Anderson localization for multi-frequency quasi-periodic operators on $\mathbb{Z}^d$

TL;DR

This paper proves Anderson localization for a broad class of multi-frequency quasi-periodic operators on multi-dimensional integer lattices, extending the understanding of localization phenomena in complex systems.

Contribution

It establishes Anderson localization for general analytic multi-frequency quasi-periodic operators on  lattices for any number of frequencies and dimensions, a significant generalization.

Findings

Proves Anderson localization in multi-frequency quasi-periodic systems.

Extends localization results to arbitrary dimensions and frequencies.

Provides a comprehensive framework for analyzing such operators.

Abstract

We establish Anderson localization for general analytic $k$-frequency quasi-periodic operators on $\mathbb{Z}^d$ for \textit{arbitrary} $k,d$.

Equations

$$|S(n,n^{\prime })|\le e^{-\rho |n-n^{\prime }|},\rho >0,$$

$$H(x)=S+\lambda v(x+n\omega ){\delta }_{n{n}^{\prime }},n,n^{\prime }\in {\mathbb{Z}}^d,$$

$$x=(x_{11},\cdots ,x_{b_{1}1},\cdots ,x_{1d},\cdots ,x_{b_{d}d})\in {\mathbb{T}}^b,$$

$$n\omega =(n_1{\omega }_{11},\cdots ,n_1{\omega }_{b_{1}1},\cdots ,n_d{\omega }_{1d},\cdots ,n_d{\omega }_{b_{d}d}).$$

$$H=S+\lambda v(x_1+n_1{\omega }_1,x_2+n_1{\omega }_2,x_3+n_2{\omega }_3){\delta }_{n{n}^{\prime }},$$

$$v(x_{11},\cdots ,x_{k1},\cdots ,x_{1d},\cdots ,x_{kd})=f(x_{11}+\cdots +x_{1d},\cdots ,x_{k1}+\cdots +x_{kd}).$$

$$H(x)=S+\lambda f(x+nA){\delta }_{n{n}^{\prime }},$$

$$\tilde{H}(x)=F+\lambda s(x+Am){\delta }_{n{n}^{\prime }},$$

$$x^j:=(x_{1j},\cdots ,x_{b_{j}j})\in {\mathbb{T}}^{b_j}(1\le j\le d).$$

$$(x^1,\cdots ,x^{j-1},x^{j+1}\cdots ,x^d)\in {\mathbb{T}}^{b-b_j},$$

$${\mathbb{T}}^{b_j}\ni \theta \mapsto v(x^1,\cdots ,x^{j-1},\theta ,x^{j+1}\cdots ,x^d)$$

$$X(x):=\{y\in {\mathbb{R}}^{d_2}:(x,y)\in X\}.$$

$$\mathrm{diam}(\Lambda )=\underset{n,n^{\prime }\in \Lambda }{\sup }|n-n^{\prime }|,\mathrm{dist}(m,\Lambda )=\underset{n\in \Lambda }{\inf }|m-n|(m\in {\mathbb{R}}^d),$$

$$G_{\Lambda }(E;x)=(R_{\Lambda }({\lambda }^{-1}H-E+i0)R_{\Lambda }{)}^{-1}.$$

$$G_{n+\Lambda }(x)=G_{\Lambda }(x+n\omega ).$$

$$Q_N=[-N,N{]}^d$$

$$Q_N=[-N,N{]}^d\setminus \{n\in {\mathbb{Z}}^d:n_i{\varsigma }_{i}0,1\le i\le d\},$$

$${\mathcal{E}}_N:=\{n+Q_N{\}}_{n\in {\mathbb{Z}}^d,Q_N\in {\mathcal{E}}_N^0}.$$

$$\mathcal{S}=\bigcup _j\bigcap _{\ell \in {\mathcal{L}}_j}\left\{x\in {\mathbb{R}}^n:P_{\ell }(x){\varsigma }_{j\ell }0\right\},$$

$$\mathrm{mes}(\mathcal{S}(y))<\eta \mathrm{for}\forall y\in [0,1{]}^r.$$

$$\left\{(x_1,\cdots ,x_{2^r})\in [0,1{]}^{d{2}^r}:\bigcap _{1\le i\le 2^r}\mathcal{S}(x_i)\ne \varnothing \right\}$$

$$B^C{\eta }^{d^{-r}{2}^{-r(r-1)/2}},$$

$$\log B\ll \log \frac{1}{\eta },$$

$${\eta }^{\frac{1}{d}}\le ϵ.$$

$$\mathcal{S}={\mathcal{S}}_1\cup {\mathcal{S}}_2$$

$${\mathrm{mes}}_{d_1}({\mathrm{Proj}}_{[0,1{]}^{d_1}}{\mathcal{S}}_1)\le B^{C(d)}ϵ,$$

$${\mathrm{mes}}_{d_2}(\mathcal{L}\cap {\mathcal{S}}_2)\le B^{C(d)}{ϵ}^{-1}{\eta }^{\frac{1}{d}},$$

$$\underset{1\le j\le d_1}{\max }|{\mathrm{Proj}}_{\mathcal{L}}(e_j)|<ϵ,$$

$$\mathrm{mes}(\mathcal{S})<\eta .$$

$$n\omega =(n_1{\omega }^1,n_2{\omega }^2,\cdots ,n_r{\omega }^r).$$

$$\underset{1\le s\le r}{\min }|n_s|>(B\underset{1\le s\le r}{\max }|m_s|{)}^C,$$

Full text

1. Introduction and main results

The theory of quasiperiodic opetarors with analytic potentials has seen dramatic advances in the last 20 years, since the development of first non-perturbative methods for control of the Green’s functions [17, 18, 8, 4, 6] that replaced earlier perturbation of eigenfunctions techniques. The most well-developed and remarkably rich theory concerns the case of one-dimensional one-frequency potentials, where powerful reducibility/dynamical techniques are particularly enhanced by the analyticity arguments. There are now non-perturbative results on both small and high coupling sides ([4, 19, 21] and references therein), global theory [1], and sharp arithmetic transitions and related universality (e.g. [2, 14, 15, 16]). However, if one increases either the dimension of the undelying torus (the number of frequencies) or, especially, the space dimension, the situation becomes significantly more complicated. First, non-perturbative results become, generally, false [4], so throwing away small measure sets of parameters where things actually do sometimes go bad, becomes a necessity. Even more importantly, one-dimensional (and therefore dynamical) techniques are not applicable in higher space dimension. The fist multi-dimensional localization was obtained by perturbative (KAM) methods by Chulaevsky-Dinaburg for $k$-frequency operators on $\ell^{2}(\mathbb{Z}^{d})$ for $k=1$ and arbitrary $d$ [11]. However, perturbative techniques have not been made to work to prove localization in the multi-frequency case, $k>1,$ even for $d=1.$ Bourgain-Goldstein-Schlag developed a way to apply some of the non-perturbative methods to the two dimensional case [7], obtaining localization at high coupling for $k=d=2.$ This was extended by Bourgain to arbitrary $k=d$ [5], where he developed a new powerful scheme that allowed to circumvent the arithmetic difficulties that restricted [7] to $k=d=2.$ In this paper we extend Bourgain’s result to the case of general $k,d$ (in fact, an even significantly more general situation).

Let $S$ be a Toeplitz (operator) matrix on $\ell^{2}(\mathbb{Z}^{d})$ satisfying,

[TABLE]

where $|n|:=\max\limits_{1\leq i\leq d}|n_{i}|$ for $n=(n_{1},n_{2},\cdots,n_{d})\in\mathbb{Z}^{d}$.

Let $v$ be a real analytic function on $\mathbb{T}^{b}$, where $b=\sum\limits_{i=1}^{d}b_{i}$ ($b_{i}\in\mathbb{N}$ for $1\leq i\leq d$).

In this paper, we consider the following operators

[TABLE]

where

[TABLE]

Example 0. Taking $b_{i}=1,i=1,...,d$ and the nearest neighbor Laplacian $S$ we obtain operators considered in [5].

Example 1. $d=2$, $b_{1}=2,$ $b_{2}=1$. $v$ is a function on $\mathbb{T}^{3}$. For $x=(x_{1},x_{2},x_{3})$ and $\omega=(\omega_{1},\omega_{2},\omega_{3})$, the operator becomes

[TABLE]

where $n=(n_{1},n_{2})$.

Example 2. $b=kd,b_{i}=k,i=1,...,d,$ $f$ is a function on $\mathbb{T}^{k}$, and

[TABLE]

Then the operator becomes

[TABLE]

where $x\in\mathbb{T}^{k},\,n\in\mathbb{Z}^{d},$ and $A$ is a $d$ by $k$ matrix of frequencies. This is the most general form of a $d$-dimensional quasiperiodic operator with a $k$-dimensional phase space. The Aubry dual family has the form

[TABLE]

where $x\in\mathbb{T}^{d},\,n\in\mathbb{Z}^{k},$ and $F,S$ are Toeplitz operators with $(n,n^{\prime})$ terms given by the $n-n^{\prime}$ Fourier coefficients of, correspondingly $f,s.$ The standard Laplacian is therefore dual to the potential given by the sum of cosines, and the dual of a general analytic potential is a Toeplitz matrix as above.

Remark 1. When considering families (1.4) with $A$ restricted to a linear submanifold of $d$ by $k$ matrices of frequencies, one needs to take $b_{i}$ equal to the number of free variables in the $i^{th}$ row of $A$ and adjust $v$ accordingly. As such, the family considered in [5] can of course also be recast in this language: it corresponds to $A$ restricted to $diag(\omega_{1},...,\omega_{d}).$

We call $x\in\mathbb{T}^{b}$ the phase, $\omega\in\mathbb{T}^{b}$ the frequency and $\lambda\geq 0$ the coupling. Let

[TABLE]

We assume $v$ satisfies the following non-degeneracy condition: for any

[TABLE]

the function

[TABLE]

is nonconstant.

Denote by ${\rm mes}$ the Lebesgue measure. We say operator $H$ satisfies Anderson localization if it has only pure point spectrum with exponentially decaying eigenfunctions.

Theorem 1.1**.**

Let $H(x)$ be given by (1.2) with $v$ satisfying the non-degeneracy condition. Then for any $\delta>0$, there is a $\lambda_{0}=\lambda_{0}(\delta,v,\rho,b,d)>0$ such that the following statement holds: for any $\lambda\geq\lambda_{0}$ and any $x\in\mathbb{T}^{b}$, there exists $\Omega=\Omega(x,\lambda v,\delta,\rho,b,d)\subset\mathbb{T}^{b}$ with $\mathrm{mes}(\mathbb{T}^{b}\setminus\Omega)\leq\delta$ such that for $\omega\in\Omega$, $H(x)$ satisfies Anderson localization.

Remarks

1. (1)

In particular, this holds for all operators (1.4) with arbitrary $k,d$ and any non-constant analytic function $f,$111As in Remark 1, the non-degenracy condition on $v$ leads to additional non-degeneracy conditions on $f$ if the number of free variables in a certain row of the submanifold is bounded by $1.$ In particular, for $A$ restricted to $diag(\omega_{1},...,\omega_{d}),$ as in [5], the required non-degeneracy condition is exactly as in [5]. which is an important building block in the proof of absolutely continuous spectrum for operators (1.5) [9] and was the key initial motivation for our work. 2. (2)

We note that our phase space dimension $b$ satisfies $b\geq d$ since $b=\sum_{i=1}^{d}b_{i}$, $b_{i}\geq 1$. This is essential for our arguments. As shown in Example 2, general quasiperiodic operators always have $b\geq d$. However, operators (1.2) with $b<d$, for example $V_{n_{1},n_{2}}=v(x+n_{1}\omega,x+n_{2}\omega)$, also appear naturally, e.g. in the study of interacting particles, and our proof does not apply in this setting. A localization result for a model with $b=1,d=2$ was recently obtained by Bourgain-Kachkovskiy [10]. 3. (3)

Previous multidimensional/multifrequency localization results [7, 5] were not only restricted to $k=d$ , but also done only for the nearest neighbor Laplacian, i.e. $S(n,n^{\prime})=\delta_{|n-n^{\prime}|,1}.$ The extension to general $S$ as in (1.1) is motivated by the Aubry duality purposes in [9]. Localization for long-range operators (general $S$) was previously obtained for $k=1$ in [11] and, nonperturbatively, for $k=d=1$ in [4, 8].

The main scheme of our proof is definitely adapted from Bourgain [5]. However, while our result is significantly more general and more technically complex, our argument can also be viewed as both a clarification and at the same time streamlining of [5]. Indeed, our proof, while including more detail and hopefully increasing the readability, is only shorter than the corresponding part of Bourgain’s. This is due to several important technical improvements that we add to Bourgain’s scheme. One important highlight is that, in the process of deterministic multi-scale analysis proceeding from scale $N_{1}$ to $N_{2}$, a chain of scales between $N_{1}$ and $N_{2}$ has always been used in the past work, [5, 7]. Here, instead of gluing “good” Green’s function at multiple scales between $N_{1}$ and $N_{2}$ to establish the “goodness” of Green’s function at scale $N_{2}$, we find a way to directly use the “good” Green’s function at scale $N_{1}$ + subexponential bound of the norm to prove the “goodness” of Green’s function at scale $N_{2}$.

Another issue we want to highlight is that the $k=d=2$ analysis of [7] required dealing with many different types of elementary regions, something that would be prohibitively difficult to carry out in higher dimensions. In dealing with higher dimensions in [5] Bourgain significantly reduces the allowed elementary regions. This comes at the price of some complications in dealing with the lattice points at the boundary of the elementary regions, which Bourgain claims can be carried out, but provides no detail. We use the same (slightly corrected) type of restriction on the elementary regions but believe this issue is not entirely trivial and tackling it requires a certain modification of the procedure, which we provide in full detail.

Non-perturbative proofs of localization for $d>1$ are in a sense a version of deterministic multi-scale analysis. The latter is a powerful method originally developed for random operators by Fröhlich and Spencer [12], that crucially relied on independence and Wegner’s Lemma that is effectively dependent on rank-one perturbations. For the deterministic version, difficulties with lack of independence/rank one perturbations are circumvented by the semi-algebraic sets considerations and subharmonicity arguments [4]. The non-perturbative proofs consist of two parts. First, one needs to obtain measure and complexity estimates for phases/frequencies with exponential off-diagonal decay and subexponential upper bounds for the matrix elements of the Green’s function for box-restricted operators for a given energy. From this, localization follows through elimination of energy via an argument involving complexity bounds on semi-algebraic sets. The second part is by now rather standard and follows the reasonably short argument in [5] essentially verbatim. In fact, it is the first part that presents the main difficulty associated with higher dimensions. Thus we focus only on the first, single energy, part here. This is also where the key difficulty in extending [7] and the key difference between [5] and [7] lies. One needs to guarantee a sublinear upper bound on the number of times the ergodic trajectory hits certain forbidden regions of given measure/algebraic complexity, without further detail on the structure of those fordbidden regions. A key argument in [7] is a Lemma that does guarantee it for $k=b=2$ under an explicit arithmetic condition on the frequencies. Roughly, it means that too many points on the trajectory of rotation close to an algebraic curve of a bounded degree would force it to oscillate more than the degree allows. However, this statement is not extendable to $d\geq 3.$ In [5] Bourgain instead developed a way to restrict to suitable frequencies already for the first step, which turned out to be a very robust approach that we also develop here. Besides the elimination of energy argument, we do not include detailed proofs of two further statements very similar to those in [5], and with proofs presented there in a very clear way. The proofs that are similar to Bourgain’s that we do present either have certain novelty or contain important technical clarifications.

In Section 2 we introduce the main concepts and also list the above mentioned results for which we do not present detailed proofs. One such concept is “property P at scale $N$” - essentially, the single energy statement one wants to establish for all large scales, that allows to streamline certain formulations. Section 3 is devoted to the main multi-scale argument: property P at scales $N,N^{c}$ implies property P at an interval of subexponentially large scales, Theorem 3.6. In section 4 we take care of the initial scale and give a very short argument to obtain the final single energy estimate, Theorem 4.1, from Theorem 3.6. In the appendix we prove a several variables matrix-valued Cartan estimate (Lemma 3.4 used in the proof of Theorem 3.6), that follows Bourgain’s one-variable argument in [4] but also uses high-dimensional Cartan sets estimates of [13].

2. Preparations

2.1. Notation

For any $x\in\mathbb{R}^{d_{1}}$ and $X\subset\mathbb{R}^{d_{1}+d_{2}}$, denote the $x$-section of $X$ by

[TABLE]

Let $\tilde{b}=\max_{i}b_{i}$. For any $x\in\mathbb{T}^{b}$ and $1\leq j\leq d$, let ${x}_{j}^{\neg}=(x^{1},\cdots,x^{j-1},x^{j+1}\cdots,x^{d})\in\mathbb{T}^{b-b_{j}}$. For $x=(x_{1},x_{2},\cdots,x_{l}),y=(y_{1},y_{2},\cdots,y_{l})\in\mathbb{R}^{l}$, let $|x-y|=\max_{i}|x_{i}-y_{i}|$.

For $\Lambda_{1},\Lambda\subset\mathbb{Z}^{d}$, we introduce

[TABLE]

and $\mathrm{dist}(\Lambda_{1},\Lambda)=\inf\limits_{n\in\Lambda_{1}}\mathrm{dist}(n,\Lambda)$.

We also use $\|\cdot\|$ as $\ell^{2}$ norm of the matrix. For convenience, in the following, we study operator $\lambda^{-1}H(x)$. We always assume $\lambda>1$. Since the spectra of $\lambda^{-1}H(x)$ are bounded by $C(S,v)$, we can further assume $E$ is bounded.

2.2. Green’s functions and elementary regions

For $\Lambda\subset\mathbb{Z}^{d}$, let $R_{\Lambda}$ be the restriction operator, i.e., $(R_{\Lambda}\xi)(n)=\xi(n)$ for $n\in\Lambda$, and $(R_{\Lambda}\xi)(n)=0$ for $n\notin\Lambda$. Denote by $H_{\Lambda}=R_{\Lambda}HR_{\Lambda}$ and the Green’s functions

[TABLE]

We will also write $G_{\Lambda}$ when there is no ambiguity. Clearly,

[TABLE]

We denote by $Q_{N}$ an elementary region of size $N$ centered at 0, which is one of the following regions,

[TABLE]

or

[TABLE]

where for $i=1,2,\cdots,d$, $\varsigma_{i}\in\{<,>,\emptyset\}^{d}$ and at least two $\varsigma_{i}$ are not $\emptyset$.

Denote by $\mathcal{E}_{N}^{0}$ the set of all elementary regions of size $N$ centered at 0. Let $\mathcal{E}_{N}$ be the set of all translates of elementary regions, namely,

[TABLE]

2.3. Semi-algebraic sets

Definition 2.1** (Chapter 9, [4]).**

A set $\mathcal{S}\subset\mathbb{R}^{n}$ is called a semi-algebraic set if it is a finite union of sets defined by a finite number of polynomial equalities and inequalities. More precisely, let $\{P_{1},\cdots,P_{s}\}\subset\mathbb{R}[x_{1},\cdots,x_{n}]$ be a family of real polynomials whose degrees are bounded by $d$. A (closed) semi-algebraic set $\mathcal{S}$ is given by an expression

[TABLE]

where $\mathcal{L}_{j}\subset\{1,\cdots,s\}$ and $\varsigma_{j\ell}\in\{\geq,\leq,=\}$. Then we say that $\mathcal{S}$ has degree at most $sd$. In fact, the degree of $\mathcal{S}$ which is denoted by $\deg(\mathcal{S})$, means the smallest $sd$ over all representations as in (2.2).

In [5], Bourgain proved a result for eliminating several variables.

Lemma 2.2** (Lemma 1.18, [5]).**

Let $\mathcal{S}\subset[0,1]^{d+r}$ be a semi-algebraic set of degree $B$ and such that

[TABLE]

Then the set

[TABLE]

is semi-algebraic of degree at most $B^{C}$ and measure at most

[TABLE]

where $C=C(d,r)>0$.

Another important fact is the following decomposition Lemma for semi-algebraic sets in the product spaces.

Lemma 2.3** ([5, 4]).**

Let $\mathcal{S}\subset[0,1]^{d=d_{1}+d_{2}}$ be a semi-algebraic set of degree $\deg(\mathcal{S})=B$ and $\mathrm{mes}_{d}(\mathcal{S})\leq\eta$, where

[TABLE]

with

[TABLE]

Then there is a decomposition of $\mathcal{S}$ as

[TABLE]

such that the projection of $\mathcal{S}_{1}$ on $[0,1]^{d_{1}}$ has small measure

[TABLE]

and $\mathcal{S}_{2}$ has the transversality property

[TABLE]

where $\mathcal{L}$ is any $d_{2}$-dimensional hyperplane in $[0,1]^{d}$ s.t.,

[TABLE]

where we denote by $e_{1},\cdots,e_{d_{1}}$ the coordinate vectors in $\mathbb{R}^{d_{1}}$.

We then have

Lemma 2.4**.**

Suppose that $\omega^{i}\in\mathbb{R}^{l_{i}}$ ($i=1,2,3,\cdots,r$) and $l=\sum_{i=1}^{r}l_{i}$. Let $\mathcal{S}\subset[0,1]^{lJ}$ be a semi-algebraic set of degree $B$ and such that

[TABLE]

For $\omega=(\omega^{1},\cdots,\omega^{r})\in[0,1]^{l}$ and $n=(n_{1},n_{2},\cdots,n_{r})\in\mathbb{Z}^{r}$, define

[TABLE]

Let $\mathcal{N}^{1},\cdots,\mathcal{N}^{J-1}\subset\mathbb{Z}^{r}$ be finite sets with the following property

[TABLE]

where $n\in\mathcal{N}^{i},m\in\mathcal{N}^{i-1}\ (2\leq i\leq J-1)$, where $C=C(J,l)$. Assume also

[TABLE]

Then

[TABLE]

where

[TABLE]

Proof.

The proof follows from Lemmas 2.2 and 2.3 just as the proof of Lemma 1.20 in [5] follows from the corresponding Lemma 1.18 and property (1.5) of semi-algebraic sets in [5]. ∎

Definition 2.5**.**

We say $(E,x)$ is $(\bar{\rho},N)$ good, if for any $Q_{N}\in\mathcal{E}_{N}^{0}$,

[TABLE]

Definition 2.6**.**

We say Green’s function satisfies property $P$ with parameters $(\gamma,\bar{\rho})$ at size $N$ if there is a semi-algebraic set $\Omega_{N}=\Omega_{N}(\lambda v,\rho,b,d)\subset\mathbb{T}^{b}$ with $\deg(\Omega_{N})\leq N^{4d}$ such that the following statement is true: for any $\omega\in\Omega_{N}$ and $E\in\mathbb{R}$, there exists a set $X_{N}=X_{N}(\lambda v,\rho,b,d,\omega,E)\subset\mathbb{T}^{b}$ such that

[TABLE]

and for any $x$ not in $X_{N}$, $(E,x)$ is $(\bar{\rho},N)$ good.

Theorem 2.7**.**

There exist small positive constants $c_{3}<c_{4}<1$, where $c_{3}$ and $c_{4}$ depend on $b,d$ such that the following statements are true. Let $c_{1}=\frac{c_{3}}{4\tilde{b}}$ and $c_{2}=c_{1}^{2}/2$. Fix a large number $N_{1}$. Let $N_{2}=N_{1}^{2/c_{1}}$ and $N_{3}=e^{N_{2}^{c_{2}}}$. Suppose the Green’s functions satisfy property P at size $N_{1}$ with parameters $(c_{1},\bar{\rho})$, and corresponding semi-algebraic sets $\Omega_{N_{1}}$. Then there exists a semi-algebraic set $\Omega_{3}\subset\Omega_{{N}_{1}}$ with $\deg(\Omega_{3})\leq{N}_{3}^{4d}$ and $\mathrm{mes}((\Omega_{{N}_{1}}\backslash\Omega_{3})\leq{N_{3}}^{-c_{3}}$ such that, if $\omega\in\Omega_{3}$, then for any $E\in\mathbb{R}$ and $x\in\mathbb{T}^{b}$, there exists ${N}_{3}^{c_{3}}<N<{N}_{3}^{c_{4}}$ such that, for all $k\in\Lambda\backslash\bar{\Lambda}$, $x+k\omega\mod\mathbb{Z}^{b}\notin X_{N_{1}}$, where

[TABLE]

Proof.

The proof is based on Lemmas 2.2 and 2.4. For details, we refer the reader to the proof of the Claim in [5, p.694]. To make it easier to check the corresponding relation between Theorem 2.7 and Claim in [5], we present the alignment of our notations with these of [5]. Let $X(B)$ denote the notation $X$ used in [5].

1. (1)

$\tilde{b}(B)=1$ since $b_{i}(B)=1$, $i=1,2,\cdots,d$. 2. (2)

$c_{1}=c_{1}(B)$, $c_{2}=c_{2}(B)$, $c_{3}=c_{5}(B)$ and $c_{4}=c_{4}(B)$. The formula before (2.8) in [5] gives the relation between $c_{1}$ and $c_{2}$. The relation between $c_{1}$ and $c_{3}$ is presented at the end of Section 2 in [5]. 3. (3)

$N_{1}=N_{2}(B)$, $N_{3}=\bar{N}(B)$ and ${N}_{3}^{c_{3}}=\bar{\bar{N}}(B)$. See (2.8), (2.11) and (2.24) in [5] for the corresponding relations. 4. (4)

$\Omega_{3}=\Omega_{\bar{N}}(B)$. See (2.25) in [5] .

∎

3. Resolvent identities and Cartan’s Lemma

Let $\Lambda_{1},\Lambda_{2}\subset\mathbb{Z}^{d}$ and $\Lambda_{1}\cap\Lambda_{2}=\emptyset$. Let $\Lambda=\Lambda_{1}\cup\Lambda_{2}$. Suppose that $R_{\Lambda}(\lambda^{-1}{H}(x)-E)R_{\Lambda}$ and $R_{\Lambda_{i}}(\lambda^{-1}{H}(x)-E)R_{\Lambda_{i}}$, $i=1,2$ are invertible. Then

[TABLE]

If $m\in\Lambda_{1}$ and $n\in\Lambda$, we have

[TABLE]

We remind

Lemma 3.1** (Schur test).**

Suppose $A=A_{ij}$ is a symmetric matrix. Then

[TABLE]

We now prove

Lemma 3.2**.**

Let $M_{0}\geq(\log N)^{2},\bar{\rho}\in[\frac{\rho}{2},\rho]$ and $M_{1}\leq N$. Let ${\rm diam}(\Lambda)\leq 2N+1$. Suppose that for any $n\in\Lambda$, there exists some $W=W(n)\in\mathcal{E}_{M}$ with $M_{0}\leq M\leq M_{1}$ such that $n\in W\subset\Lambda$, ${\rm dist}(n,\Lambda\backslash W)\geq\frac{M}{2}$ and

[TABLE]

We assume further that $N$ is large enough so that

[TABLE]

Then

[TABLE]

Proof.

For simplicity, we drop the dependence on $E$ and $x$. Under the assumption of (3.4), it is easy to check that for all $M_{0}\leq M\leq M_{1}$,

[TABLE]

By (3.2) and (3.3), one has

[TABLE]

For each $n\in\Lambda$, applying (3.1) with $\Lambda_{1}=W(n)$, one has

[TABLE]

By (3.6) and the fact that $|W(n)|\leq(2M+1)^{d}$, one has

[TABLE]

where the last inequality holds by the assumption ${\rm dist}(n,\Lambda\backslash W(n))\geq\frac{M}{2}$.

Summing over $n^{\prime}\in\Lambda$ in (3.8) and noticing (3.5) yields

[TABLE]

Now the lemma follows from Lemma 3.1.

∎

Theorem 3.3**.**

Let ${\rm diam}(\Lambda)\leq 2N+1$ and ${\rm diam}(\Lambda_{1})\leq N^{\frac{1}{2d}}$. Let $M_{0}\geq(\log N)^{2},\bar{\rho}\in[\frac{\rho}{2},\frac{4\rho}{5}]$. Suppose that for any $n\in\Lambda\backslash\Lambda_{1}$, there exists some $W=W(n)\in\mathcal{E}_{M}$ with $M\geq M_{0}$ such that $n\in W$, ${\rm dist}(n,\Lambda\backslash\Lambda_{1}\backslash W)\geq\frac{M}{2}$, $W\subset\Lambda\backslash\Lambda_{1}$ and

[TABLE]

Suppose that

[TABLE]

Then

[TABLE]

Proof.

As usual, we drop the dependence on $E$ and $x$. Suppose $|n-n^{\prime}|\geq N^{\frac{1}{d}}+1$. Obviously, one of $n$ and $n^{\prime}$ is not in $\Lambda_{1}$. By the self-adjointness of Green’s functions, we can assume $n\notin\Lambda_{1}$.

Applying (3.1) with $\Lambda_{1}=W=W(n)$, one has

[TABLE]

It implies (since $\lambda>1$)

[TABLE]

where the third inequality holds because of $\bar{\rho}\leq\frac{4}{5}\rho$ and $|n-n_{2}|\geq\frac{M}{2}$.

Iterating (3.11) until $|n_{2}-n^{\prime}|\leq N^{\frac{1}{2}}$ (but at most $\frac{2|n-n^{\prime}|}{M_{0}}$ times), we have $|n-n^{\prime}|\geq\frac{N}{10}$,

[TABLE]

∎

Lemma 3.4** (Several variables matrix-valued Cartan estimate).**

Let $T(x)$ be a self-adjoint $N\times N$ matrix function of a parameter $x\in[-\delta,\delta]^{J}$ ($J\in\mathbb{Z}^{+}$) satisfying the following conditions:

• (i)

$T(x)$* is real analytic in $x\in[-\delta,\delta]^{J}$ and has a holomorphic extension to*

[TABLE]

satisfying

[TABLE]

• (ii)

For all $x\in[-\delta,\delta]^{J}$, there is subset $V\subset[1,N]$ with

[TABLE]

and

[TABLE]

• (iii)

[TABLE]

Let

[TABLE]

Then

[TABLE]

where $C=C(J,B_{1}),c=c(J,B_{1})>0$.

Proof.

The proof is similar to that of the case $J=1$ in Chapter 14 of [4] (see also Remark 3 there). We use the higher dimensional Cartan sets techniques of [13]. For convenience, we give the details in the Appendix. ∎

Theorem 3.5**.**

Under the assumptions of Theorem 2.7, let $\omega\in\Omega_{N_{2}}\cap\Omega_{N_{1}}$. We assume for some $x=(x^{j},x_{j}^{\neg})\in\mathbb{T}^{b}$, there exist $N\in[\frac{1}{4}N_{3}^{c_{3}},N_{3}^{c_{4}}]$ and $\bar{\Lambda}\subset\Lambda\in\mathcal{E}_{N}$ with ${\rm diam}(\bar{\Lambda})\leq 10N^{\frac{1}{10d}}$ such that, for any $k\in\Lambda\backslash\bar{\Lambda}$, there exists some $k\in W\in\mathcal{E}_{N_{1}}$,$W\subset\Lambda\backslash\bar{\Lambda}$ such that ${\rm dist}(k,\Lambda\backslash\bar{\Lambda}\backslash W)\geq\frac{N_{1}}{2},$ and $x+k\omega\mod\mathbb{Z}^{b}\notin X_{N_{1}}$. Let

[TABLE]

Then

[TABLE]

Proof.

Without loss of generality, we assume $j=1$. Fix $x^{1}\in\mathbb{T}^{b_{1}}$ and $x_{1}^{\neg}\in\mathbb{T}^{b-b_{1}}$.

Let $\mathcal{D}$ be the $e^{-\rho N_{1}}$ neighbourhood of $x^{1}$ in the complex plane, i,e.,

[TABLE]

By the assumption of Theorem 3.5, one has for all $k\in\Lambda\backslash\bar{\Lambda}$ and $Q_{N_{1}}\in\mathcal{E}_{N_{1}}^{0}$,

[TABLE]

By standard perturbation arguments222See e.g. the proof of Theorem 4.3., (3.18) and (3.19), we have for any $y\in\mathcal{D}$, $Q_{N_{1}}\in\mathcal{E}_{N_{1}}^{0}$, and $k\in\Lambda\backslash\bar{\Lambda}$,

[TABLE]

Substituting $\Lambda$ with $\Lambda\backslash\bar{\Lambda}$ in Lemma 3.2, one has for any $y\in\mathcal{D}$,

[TABLE]

We want to use Lemma 3.4. For this purpose, let

[TABLE]

Now we are in the position to check the assumptions of Lemma 3.4. Obviously, $B_{1}=O(1)$ since $\lambda>1$ and $E$ is bounded.

Let $V=\bar{\Lambda}$. By (3.22), one has

[TABLE]

By the fact that the Green’s functions satisfy property P and (2.7), one has that both (2.5) and (2.6) hold at scale $N_{2}$ for all $y$ except a set of $y\in\mathbb{T}^{b_{1}}$ with measure less than $e^{-{N_{2}^{c_{1}}}}$. It implies both (2.5) and (2.6) holds at scale $N_{2}$ for all $x+k\omega$ with $|k|\leq N_{3}$ except a set of measure less than $(2N_{3}+1)^{d}e^{-{N_{2}^{c_{1}}}}$.

Applying Lemma 3.2 with $M_{0}=M_{1}=N_{2}$ and (2.1), one has

[TABLE]

except on a set of $y\in\mathbb{T}^{b_{1}}$ with measure less than $(2N_{3}+1)^{d}e^{-{N_{2}^{c_{1}}}}$.

Since $N_{2}=N_{1}^{\frac{2}{c_{1}}}$, direct computation shows that

[TABLE]

This verifies (iii) in Lemma 3.4.

For $\epsilon=e^{-\sqrt{N}}$, by (3.23), one has

[TABLE]

By (3.16) of Lemma 3.4,

[TABLE]

∎

Theorem 3.6**.**

Let $c_{1},c_{2},c_{3},c_{4},N_{1},N_{2},N_{3},\Omega_{3}$ be given by Theorem 2.7, so in particular, Green’s functions satisfy property P at $N_{1},N_{2}$ with parameters $(c_{1},\bar{\rho})$. Then for all $N_{3}\leq N\leq N_{3}^{2}$, Green’s functions satisfy property P at size ${N}$ with parameters $(c_{1},\bar{\rho}-\frac{O(1)}{N_{1}^{1/2}})$ and $\Omega_{N}=\Omega_{3}\cap\Omega_{N_{2}}$, where $O(1)$ only depends on $d$.

Proof.

We fix ${N}\in[{N}_{3},{N}_{3}^{2}]$ and $Q_{N}\in\mathcal{E}_{N}^{0}$. Let $\omega\in\Omega_{N_{3}}$.

For any $n\in Q_{N}$, replacing $x$ with $x+n\omega$ in Theorem 2.7, there exits ${N}_{3}^{c_{3}}<\bar{N}<{N}_{3}^{c_{4}}$ such that, for all $k\in(n+\Lambda)\backslash(n+\bar{\Lambda})$, $x+k\omega\mod\mathbb{Z}^{b}\notin X_{N_{1}}$, where

[TABLE]

and $n+\Lambda$, $n+\bar{\Lambda}$ are the shift of $\Lambda$ and $\bar{\Lambda}$ by $n$.

We are going to possibly shrink the $n+\Lambda$ a little bit so that it is in $Q_{N}$. More precisely, we claim that for any $n\in Q_{N}$, there exist

[TABLE]

$\Lambda_{\rm new}\in\mathcal{E}_{\tilde{N}}$ and $\bar{\Lambda}_{\rm new}$, such that

[TABLE]

[TABLE]

and

[TABLE]

Also for any $k\in\Lambda_{\rm new}\backslash\bar{\Lambda}_{\rm new}$, there exists some $\mathcal{E}_{N_{1}}\ni W\subset\Lambda_{\rm new}\backslash\bar{\Lambda}_{\rm new}$ such that

[TABLE]

We split the proof into three cases.

Case 1: $n+\Lambda\subset Q_{N}$. In this case, let $\Lambda_{\rm new}=n+\Lambda$ and $\bar{\Lambda}_{\rm new}=n+\bar{\Lambda}$. See Case 1 of Fig.1.

Case 2: $(n+\Lambda)\cap(\mathbb{Z}^{d}\backslash Q_{N})$ is non-empty and ${\rm dist}(n+\bar{\Lambda},\partial Q_{N})\geq 2N_{1}$. See Case 2 of Fig.1. In this case, let $\bar{\Lambda}_{\rm new}=n+\bar{\Lambda}$ (the red square). By shrinking $n+\Lambda$ a little bit, we can obtain proper ${\Lambda}_{\rm new}\subset(n+\Lambda)\cap Q_{N}$ satisfying (3.28). Since ${\rm dist}(n+\bar{\Lambda},\partial Q_{N})\geq 2N_{1}$, we can also guarantee (3.30) holds.

Case 1****Case 2$\Lambda_{\rm new}$Fig.1

Case 3: $(n+\Lambda)\cap(\mathbb{Z}^{d}\backslash Q_{N})$ is non-empty and ${\rm dist}(n+\bar{\Lambda},\partial Q_{N})\leq 2N_{1}$. In this case, making $(n+\bar{\Lambda})\cap Q_{N}$ possibly larger, we obtain $\bar{\Lambda}_{\rm new}\subset Q_{N}$. We can also make sure for any $k\in Q_{N}\backslash\bar{\Lambda}_{\rm new}$, there exists some $W\in\mathcal{E}_{N_{1}}\subset Q_{N}\backslash\bar{\Lambda}_{\rm new}$

[TABLE]

See Fig.2. For B, $\bar{\Lambda}_{\rm new}=(n+\bar{\Lambda})\cap Q_{N}$ (the red part). For A and C, $(n+\bar{\Lambda})\cap Q_{N}$ is the red part, and $\bar{\Lambda}_{\rm new}$ is union of the red part and the blue part. Shrinking $n+\Lambda$, we can obtain proper ${\Lambda}_{\mathrm{new}}$ satisfying (3.28). This implies (3.30) by (3.31).

B$\Lambda_{\rm new}$C$\Lambda_{\rm new}$A$\Lambda_{\rm new}$Fig.2: Case 3

Fix $x_{j}^{\neg}$. Divide $\mathbb{T}^{b_{j}}$ into $e^{2b_{j}\rho N_{1}}$ cubes of size $e^{-\rho N_{1}}$.

Applying Theorem 3.5 in each cube, ((3.27), (3.29) and (3.30) ensure we can use Theorem 3.5), there exists a set $Y_{\tilde{N}}(x_{j}^{\neg})$ such that

[TABLE]

and for $x=(x^{j},x_{j}^{\neg})$ with $x^{j}\notin Y_{\tilde{N}}(x_{j}^{\neg})$,

[TABLE]

Setting $M_{0}=N_{1}$, $\Lambda=\Lambda_{\rm new}$ and $\Lambda_{1}=\bar{\Lambda}_{\rm new}$ in Theorem 3.3 ((2.5), (2.6), (3.27), (3.29), (3.30) and (3.33) ensure we can use Theorem 3.3), we have for such $x$,

[TABLE]

Let

[TABLE]

By (3.32), (3.35) and since $c_{1}=c_{3}/4\tilde{b}$, one has for any $j$ and $x_{j}^{\neg}\in\mathbb{T}^{b-b_{j}}$,

[TABLE]

Suppose $x^{j}\notin{B}_{{N}}(x_{j}^{\neg})$. Applying $\Lambda=Q_{N}$, $M_{0}=\frac{1}{4}N_{3}^{c_{3}}$ and $M_{1}=N_{3}^{c_{4}}$ in Lemma 3.2 since $N\in[N_{3},N_{3}^{2}]$ ((3.26), (3.28), (3.33) and (3.34) ensure the assumption of Lemma 3.2), one has

[TABLE]

Applying $\Lambda=Q_{N}\in\mathcal{E}_{N}^{0}$, $M_{0}=\frac{1}{4}N_{3}^{c_{3}}$ and $\Lambda_{1}=\emptyset$ in Theorem 3.3, by (3.33), (3.34) and (3.37), we have

[TABLE]

Let

[TABLE]

The theorem follows from (3.38), (3.37) and (3.36). ∎

4. Large deviation theorem for Green’s functions and proof of Theorem 1.1

The main result of this section is the following large deviation theorem (LDT) for Green’s functions.

Theorem 4.1** (LDT).**

There exist constants $\gamma=\gamma(b,d)\in(0,1)$, $N_{0}=N_{0}(v,\rho,b,d)$ and $\lambda_{0}=\lambda_{0}(v,\rho,b,d)$, such that for all $N\geq N_{0}$ and $\lambda\geq\lambda_{0}$, the Green’s functions satisfy property P with parameters $(\gamma,\frac{\rho}{2})$ at size $N$, and the corresponding semi-algebraic set $\Omega_{N}$ satisfying

[TABLE]

*as $\lambda\to\infty$. *

Compactness arguments and Theorem 8 in [20] immediately imply

Lemma 4.2** (Łojasiewicz type Lemma).**

For $E\in\mathbb{R},\delta>0$, define

[TABLE]

Then there are constants $C(v),a(v)>0$ such that

[TABLE]

Theorem 4.3**.**

Let ${X}$ be as in Lemma 4.2 and

[TABLE]

Then we have

[TABLE]

Moreover, if

[TABLE]

then for any $x\notin X_{N},\ \omega\in\mathbb{T}^{b}$, we have for $Q_{N}\in\mathcal{E}_{N}^{0}$,

[TABLE]

Proof.

The bound (4.3) follows from Lemma 4.2 immediately.

Let $x\notin X_{N}$ and fix $Q_{N}\in\mathcal{E}_{N}^{0}$. Let $A=\lambda^{-1}R_{Q_{N}}SR_{Q_{N}}$, with the kinetic term $S$ being given by (1.1). Let $B$ be diagonal part of the restriction of $\lambda^{-1}H-E$ on $Q_{N}$, namely,

[TABLE]

By (4.2), one has

[TABLE]

It leads to

[TABLE]

Since $|S(n,n^{\prime})|\leq e^{-\rho|n-n^{\prime}|}$ for all $n,n^{\prime}$, by Lemma 3.1 again, one has for $N\geq N(\rho,d)$,

[TABLE]

By (4.4),

[TABLE]

Combining with (4.7) and (4.8), we have the following Neumann series expansion

[TABLE]

Thus one has

[TABLE]

It implies (4.5). In particular, (4.6) is also true for $n=n^{\prime}$.

For $n\neq n^{\prime}$, by (1.1), (4.9) and the fact that $B$ is diagonal, we have

[TABLE]

where the last inequality holds by (4.4).

∎

Proof of Theorem 4.1.

Denote the relation between $N_{1}$ and ${N}_{3}$ in Theorem 2.7 by $f$, i.e., $f(x)=e^{x^{c_{1}}}$. Let $N_{0}=N_{0}(v,\rho,b,d)$ be sufficiently large. Denote by $f^{(n)}(x)$ the $n$th iteration of $f$, namely, $f^{(n)}(x)=f(f(f(\cdots x\cdots)))$. Let $g(x)=f^{2}(x)$. Clearly, $g(x)\geq f(x+1)$ for large $x$.

By letting $\delta=\frac{1}{2}e^{-\bar{N}^{1/2}}$ and Theorem 4.3, since $c_{1}<1/2$, the Green’s functions satisfy property P with parameters $(c_{1},\frac{4\rho}{5})$ for $N_{0}\leq N\leq\bar{N}$ and $\Omega_{N}=\mathbb{T}^{b}$ if $\lambda\geq 4e^{\bar{N}^{1/2}}(2\bar{N}+1)^{d}$.

Theorem 3.6 allows us to proceed from scales $N$, $N^{\frac{2}{c_{1}}}$ to scales $[f(N),g(N)]$. Since we want to cover all scales, our initial step will consist of property P at the interval of scales $[N_{1},f(N_{1})]$. For this reason, we need to take $N_{1}=\log\log\lambda$.

Initial step: For large $\lambda$, the Green’s functions satisfy property P with parameters $(c_{1},\rho_{0})$ for all $N_{0}\leq N\leq g(\log\log\lambda)$ and $\Omega_{N}=\mathbb{T}^{b}$, where $\rho_{0}=\frac{4\rho}{5}$.

Let

[TABLE]

Applying Theorem 3.6 to $N_{1}=\log\log\lambda,\log\log\lambda+1,\log\log\lambda+2,\cdots,f(\log\log\lambda)$, the Green’s functions satisfy property P with parameters $(c_{1},\rho_{1})$ for all $g(\log\log\lambda)\leq N\leq g(f(\log\log\lambda))$ since $g(x)\geq f(x+1)$. Moreover,

[TABLE]

Applying Theorem 3.6 to $N_{1}=f(\log\log\lambda),f(\log\log\lambda)+1,f(\log\log\lambda)+2,\cdots,f^{(2)}(\log\log\lambda)$, the Green’s functions satisfy property P with parameters $(c_{1},\rho_{2})$ for all $g(f(\log\log\lambda))\leq N\leq g(f^{(2)}(\log\log\lambda))$. Moreover,

[TABLE]

By induction, we have the Green’s functions satisfy property P with parameters $(c_{1},\rho_{i})$ for $g(f^{(i-1)}(\log\log\lambda))\leq N\leq g(f^{(i)}(\log\log\lambda))$, $i=1,2,\cdots$. Moreover,

[TABLE]

Now Theorem 4.1 follows from (4.11) and (4.13). ∎

Proof of Theorem 1.1.

With Theorem 4.1 at hand, the proof Theorem 1.1 is rather standard. We refer the readers to [4, Section 3] or [7, Section 6] for details. ∎

Appendix A

In the following, we will prove the several variables matrix-valued Cartan estimate, i.e., Lemma 3.4. The proof is similar to that in [4, 3]. Before going to the details, we recall some useful lemmas. The first result is the standard Schur’s complement theorem. For convenience, we include a proof here.

Lemma A.1**.**

Let $T$ be the matrix

[TABLE]

where $T_{1}$ is an invertible $n\times n$ matrix , $T_{2}$ is an $n\times k$ matrix and $T_{3}$ is a $k\times k$ matrix. Let

[TABLE]

Then $T$ is invertible if and only if $S$ is invertible, and

[TABLE]

where $C$ depends only on $\|T_{2}\|$.

Proof.

It is easy to check that

[TABLE]

It implies $T$ is invertible if and only if $S$ is invertible and also the second inequality of (A.1). By (A.2), one has

[TABLE]

implying the first inequality of (A.1). ∎

We then introduce the higher dimensional Cartan sets Lemma of Goldstein-Schlag [13]. We denote by $\mathcal{D}(z,r)$ the standard disk on $\mathbb{C}$ of center $z$ and radius $r>0$.

Lemma A.2**.**

[13, Lemma 2.15]** Let $f(z_{1},\cdots,z_{J})$ be an analytic function defined in a ploydisk $\mathcal{P}=\prod\limits_{1\leq i\leq J}\mathcal{D}(z_{i,0},1/2)$ and $\phi=\log|f|$. Let $\sup\limits_{\underline{z}\in\mathcal{P}}\phi(\underline{z})\leq M,m\leq\phi(\underline{z}_{0})$, $\underline{z}_{0}=(z_{1,0},\cdots,z_{J,0})$. Given $F\gg 1$, there exists a set $\mathcal{B}\subset\mathcal{P}$ such that

[TABLE]

and

[TABLE]

Proof of Lemma 3.4.

The proof is similar to that of Proposition 14.1 in [4] in case $J=1$ and Lemma 1.43 in [3] without explicit bounds. In the following proof, $C=C(B_{1},J)$ and $c=c(B_{1},J)$.

Let

[TABLE]

Fix

[TABLE]

and consider $T(z)$ with $|z-x_{0}|=\sup\limits_{1\leq i\leq J}|z_{i}-x_{0,i}|<\mu$. Thanks to Cauchy’s estimate and (3.12), one obtains for $|z-x_{0}|<\mu$,

[TABLE]

which implies

[TABLE]

From the assumption (ii) of Lemma 3.4, we can find $V=V(x_{0})$ so that $|V|\leq M$ and (3.13) is satisfied. Denote by $V^{c}=[1,N]\setminus V$. Thus using the standard Neumann series argument and (3.13), one has

[TABLE]

We define for $|z-x_{0}|<\mu$ the analytic self-adjoint function

[TABLE]

Then by (A.8) and (A.9), we have

[TABLE]

Recalling Lemma A.1, if $S(z)$ is invertible, so is $T(z)$ and by (A.1),

[TABLE]

For $x\in\mathbb{R}^{J}$, one has

[TABLE]

By (A.10), one has

[TABLE]

Let

[TABLE]

Then by (A.12) and (A.10),

[TABLE]

By (3.14) and the definition of $\mu$, there is some $x_{1}$ with $|x_{0}-x_{1}|<\mu/10$ such that

[TABLE]

Hence by (A.11), $\|S^{-1}(x_{1})\|\leq CB_{3}$, and from (A.12),

[TABLE]

where $a=\frac{x_{1}-x_{0}}{\mu}$, so $|a|<1/10$. Let

[TABLE]

Then one has

[TABLE]

Applying Lemma A.2 and recalling (A.6), (A.7), for any $F\gg 1$, there is some set $\mathcal{B}\subset\prod\limits_{1\leq i\leq J}\mathcal{D}(a_{i},{1}/{4})$ with

[TABLE]

and

[TABLE]

For $0<\epsilon<1$, let

[TABLE]

Then by (A.17) and (A.18),

[TABLE]

Since $|x_{0}-x_{1}|<\mu/10$, we have

[TABLE]

Recalling (A.11), (A.13) and (3.15), one has for $|x-x_{0}|<\mu/8$ and $|\det S(x)|\geq\epsilon$,

[TABLE]

Covering $[-\frac{\delta}{2},\frac{\delta}{2}]^{J}$ by cubes of side $\mu/4$, and combining (A.19) and (A.20), one has

[TABLE]

∎

Acknowledgments

We are grateful to Jean Bourgain for his encouragement. This research was supported by NSF DMS-1401204, DMS-1901462, and DMS-1700314.