in [11]. It says that if Ω ⊂ T is an arbitrary ﬁnite set, then for any 0 < σ < 1, Z 2σ exp (σ| log dist(x, Ω)|) dx ≤ (]Ω)σ. (2.6) 1 − σ T Set Ω = {mω : 0 ≤ m < lsqs}. Then ]Ω = lsqs and dist(x−ξ, Ω) = D(x−ξ, ω, lsqs). Thus, by (2.6), Z Z 2σ exp (σ| log D(x − ξ, ω, l q )|) dx = exp (σ| log dist(x, Ω)|) dx ≤ (l q )σ. s s 1 − σ s s T T By Lemma 2.1, we have Z

exp (σ|Flsqs,ζ (x) − lsqsI(ζ)|) dx T 5314 WENMENG GENG AND KAI TAO

≤ exp (2Cσ log(lsqs) + Cσls log qs + 2σls log qs+1) < exp (5σls log qs+1) . n Now for any n, there exist qs and qs+1 such that qs ≤ n < qs+1. Let ls = [ ] and qs qi+1 li = [ ] for i < s. Then there exists rs−2m+1 satisfying 0 ≤ rs−2m+1 < qs−2m+1 qi such that

n = lsqs + ls−1qs−1 + ··· + ls−2m+1qs−2m+1 + rs−2m+1. Deﬁne n¯ = lsqs + ls−1qs−1 + ··· + ls−2m+1qs−2m+1. (2.7) Then we have

Lemma 2.3. For any compact Ω ⊂ C, there exist constants c˜ =c ˜(ω) such that for any 0 < σ ≤ c˜, we have Z 1 ˜ ˘(n) exp(σ|Fn,ζ¯ (x) − nI¯ (ζ)|)dx ≤ exp Cσnδ0 , (2.8) 0 where  ˘ log ∆(n) δH.1(n) := 1− , if ω satisfies H.1,  [∆−1(C n)]  ω (n) ˘ 1 δ˘ = δH.2(n) := , if ω satisfies H.2, (2.9) 0 [log(∆−1(C n))]1−  ω ˘ log(Cωn) δH.3(n) := , if ω satisfies H.3.  −1 1− [∆ (Cωn)]

Proof. Let rs =n ¯ − lsqs. Due to the H¨olderinequality and Lemma 2.2, Z 1 exp(σ|Fn,ζ¯ (x) − nI¯ (ζ)|)dx 0 1 1 Z 1 2 Z 1 2

≤ exp(2σ|Flsqs,ζ (x) − lsqsI(ζ)|)dx × exp(2σ|Frs,ζ (x) − rsI(ζ)|)dx 0 0 1 Z 1 2

≤ exp(5σls log qs+1) exp(2σ|Frs,ζ (x) − rsI(ζ)|)dx . 0 rs−i+1 Let rs−i+1 = ls−iqs−i + rs−i, where ls−i = [ ], 0 ≤ rs−i = rs−i+1 − ls−iqs−i < qs−i qs−i. Then 1 Z 1 2i i exp(2 σ|Frs−i+1,ζ (x) − rs−i+1I(ζ)|)dx 0 1 Z 1 2i+1 i+1 ≤ exp(2 σ|Fls−iqs−i,ζ (x) − ls−iqs−iI(ζ)|)dx 0 1 Z 1 2i+1 i+1 × exp(2 σ|Frs−i,ζ (x) − rs−iI(ζ)|)dx 0 1 Z 1 2i+1 i+1 ≤ exp (5σls−i log qs−i+1) × exp(2 σ|Frs−i,ζ (x) − rs−iI(ζ)|)dx . 0 Therefore, Z 1 exp(σ|Fn,ζ¯ (x) − nI¯ (ζ)|)dx 0 ≤ exp [5σ (ls log qs+1 + ls−1 log qs + ··· + ls−2m+1 log qs−2m+2)] LDT DIRICHLET DETERMINANTS WITH BRJUNO-RUSSMANN¨ FREQUENCY 5315

  2m−1   n X qs−j+1  ≤ exp 5σ log q + log q . (2.10) q s+1 q s−j+1  s j=1 s−j  t log t Assume that ω satisﬁes Hypothesis H.1. Then, we assert that −1 is mono- ∆ (Cω t) −1 tone increasing. Indeed, let y = ∆ (Cωt). Then, due to the hypothesis that y∆0(y) > ∆(y), it yields that C t ∆−1 (C t) ∆0 ∆−1 (C t) > C t ≥ C t − ω . ω ω ω ω log t + 1 Combining it with the fact that 1 = ∆0 ∆−1 (C t) , −1 0 ω (∆ (Cωt)) we have −1 −10 ∆ (Cωt) (log t + 1) > Cωt log t ∆ (Cωt) . (2.11) Now we ﬁnish the proof of the assertion as (2.11) shows that the numerator of the t log t derivative of −1 is positive. ∆ (Cω t) Due to (1.15) and (1.11), −1 Cωqi < ∆(qi−1) and qi−1 > ∆ (Cωqi). (2.12) Therefore, qi qi log qi < −1 log qi. qi−1 ∆ (Cωqi) We apply the assertion and obtain 2m−1 X qs−j+1 n log q < (2m − 1) log n. (2.13) q s−j+1 ∆−1(C n) j=1 s−j ω Recall that ∆(t) is monotone increasing and continuous. And so is ∆−1(t). Com- bining it with (2.12), we have −1 −1 qs > ∆ (Cωqs+1) > ∆ (Cωn), (2.14) and for any n > n0(ω), n ∆(qs) 2n log ∆(qs) 2n log ∆(qs) 2n log ∆(n) ls log qs+1 ≤ log ≤ ≤ −1 ≤ −1 . (2.15) qs Cω qs ∆ (Cωn) ∆ (Cωn) Choosing −1 m = mH.1 = log2 ∆ (Cωn), (2.16) and due to (2.10), (2.13), (2.15), we have Z 1 exp(σ|Fn,ζ¯ (x) − nI¯ (ζ)|)dx 0 2n log ∆(n) −1 n ≤ exp 5σ −1 + 2 log2 ∆ (Cωn) −1 log n ∆ (Cωn) ∆ (Cωn) −1 logC¯ ∆ (Cωn) ≤ exp 20σn log ∆(n) −1 ∆ (Cωn) ( ) log ∆(n) ≤ exp 20σn . (2.17) −1 1− (∆ (Cωn)) 5316 WENMENG GENG AND KAI TAO

t Assume that ω satisﬁes Hypothesis H.2 which implies that ∆(t) < exp( log t ) holds. Then

log ∆(qs) 1 log n 1 < and −1 < −1 . (2.18) qs log qs ∆ (Cωn) log ∆ (Cωn) Combining them with (2.13) and (2.14),

2n log ∆(qs) 2n 2n ls log qs+1 < < < −1 , qs log qs log(∆ (Cωn)) and 2m−1 X qs−j+1 qs log q <(2m − 1) log q q s−j+1 ∆−1(C q ) s j=1 s−j ω s n <(2m − 1) −1 log n ∆ (Cωn) n <(2m − 1) −1 . log(∆ (Cωn)) Letting −1 m = mH.2 = log2 log ∆ (Cωn), (2.19) we obtain that Z 1 −1 n exp(σ|Fn,ζ¯ (x) − nI¯ (ζ)|)dx ≤ exp 20σ log2 log ∆ (Cωn) −1 0 log ∆ (Cωn) ( ) n ≤ exp 20σ . (2.20) −1 1− [log(∆ (Cωn))]

log ∆(t) Assume the ω satisﬁes Hypothesis H.3 which implies that t is non-increasing. Due to (2.14) and (2.15),

−1 2n log ∆(qs) n log ∆(∆ (Cωn)) n log(Cωn) ls log qs+1 ≤ ≤ −1 = −1 . (2.21) qs ∆ (Cωn) ∆ (Cωn) By (2.13),

2m−1 X qs−j+1 qs log q <(2m − 1) log q q s−j+1 ∆−1(C q ) s j=1 s−j ω s n ≤(2m − 1) −1 log n. (2.22) ∆ (Cωn) Choosing −1 m = mH3 = log2(∆ (Cωn)), (2.23) we have Z 1 ! n log(Cωn) exp(σ|Fn,ζ¯ (x) − nI¯ (ζ)|)dx ≤ exp 20σ . (2.24) −1 1− 0 [∆ (Cωn)]

To ﬁnish the proof of Theorem 1.4, we need the following Riesz’s theorem proved in [12]: LDT DIRICHLET DETERMINANTS WITH BRJUNO-RUSSMANN¨ FREQUENCY 5317

Lemma 2.4. Let u :Ω → R be a subharmonic function on a domain Ω ⊂ C. Suppose that ∂Ω consists of ﬁnitely many piece-wise C1 curves. There exists a positive measure µ on Ω such that for any Ω1 b Ω (i.e., Ω1 is a compactly contained subregion of Ω), Z u(z) = log |z − ζ| dµ(ζ) + h(z), (2.25) Ω1 where h is harmonic on Ω1 and µ is unique with this property. Moreover, µ and h satisfy the bounds

µ(Ω1) ≤ C(Ω, Ω1) (sup u − sup u), (2.26) Ω Ω1

∞ kh − sup ukL (Ω2) ≤ C(Ω, Ω1, Ω2) (sup u − sup u) (2.27) Ω1 Ω Ω1 for any Ω2 b Ω1. What’s more, we can obtain the following strong Birkhoff Ergodic Theorem for this harmonic function h with the Brjuno-Rüssmanfrequency easily: Lemma 2.5. Let h be a 1−periodic harmonic function defined on a neighborhood of the real axis. There exists a constant C depending only on h such that for any integer n and Brjuno-Rüssman ω, n X Z 1 C h(x + kω) − n h(y)dy < , Cω k=1 0 where Cω comes from (1.15). Proof.

n Z 1 X h(x + kω) − n h(y)dy k=1 0

n ∞ n ! X X 2πij(x+kω) X X 2πijkω = hˆ(j)e − nhˆ(0) ≤ hˆ(j) e ,

k=1 j=−∞ j6=0 k=1 where hˆ(j) is the j−th Fourier coeﬃcients of h. Due to its harmonicity, there exists two constants Cˇh and ρ depending only on h such that ˆ |h(j)| ≤ Cˇh exp (−ρj) . (2.28) Easy computation shows that n X 2πijkω exp(2πijω) · (1 − exp(2πinjω)) 1 e = ≤ . 1 − exp(2πijω) 2kjωk k=1 P∞ log ∆(j) Therefore, we obtain this lemma by (2.28) and j=1 j2 < ∞ from (1.16). The proof of Theorem 1.4. Notice that the ergodic measure for the shift on the Torus is the Lebesgue measure and m( ) = 1. Then, hui = R u(x)dx, and T T n n X X Z Z u(x + kω) − nhui = log |{x + kω} − ζ|dµ(ζ) − n I(ζ)dµ(ζ) k=1 k=1 Ω1 Ω1 n X Z 1 + h({x + kω}) − n h(y)dy. k=1 0 5318 WENMENG GENG AND KAI TAO

Recall that n X Z Z log |{x + kω} − ζ|dµ(ζ) = Fn,ζ (x)dµ(ζ). k=1 Ω1 Ω1 1 Thus, due to Lemma 2.5, it yields that for any n and δ n , ( n )

X x ∈ T : u(x + kω) − nhui > δn k=1 Z δn ⊆ x ∈ T : (Fn,ζ (x) − nI(ζ))dµ(ζ) > . (2.29) Ω1 2 To estimate the measure of the upper set, we use the Markov’s inequality: For any measurable extended real-valued function f(x) and > 0,we have 1 Z mes ({x ∈ : |f(x)| ≥ }) ≤ |f|dx. X X R Let f(x) = exp σ (Fn,ζ (x) − nI(ζ))dµ(ζ) and = exp(σδn/2), then Ω1 Z δn mes x ∈ T : (Fn,ζ (x) − nI(ζ))dµ(ζ) > Ω1 2 Z 1 Z σδn ≤ exp − exp σ (Fn,ζ (x) − nI(ζ))dµ(ζ) . (2.30) 2 0 Ω1 Due to the H¨olderinequality, Z 1 Z

exp σ (Fn,ζ (x) − nI(ζ))dµ(ζ) dx 0 Ω1 1 Z 1 Z 2

≤ exp 2σ (Fn−n,ζ¯ (x) − (n − n¯)I(ζ))dµ(ζ) dx 0 Ω1 1 Z 1 Z 2

× exp 2σ (Fn,ζ¯ (x) − nI¯ (ζ))dµ(ζ) dx , (2.31) 0 Ω1 wheren ¯ comes from (2.7). Since exp(σ·) is a convex function, the Jensen’s inequality and Lemma 2.3 imply that Z 1 Z

exp σ (Fn,ζ¯ (x) − nI¯ (ζ))dµ(ζ) dx 0 Ω1 Z 1 Z dµ(ζ) ≤ exp (σµ(Ω1) |Fn,ζ¯ (x) − nI¯ (ζ)|) dx 0 Ω1 µ(Ω1) Z Z 1 dµ(ζ) = exp (σµ(Ω1) |Fn,ζ¯ (x) − nI¯ (ζ)|) dx Ω1 0 µ(Ω1) Z ˘(n) dµ(ζ) ˘(n) ≤ exp 20σµ(Ω1)nδ0 ≤ exp 20σµ(Ω1)nδ0 . (2.32) µ(Ω1) On the other hand, due to the facts that it is integrable for log |z| on the disc |z| < r and |I(ζ)| ≤ |log (Imζ)| if |Imζ| is close to 0, it is obvious that there exists a constant Cˆ = Cˆ(Ω1) such that Z Z dµ(ζ) ˆ (log |x − ζ| − I(ζ)) dµ(ζ) ≤ µ(Ω1) (log |x − ζ| − I(ζ)) ≤ µ(Ω1)C. Ω1 Ω1 µ(Ω1) LDT DIRICHLET DETERMINANTS WITH BRJUNO-RUSSMANN¨ FREQUENCY 5319

Thus,

1 Z 1 Z 2

exp 2σ (Fn−n,ζ¯ (x) − (n − n¯)I(ζ))dµ(ζ) dx 0 Ω1 ≤ exp σµ(Ω1)Cˆ(n − n¯) ≤ exp σµ(Ω1)Crˆ s−2m+1 . (2.33) Note that for any irrational ω, the denominators of its continued fraction approximates satisfy qt+1 = at+1qt + qt−1 > 2qt−1. Thus m −m qt > 2 qt−2m and rs−2m+1 < 2 n. −1 Recall that in the proof of Lemma 2.3, we choose m = mH.1 = log2 ∆ (Cωn) when ω satisfy the hypothesis H.1. Then ˆ ˆ n ˇ ˇ(n) Crs−2m+1 < C −1 nδH.1(n) = nδ0 . ∆ (Cωn)

Similarly, Crˆ s−2m+1 also has the same upper bound in the other two hypothesises. Combined it with (2.30), (2.31), (2.32) and (2.33), we have that for any 0<σ ≤ c˜ , µ(Ω1) Z δn mes x ∈ T : (Fn,ζ (x) − nI(ζ))dµ(ζ) > Ω1 2 σδn ≤ exp − + 21σµ(Ω )nδ˘(n) . 2 1 0

Thus, we ﬁnish this proof by (2.29) and setting C˘ = 100µ(Ω1) with the fact that (n) 1 δ0 n .

a 3. Large deviation theorems for fn (x, E, ω). To apply Theorem 1.4, we first need to define some subharmonic functions. Let  n  a Y Mn (x, E, ω) :=  a(x + jω) Mn(x, E, ω) j=1 n Y v(x + jω) − E a(x + jω) = . (3.1) a(x + (j + 1)ω) 0 j=1 Note that a real function f(x) on T has its complex analytic extension f(z) on the complex strip Tρ = {z : |Imz| < ρ} and the complex analytic extension ofa ¯(x) should be defined on Tρ by 1 a˜(z) := a( ). z a Then, the extension of Mn (x, E, ω) is n Y v(z + jω) − E a˜(z + jω) M a(z, E, ω) = , (3.2) n a(z + (j + 1)ω) 0 j=1 where z + ω means z exp (2πiω) here. Moreover, simple computations yield that a a a fn (z, E, ω)a ˜(z)fn−1(z + ω, E, ω) Mn (z, E, ω) = a a , a(z + nω)fn−1(z, E, ω) −a˜(z)a(z + nω)fn−2(z + ω, E, ω) (3.3) 5320 WENMENG GENG AND KAI TAO where a fn (z, E, ω) = det Hn(z, ω) − E v z + ω − E −a z + 2ω 0 ··· 0

−a˜ z + 2ω v z + 2ω − E −a z + 3ω ··· 0

= ......

0 ··· 0 −a˜z + nω vz + nω − E

Note that if Imz = 0, then Hn(z, ω) = Hn(x, ω) is Hermitian. Now with ﬁxed E 1 a and ω, the function N log kMN (z, E, ω)k is subharmonic. In this paper, we only need to consider E ∈ E , where

∞ ∞ ∞ ∞ E := [−2ka(x)kL (T) − kv(x)kL (T), 2ka(x)kL (T) + kv(x)kL (T)], as the spectrum Sω ⊂ E . Thus, for any irrational ω and 1 ≤ n ∈ N,

1 a sup log kMN (z, E, ω)k ≤ M0, (3.4) E∈E ,x∈T N where M := log 3kak ∞ + 2kvk ∞ . 0 L (Tρ) L (Tρ) We also need to deﬁne the unimodular matrix

u Mn(x, E, ω) Mn (x, E, ω) := 1 , (3.5) | det Mn(x, E, ω)| 2 which makes sense a.e. x ∈ T and has the relationship Z 1 log kM u(x, E, ω)kdx = L (E, ω). n n n T Then, we have the LDTs for the matrices as follows: Lemma 3.1. Let ω be the Brjuno-R¨ussmannnumber satisfying Hypothesis H.1, or H.2, or H.3 and L(E, ω) > 0. There exist cˆ =c ˆ(v, a, ω) and cˇ =c ˇ(v, a, ω) such that (n) for any n ≥ 0 and δ > δ0 , mes {x : |u(x, E, ω) − hui| > δ} < exp (−cδnˆ ) + exp(−cδˇ 2n), (3.6) 1 a 1 1 u where u(x, E, ω)can be n logkMn (x, E, ω)k, n logkMn(x, E, ω)k, n logkMn (x, E, ω)k. 1 What’s more, there exists c¯ =c ¯(a, v, ω) such that if δ = κL(E, ω) with κ < 10 , then the exception measure in (3.6) will be less than exp −cκ¯ 2L(ω, E)n.

1 a Proof. When u = n log kMn (x, E, ω)k, The LDT (3.6) is about the analytic matrix; 1 when u = n log kMn(x, E, ω)k, The LDT (3.6) is about the Jacobi cocycles; when 1 u u = n log kMn (x, E, ω)k, The LDT (3.6) is about the unimodular matrix to satisfy the hypothesises of Lemma 3.8, Lemma 3.9 and the Avalanche Principle(Proposition 1). In [21], our second author obtained these LDTs with ﬁnite Liouville frequency, which means that β(ω) < ∞. The proofs in that paper are also available here for (n) β = 0 and δ > δ0 . a What’s more, the following lemma shows that Ln(E, ω) and Ln(E, ω), which is deﬁned by Z 1 La (E, ω) = log kM a(x, E, ω)kdx, n n n T LDT DIRICHLET DETERMINANTS WITH BRJUNO-RUSSMANN¨ FREQUENCY 5321 in the above LDTs can be exchanged by L(E, ω) and La(E, ω), respectively. Here,

a a L (E, ω) = lim Ln(E, ω) = L(E, ω) + D, (3.7) n→∞ and Z Z D := log |a(x)|dx = log |a¯(x)|dx. (3.8) T T Lemma 3.2. Let L(E, ω) > 0. For any integer n > 1, we have

(log n)2 0 ≤ L − L = Lu − Lu = La − La < C n n n 0 n where C0 = C0 (a, v, ω, E). Proof. It is the same as Lemma 3.9 in [8], which was for the strong Diophantine ω. (n) (log n)A They applied the same LDTs, whose δ0 = n with that frequency, to obtain its proof. It is available here, since it only need the fact, which our LDTs also (n) satisfy, that δ0 is much less than the positive Lyapunov exponent.

Although the details of the proof of Lemma 3.1 can be found in [21], we still give a brief introduction here, to make the readers understand the methods we apply in this section to obtain Theorem 1.1-1.3. Easy computations show that 1 a 1 the diﬀerences between N log kMN (x, E, ω)k and N log kMN (x, E, ω)k and between 1 a 1 u N log kMN (x, E, ω)k and N log kMn (x, E, ω)k are constructed by the combination 1 PN 1 PN of N j=1 log |a(x + jω)| and N j=1 log |a¯(x + jω)|, whose complex extensions can be estimated by our Theorem 1.4 easily. Therefore, we only need to prove the 1 a LDT for N log kMn (x, E, ω)k, which also has a subharmonic extension. Due to this subharmonicity,   n  1 X a a  mes x : log kM (x + jω, E, ω)k − L (E, ω) > δ < exp (−cδn) . (3.9) n n n  j=1  On the other hand, for any k ∈ Z, k−1 2M0k X k − j − + d(x + jω) n nk j=0 k 1 1 X ≤ log kM a(x, E, ω)k − log kM a(x + jω, E, ω)k n n kn n j=1 k−1 2M0k X k − j ≤ − d(x + (n + j − 1)ω), n nk j=0 where d(x) = log |a(x + ω)¯a(x)|. Obviously, it also can be solved by our Theorem 1.4. Now, we can explain why we apply the BMO norm and the John-Nirenberg a inequality, not the method for (3.6), to obtain the LDTs for fn (x, E, ω). The reason a is that Theorem 1.4 holds for fn (x, E, ω), but we can not handle the diﬀerence 1 a 1 Pk a between n log |fn (x, E, ω)| and kn j=1 log |fn (x + jω, E, ω)|. a Next, we will apply the analyticity of fn (x, E, ω) and the subharmonicity of 1 a n log |fn (x, E, ω)| via the following lemmas in this paper. 5322 WENMENG GENG AND KAI TAO

Deﬁnition 3.3. Let H > 1. For any arbitrary subset B ⊂ D(z0, 1) ⊂ C we say j0 B ∈ Car1(H,K) if B ⊂ ∪j=1D(zj, rj) with j0 < K, and X −H rj < e . (3.10) j Here D(z, r) means the complex platform center at z with radius r. If d is a Qd d positive integer greater than one and B ⊂ i=1 ⊂ C then we deﬁne inductively (j) that B ∈ Card(H,K) for any z ∈ C\Bj, here Bz = {(z1, ··· , zd) ∈ B : zj = z}.

Lemma 3.4 (Cartan estimate, Lemma 2.4 in [13]). Let φ(z1, ··· , zd) be an an- Qd alytic function deﬁned in a polydisk P = j=1 D(zj,0, 1), zj,0 ∈ C. Let M ≥ supz∈P log |φ(z)|, m ≤ log |φ(z0)|, z0 = (z1,0, ··· , zd,0). Given H 1 there exists 1 a set B ⊂ P, B ∈ Card(H d ,K),K = CdH(M − m), such that

log |φ(z)| > M − CdH(M − m) Qd 1 for any z ∈ j=1 D(zj,0, 6 )\B.

Lemma 3.5 (Lemma 2.4 in [12]). Let u be a subharmonic function defined on Aρ such that supAρ u ≤ M. There exist constants C1 = C1 (ρ) and C2 such that, if for some 0 < δ < 1 and some L we have mes {x ∈ T : u (x) < −L} > δ, then L sup u ≤ C1M − . T C1 log (C2/δ) a u Recalling the definitions of Mn(x, E, ω), Mn (x, E, ω), Mn (x, E, ω) and the ex- pression (3.3), we have  a˜(z)  fn(z, E, ω) − fn−1(z + ω, E, ω) M (z, E, ω) =  a(z + ω)  , (3.11) n  a˜(z)   f (z, E, ω) − f (z + ω, E, ω)  n−1 a(z + ω) n−2 and u Mn (z, E, ω) = 1 a˜(z) a(z+ω) 2 f u(z, E, ω) − f u (z+ω, E, ω) ! n a(z+ω) a˜(z) n−1 1 1 , a(z+nω) 2 a˜(z) a(z+nω)a(z+ω) 2 f u (z, E, ω) − f u (z+ω, E, ω) a˜(z+(n−1)ω) n−1 a(z+ω) a˜(z)ã(z+(n−1)ω) n−2 where 1 a fn(z, E, ω) = Qn fn (z, E, ω), (3.12) j=1 a(z + jω) and

u 1 a fn (z, E, ω) = 1 fn (z, E, ω) 2 Qn−1 j=0 a(z + (j + 1)ω)˜a(z + jω) n−1 1  2 Y a(z + (j + 1)ω) =   fn(z, E, ω). (3.13) a˜(z + jω) j=0 LDT DIRICHLET DETERMINANTS WITH BRJUNO-RUSSMANN¨ FREQUENCY 5323

AssumeL(E, ω) = γ > 0. Then,we can obtain a particular deviation theorem as follow:

Lemma 3.6. There exists l0 = l0 (a, v, γ) such that 3 mes x ∈ T : |fl (x)| ≤ exp −l ≤ exp (−l) for all l ≥ l0. Proof. It is the same as Lemma 4.2 in [8], which was for the strong Diophantine ω. We have the same reason which we just stated in the proof of Lemma 3.2 to omit this proof.

Note that in order to simplify the notation, we suppressed the dependence on E and ω. We will be doing this throughout this paper if there is no confusion. According to Lemma 3.5 and 3.6, we can have more choices of the deviation and the exceptional measure.

Lemma 3.7. Let σ > 0 and g(n) > 0. There exist constants l0 = l0 (a, v, γ) and n0 = n0 (a, v, γ) such that −3 mes {x ∈ T : |fl (x)| ≤ exp (−g(n))} ≤ exp −g(n)l for any n ≥ n0 and for any l0 ≤ l . g(n). The same result, but with possibly u diﬀerent l0 and n0, holds for fl . Proof. Assume −3 mes {x ∈ T : |fl (x)| ≤ exp (−g(n))} > exp −g(n)l . We have that l Y 1 |f a (x)| = |f (x)| |a (x + jω)| ≤ exp (−g(n)) Cl−1 ≤ exp − g(n) l l 2 j=1 on a set of measure greater than exp −g(n)l−3. By Lemma 3.5, it implies that for any x ∈ T, a g(n) 3 |fl (x)| ≤ exp C1l − −3 ≤ exp −Cl . 2C1 log (C2 exp (g(n)l )) Due to Theorem 1.4, ( l )! X mes x ∈ T : | log |a(x + kω) − lD| > l ≤ exp (−cl) . k=1 Therefore, recalling (3.12), we have 0 3 3 |fl (x)| ≤ exp l(1 − D) − C l ≤ exp −Cl for all x except for a set of measure less than exp (−cl). It contradicts with the u previous lemma. At last, by (3.13), we can prove the result for fl by similar methods.

Now we need some facts about stability of contracting and expanding directions of unimodular matrices. It follows from the polar decomposition that if A ∈ SL (2, C) + − + − + + then there exist unit vectors uA ⊥ uA and vA ⊥ vA such that AuA = kAk vA and − −1 − AuA = kAk vA . 5324 WENMENG GENG AND KAI TAO

Lemma 3.8 (Lemma 2.5 in [12]). For any A, B ∈ SL (2, C) we have − − −2 − − −2 2 BuAB ∧ uA ≤ kAk kBk , uBA ∧ uA ≤ kAk kBk + + −2 2 + + −2 vAB ∧ vA ≤ kAk kBk , vBA ∧ BvA ≤ kAk kBk .

Lemma 3.9 (Lemma 4.5 in [8]). If A ∈ SL (2, C) and w1, w2, and w3 are unit vectors in the plane then √ −1 |w1 ∧ Aw2| ≤ |w1 ∧ Aw3| + 2 A |w2 ∧ w3| and √ |w1 ∧ Aw2| ≤ |w3 ∧ Aw2| + 2 kAk |w1 ∧ w3| Now, we can improve the Lemma 3.7, but the LDT is about three determinants. Lemma 3.10. There exist constants 0 < κ = κ(ω) < 1, 0 < τ = τ(ω) < 1, l0 = l0 (a, v, γ) and n0 = n0 (a, v, γ) such that

n u u u (n)o mes x ∈ T : |fn (x)| + |fn (x + j1ω)| + |fn (x + j2ω)| ≤ exp nLn − 100nδ0 < exp −n1−κ (3.14) τ for any l0 ≤ j1 ≤ j1 + l0 ≤ j2 ≤ n and n ≥ n0.

(n) − 1 Proof. Here we assume δ0 ≥ n 3 , since the proof of Lemma 4.6 in [8] can be − 1 applied without any change when δ ≤ n 3 . n (n) For any 1 ≤ j ≤ n, due to Lemma 3.1 and 3.2, choose the deviation δ = j δ0 > (j) δ0 and then n u (n)o mes x : |log kMl (x)k − jL| > nδ0 2 2 (n) < exp (−cδjˆ ) + exp −cδˇ j ≤ 2 exp −cˇ δ0 n . (3.15)

Let Gn be the set of points x ∈ T such that for any 1 ≤ j ≤ n and |l| ≤ 2n, u (n) log Mj (x + lω) − jL ≤ nδ0 , and (n) |log |a (x + jω)| − D| ≤ nδ0 . Due to (3.15) and Theorem 1.4 for log |a(x)|, we have that 2 2 2 (n) (n) mes (T \Gn) ≤ 4n exp −cˇ δ0 n ≤ exp −c δ0 n .

u Note that det Ml (x, E, ω) ≡ 1. Therefore, for any x, E and ω,

u u −1 kMl (x, E, ω)k = (Ml ) (x, E, ω) . 2 + 1 + Let {e1, e2} be the standard basis of R and for any integer j, uj , uj , vj and − + − + − u + u + vj be the unit vectors satisfying uj ⊥ uj , vj ⊥ vj , Mj uj = Mj vj and u − u −1 − Mj uj = Mj vj . Then u u u + + − − fn (x) =Mn (x) e1 ∧ e2 = Mn (x) un (x) · e1 un (x) + un (x) · e1 un (x) ∧ e2 + u + − u −1 − = un (x) · e1 kMn (x)k vn (x) ∧ e2 + un (x) · e1 kMn (x)k vn (x) ∧ e2. LDT DIRICHLET DETERMINANTS WITH BRJUNO-RUSSMANN¨ FREQUENCY 5325

u (n) If |fn (x)| ≤ exp nLn − 100nδ0 , then

u + + u −1 − − kMn (x)k un (x) · e1 vn (x) ∧ e2 − kMn (x)k un (x) · e1 vn (x) ∧ e2 (n) ≤ exp nLn − 100nδ0 .

Due to Lemma 3.2, for any x ∈ Gn, − + u −1 (n) u −2 un (x) ∧ e1 vn (x) ∧ e2 ≤ kMn (x)k exp nLn − 100nδ0 + kMn (x)k

(n) (n) ≤ exp n (Ln − L) − 99nδ0 + exp 2nδ0 − 2nL

(n) ≤ exp −90nδ0 . Hence,

− (n) + (n) un (x) ∧ e1 ≤ exp −40nδ0 or vn (x) ∧ e2 ≤ exp −40nδ0 . (3.16)

(n) 1−2σ Suppose (3.14) fails. Let σ < κ < 1/2. Recall nδ0 ≥ n and set ˜ n u u u (n)o Gn = x ∈ Gn : |fn (x)|+|fn (x + j1ω)|+|fn (x + j2ω)|≤exp nLn −100nδ0 . We have 1 mes G˜ > exp −n1−κ − exp −nδ(n) > exp −n1−κ . n 0 2

If x ∈ G˜n, then either − (n) un (x) ∧ e1 ≤ exp −40nδ0 or + (n) vn (x) ∧ e2 ≤ exp −40nδ0 has to hold for two of the points x, x + j1ω, x + j2ω. We ﬁrst assume that − − (n) un (x + j1ω) ∧ e1 , un (x + j2ω) ∧ e1 ≤ exp −40nδ0 . (3.17)

From Lemma 3.9 and Lemma 3.8, we have that if x ∈ Gn, then u− (x + j ω) ∧ M u (x + j ω) u− (x + j ω) n 2 j2−j1 1 n 1 ≤ u− (x + j ω) ∧ M u (x + j ω) u− (x + j ω) n 2 j2−j1 1 n+j2−j1 1 u −1 − + C M (x + j1ω) un+j −j (x + j1ω) ∧ u (x + j1ω) j2−j1 2 1 n = u− (x + j ω) ∧ M u (x + j ω) u− (x + j ω) n 2 j2−j1 1 n+j2−j1 1 + C M u (x + j ω) u− (x + j ω) ∧ u− (x + j ω) j2−j1 1 n+j2−j1 1 n 1 ≤ kM u (x + j ω)k−2 M u (x + j ω) n 2 j2−j1 1 2 + C M u (x + j ω) kM u (x + j ω)k−2 M u (x + (n + j ) ω) j2−j1 1 n 1 j2−j1 1 (n) ≤ exp (−2n + j2 − j1) L + 3nδ0

(n) + C exp (−2n + 3 (j2 − j1)) L + 5nδ0 ≤ exp (−nL) . Combined it with Lemma 3.9 and (3.17), we obtain e ∧ M u (x + j ω) e 1 j2−j1 1 1 5326 WENMENG GENG AND KAI TAO

≤ e ∧ M u (x + j ω) u− (x + j ω) 1 j2−j1 1 n 1 u −1 − + C M (x + j1ω) e1 ∧ u (x + j1ω) j2−j1 n ≤ u− (x + j ω) ∧ M u (x + j ω) u− (x + j ω) n 2 j2−j1 1 n 1 + C M u (x + j ω) e ∧ u− (x + j ω) j2−j1 1 1 n 2 u −1 − + C M (x + j1ω) e1 ∧ u (x + j1ω) j2−j1 n (n) (n) ≤ exp (−nL) + C exp (j2 − j1) L − 39nδ0 + C exp −39nδ0

(n) ≤ exp −30nδ0 . Due to the fact that a (x + j ω) 1/2 e ∧ M u (x + j ω) e = 2 f u (x + j ω) , 1 j2−j1 1 1 j2−j1−1 1 a (x + (j2 − 1) ω) and the setting of Gn, we have u 1 (n) (n) (n) f (x + j1ω) ≤ C exp nδ − D − 30nδ ≤ exp −20nδ . j2−j1−1 2 0 0 0 Similarly, we can obtain f u (x + (n + j + 1) ω) ≤ exp −20nδ(n) , j2−j1−1 1 0 if we assume that + + (n) vn (x + j1ω) ∧ e2 , vn (x + j2ω) ∧ e2 ≤ exp −40nδ0 . (3.18)

What’s more, the same type of estimates are obtained if we replace (j1, j2) in (3.17) and (3.18) with (0, j1) or (0, j2). In conclusion n o 1 mes x ∈ : |f u (x)| ≤ exp −20nδ(n) > exp −n1−κ T l 0 2 (n) for some choice of l from j1 −1, j2 −1, j2 −j1 −1. However, choosing g(n) = 20nδ0 κ−σ in Lemma 3.7 and τ = 4 in the hypothesis of this lemma, we have n u (n)o mes x ∈ T : |fl (x)| ≤ exp −20nδ0

(n) −3 ≤ exp −20nδ0 l exp −cn1−κ . Thus, we complete the proof by this contradiction. One of our methods to obtain a large deviation estimate for a single determinant is the BMO(T) norm. BMO(T) is the space of functions of bounded mean oscillation on T. Identifying functions that diﬀer only by an additive constant, then norm on BMO(T) is given by 1 Z kfkBMO(T) := sup |f − hfiI |dx, (3.19) I⊂T |I| I R where hfiI := I f(x)dx. Applying the previous lemma, we obtain the following 1 u lower bound of the mean value of n |fn (x)|, which will help us estimate the BMO norm. LDT DIRICHLET DETERMINANTS WITH BRJUNO-RUSSMANN¨ FREQUENCY 5327

τ 1−κ 1 1−κ Then, mes (T \ Ωn) ≤ n exp −n < exp − 2 n . u u h nτ i τ Deﬁne ν (x) = log |f (x)| /n and set M = ≥ n 2 for large n. For any n n l0 u (n) log 3 x ∈ Ωn we have that νn (x + kl0ω) > Ln − 100δ0 − n for all but at most two k’s, 1 ≤ k ≤ M. We have M Z 1 X Z hνui = νu (x) dx = u (x + kl ω) dx n n M 0 T k=1 T Z M − 2 (n) log 3 2 u ≥ Ln − 100δ0 − + inf νn (x + kl0ω) dx 1≤k≤M Ωn M n M M 1 X Z + νu (x + kl ω) dx. (3.20) M n 0 k=1 T\Ωn a a a Deﬁne νn (x) = log |fn (x)| /n. Note that νn(x) can be extended to the complex a trip Tρ where νn(z) is subharmonic. Due to (3.3), we have that 1 S := sup νa (z) ≤ sup log kM a (z)k < M . n n n 0 z∈Aρ0 z∈Aρ0 a a Applying Cartan’s estimate, Lemma 3.4, to fn (z) with M = Sn, m =< νn > n and τ H = n 4 , we have τ τ a 4 a 4 inf νn (x + kl0ω) ≥ S − C (S − hvi) n > −C (2|S| − hνni) n (3.21) 1≤k≤M τ up to a set not exceeding CM exp −n 4 in measure. Combining it with the relationship that

 n n−1 1 X X νu (x) = νa (x) − + log |a(x + jω)| (3.22) n u 2n   j=1 j=0

τ τ 4 1 4 up to a set Bn not exceeding CM exp −n + exp(−cˆ|D|n) < exp − 2 n in measure. Therefore,

0 τ M Z u 2 (n) log 3 C n 4 2 X u hνn i ≥ 1 − Ln − 100δ0 − − − |νn (x + kl0ω)| . M n M M c k=1 Ωn∪Bn 5328 WENMENG GENG AND KAI TAO

3 u 3 Let g(n) = n in Lemma 3.7. Then simple calculations shows that kν k 2 ≤ Cn . n L (T) Thus, Z u c 1/2 |νn (x + kl0ω)| dx ≤ (mes {Ωn ∪ Bn}) kukL2( ) c T Ωn∪Bn 3 1 τ 1 τ ≤Cn exp − n 4 ≤ C exp − n 4 . 4 8 Above all, c u (n) 2 0 −τ 1 τ (n) 0 hν i≥L −100δ − L −C n 4 −C exp − n 4 ≥L − δ . (3.23) n n 0 M n 8 n 0

Remark 5. Due to the setting of τ and (3.23), easy computations shows that ( 1, if ∆(t) > t5; c = A 0 , if ∆(t) ∼ tA, 1 < A < 5. 5 a We will show that the supermum of the subharmonic function un(z, E, ω) on T is closed to its mean value. Here, we will apply the property that a subharmonic function at a point is small than the its integration on the platform center at that point. From the proof of Theorem 1.4, it is easily seen that the sharp LDT for a un(x) can been extended to the complex region Tρ: a a mes {x : |un(re(x), E, ω) − Ln(r, E, ω)| > δ} 2 (n) < exp (−cδnˆ ) + exp −cδˇ n , ∀δ > δ0 , (3.24) where Z a a Ln(r, E, ω) = un(re(x), E, ω)dx. T Lemma 4.1 in [12] proved that there exists C0 = C0 (M0, ρ) such that for any r1, r2 ∈ (1 − ρ, 1 + ρ) we have a a |Ln(r1) − Ln(r2)| ≤ C0|r1 − r2|. (3.25) Lemma 3.12. For any integer n > 1 we have that a a (n) sup log kMn (x)k ≤ nLn + 2nδ0 . x∈T (n) Proof. Due to (3.24) with δ = δ0 , we have a a (n) log kMn (re(x))k − nLn (r) ≤ nδ0 (n) (n) 2 except for a set of measure less than exp(−cnδˆ 0 ) + exp(−cˇ(δ0 ) n). By the a subharmonicity of log kMn (z)k we have Z a a 1 a a log kMn (x)k − nLn ≤ −2 (log kMn (z)k − nLn) dA (z) πn D(x,n−1) Z 1+n−1 Z x+2n−1 1 a a ≤ −2 |log kMn (ry)k − Ln| rdydr. (3.26) πn 1−n−1 x−2n−1 For r ∈ 1 − n−1, 1 + n−1 we have Z x+2n−1 a a |log kMn (ry)k − Ln| dy x−2n−1 LDT DIRICHLET DETERMINANTS WITH BRJUNO-RUSSMANN¨ FREQUENCY 5329

Z x+2n−1 a a a a ≤ |log kMn (ry)k − Ln (r)| dy + |Ln − Ln (r)| x−2n−1 cˆ cˇ 2 ≤nδ(n) + C n exp − nδ(n) + exp − δ(n) n + C n−1 < 2nδ(n). 0 a 2 0 2 0 3 0 Then, we will use the following lemma proved by Bourgain, Goldstein and Schlag in [7], not the deﬁnition, to calculate the BMO norm of subharmonic functions.

Lemma 3.13 (Lemma 2.3 in [7]). Suppose u is subharmonic on Tρ, with µ(Tρ) + sup h(z) ≤ n where µ( ) and h(z) comes from Lemma 2.4. Furthermore, z∈Tρ Tρ assume that u = u0 + u1, where

∞ 1 ku0 − hu0ikL (T) ≤ 0 and ku1kL (T) ≤ 1. (3.27)

Then for some constant Cρ depending only on ρ, n √ kukBMO(T) ≤ Cρ 0 log + n1 . 1

Lemma 3.14. There exist constant c1 = c1(a, v, E, ρ, γ) and absolute constant C such that for every integer n and any δ > 0 we have a a (n) −c0 mes {x ∈ T : |log |fn (x)| − hlog |fn |i| > nδ} ≤ C exp −c1δ(δ0 ) . where c0 comes from Remark5. The same estimate with possibly diﬀerent c1 holds u for fn . Proof. It is enough to establish the estimate for n large enough. By Lemma 3.11 and Lemma 3.12, c0 ( a a (n) hνni ≥ Ln − δ0 sup νa ≤ La + 2δ(n). T n n 0 This implies that c a a (n) 0 kν − hν ik 1 ≤ 3 δ . n n L (T) 0

Due to Lemma 3.13 with setting 0 = 0, we have c a a a a a 1/2 (n) 0 kν k = kν − hν ik ≤ Cρ kν − hν ik 1 ≤ 3Cρ δ . n BMO(T) n n BMO(T) n n L (T) 0 Then, the well-known John-Nirenberg inequality tells us how to apply this MBO norm to obtain the large deviation theorem: Let f be a function of bounded mean oscillation on T. Then there exist the absolute constants C and c such that for any γ > 0 cγ meas{x ∈ T : |f(x) − hfi| > γ} ≤ C exp − . (3.28) kfkBMO Thus, a a (n) −c0 mes {x ∈ T : |νn (x) − hνni| > δ} ≤ C exp −c1δ(δ0 ) .

Now, due to the above proof and Remark5, to prove Theorem 1.1-1.3, the only 1 a (n) A thing we need to do is obtain k n log |fn |kBMO = O(δ0 ), when ∆(t) ∼ t and 1 < A < 5. In the following proof, we will use the Avalanche Principle to reﬁne the previous estimation: 5330 WENMENG GENG AND KAI TAO

Proposition 1 (Avalanche Principle). Let A1,...,An be a sequence of 2×2–matrices whose determinants satisfy

max | det Aj| ≤ 1. (3.29) 1≤j≤n Suppose that

min kAjk ≥ H > n and (3.30) 1≤j≤n 1 max [log kAj+1k + log kAjk − log kAj+1Ajk] < log H. (3.31) 1≤j Ll < exp (−cLll) . l 20 Due to the fact that u u u log |fl (x)| ≤ log kA1 (x)k ≤ log kMl (x)k , Lemma 3.14, 3.11 and 3.1, we have 1 1 l −c0 mes x : log kA1(x)k − Ll > Ll < exp −cLl δ , l 10 0 u and an analogous estimate for log kAmk. Now the hypothesis of Avalanche Principle are satisﬁed and hence m−1 m−1 u X u X u u 1 log kM (x)k + log A (x) − log A (x) A (x) = O (3.33) n j j+1 j l j=2 j=1

l −c0 u up to a set of measure less than 3m exp(−cLl δ0 ). By the deﬁnitions of Mn a and Mn , easy computations show that m−1 m−1 u X u X u u log kMn (x)k + log Aj (x) − log Aj+1 (x) Aj (x) j=2 j=1 LDT DIRICHLET DETERMINANTS WITH BRJUNO-RUSSMANN¨ FREQUENCY 5331

m−1 m−1 a X a X a a = log kMn (x)k + log Aj (x) − log Aj+1 (x) Aj (x) . j=2 j=1 a Thus, (3.33) also holds for Mn. If we set a a a a u0 (x) = log Am (x) Am−1 (x) + log kA2 (x) A1 (x)k , then the previous relation can be rewritten as

m−1 a X a log kMn (x)k + log kMl (x + (j − 1) lω)k j=2 m−2 X 1 − log kM a (x + (j − 1) lω)k − u (x) = O . 2l 0 l j=2 Similarly,for any 0 ≤ k < l − 1,

m−1 a X a log kMn (x)k + log kMl (x + kω + (j − 1) lω)k j=2 m−2 X 1 − log kM a (x + kω + (j − 1) lω)k − u (x) = O , 2l k l j=2 where 1 0 a a uk(x) = log M 0 (x + kω + (m − 1)lω) · A (x + kω) 0 0 l −k m−1 a a 1 0 + log A (x + kω) · M (x) , 2 l+k 0 0 a a which means that we decrease the length of Am by k and increase the length of A1 by k. Adding these equations and dividing by l yields

(m−1)l−1 X 1 log kMa (x)k + log kM a (x + jω)k n l l j=l (m−2)l−1 l−1 X 1 X 1 1 − log kM a (x + jω)k − u (x) = O l 2l l k l j=l k=0

l −c0 up to a set of measure less than 3n exp −cLl(δ0) . Note that Theorem 1.4 can P(m−1)l−1 1 a P(m−2)l−1 1 a be applied for j=l l log kMl (x + jω)k and j=l l log kM2l(x + jω)k (n) and ml ∼ n. So, the deviation δ is the smallest deviation δ0 we can choose here. Then,

(m−1)l−1 (m−2)l−1 X 1 X 1 log kM a (x + jω)k − log kM a (x + jω)k l l l 2l j=l j=l a a (n) = (m − 2) lLl − (m − 3) lL2l + O nδ0

(n) up to a set of measure less than exp(−cnδ0 ). Note that uk, k = 0, . . . , l − 1 have uk the subharmonic extensions. Therefore, for any l , Theorem 1.4 can be applied 5332 WENMENG GENG AND KAI TAO

(n) nδ0 with n = 1 and δ = l , and obtain that l−1 l−1 X 1 X 1 (n) u (x) − hu i = O nδ l k l k 0 k=0 k=0

1−c0 (n) up to a set of measure less than l exp(−cn δ0 ). Thus, combining these equations, we have that

l−1 X 1 (n) log |f a (x)| + (m − 2) lLa − (m − 3) lLa − hu i = O nδ (3.34) n l 2l l k 0 k=0 up to a set B, satisfying that

l −c0 1−c (n) (n) mes (B) ≤ 3n exp −cLl δ0 + l exp −cn δ0 + exp −cnδ0 .

(n) 1 0 c0 − A + 1 c 00 2 Recalling that δ0 = Cωn , c0 = 5 and l ∼ n , we have mes (B) ≤ exp(−c n ), where c00 is a small constant depending on a, v, ω and E. Integrating (3.34) and a using the fact that klog |f |k 2 ≤ Cn, yields n L (T) l−1 X 1 hlog |f a (x)|i + (m − 2) lLa − (m − 3) lLa − hu i n l 2l l k k=0 (n) c0 (n) 00 2 =O nδ0 + Cn exp −c n = O nδ0 . Combining it with (3.34), we have

a a (n) log |fn (x)| − hlog |fn |i = O nδ0 (3.35) up to B. Deﬁne 1 1 log |f a| − log |f a| = u + u n n n n 0 1 (n) where u0 = 0 on B and u1 = 0 on \B. Obviously, ku0 − hu0ik ∞ = O(δ ) T L (T) 0 and c0 p 00 2 ku1k 2 ≤ C mes (B) ≤ C exp −c n . L (T) Due to Lemma 3.13, klog |f a|k = O nδ(n) . n BMO(T) 0

1 a Similar to Lemma 3.2, we also can prove that h n log |fn |i in Theorems 1.1-1.3 can be exchanged by La.

Lemma 3.15. There exists a constant C0 = C0 (a, v, E, ω, γ) such that a a |hlog |fn |i − nLn| ≤ C0 for all integers. Proof. Recall that m−1 m−1 a X a X a a 1 log kM (x)k + log A (x) − log A (x) A (x) = O n j j+1 j l j=2 j=1 LDT DIRICHLET DETERMINANTS WITH BRJUNO-RUSSMANN¨ FREQUENCY 5333

l −c0 up to a set of measure less than 3m exp(−cLl(δ0) ). Similarly, m−1 m−1 a X a X a a log kMn (x)k + log Aj (x) − log Aj+1 (x) Aj (x) j=2 j=1

4. The proof of Theorem 1.5. We used the LDTs and the Avalanche Principle together in the above two proofs. As we have mentioned in the introduction, this method was ﬁrst created in [11] to prove the H¨oldercontinuity of Lyapunov exponent Ls(E, ω) in E with the strong Diophantine ω. Recently, our second author also applied it to obtain the same continuity of L(E, ω) with any irrational ω in [21]. Our proof of Theorem 1.5 needs this result. Therefore, we list it as a lemma:

Lemma 4.1. Assume β(ω) = 0 and L(E0, ω) > 0. There exists rE = rE(a, v, E0, ω) such that for any |E − E0| ≤ rE, 3 5 L(E , ω) < L(E, ω) < L(E , ω). 4 0 4 0 Furthermore, there exists a constant h = h(a, v) called H¨olderexponent such that for any E1,E2 ∈ [E0 − rE,E0 + rE], h |L(E1, ω) − L(E2, ω)| < |E1 − E2| . (4.1) The proof of Theorem 1.5. From Theorem 1.1-1.3 and Lemma 3.15, we have that (n) (n) −1 for any δ > δ0 and (x, E) ∈ T × E except for a set of measure C exp(−cδ(δ0 ) ), a a |log |fn (x, E, ω)| − nLn (E, ω)| ≤ nδ. (4.2)

Then, due to Fubini’s Theorem and Chebyshev’s inequality, there exists a set Bn,δ ⊂ (n) −1 T with mes Bn,δ < C exp(−cδ(δ0 ) ), such that for each x ∈ T \Bn,δ there exists (n) −1 En,δ,x ⊂ D, with mes En,δ,x < C exp(−cδ(δ0 ) ) such that (4.2) holds for any E ∈ E \ En,δ,x. Therefore, there exist x1,E1 satisfying − 1 (n) 2 |x1 − x0| ≤ C exp −c δ0 , and − 1 (n) 2 |E1 − E0| ≤ C exp −c δ0 , such that 1 a (n) 2 log |fn (x1,E1)| ≥ nLn (E1) − n δ0 . (4.3) Deﬁne 1 − 1 (n) h (n) 2 R := δ0 C exp −c0 δ0 , 5334 WENMENG GENG AND KAI TAO and a 0 0 Nx,E (r) = # {E : fn (x, E ) = 0, |E − E| ≤ r} . The Jensen formula states that for any function f analytic on a neighborhood of D(z0,R), see [19], Z 1 X R log |f(z0 + Re(θ))| dθ − log |f(z0)| = log (4.4) |ζ − z0| 0 ζ:f(ζ)=0 provided f(z0) 6= 0. Thus, we have that Z 2π 1 a iθ a Nx1,E1 (3R) ≤ log fn x1,E1 + 4Re dθ − log |fn (x1,E1)| . (4.5) 2π 0 By Lemma 3.12, it yields ! a a (n) Nx1,E1 (3R) ≤ sup (n (Ln (E) − Ln (E1))) + 3nδ0 . |E−E1|=4R

Due to Lemma 4.1, if |E1 − E2| < 4R, then h (n) |L(E1) − L(E2)| < |E1 − E2| < 4δ0 . (n) (log n)2 Combining it with Lemma 3.2 and the fact that δ0 n , we have ! a a (n) sup (n (Ln (E) − Ln (E1))) < 10nδ0 . |E−E1|=4R Thus, (n) Nx1,E1 (3R) ≤ 13nδ0 .

Recalling that |E0 − E1| R, we have (n) Nx1,E0 (2R) ≤ Nx1,E1 (3R) ≤ 13nδ0 . (4.6)

Note that Hn(x, ω) is Hermitian. Thus, by the Mean Value Theorem,

− 1 (n) 2 kHn (x0, ω) − Hn (x1, ω)k ≤ C |x0 − x1| ≤ C exp −c0 δ0 .

(n) Let Ej (x, ω), j = 1, . . . , n be the eigenvalues of Hn (x, ω) ordered increasingly. Then, 1 − 2 (n) (n) (n) Ej (x0) − Ej (x1) ≤ C exp −c0 δ0 . This implies that (n) Nx0,E0 (R) ≤ Nx1,E0 (2R) < 13nδ0 . Remark 6. Similarly, 1 a (n) h (n) # z ∈ C : fn (z, E0, ω) = 0, |z − x0| < δ0 ≤ 13nδ0 .

Acknowledgments. We would like to thank the referees very much for their valu- able comments and suggestions. LDT DIRICHLET DETERMINANTS WITH BRJUNO-RUSSMANN¨ FREQUENCY 5335

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