2.1 The operator on the lattice

For a vector $\boldsymbol {x}\in \mathbb R^r$ , we denote by $|\boldsymbol {x}|$ (resp. $|\boldsymbol {x}|_2$ ) the supremum norm (resp. the Euclidean norm). For an operator (or a matrix), we denote by $\|\cdot \|$ its operator norm.

We define the lattice:

$$ \begin{align*}\mathbb Z^{b+d}_{\mathrm{pm}}=\mathbb Z^{b+d}\times\{+, -\}.\end{align*} $$

Let $\mathcal {S}_+=\{(\boldsymbol {e}_\ell , \boldsymbol {n}_\ell )_{\ell =1}^b\}\subset \mathbb Z^{b+d}$ and we identify $ \mathcal {S}_+$ with $\{(\boldsymbol {e}_\ell , \boldsymbol {n}_\ell )_{\ell =1}^b\}\times \{+\}\subset \mathbb Z^{b+d}_{\mathrm {pm}}$ . Similarly, let $\mathcal {S}_-=\{(-\boldsymbol {e}_\ell , \boldsymbol {n}_\ell )_{\ell =1}^b\}\subset \mathbb Z^{b+d}$ , which is then identified with $\{(-\boldsymbol {e}_\ell , \boldsymbol {n}_\ell )_{\ell =1}^b\}\times \{-\}\subset \mathbb Z^{b+d}_{\mathrm {pm}}$ . We let $\mathcal S=\mathcal S_+\cup \mathcal S_-$ and

$$ \begin{align*}\mathbb Z^{b+d}_{\mathrm{pm}, *}=\mathbb Z_{\mathrm{pm}}^{b+d}\setminus \mathcal{S}.\end{align*} $$

In the following, we study the operator on $\mathbb Z^{b+d}_{\mathrm {pm},*}$ the operator

(2.1) $$ \begin{align} {H}(\sigma)={D}(\sigma)+\varepsilon (\Delta\oplus\Delta)+\delta S,\ \sigma\in\mathbb R, \end{align} $$

where

$$ \begin{align*} {D}(\sigma)&=\mathrm{diag}_{(\boldsymbol{k},\boldsymbol{n})\in\mathbb Z^{b+d}}\left(\begin{array}{cc} {-\sigma-\boldsymbol{k}\cdot\boldsymbol{\omega}+\mu_{\boldsymbol{n}}} & {0} \\ {0}& {\sigma+\boldsymbol{k}\cdot\boldsymbol{\omega}+\mu_{\boldsymbol{n}}} \\ \end{array} \right),\ \mu_{\boldsymbol{n}}=V(\boldsymbol{n}\boldsymbol{\alpha}+\boldsymbol{\theta})\\ \end{align*} $$

and the operator S refers typically to the linearized operator of the nonlinear perturbation. So we may let S satisfy the following properties:

• ○ First

  $$ \begin{align*}S:\ \ell^2(\mathbb Z^{b+d}_{\mathrm{pm}})\to \ell^2(\mathbb Z^{b+d}_{\mathrm{pm}})\end{align*} $$
  is a bounded self-adjoint operator.

• ○ For all $\boldsymbol {k},\boldsymbol {k}', \boldsymbol {k}''\in \mathbb Z^b$ , $\boldsymbol {n},\boldsymbol {n}'\in \mathbb Z^d$ and $\xi ,\xi '\in \{+, -\}$ , we have the Töplitz property in the $\boldsymbol {k}$ -variable:

  $$ \begin{align*} S((\boldsymbol{k}'+\boldsymbol{k}, \boldsymbol{n}, \xi); (\boldsymbol{k}"+\boldsymbol{k}, \boldsymbol{n}', \xi'))=S((\boldsymbol{k}', \boldsymbol{n}, \xi ); (\boldsymbol{k}", \boldsymbol{n}', \xi')). \end{align*} $$

• ○ We have for some $C_2>0$ and $\gamma \in (\frac 12, 10), $

  $$ \begin{align*} |S((\boldsymbol{k}, \boldsymbol{n}, \xi); (\boldsymbol{k}', \boldsymbol{n}', \xi'))|\leq C_2(1+|\boldsymbol{k}-\boldsymbol{k}'|)^{C_2} e^{-\gamma|\boldsymbol{k}-\boldsymbol{k}'|-\gamma|\boldsymbol{n}|}\delta_{\boldsymbol{n},\boldsymbol{n}'}. \end{align*} $$

The frequency $\boldsymbol {\omega }$ satisfies

(2.2) $$ \begin{align} \boldsymbol{\omega}\in\Omega=\boldsymbol{\omega}^{(0)}+[-C\delta, C\delta]^b, \end{align} $$

where $C>0$ and $\boldsymbol {\omega }^{(0)}=\boldsymbol {\omega }^{(0)}(\boldsymbol {\alpha }, \boldsymbol {\theta })$ is defined in Theorem 1.1.

2.2 The (generalized) elementary regions on the lattice

For some technical reasons, we will introduce the definitions of elementary regions and generalized elementary regions on $\mathbb Z^{b+d}$ originating from [Reference Bourgain, Goldstein and SchlagBGS02, Reference BourgainBou07]. We refer to [Reference LiuLiu22] (cf. Section 2 in [Reference LiuLiu22]) for some important clarifications and improvements on those concepts.

Denote by $\Lambda _N(\boldsymbol {x})$ ( $\boldsymbol {x}\in \mathbb Z^{r},\ r=b+d,d$ ),

$$ \begin{align*}\Lambda_N(\boldsymbol{x})=\{\boldsymbol{y}\in\mathbb Z^{r}:\ |\boldsymbol{y}-\boldsymbol{x}|\leq N\},\end{align*} $$

the cube of radius N centered at $\boldsymbol {x}$ . In particular, we write $\Lambda _N=\Lambda _N(\boldsymbol {0})$ . For $\Lambda \subset \mathbb Z^{r}$ , we define its diameter to be $\mathrm {diam}\ \Lambda =\sup \limits _{\boldsymbol {x}, \boldsymbol {y}\in \Lambda }|\boldsymbol {x}-\boldsymbol {y}|$ . For $\boldsymbol {x}\in \mathbb Z^{r}$ and $\boldsymbol {w}\in \mathbb R_+^{r}$ , let $R_{\boldsymbol {w}}(\boldsymbol {x})=\{\boldsymbol {y}\in \mathbb Z^{r}:\ |y_\ell -x_\ell |\leq w_l , \, 1\leq \ell \leq r\}$ denote the rectangle.

A generalized elementary region is defined to be a set $\Lambda $ of the form

$$ \begin{align*}\Lambda=R_{\boldsymbol{w}}(\boldsymbol{x})\setminus(R_{\boldsymbol{w}}(\boldsymbol{x})+\boldsymbol{z}),\end{align*} $$

where $\boldsymbol {z}\in \mathbb Z^{r}$ is arbitrary. The size of a generalized elementary region is simply its diameter. The set of all generalized elementary regions of size at most M will be denoted by $\mathcal {E}_M$ .

An elementary region $Q_N$ of size N and centered at $\boldsymbol {0}$ is one of the following regions

$$ \begin{align*}Q_N=\Lambda_N\ \mathrm{or}\ Q_N=\Lambda_N\setminus\{\boldsymbol{x}\in\mathbb Z^r:\ x_\ell \square_\ell 0,\ 1\leq\ell\leq r\},\end{align*} $$

where $\square _\ell \in \{<,>, \emptyset \}$ and at least two $\square _\ell $ ’s are not $\emptyset .$ Denote by $\mathcal {ER}_{\boldsymbol {0}}(N)$ the set of all elementary regions of size N and centered at $\boldsymbol {0}$ . Let

$$ \begin{align*}\mathcal {ER}(N)=\{\boldsymbol{x}+Q_N:\ \boldsymbol{x}\in\mathbb Z^r,\ Q_N\in \mathcal{ER}_{\boldsymbol{0}}(N)\}.\end{align*} $$

Let $\mathcal E_{M}^{L}$ denote the set of all M-size generalized elementary regions of width at least $L\leq M$ , namely, $\Lambda \in \mathcal {E}_{M}^{L}$ iff $\Lambda \in \mathcal {E}_{M}$ and, for any $\boldsymbol {x}\in \Lambda , 0<L'<L$ , there is some $\Lambda '\in \mathcal {ER}(L')$ so that $\boldsymbol {x}\in \Lambda '\subset \Lambda $ , $\mathrm {dist}(\boldsymbol {x}, \Lambda \setminus \Lambda ')\geq L'/2$ , where $\mathrm {dist} (\cdot , \cdot )$ is induced by the supremum norm. So we have $\mathcal {ER}(N) \subset \mathcal E_{2N}^{N}$ .

With a slight abuse of notation, we also use $\mathcal {ER}_{\boldsymbol {0}}(N), \mathcal {ER}(N), \mathcal E_N, \mathcal E_N^L , \Lambda _N(\boldsymbol {x}),\ \Lambda _N$ to denote $\mathcal {ER}_{\boldsymbol {0}}(N)\times \{+,-\}, \mathcal {ER}(N)\times \{+,-\}, \mathcal E_N\times \{+,-\}, \mathcal E_N^L \times \{+,-\}, \Lambda _N(\boldsymbol {x})\times \{+,-\}, \Lambda _N\times \{+,-\}$ . Similarly, for any $\Lambda \subset \mathbb Z^r$ , denote by $R_\Lambda $ the restriction to $\Lambda \times \{+,-\}$ .

2.3 Green’s functions and LDE

We now define Green’s functions on subsets of $\mathbb Z_{\mathrm {pm},*}^{b+d}$ . Let $\Lambda \subset \mathbb Z_{\mathrm {pm},*}^{b+d}$ be a nonempty set, $R_\Lambda $ the restriction operator to $\Lambda $ , and define the $(\# \Lambda ) \times (\#\Lambda )$ matrix:

(2.3) $$ \begin{align} (R_\Lambda {H}(\sigma)R_\Lambda)((\boldsymbol{x}, \xi);(\boldsymbol{x}', \xi'))={H}(\sigma)((\boldsymbol{x}, \xi);(\boldsymbol{x}', \xi')), \ (\boldsymbol{x}, \xi), (\boldsymbol{x}', \xi')\in\Lambda, \end{align} $$

where $H(\sigma )$ is defined by (2.1). Define the Green’s function to be (if it exists)

$$ \begin{align*}G_\Lambda(\sigma)=(R_\Lambda {H}(\sigma)R_\Lambda)^{-1}.\end{align*} $$

Before addressing the LDT for Green’s functions, let us first give the definition of large deviation estimates (LDE):

Definition 2.1 (LDE)

Let $\rho>0$ , $0<\gamma '<\gamma $ and $M\in \mathbb {N}$ . We say ${H}(\sigma )$ defined by (2.1) satisfies the $(\rho , \gamma ',M)$ -LDE if there exists a set $\Sigma _M\subset \mathbb R$ with

$$ \begin{align*}\mathrm{meas}(\Sigma_M)\leq e^{-M^{\rho}},\end{align*} $$

so that for $\sigma \notin \Sigma _M$ the following estimates hold: If $\Lambda \in (\boldsymbol {0},\boldsymbol {n})+ \mathcal {ER}_{\boldsymbol {0}}(M)$ satisfies $|\boldsymbol {n}|\leq 10M$ , then

$$ \begin{align*} \|G_{\Lambda}(\sigma)\|&\leq e^{M^{\frac34}},\\ |G_{\Lambda}(\sigma)((\boldsymbol{x}, \xi); (\boldsymbol{x}', \xi'))| &\leq e^{-\gamma'|\boldsymbol{x}-\boldsymbol{x}'|}\ \mathrm{for}\ |\boldsymbol{x}-\boldsymbol{x}'|\geq M^{\frac 89}\ \mathrm{and }\ \xi,\xi'\in\{\pm\}, \end{align*} $$

where $\boldsymbol {x}=(\boldsymbol {k}',\boldsymbol {n}'), \boldsymbol {x}'=(\boldsymbol {k}'',\boldsymbol {n}'')$ .

We will show in this section that the $(\rho , \frac {\gamma }{2},N)$ -LDE hold for some small

$$ \begin{align*}\rho=\rho(b,d)>0\end{align*} $$

and all $N\gg 1$ , under certain nonresonant conditions on $\boldsymbol {\alpha }, \boldsymbol {\theta }$ and $\boldsymbol {\omega }$ . This leads to the LDT, Theorem 2.11, below. Since we view $\boldsymbol {\omega }$ as an $O(\delta )$ perturbation of $\boldsymbol {\omega }^{(0)}$ , the properties of $\boldsymbol {\omega }^{(0)}$ become essential in the proof of LDT. So, in the following, we first establish nonresonant properties (called Diophantine estimates) of $\boldsymbol {\omega }^{(0)}$ . Then the proof of LDT will follow from a multi-scale analysis scheme in the spirit of [Reference Bourgain, Goldstein and SchlagBGS02, Reference BourgainBou05a, Reference BourgainBou07].

2.4 Diophantine estimates

Recall that

$$ \begin{align*} \boldsymbol{\omega}^{(0)}&=\boldsymbol{\omega}^{(0)}(\boldsymbol{\alpha},\boldsymbol{\theta})=(V(\boldsymbol{n}_\ell\boldsymbol{\alpha}+\boldsymbol{\theta}))_{\ell=1}^b,\\ \mu_{\boldsymbol{n}}&=\mu_{\boldsymbol{n}}(\boldsymbol{\alpha},\boldsymbol{\theta})=V(\boldsymbol{n}\boldsymbol{\alpha}+\boldsymbol{\theta}),\ \boldsymbol{n}\in\mathbb Z^d. \end{align*} $$

The main purpose of this section is to obtain lower bounds on

$$ \begin{align*} &|\boldsymbol{k}\cdot\boldsymbol{\omega}^{(0)}|\ \mathrm{for}\ \boldsymbol{k}\in\mathbb Z^b\setminus\{\boldsymbol{0}\},\\ &|\pm\boldsymbol{k}\cdot\boldsymbol{\omega}^{(0)}+\mu_{\boldsymbol{n}}| \ \mathrm{for}\ (\boldsymbol{k},\boldsymbol{n})\in\mathbb Z^{b+d}\backslash {\mathcal S} ,\\ &|\boldsymbol{k}\cdot\boldsymbol{\omega}^{(0)}+\mu_{\boldsymbol{n}}-\mu_{\boldsymbol{n}'}| \ \mathrm{for} \ \boldsymbol{n}\neq \boldsymbol{n}'\in\mathbb Z^d, \end{align*} $$

under certain restrictions on $(\boldsymbol {\alpha }, \boldsymbol {\theta })$ . As mentioned in the Introduction, due to the d-dimensional phase space, such Diophantine-type estimates are highly nontrivial. We use a generalized Wronskian approach and work out the details in the Appendix (cf. Theorem A.2 and Corollary A.4). These estimates can be read independently, and are key to the existence of quasi-periodic solutions to the nonlinear equation.

Applying Corollary A.4 then leads to

Theorem 2.2. There exist some $0<c_1=c_1(b,d,K,V)<\frac {1}{100b}, C_1=C_1(b,d, K, V)>1$ such that, if $0<\varepsilon +\delta \leq \delta _0(b,d,K,V,\max \limits _{1\leq \ell \leq b}|\boldsymbol {n}_\ell |)\ll 1$ , then there is some $\mathcal M \subset [0, 1]^{2d}$ satisfying

(2.4) $$ \begin{align} \mathrm{meas}([0,1]^{2d}\setminus\mathcal M)\leq \log^{-1}\frac{1}{\varepsilon+\delta}, \end{align} $$

so that the following properties hold true for $(\boldsymbol {\alpha }, \boldsymbol {\theta })\in \mathcal M$ and $L_{\varepsilon ,\delta }:=100(\varepsilon +\delta )^{-c_1}$ .

1. (1) For all $\log \frac {1}{\varepsilon +\delta }\leq L\leq L_{\varepsilon ,\delta }$ and all $(\boldsymbol {k},\boldsymbol {n})\in \Lambda _{L}\setminus \mathcal S,$ we have

   (2.5) $$ \begin{align} \min_{\xi=\pm1}|\xi\boldsymbol{k}\cdot\boldsymbol{\omega}^{(0)}+\mu_{\boldsymbol{n}}|>L^{-C_1}. \end{align} $$

2. (2) We have for any $\xi =\pm 1,$

   (2.6) $$ \begin{align} \sup_{\sigma\in\mathbb R}\#\left\{(\boldsymbol{k},\boldsymbol{n})\in \Lambda_{L_{\varepsilon,\delta}}:\ |\xi(\sigma+\boldsymbol{k}\cdot\boldsymbol{\omega}^{(0)})+\mu_{\boldsymbol{n}}|<\frac{(\varepsilon+\delta)^{\frac{1}{8b}}}{4}\right\}\leq b. \end{align} $$

Proof. The proof is similar to that in [Reference Shi and WangSW23]. First, the measure estimate (2.4) is a consequence of Corollary A.4. Next, the inequality (2.5) in (1) follows directly from (iii) of Corollary A.4.

So, it suffices to establish (2.6) of (2). We prove it by the contradiction. Without loss of generality, we consider the case $\xi =+1$ . Assume that there are $b+1$ distinct $\{(\boldsymbol {k}_\ell , \boldsymbol {n}_{\ell }')\}_{1\leq \ell \leq b+1}$ satisfying

(2.7) $$ \begin{align} |\sigma+\boldsymbol{k}_\ell\cdot\boldsymbol{\omega}^{(0)}+\mu_{\boldsymbol{n}_{\ell}'}|<\frac{(\varepsilon+\delta)^{\frac{1}{8b}}}{4}. \end{align} $$

We claim that all $\boldsymbol {k}_{\ell }$ ( $1\leq \ell \leq b+1$ ) are distinct. Actually, if there are $\boldsymbol {k}_{\ell _1}=\boldsymbol {k}_{\ell _2}$ for some $1\leq \ell _1\neq \ell _2\leq b+1$ , then it must be that $\boldsymbol {n}_{\ell _1}'\neq \boldsymbol {n}_{\ell _2}'$ . As a result, we obtain using (2.7) that

$$ \begin{align*} |\mu_{\boldsymbol{n}_{\ell_1}'}-\mu_{\boldsymbol{n}_{\ell_2}'}|&\leq |\sigma+\boldsymbol{k}_{\ell_1}\cdot\boldsymbol{\omega}^{(0)}+\mu_{\boldsymbol{n}_{\ell_1}'}|\\ &\ \ \ +|\sigma+\boldsymbol{k}_{\ell_2}\cdot\boldsymbol{\omega}^{(0)}+\mu_{\boldsymbol{n}_{\ell_2}'}|\\ &\leq \frac{(\varepsilon+\delta)^{\frac{1}{8b}}}{2}, \end{align*} $$

which contradicts (i) of Corollary A.4. Next, we claim that $\{\boldsymbol {n}_{\ell }'\}_{1\leq \ell \leq b+1}\subset \{\boldsymbol {n}_{\ell }\}_{1\leq \ell \leq b}.$ In fact, if $\boldsymbol {n}_{\ell _1}'\notin \{\boldsymbol {n}_{\ell }\}_{1\leq \ell \leq b}$ for some $1\leq \ell _1\leq b+1$ , we can assume that there is $\ell _2\neq \ell _1$ so that $\boldsymbol {n}_{\ell _2}'\neq \boldsymbol {n}_{\ell _1}'$ . Otherwise, we may have

$$ \begin{align*} |(\boldsymbol{k}_{\ell_1}-\boldsymbol{k}_{\ell_2})\cdot\boldsymbol{\omega}^{(0)}|\leq\frac{(\varepsilon+\delta)^{\frac{1}{8b}}}{2}, \end{align*} $$

which contradicts (ii) of Corollary A.4, since $\boldsymbol {k}_{\ell _1}\neq \boldsymbol {k}_{\ell _2}$ as claimed above. So we obtain for some $\ell _2\neq \ell _1$ and $1\leq \ell _2\leq b+1$ with $\boldsymbol {n}_{\ell _2}'\neq \boldsymbol {n}_{\ell _1}'$ that

$$ \begin{align*} |(\boldsymbol{k}_{\ell_1}-\boldsymbol{k}_{\ell_2})\cdot\boldsymbol{\omega}^{(0)}+\mu_{\boldsymbol{n}_{\ell_1}'}-\mu_{\boldsymbol{n}_{\ell_2}'}|\leq\frac{(\varepsilon+\delta)^{\frac{1}{8b}}}{2}, \end{align*} $$

which contradicts (iv) of Corollary A.4. So it suffices to assume $\{\boldsymbol {n}_{\ell }'\}_{1\leq \ell \leq b+1}\subset \{\boldsymbol {n}_{\ell }\}_{1\leq \ell \leq b}.$ However, in this case, we have by the pigeonhole principle that there are $1\leq \ell _1\neq \ell _2\leq b+1$ so that $\boldsymbol {n}_{\ell _1}'=\boldsymbol {n}_{\ell _2}'$ . Then we get

$$ \begin{align*} |(\boldsymbol{k}_{\ell_1}-\boldsymbol{k}_{\ell_2})\cdot \boldsymbol{\omega}^{(0)}|\leq \frac{(\varepsilon+\delta)^{\frac{1}{8b}}}{2}, \end{align*} $$

which contradicts (ii) of Corollary A.4 as shown above. This proves (2.6).

2.5 Bourgain’s geometric lemma

To proceed with the analysis, we will also need Anderson localization-type properties for the linear quasi-periodic Schrödinger operator $\mathcal H$ in (1.3), namely,

$$ \begin{align*}\mathcal H=\mathcal H(\boldsymbol{\alpha}, \boldsymbol{\theta})=\varepsilon\Delta+V(\boldsymbol{\theta}+\boldsymbol{n}\boldsymbol{\alpha})\delta_{\boldsymbol{n}, \boldsymbol{n}'}.\end{align*} $$

This was proven by Bourgain in the remarkable work [Reference BourgainBou07], where he established Anderson localization for general quasi-periodic Schrödinger operators on $\mathbb Z^d$ with nondegenerate analytic potentials on $\mathbb {T}^d$ (identified with $[0,1]^d$ ). This result was later significantly extended by Jitomirskaya-Liu-Shi [Reference Jitomirskaya, Liu and ShiJLS20] (cf. also [Reference ShiShi22, Reference LiuLiu22]) to potentials on $\mathbb {T}^b$ for $b\geq d.$

In the present paper, due to the d-dimensional phase space, we need to apply Bourgain’s arguments. Using short-range property of S in the $\boldsymbol {n}$ -variable, Bourgain’s geometric lemma is made available to handle LDE for Green’s functions at large scales. This is a main new aspect not present in [Reference Shi and WangSW23]. The general scheme is similar to that in Chapter 19 of [Reference BourgainBou05a], where Bourgain made a great breakthrough and first proved the existence of quasi-periodic solutions for the nonlinear Schrödinger equation on arbitrarily dimensional torus (a KAM theorem was later established by Eliasson-Kuksin in an important paper [Reference Eliasson and KuksinEK10]). Among others, Bourgain [Reference BourgainBou05a] employed heavily the separation property of eigenvalues of the Laplace operator on the higher dimensional torus, while the separation property is missing in the present setting (indeed, the spectrum of $\mathcal H$ may contain an interval).

We start with introducing the definition of semi-algebraic sets.

Definition 2.4. A set $X\subset \mathbb {R}^r$ is called semi-algebraic if it is a finite union of sets defined by a finite number of polynomial equalities and inequalities. More precisely, let $\{P_1,\cdots ,P_k\}\subset \mathbb {R}[x_1,\cdots ,x_r]$ be a family of real polynomials whose degrees are bounded by p. A (closed) semi-algebraic set ${X}$ is given by an expression

(2.8) $$ \begin{align} {X}=\bigcup\limits_{s}\bigcap\limits_{\ell\in\mathcal{K}_s}\left\{\boldsymbol{x}\in\mathbb{R}^r: \ P_{\ell}(\boldsymbol{x})\varsigma_{s\ell}0\right\}, \end{align} $$

where $\mathcal {K}_s\subset \{1,\cdots ,k\}$ and $\varsigma _{s\ell }\in \{\geq ,\leq ,=\}$ . Then we say that ${X}$ has degree at most $kp$ . In fact, the degree of ${X}$ which is denoted by $\deg {X}$ , means the smallest $kp$ over all representations as in (2.8).

Denote by $\mathcal {ER}_{\mathbb Z^d}(M)$ (resp. $\mathcal {ER}_{\mathbb Z^d, \boldsymbol {0}}(M)$ ) the set of all M-size elementary regions on $\mathbb Z^d$ (resp. centered at $\boldsymbol {0}$ ). We have

Lemma 2.5 (cf. the Claim (page 694) in [Reference BourgainBou07] and Theorem 2.7 in [Reference Jitomirskaya, Liu and ShiJLS20])

There are small constants $0<\kappa _1<\kappa _2<1$ depending only on d so that the following holds true. For $0<\varepsilon \leq \varepsilon _0(V,d)\ll 1 $ and $N\geq N_1:=10^{-\frac {32}{\kappa _1^2}}\log ^{\frac {4}{\kappa _1}}(\log \frac 1\varepsilon )$ , there is a semi-algebraic set $\mathcal {A}_N\subset [0,1]^d$ satisfying

$$ \begin{align*}\deg \mathcal A_N\leq N^{4d},\ \mathrm{meas}(\bigcap_{N\geq N_1}\mathcal{A}_N)=1-O(\log^{-c}\frac1\varepsilon),\ c=c(d)>0,\end{align*} $$

so that the following properties hold true for $\boldsymbol {\alpha }\in \mathcal {A}_N.$

1. (1) For all $E\in \mathbb R$ and all $\boldsymbol {\theta }\in [0,1]^d$ , there is $L\in [N^{\kappa _1}, N^{\kappa _2}]$ so that, for all $Q\in \mathcal {ER}_{\mathbb Z^d}(L_1) $ satisfying $L_1\sim (\log N)^{\frac {4}{\kappa _1}}$ and

   $$ \begin{align*}Q\subset [-L,L]^d\setminus[-L^{\frac{1}{10d}}, L^{\frac{1}{10d}}]^d,\end{align*} $$
   one has
   $$ \begin{align*} \|\mathcal{H}^{-1}_{Q}(E;\boldsymbol{\theta})\|&\leq e^{\sqrt{L_1}},\\ |\mathcal{H}^{-1}_{Q}(E;\boldsymbol{\theta})(\boldsymbol{n}; \boldsymbol{n}')|&\leq e^{-\frac12|\log\varepsilon|\cdot|\boldsymbol{n}-\boldsymbol{n}'|}\ \mathrm{for}\ |\boldsymbol{n}-\boldsymbol{n}'|\geq L_1^{\frac 89}, \end{align*} $$
   where
   $$ \begin{align*} \mathcal{H}_{Q}(E;\boldsymbol{\theta})&=R_{Q}\left(\varepsilon\Delta+V(\boldsymbol{\theta}+\boldsymbol{n}\boldsymbol{\alpha})\delta_{\boldsymbol{n}, \boldsymbol{n}'}\right)R_{Q}-E\\ &:=\mathcal{L}_{Q}(\boldsymbol{\theta})-E. \end{align*} $$

2. (2) There is some $\Theta _N=\Theta _N(\boldsymbol {\alpha })\subset [0,1]^d$ satisfying

   $$ \begin{align*}\mathrm{meas}([0, 1]^d\setminus\Theta_N)\leq e^{-N^{\frac{\kappa_1}{4}}}\end{align*} $$
   so that, if $\boldsymbol {\theta }\in \Theta _N$ , then one has for all $\Lambda \in \mathcal {ER}_{\mathbb Z^d, \boldsymbol {0}}(N),$
   $$ \begin{align*} \|\mathcal{H}^{-1}_{\Lambda}(\boldsymbol{\alpha}, \boldsymbol{\theta})\|&\leq e^{\sqrt{N}},\\ |\mathcal{H}^{-1}_{\Lambda}(\boldsymbol{\alpha}, \boldsymbol{\theta})(\boldsymbol{n}; \boldsymbol{n}')|&\leq e^{-\frac12|\log\varepsilon|\cdot|\boldsymbol{n}-\boldsymbol{n}'|}\ \mathrm{for}\ |\boldsymbol{n}-\boldsymbol{n}'|\geq {N^{\frac 89}}. \end{align*} $$

   Denote

   (2.9) $$ \begin{align} \nonumber&\mathcal{A}=\bigcap_{N\geq N_1}\mathcal{A}_N,\ \tilde\Theta_N(\boldsymbol{\alpha})=\bigcap_{|\boldsymbol{n}|\leq e^{N^{\frac{\kappa_1}{5}}}}\left\{\boldsymbol{\theta}:\ \boldsymbol{\theta}+\boldsymbol{n}\boldsymbol{\alpha}\in\Theta_N\right\}, \\ &\mathcal W'=\left\{(\boldsymbol{\alpha}, \boldsymbol{\theta})\in[0,1]^{2d}:\ \boldsymbol{\alpha}\in\mathcal A,\ \boldsymbol{\theta}\in\bigcap_{N\geq N_1}\tilde\Theta_N(\boldsymbol{\alpha})\right\}. \end{align} $$

   Then

   (2.10) $$ \begin{align} \mathrm{meas}(\mathcal W')\geq 1-\log^{-c}\frac1\varepsilon. \end{align} $$

Remark 2.8. We mention that the set $\mathcal W$ on which Theorem 1.1 holds satisfies

(2.11) $$ \begin{align} \mathcal W=\mathcal W'\cap\mathcal M, \end{align} $$

where $\mathcal M$ is the set on which the Diophantine estimates in Theorem 2.2 hold, and $\mathcal W'$ is defined by (2.9). So, we have

$$ \begin{align*}\mathrm{meas}(\mathcal W)\geq 1-\log^{-c}\frac{1}{\varepsilon+\delta},\ c=c(d)>0\end{align*} $$

as shown in Theorem 1.1. For $(\boldsymbol {\alpha }, \boldsymbol {\theta })\in \mathcal W$ , the conclusion in (2) plays an essential role in the nonlinear analysis.

Proof. It suffices to establish the measure bounds. The proof is a small modification of that of Theorem 4.1 in [Reference Jitomirskaya, Liu and ShiJLS20] (cf. pages 475–476). Indeed, the only difference is that we will apply Theorem 3.7 in [Reference Jitomirskaya, Liu and ShiJLS20] starting from $N_1=10^{-\frac {32}{\kappa _1^2}}\log ^{\frac {4}{\kappa _1}}(\log \frac 1\varepsilon )$ rather than $\log \log \frac 1\varepsilon $ as in [Reference Jitomirskaya, Liu and ShiJLS20]. This change makes sense since the large deviation estimates in [Reference Jitomirskaya, Liu and ShiJLS20] hold for the initial scales of $N_0(V,d)\leq N\leq N_2:=\frac {1}{10}\log ^2\frac 1 \varepsilon $ (via the Neumann series argument), and for $f(x)=e^{x^{\frac {\kappa _1}{4}}},$

$$ \begin{align*}[N_1, (f(N_1))^{\frac {8}{\kappa_1}}]\subset [N_0, N_2].\end{align*} $$

This modification then leads to the following measure estimate:

$$ \begin{align*} \mathrm{meas}(\bigcap_{N\geq N_1}\mathcal{A}_N)& \geq 1-\sum_{N\geq N_1} f(N)^{-\kappa_1}\\ &\geq 1-\sum_{N\geq N_1} e^{-\kappa_1N^{\frac{\kappa_1}{4}}}\\ &\geq 1- C(d)e^{-\frac{\kappa_1}{2}N_1^{\frac{\kappa_1}{4}}}\\ &\geq 1-C(d) \log^{-\frac{\kappa_1}{2}\times10^{-\frac{8}{\kappa_1}}} \frac1\varepsilon. \end{align*} $$

For the proof of (2.10), it follows directly from applying Fubini’s theorem and

$$ \begin{align*}\mathrm{meas}(\bigcap_{N\geq N_1}\tilde\Theta_N(\boldsymbol{\alpha}))\geq 1-C(d) \log^{-\frac{1}{5}\times10^{-\frac{8}{\kappa_1}}} \frac1\varepsilon.\\[-46pt]\end{align*} $$

2.6 Large deviation theorem

In this section, we establish that the LDE (cf. Definition 2.1) hold for all sufficiently large scales by a multi-scale analysis. From (2.2), however, the variation of the frequency, $\boldsymbol {\omega }$ is only $O(\delta )$ and not $O(1)$ , hence as we will see shortly, the proof of LDT will be accomplished in three steps instead of the usual two:

• ○ The first step deals with the small scales, for which we establish LDE for all $\boldsymbol {\omega }\in \Omega $ and scales

  $$ \begin{align*}N_0\leq N \leq 10^{-\frac{1}{\rho}}\log^{\frac{1}{\rho}} \frac{1}{\varepsilon+\delta},\end{align*} $$
  for some $\rho>0.$ In this step, we only use the Neumann series argument, which leads to the above scales (cf. Lemma 2.15).

• ○ Next, we establish intermediate scales LDE, that is, scales in the range

  $$ \begin{align*}10^{-\frac{1}{\rho}}\log^{\frac{1}{\rho}} \frac{1}{\varepsilon+\delta}\leq N\leq (\varepsilon+\delta)^{-c_1}\end{align*} $$
  for all $\boldsymbol {\omega }\in \Omega $ , where $c_1$ is defined in Theorem 2.2. Note that Theorem 2.2 is available in the above interval of scales. The proof of such intermediate scales LDE is based on preparation type theorem together with clustering properties of the spectrum of the diagonal matrix $D(0)$ (cf. (2.6), Theorem 2.2).

  Remark 2.9. The intermediate scales are needed here because for a fixed $(\boldsymbol {\alpha },\boldsymbol {\theta })$ , hence fixed $\boldsymbol {\omega }^{(0)}$ , $\boldsymbol {\omega }$ only varies in an interval of size $O(\delta )$ , see (2.2), as mentioned above. Consequently, there is a gap in the range of scales before Bourgain-type analysis becomes applicable. Fortunately, the Diophantine estimates for $(\boldsymbol {\alpha },\boldsymbol {\theta })\in \mathcal M$ can control the resonances directly as in (2.6), leading to LDE for the range of scales in the gap, which we term the intermediate scales.

• ○ For the large scales (i.e., $N>(\varepsilon +\delta )^{-c_1}$ ) LDE, we will apply matrix-valued Cartan’s lemma and semi-algebraic sets theory of [Reference Bourgain, Goldstein and SchlagBGS02, Reference BourgainBou07]. Of particular importance is Bourgain’s geometric lemma (cf. Lemma 2.5). This step requires both the Diophantine condition and the so-called weak second Melnikov’s condition on $\boldsymbol {\omega }.$

  To be coherent, since the variation in $\boldsymbol {\omega }$ is only $O(\delta )$ , the excised measure in $\Omega $ needs to be less than $\delta ^b$ . So we make a particular choice of the exponent and impose the following Diophantine condition:

  (2.12) $$ \begin{align} \nonumber&\ \ \ \mathrm{DC}_{\boldsymbol{\omega}}(N)\\ &=\left\{\boldsymbol{\omega}\in\Omega:\ |\boldsymbol{k}\cdot\boldsymbol{\omega}|\geq e^{-N^{\rho^4}} \ \mathrm{for}\ \forall\ 0<|\boldsymbol{k}|\leq 100N^2\right\}. \end{align} $$

  Then we have for $\varepsilon \leq \delta $ ,

  (2.13) $$ \begin{align} \mathrm{meas}\ \left(\Omega\setminus\left(\bigcap_{N>(\varepsilon+\delta)^{-c_1} }\mathrm{DC}_{\boldsymbol{\omega}}(N)\right)\right) \ll\delta^b. \end{align} $$

  The weak second Melnikov’s condition is imposed in order to restrict the time $\boldsymbol {k}$ -direction where resonances could possibly occur. This greatly reduces the number of resonances, and paves the way for the application of Lemma 2.25.

  Remark 2.10. The condition on the size of $\varepsilon $ relative to $\delta $ stems again from the modulation in $\boldsymbol {\omega }$ being $O(\delta )$ , while the excised measure depends on both $\varepsilon $ and $\delta $ , as we will see in the proofs. This also ensures, in addition, that the amplitude-frequency map: $\boldsymbol {a}\to \boldsymbol {\omega }(\boldsymbol {a})$ is nondegenerate in the nonlinear analysis, so that we may indeed vary $\boldsymbol {\omega }$ in the linear analysis.

Before stating the LDT, let us summarize various parameters and the relation between $\varepsilon , \delta $ .

• ○ Throughout this paper, we first fix constants

  $$ \begin{align*}\kappa_1=\kappa_1(d), \kappa_2=\kappa_2(d)\in(0,1)\end{align*} $$
  given in Bourgain’s geometric lemma (cf. Lemma 2.5).

• ○ The most important parameter $\rho>0$ is then fixed so that it depends only on $b,d$ and satisfies

  (2.14) $$ \begin{align} \rho=c_2 \kappa_1\leq \frac{\kappa_1}{10^4(b+d)^2}, \end{align} $$
  where $c_2=c_2(b,d)$ is given by Lemma 2.22.

• ○ The parameters

  $$ \begin{align*}0<c_1=c_1(b,d, K, V)<\frac{1}{100b}, \ C_1=C_1(b,d,K,V)>1\end{align*} $$
  are fixed, which appear in the Diophantine estimates (cf. Theorem 2.2). Also, the parameter $c_1$ is the power-law exponent of the upper bound of intermediate scales.

• ○ Fix

  $$ \begin{align*}\kappa=\frac12\min\left\{\vartheta, \ \frac{1}{10}\right\}\leq \frac{1}{20},\end{align*} $$
  where $\vartheta>0$ is an absolute constant appearing in the coupling lemma of [Reference LiuLiu22] (cf. Lemma B.3). Then $\kappa $ (cf. (2.35)) describes the changes of off-diagonal exponential decay rates of Green’s functions along the multi-scale iterations.

• ○ For convenience, we assume the decay rate of S satisfies $\gamma \in (\frac 12, 10)$ . It turns out that the $(\rho , \gamma _{\infty }, N)$ -LDE hold true for all $N\geq \log ^{3}\frac {1}{\varepsilon +\delta }$ and for some $\gamma _\infty \in [\frac \gamma 2, \gamma ].$ The constant $C_2>0$ is the power-law index in $S.$

• ○ With these parameters having been fixed, we aim to find

  $$ \begin{align*}\delta_0=\delta_0(b, d, K, V, \max\limits_{1\leq \ell\leq b}|\boldsymbol{n}_\ell|, \kappa_1, \kappa_2, \rho, c_1, C_2)>0,\end{align*} $$
  so that, if $0<\varepsilon +\delta \leq \delta _0$ , then LDT can be established. Since we will use the multi-scale analysis method to prove LDT, various conditions on $\delta _0$ are required, and it is unclear a priori if such restrictions are meaningful. Fortunately, we will divide the LDT proof into 3 steps, and at each step, we can find a desired $\delta _\ell $ ( $\ell =1,2,3$ ). Finally, it suffices to take $\delta _0=\min \{\delta _1, \delta _2, \delta _3\}.$

• ○ As we will see later, we further assume $\delta \leq \log ^{-1}\frac 1\varepsilon $ when proving large scales LDE.

Let $\rho , \kappa _1, \kappa _2, c_1, \gamma , C_2$ be fixed as above. We have

Theorem 2.11 (LDT)

There is some

$$ \begin{align*}\delta_0=\delta_0(b, d, K, V, \max\limits_{1\leq \ell\leq b}|\boldsymbol{n}_\ell|, \kappa_1, \kappa_2, \rho, c_1, C_2)>0\end{align*} $$

so that the following holds true for $0<\varepsilon +\delta \leq \delta _0.$

1. (1) For $\log ^{3}\frac {1}{\varepsilon +\delta }< N\leq 10^{-\frac 1\rho }\log ^{\frac {1}{\rho }}\frac {1}{\varepsilon +\delta },$ the $(\rho , \gamma _N, N)$ -LDE hold true for all $(\boldsymbol {\alpha }, \boldsymbol {\theta })\in [0,1]^{2d}$ (and thus for all $\boldsymbol {\omega }\in \Omega $ ) with $\gamma _N\equiv \gamma _0=\gamma -\log ^{-2}\frac {1}{\varepsilon +\delta }\in [\frac \gamma 2, \gamma ].$

2. (2) For $10^{-\frac 1\rho }\log ^{\frac {1}{\rho }}\frac {1}{\varepsilon +\delta }\leq N\leq (\varepsilon +\delta )^{-c_1},$ the $(\rho , \gamma _N, N)$ -LDE hold true for $(\boldsymbol {\alpha }, \boldsymbol {\theta })\in \mathcal W$ (cf. (2.11)) and all $\boldsymbol {\omega }\in \Omega $ (depending on $(\boldsymbol {\alpha }, \boldsymbol {\theta })$ ) with $\gamma _N=\gamma _0-N^{-\vartheta }\in [\frac \gamma 2,\gamma ].$

3. (3) Fix $(\boldsymbol {\alpha }, \boldsymbol {\theta })\in \mathcal W $ (cf. (2.11)) and let $\varepsilon \leq \delta \leq \log ^{-1}\frac 1\varepsilon $ . Let $N>(\varepsilon +\delta )^{-c_1}$ , $N_1\sim N^{\rho ^2}$ and $N_2\sim N^{2\rho }$ . Then there are some $\widetilde \Omega _N\subset \Omega $ (independent of S) satisfying $\mathrm {meas}(\Omega \setminus \widetilde \Omega _N)\leq e^{-\frac 15 N^{\frac {3\kappa _1\rho ^2}{4}}}$ and some $\gamma _N\in [\frac \gamma 2, \gamma ]$ so that the $(\rho , \gamma _N, N)$ -LDE hold true for $\boldsymbol {\omega }\in \Omega _N:=\widetilde \Omega _N\cap \mathrm {DC}_{\boldsymbol {\omega }}(N)\cap \Omega _{N_2}$ with $\mathrm {DC}_{\boldsymbol {\omega }}(N)$ given by (2.12) and

   $$ \begin{align*} \widetilde\Omega_N=\bigcap_{1\leq \ell,\ell'\leq N^{4(b+d)}, \xi=\pm1, \xi'=\pm1, 0<|\boldsymbol{k}|\leq 2N^2}\widetilde\Omega_{\boldsymbol{k},\ell,\ell',\xi,\xi'}, \end{align*} $$
   where
   $$ \begin{align*} \widetilde\Omega_{\boldsymbol{k},\ell,\ell',\xi,\xi'}:=\left\{\boldsymbol{\omega}\in\Omega:\ |\boldsymbol{k}\cdot\boldsymbol{\omega}+\xi {\lambda_\ell}-\xi'{\lambda_{\ell'}}|>10e^{-\frac 14 N^{\frac{3\kappa_1\rho^2}{4}}}\right\}, \end{align*} $$
   $\gamma _N\geq \gamma _{N_1}-N^{-\kappa }$ and $\lambda _\ell = \lambda _\ell (\boldsymbol {\alpha }, \boldsymbol {\theta })$ ( ${1\leq \ell \leq N^{4(b+d)}})$ are all eigenvalues of the operators $\mathcal H_{Q}(\boldsymbol {\alpha }, \boldsymbol {\theta })$ for all $Q\subset [-100N^2, 100N^2]^d$ satisfying
   $$ \begin{align*}Q\in\bigcup_{\tilde N\in [\frac14N^{\kappa_1\rho^2}, N^{\kappa_2\rho^2}]}\mathcal{ER}_{\mathbb Z^d}(\tilde N).\end{align*} $$

Remark 2.13. One may observe that this LDT is for a fixed operator H; while the linearized operator is varying in a Newton scheme. This issue, however, can be remedied by using that the Green’s function estimates in LDE (at scale N) are stable under perturbations of order $e^{-N^2}$ using Lemma B.1. The sup-exponentially decaying error (cf. (3.14)) in the Newton scheme can ensure this. More precisely, let

$$ \begin{align*} &\gamma\geq \gamma_1\geq \gamma_2\geq \cdots\geq \frac\gamma2,\\ &\gamma\geq \gamma_1'\geq \gamma_2'\geq\cdots\geq \frac\gamma2,\\ &\gamma_{\ell}'\leq \min\{\gamma_{[\ell^{\rho^2}]+1}, \gamma_{[\ell^{\rho^2}]+1}'\}-\ell^{-\kappa}. \end{align*} $$

We can replace H in LDT with a sequence of operators $H^{(\ell )}(\sigma )=D{(\sigma )}+\delta S_\ell $ ( $\ell \geq 1$ ) satisfying (we hide the dependence on $\xi ,\xi '\in \{\pm \}$ )

$$ \begin{align*} |S_\ell((\boldsymbol{k} ,\boldsymbol{n}); (\boldsymbol{k}', \boldsymbol{n}'))| &\leq C_2(1+|\boldsymbol{k}-\boldsymbol{k}'|)^{C_2}e^{-\gamma_\ell|\boldsymbol{k}-\boldsymbol{k}'|-\gamma_{\ell}|\boldsymbol{n}|}\delta_{\boldsymbol{n}, \boldsymbol{n}'},\\ \|H^{(\ell+1)}-H^{(\ell)}\|&\leq e^{-\ell^{\frac{2}{\rho^2}}}. \end{align*} $$

It is remarkable that in our LDT, $\Omega _N$ does not depend on S. So with this modification, the conclusion of LDT becomes for $\boldsymbol {\omega }\in \Omega _N, H^{(\ell )}$ has $(\rho , \gamma _N', N)$ -LDE for all $\ell \geq N\geq \log ^3{\frac {1}{\varepsilon +\delta }}$ ( $\Sigma _N$ depending only on $S_\ell $ , $1\leq \ell \leq N$ ).

Indeed, assume we have established the LDE at scale $N_1=N^{\rho ^2}$ for all $H^{(\ell )}$ with $\ell \geq N_1$ . Then it suffices to establish $(\rho , \gamma _N', N)$ -LDE for $H^{(N)}$ , since for all $\ell>N$ ,

$$ \begin{align*}\|H^{(\ell)}-H^{(N)}\|\leq \sum\limits_{\ell\geq N}e^{-\ell^{\frac{2}{\rho^2}}}<e^{-\frac{N^{\frac{2}{\rho^2}}}{2}}\ll e^{-100N},\end{align*} $$

and $(\rho , \gamma _N', N)$ -LDE for $H^{(\ell )}$ can be obtained via the perturbation argument (cf. Lemma B.1). Next, we fix $S_\ell \equiv S_{N_1}$ for $\ell \in [N_1, N]$ . Similar to the proof of Lemma 2.20, we can show using the modified resolvent identities Lemmas B.3, B.5, B.6 (e.g., taking into account the power-law factor in the estimate of $S_\ell $ ) that $H^{(N_1)}$ satisfies $(\rho , \gamma _N'', N)$ -LDE with

$$ \begin{align*} \gamma_N"\geq\min\{\gamma_{N_1}, \gamma_{N_1}'\}-N^{-\kappa}\geq \gamma_{N}'. \end{align*} $$

Finally, using again the perturbation argument Lemma B.1 and

$$ \begin{align*}\|H^{(N_1)}-H^{(N)}\|\leq e^{-\frac{N_1^{\frac{2}{\rho^2}}}{2}}\leq e^{-\frac{N^2}{5}}\ll e^{-100N},\end{align*} $$

we get that $H^{(N)}$ satisfies $(\rho , \gamma _N', N)$ -LDE by the assumptions on $\gamma _\ell ,\gamma _\ell '$ .

This LDT will be proved in detail in the following three sections.

2.6.1 The small scales LDE

This section is devoted to the proof of LDE for small scales, that is,

$$ \begin{align*} \log^{3}\frac{1}{\varepsilon+\delta}\leq N\leq 10^{-\frac1\rho}\log^{\frac{1}{\rho}}\frac{1}{\varepsilon+\delta}. \end{align*} $$

In this case, we only use the standard Neumann series argument. As a result, we can establish LDE for all $\boldsymbol {\omega }\in \Omega .$ We have

Lemma 2.15. Let

$$ \begin{align*} \Sigma_N &=\bigcup_{\Lambda\in (\boldsymbol{0},\boldsymbol{n})+\mathcal{ER}_{\boldsymbol{0}}(N),\ |\boldsymbol{n}|\leq 10N}\left\{\sigma\in\mathbb R:\ \min_{ (\boldsymbol{k},\boldsymbol{n}, \xi)\in\Lambda}|\xi(\sigma+\boldsymbol{k}\cdot\boldsymbol{\omega})+\mu_{\boldsymbol{n}}|\leq e^{-2N^{\rho}}\right\}. \end{align*} $$

Let $0<\rho <\frac {1}{10^4}$ . Then there is some $\delta _1=\delta _1(b,d,C_2, \rho )>0$ so that if $0<\varepsilon +\delta \leq \delta _1,$ for

$$ \begin{align*}\log^{3}\frac{1}{\varepsilon+\delta}\leq N\leq 10^{-\frac1\rho} \log^{\frac{1}{\rho}}\frac{1}{\varepsilon+\delta},\end{align*} $$

we have

(2.15) $$ \begin{align} \mathrm{meas}(\Sigma_N)\leq e^{-N^{\rho}}. \end{align} $$

Moreover, if $\sigma \notin \Sigma _N$ , we have for all $\Lambda \in (\boldsymbol {0},\boldsymbol {n})+\mathcal {ER}_{\boldsymbol {0}}(N)$ with $|\boldsymbol {n}|\leq 10N$ ,

(2.16) $$ \begin{align} \|G_{\Lambda}(\sigma)\|&\leq e^{N^{\frac34}}, \end{align} $$

(2.17) $$ \begin{align} |G_{\Lambda}(\sigma)((\boldsymbol{x},\xi);(\boldsymbol{x}',\xi'))|&\leq e^{-\gamma_0|\boldsymbol{x}-\boldsymbol{x}'|}\ \mathrm{for}\ |\boldsymbol{x}-\boldsymbol{x}'|\geq N^{\frac89}, \end{align} $$

where

(2.18) $$ \begin{align} \gamma_0=\gamma-\log ^{-2}\frac{1}{\varepsilon+\delta}\in[\frac\gamma2,\gamma]. \end{align} $$

Remark 2.16. In this lemma, we make no restrictions on $\boldsymbol {\omega }.$

Proof. We first establish the measure bound (2.15). We have

$$ \begin{align*} \mathrm{meas}(\Sigma_N)&\leq C(b,d)N^{C(b,d)}e^{-2N^{\rho}} \leq e^{-N^{\rho}}, \end{align*} $$

where in the last inequality we use $0<\varepsilon +\delta \leq \delta _1^{(1)}(b,d,\rho )\ll 1$ to ensure $N\geq \log ^{3}\frac {1}{\varepsilon +\delta }\geq N_0(b,d,\rho )\gg 1.$

Next, we let $\sigma \notin \Sigma _N$ and fix any $\Lambda \in (\boldsymbol {0}, \boldsymbol {n})+\mathcal {ER}_{\boldsymbol {0}}(N)$ with $|\boldsymbol {n}|\leq 10N$ . Let $A=D_{\Lambda }(\sigma ), B=\delta S_\Lambda $ . We have

$$ \begin{align*}|A^{-1}(\boldsymbol{x}, \boldsymbol{x}')|\leq e^{N^\rho}\delta_{\boldsymbol{x}, \boldsymbol{x}'},\ |B(\boldsymbol{x}, \boldsymbol{x}')|\leq C_2\delta (1+|\boldsymbol{x}-\boldsymbol{x}'|)^{C_2}e^{-\gamma |\boldsymbol{x}-\boldsymbol{x}'|}.\end{align*} $$

Note that

$$ \begin{align*} CN^{2(b+d)+C_2} \delta e^{N^\rho}&\leq C(b,d, C_2)N^{2(b+d)+C_2}(\varepsilon+\delta)^{\frac{9}{10}}\\ &\leq C(b,d, C_2)10^{-\frac{2(b+d)+C_2}{\rho}}(\varepsilon+\delta)^{\frac{9}{10}}\log^{-\frac{2(b+d)+C_2}{\rho}}\frac{1}{\varepsilon+\delta}\\ &\leq \frac{1}{2} \end{align*} $$

assuming $0<\varepsilon +\delta \leq \delta _1^{(2)}(b,d, C_2, \rho )\ll 1.$ So we can apply Lemma B.1 with $A=D_{\Lambda }(\sigma ), B=\delta S_\Lambda , \epsilon _2=C_2\delta , \epsilon _1=e^{-N^\rho }, M=0, X=\Lambda $ to obtain (we hide the dependence on $\xi , \xi '$ )

$$ \begin{align*} \|G_\Lambda(\sigma)\|&\leq 2e^{N^\rho}<e^{N^{\frac34}}\ (\mathrm{since}\ 0<\rho<\frac{1}{10^4}),\\ |G_\Lambda(\sigma)(\boldsymbol{x}, \boldsymbol{x}')|&\leq e^{N^{\rho}}e^{-\gamma|\boldsymbol{x}-\boldsymbol{x}'|}. \end{align*} $$

Then for $|\boldsymbol {x}-\boldsymbol {x}'|\geq N^{\frac 89},$

$$ \begin{align*} |G_\Lambda(\sigma)(\boldsymbol{x}, \boldsymbol{x}')|&\leq e^{-|\boldsymbol{x}-\boldsymbol{x}'|(\gamma-\frac{N^\rho}{|\boldsymbol{x}-\boldsymbol{x}'|})}\\ &\leq e^{-|\boldsymbol{x}-\boldsymbol{x}'|(\gamma-N^{-\frac{7}{8}})}\ \ (\mathrm{since}\ 0<\rho<\frac{1}{10^4})\\ &:=e^{-\gamma'|\boldsymbol{x}-\boldsymbol{x}'|}, \end{align*} $$

which implies

$$ \begin{align*}\gamma'>\gamma_0= \gamma-\log^{-2}\frac{1}{\varepsilon+\delta}.\end{align*} $$

Then both (2.16) and (2.17) hold true if we assume that

$$ \begin{align*}0<\varepsilon+\delta\leq \delta_1=\min\{\delta_1^{(1)}(b,d,C_2, \rho), \delta_1^{(2)}(b,d,C_2, \rho)\},\end{align*} $$

and $\sigma \notin \Sigma _N.$

2.6.2 The intermediate scales LDE

This section and the next deal with intermediate-scale LDE. We will show that the LDE hold true for all $\boldsymbol {\omega }\in \Omega $ and scales in the range

$$ \begin{align*}10^{-\frac1\rho}\log^{\frac{1}{\rho}}\frac{1}{\varepsilon+\delta}\leq N\leq (\varepsilon+\delta)^{-c_1},\end{align*} $$

where $c_1$ is given by Theorem 2.2. In this case, the analysis in Section 2.4 becomes essential.

Lemma 2.17. Let $0<\rho <\frac {1}{10^4}$ . There is some $\delta _2=\delta _2(b,d, K, V, \rho , c_1, C_2)>0$ so that the following holds for $0<\varepsilon +\delta \leq \delta _2.$ Assume $(\boldsymbol {\alpha }, \boldsymbol {\theta })\in \mathcal {M}$ (cf. Theorem 2.2) and fix

$$ \begin{align*}10^{-\frac1\rho}\log^{\frac{1}{\rho}}\frac{1}{\varepsilon+\delta}\leq N\leq (\varepsilon+\delta)^{-c_1}.\end{align*} $$

Then for each $\boldsymbol {\omega }\in \Omega $ , there exists a set $\Sigma _N\subset \mathbb R$ with

$$ \begin{align*} \mathrm{meas}(\Sigma_N)\leq (\varepsilon+\delta)^{-\frac{9}{10}}e^{-\frac{1}{2b}N^{\frac{3}{8}}}<e^{-N^{\rho}}, \end{align*} $$

so that if $\sigma \notin \Sigma _N$ , then for all $\Lambda \in (\boldsymbol {0}, \boldsymbol {n})+\mathcal {ER}_{\boldsymbol {0}}(N)$ with $|\boldsymbol {n}|\leq 10N$ ,

$$ \begin{align*} \|G_{\Lambda}(\sigma)\|&\leq e^{N^{\frac34}},\\ |G_{\Lambda}(\sigma)((\boldsymbol{x},\xi);(\boldsymbol{x}',\xi'))|&\leq e^{-\gamma_N|\boldsymbol{x}-\boldsymbol{x}'|}\ \mathrm{for}\ |\boldsymbol{x}-\boldsymbol{x}'|\geq N^{\frac89}, \end{align*} $$

where

$$ \begin{align*}\gamma_N=\gamma_0-N^{-\vartheta}\in[\frac\gamma2,\gamma],\end{align*} $$

with $\gamma _0$ given by (2.18) and $\vartheta>0$ by Lemma B.3.

Remark 2.18. Note that this lemma also holds for all $\boldsymbol {\omega }\in \Omega $ .

Proof. The proof is based on the clustering property (cf. (2) of Theorem 2.2), Schur’s complement argument, Rouché’s theorem and the iteration of resolvent identities. So we first choose $0<\varepsilon +\delta \leq c(b,d,K,V, \max \limits _{1\leq \ell \leq b}|\boldsymbol {n}_\ell |)\ll 1$ so that Theorem 2.2 holds true. Then the proof can be decomposed into two steps.

Step 1. We define

(2.19) $$ \begin{align} \Sigma_N=\bigcup_{\sqrt{N}\leq N'\leq N,\ \Lambda\in (\boldsymbol{0}, \boldsymbol{n})+\mathcal{ER}_{\boldsymbol{0}}(N'),\ |\boldsymbol{n}|\leq 10N'}\Sigma_\Lambda, \end{align} $$

where

$$ \begin{align*} \Sigma_\Lambda=\left\{\sigma\in\mathbb R:\ \|G_\Lambda(\sigma)\|\geq e^{(\mathrm{diam}\ \Lambda)^{\frac34}}\right\}. \end{align*} $$

We first show that

(2.20) $$ \begin{align} \Sigma_\Lambda\subset \bigcup_{\xi=\pm 1, \boldsymbol{i}=(\boldsymbol{k}, \boldsymbol{n})\in\Lambda} I_{\xi, \boldsymbol{i}}, \end{align} $$

where

$$ \begin{align*} I_{\xi, \boldsymbol{i}}=\left\{\sigma\in\mathbb R:\ |\xi(\sigma+\boldsymbol{k}\cdot\boldsymbol{\omega}^{(0)})+\mu_{\boldsymbol{n}}|\leq \frac{1}{100}(\varepsilon+\delta)^{\frac{1}{7b}}\right\}. \end{align*} $$

In fact, if $\sigma \notin \bigcup \limits _{(\xi ,\boldsymbol {i})\in \Lambda } I_{\xi , \boldsymbol {i}},$ then for each $\boldsymbol {\omega }\in \Omega ,$

$$ \begin{align*} &\ \ \ \ \inf_{(\xi,\boldsymbol{i})\in\Lambda} |\xi(\sigma+\boldsymbol{k}\cdot\boldsymbol{\omega})+\mu_{\boldsymbol{n}}|\\ &\geq \inf_{(\xi,\boldsymbol{i})\in\Lambda} |\xi(\sigma+\boldsymbol{k}\cdot\boldsymbol{\omega}^{(0)})+\mu_{\boldsymbol{n}}|-\sup_{\boldsymbol{k}\in\Lambda}|\boldsymbol{k}\cdot(\boldsymbol{\omega}-\boldsymbol{\omega}^{(0)})|\\ &\geq \frac{1}{200}(\varepsilon+\delta)^{\frac{1}{7b}}. \end{align*} $$

Since $N\geq 10^{- \frac 1\rho } \log ^{\frac {1}{\rho }} \frac {1}{\varepsilon +\delta }$ , the Neumann series argument implies

$$ \begin{align*} \|G_\Lambda(\sigma)\|\leq C(\varepsilon+\delta)^{-\frac{2}{7b}}<e^{(\mathrm{diam}\ \Lambda)^{\frac34}} \end{align*} $$

when $\sigma \notin \bigcup \limits _{(\xi ,\boldsymbol {i})\in \Lambda } I_{\xi , \boldsymbol {i}}.$ This proves (2.20) and

$$ \begin{align*} \Sigma_\Lambda=\bigcup_{(\xi,\boldsymbol{i})\in\Lambda}(\Sigma_\Lambda\cap I_{\xi,\boldsymbol{i}}). \end{align*} $$

In the following, we estimate $\mathrm {meas}(\Sigma _\Lambda \cap I_{\xi , \boldsymbol {i}}).$ For this purpose, we fix any $\xi ^*\in \{+,-\}$ , $\boldsymbol {i}^*=(\boldsymbol {k}^*, \boldsymbol {n}^*)$ and denote

$$ \begin{align*}I^*=I_{\xi^*, \boldsymbol{i}^*}=\left[\sigma^*-\frac{1}{100}(\varepsilon+\delta)^{\frac{1}{7b}}, \sigma^*+\frac{1}{100}(\varepsilon+\delta)^{\frac{1}{7b}}\right],\end{align*} $$

where

(2.21) $$ \begin{align} \sigma^*=-\boldsymbol{k}^*\cdot\boldsymbol{\omega}^{(0)}-{\xi^*}\mu_{\boldsymbol{n}^{*}}. \end{align} $$

We will study $G_\Lambda (\sigma )$ for $\sigma \in I^*.$ From (2) of Theorem 2.2, we have for every $\sigma ^*\in \mathbb R$ , there exists $B_*=B_*^+\cup B_*^-\subset \Lambda $ with $\# B_*^{\pm }\leq b$ so that

$$ \begin{align*} \inf_{(\xi,\boldsymbol{i})\in\Lambda\setminus B_*}|\xi(\sigma^*+\boldsymbol{k}\cdot\boldsymbol{\omega}^{(0)})+\mu_{\boldsymbol{n}}|\geq \frac{(\varepsilon+\delta)^{\frac{1}{8b}}}{4}. \end{align*} $$

Similar to the proof of (2.20), if (we denote $\mathbb D_r(z):=\{w\in \mathbb C:\ |w-z|\leq r\}$ ) $z\in \mathbb {D}_{(\varepsilon +\delta )^{\frac {1}{8b}}/10}(\sigma ^*),$ then

$$ \begin{align*} \inf_{(\xi,\boldsymbol{i})\in\Lambda\setminus B_*}|\xi(z+\boldsymbol{k}\cdot\boldsymbol{\omega})+\mu_{\boldsymbol{n}}|\geq \frac{(\varepsilon+\delta)^{\frac{1}{8b}}}{8}. \end{align*} $$

So we have an analytic extension of ${H}_\Lambda (z)$ to $z\in \mathbb {D}_{(\varepsilon +\delta )^{\frac {1}{8b}}/10}(\sigma ^*)$ and

$$ \begin{align*} \sup_{(\boldsymbol{x}, \xi),(\boldsymbol{x}', \xi')\in \Lambda,\ z\in\mathbb{D}_{(\varepsilon+\delta)^{\frac{1}{8b}}/10}(\sigma^*)}|{D}_\Lambda(z)((\boldsymbol{x},\xi); (\boldsymbol{x}', \xi'))|\leq C(b)(\varepsilon+\delta)^{-c_1}. \end{align*} $$

More importantly, if we denote $\Lambda ^c=\Lambda \setminus B_*$ , then by the Neumann series argument, we have for all $z\in \mathbb {D}_{(\varepsilon +\delta )^{\frac {1}{8b}}/10}(\sigma ^*),$

$$ \begin{align*} \|G_{\Lambda^c}(z)\|\leq (\varepsilon+\delta)^{-\frac{1}{4b}}. \end{align*} $$

In the following, we estimate $G_{\Lambda }(z)$ using the Schur complement reduction. We define

$$ \begin{align*} S(z)=R_{B_*}{H}(z)R_{B_*}-R_{B_*}{H}(z)R_{\Lambda^c}G_{\Lambda^c}(z)R_{\Lambda^c}{H}(z)R_{B_*}. \end{align*} $$

Obviously, we have

(2.22) $$ \begin{align} R_{B_*}{H}(z)R_{B_*} &=R_{B_*}{D}(z)R_{B_*}+O(\varepsilon+\delta) \end{align} $$

and

(2.23) $$ \begin{align} R_{B_*}{H}(z)R_{\Lambda^c}G_{\Lambda^c}(z)R_{\Lambda^c}{H}(z)R_{B_*}=O((\varepsilon+\delta)^{2-\frac{1}{4b}}). \end{align} $$

Now we consider $\det S(z)$ . Combining (2.22) and (2.23) yields

$$ \begin{align*} &\ \ \ \det S(z)\\ &= \prod_{ \boldsymbol{i}=(\boldsymbol{k}, \boldsymbol{n})\in B_*^+}(z+\boldsymbol{k}\cdot\boldsymbol{\omega}+\mu_{\boldsymbol{n}})\prod_{\boldsymbol{i}=(\boldsymbol{k},\boldsymbol{n})\in B_*^-}(-z-\boldsymbol{k}\cdot\boldsymbol{\omega}+\mu_{\boldsymbol{n}})\\ &\ \ \ +O((\varepsilon+\delta)^{3/4})\\ &=(-1)^{\#B_*^-}\prod_{\ell=1}^{\# B_*}(z-\sigma_\ell)+O((\varepsilon+\delta)^{3/4}), \end{align*} $$

where $\sigma _\ell \in \{-\boldsymbol {k}\cdot \boldsymbol {\omega }- \mu _{\boldsymbol {n}}\}_{(\boldsymbol {k}, \boldsymbol {n})\in B_*^+}\cup \{-\boldsymbol {k}\cdot \boldsymbol {\omega }+\mu _{\boldsymbol {n}}\}_{(\boldsymbol {k}, \boldsymbol {n})\in B_*^-}.$ Recalling the definition of $B_*$ , we must have

$$ \begin{align*} |z-\sigma_\ell|\leq \frac45(\varepsilon+\delta)^{\frac{1}{8b}}. \end{align*} $$

From (2.21), there exists at least one $\sigma _\ell $ so that $|\sigma _\ell -\sigma ^*|=O((\varepsilon +\delta )^{1-c_1}).$ At this stage, we need a useful result.

Lemma 2.19. Let $\varphi =\left (\frac {14}{15}\right )^{\frac {1}{5b}}<1$ . There is $0\leq \ell _*\leq 5b-1$ so that

$$ \begin{align*} \left(\mathbb{A}_{\ell_*+1}\setminus\mathbb{A}_{\ell_*}\right)\cap \{\sigma_\ell\}_{1\leq \ell\leq \# B_*}=\emptyset, \end{align*} $$

where for $0\leq \ell \leq 5b$ ,

$$ \begin{align*} \mathbb{A}_\ell=\mathbb{D}_{(\varepsilon+\delta)^{\frac{\varphi^{\ell}}{7b}}}(\sigma^*). \end{align*} $$

Proof of Lemma 2.19

It suffices to note that

$$ \begin{align*} \mathbb{A}_{5b}\setminus\mathbb{A}_{0}=\bigcup_{\ell=0}^{5b-1}\left(\mathbb{A}_{\ell+1}\setminus \mathbb{A}_{\ell}\right). \end{align*} $$

The proof then follows from the pigeonhole principle since $\#B_*\leq 2b$ .

Let $r_*=(\varepsilon +\delta )^{\frac {\varphi ^{\ell _*}}{7b}}$ , where $\varphi $ and $\ell _*$ are defined in Lemma 2.19. From Lemma 2.19, we can restrict our considerations on $\mathbb {A}_*$ with

$$ \begin{align*} \mathbb{A}_*=\mathbb{D}_{\frac{r_*+r_*^\varphi}{2}}(\sigma^*). \end{align*} $$

Then

$$ \begin{align*} \mathbb{D}_{(\varepsilon+\delta)^{\frac{1}{7b}}}(\sigma^*)\subset\mathbb{A}_{\ell_*}\subset\mathbb{A}_*\subset \mathbb{D}_{(\varepsilon+\delta)^{\frac{1}{7.5b}}}(\sigma^*). \end{align*} $$

Let

$$ \begin{align*} \mathcal K=\{1\leq \ell\leq \# B_*:\ \sigma_\ell\in \mathbb{D}_{10r_*}(\sigma^*)\}. \end{align*} $$

We have $\mathcal K\neq \emptyset $ and for $\ell \not \in \mathcal K$ ,

$$ \begin{align*} \sigma_\ell\notin\mathbb{A}_*,\ \mathrm{dist}(\sigma_\ell,\partial \mathbb{D}_{20r_*}(\sigma^*))\geq\mathrm{dist}(\sigma_\ell, \partial\mathbb{A}_*)\geq \frac{r_*^\varphi}{4}\gg(\varepsilon+\delta)^{\frac{1}{7b}}. \end{align*} $$

As a result, we can write for $z\in \mathbb {A}_*,$

(2.24) $$ \begin{align} \frac {\det S(z)}{\prod_{\ell\notin \mathcal K}(z-\sigma_\ell)}=\pm{\prod_{\ell\in \mathcal K}(z-\sigma_\ell)}+O((\varepsilon+\delta)^{\frac13}). \end{align} $$

It holds that

$$ \begin{align*} {\prod_{\ell\in \mathcal K}|z-\sigma_\ell|}\big|_{\partial \mathbb{D}_{20r_*}(\sigma^*)}\geq (\varepsilon+\delta)^{\frac{\#\mathcal K}{7b}}\geq (\varepsilon+\delta)^{2/7}\gg (\varepsilon+\delta)^{\frac13}. \end{align*} $$

Applying Rouché’s theorem shows that the function $\frac {\det S(z)}{\prod _{\ell \notin \mathcal K}(z-\sigma _\ell )}$ has exactly $\#\mathcal K$ zeros (denoted by $z_\ell ,\ \ell \in \mathcal K$ ) in $ \mathbb {D}_{20r_*}(\sigma ^*).$ From $\sigma _\ell \in \mathbb {D}_{10r_*}(\sigma ^*)$ , we also get

(2.25) $$ \begin{align} |\sigma_\ell-z_\ell|\le 30r_*. \end{align} $$

From similar analysis, we can obtain that (2.24) also has exactly $\#\mathcal {K}$ zeros in $\mathbb {A}_*$ , which implies that $\{z_\ell \}_{\ell \in \mathcal {K}}$ are all zeros of (2.24) in $\mathbb {A}_*$ . So on $\mathbb {A}_*$ , we have

(2.26) $$ \begin{align} \frac {\det S(z)}{\prod_{\ell\notin \mathcal K}(z-\sigma_\ell)}=g(z)\prod_{\ell\in \mathcal K}(z-z_\ell), \end{align} $$

where g is analytic on $\mathbb {A}_*$ with $\inf \limits _{z\in \mathbb {A}_*}|g(z)|>0.$ Using (2.25) and (2.26), we have

$$ \begin{align*} g\big|_{\partial \mathbb{A}_*}=1+O((\varepsilon+\delta)^{\frac{1}{21}}). \end{align*} $$

Using the maximum principle then leads to

$$ \begin{align*} g\big|_{ \mathbb{A}_*}=1+O((\varepsilon+\delta)^{\frac{1}{21}}). \end{align*} $$

We have established on $\mathbb {A}_*,$

$$ \begin{align*} \det S(z)=g(z)\prod_{\ell\notin \mathcal K}(z-\sigma_\ell)\prod_{\ell\in \mathcal K}(z-z_\ell). \end{align*} $$

In particular, one has for all $z\in \mathbb {A}_*,$

$$ \begin{align*} |\det S(z)|&\geq \frac 45 (\varepsilon+\delta)^{\frac{\#B_*-\#\mathcal K}{7b}}\prod_{\ell\in \mathcal K}|z-z_\ell|\\ &\geq (\varepsilon+\delta)^{\frac{2}{7}}\prod_{\ell\in \mathcal K}|z-z_\ell| \end{align*} $$

assuming $0<\varepsilon +\delta \leq c(b)\ll 1.$ Thus combining Hadamard’s inequality and Cramer’s rule implies for $z\in \mathbb {A}_*,$

$$ \begin{align*} \|S^{-1}(z)\|&\leq C(b) (\varepsilon+\delta)^{-{2bc_1}}(\varepsilon+\delta)^{-\frac{2}{7}}\prod_{\ell\in \mathcal K}|z-z_\ell|^{-1}\\ &\leq (\varepsilon+\delta)^{-\frac{1}{3}}\prod_{\ell\in \mathcal K}|z-z_\ell|^{-1}\ (\mathrm{since}\ c_1<\frac{1}{100b}). \end{align*} $$

Applying the Schur complement argument shows that for $z\in \mathbb {A}_*,$

$$ \begin{align*} \|G_\Lambda(z)\|&\leq 4(1+\|G_{\Lambda^c}(z)\|)^2(1+\|S^{-1}(z)\|)\\ &\leq (\varepsilon+\delta)^{-\frac{7}{8}}\prod_{\ell\in \mathcal K}|z-z_\ell|^{-1}. \end{align*} $$

Recall that $I^*\subset \mathbb {A}_*$ . So for $\sigma \in I^*,$ we have

$$ \begin{align*} \|G_\Lambda(\sigma)\|&\leq (\varepsilon+\delta)^{-\frac{7}{8}}\prod_{\ell\in \mathcal K}|\sigma-z_\ell|^{-1} \end{align*} $$

and

$$ \begin{align*} \Sigma_\Lambda\cap I^*\subset \bigcup_{\ell\in \mathcal K}\left\{\sigma\in I^*:\ |\sigma-z_\ell|\leq (\varepsilon+\delta)^{-\frac{7}{8}}e^{-\frac{1}{2b}(\mathrm{diam}\ \Lambda)^{\frac34}}\right\}. \end{align*} $$

This implies

$$ \begin{align*} \mathrm{meas}(\Sigma_\Lambda\cap I^*)\leq C(\varepsilon+\delta)^{-\frac{7}{8}}e^{-\frac{1}{2b}(\mathrm{diam}\ \Lambda)^{\frac34}}, \end{align*} $$

and subsequently

$$ \begin{align*} \mathrm{meas}(\Sigma_\Lambda)\leq (\varepsilon+\delta)^{-\frac{8}{9}}e^{-\frac{1}{2b}(\mathrm{diam}\ \Lambda)^{\frac34}}, \end{align*} $$

which combined with (2.19) shows

$$ \begin{align*} \mathrm{meas}(\Sigma_N)\leq (\varepsilon+\delta)^{-\frac{9}{10}}e^{-\frac{1}{2b}N^{\frac{3}{8}}}. \end{align*} $$

Step 2. In this step, we prove the off-diagonal exponential decay of $G_\Lambda (\sigma )((\boldsymbol {x},\xi );(\boldsymbol {x}', \xi '))$ assuming $\sigma \not \in \Sigma _{N}$ and $\Lambda \in (\boldsymbol {0}, \boldsymbol {n})+\mathcal {ER}_{\boldsymbol {0}}(N)$ with $|\boldsymbol {n}|\leq 10N.$ By Theorem 2.2, we know that there exists $B\subset \Lambda $ so that

$$ \begin{align*} \inf_{ (\xi, \boldsymbol{k}, \boldsymbol{n})\in\Lambda\setminus B}|\xi(\sigma+\boldsymbol{k}\cdot\boldsymbol{\omega})+\mu_{\boldsymbol{n}}|\geq \frac{(\varepsilon+\delta)^{\frac{1}{8b}}}{8}. \end{align*} $$

Using the Neumann series argument (cf. Lemma B.1) yields that for any $\Lambda '\subset \Lambda $ with $\Lambda '\cap B=\emptyset ,$

(2.27) $$ \begin{align} \|G_{\Lambda'}(\sigma)\|&\leq 50(\varepsilon+\delta)^{-\frac{1}{4b}}, \end{align} $$

(2.28) $$ \begin{align} |G_{\Lambda'}(\sigma)((\boldsymbol{x},\xi);(\boldsymbol{x}', \xi'))|&\leq 50(\varepsilon+\delta)^{-\frac{1}{4b}}e^{-\gamma|\boldsymbol{x}-\boldsymbol{x}'|}\ \mathrm{for}\ \boldsymbol{x}\neq \boldsymbol{x}'. \end{align} $$

We say that $\mathcal {ER}(\sqrt {N})\ni Q,\ Q\subset \Lambda $ is $\sqrt {N}$ -regular if both (2.27) and (2.28) hold with $\Lambda '=Q.$ Otherwise, Q is called $\sqrt {N}$ -singular. Let $\mathcal {F}$ be any family of pairwise disjoint $\sqrt {N}$ -singular sets contained in $\Lambda $ . We obtain

$$ \begin{align*} \#\mathcal{F}\leq 2b. \end{align*} $$

We have established the sublinear bound. Then applying the coupling lemma, that is, Lemma B.3 of [Reference LiuLiu22], shows

$$ \begin{align*} |G_\Lambda(\sigma)((\boldsymbol{x},\xi);(\boldsymbol{x}', \xi'))|\leq e^{-\gamma_N|\boldsymbol{x}-\boldsymbol{x}'| } \ \mathrm{for}\ |\boldsymbol{x}-\boldsymbol{x}'|>N^{\frac89}, \end{align*} $$

where

$$ \begin{align*} \gamma_N&\geq \gamma- N^{-\frac89}\cdot \log\frac{1}{\varepsilon+\delta}-N^{-\vartheta}\\ &\geq \gamma- {\log^{-\frac{8}{9\rho}+1} \frac{1}{\varepsilon+\delta}}-N^{-\vartheta}\\ &\geq \gamma_0-N^{-\vartheta}\ (\mathrm{since}\ 0<\rho<\frac{1}{10^4}). \end{align*} $$

This proves the off-diagonal exponential decay estimates.

We have completed the proof of Lemma 2.17.

2.6.3 The large scales LDE

In this section, we will prove the LDE for

$$ \begin{align*} (\varepsilon+\delta)^{-c_1}<N<\infty. \end{align*} $$

To perform the multi-scale scheme, we need two small scales $N_1<N_2$ with

$$ \begin{align*} N_2=N_1^{\frac{2}{\rho}},\ N=N_1^{\frac{1}{\rho^2}}. \end{align*} $$

We assume the LDE hold for $\boldsymbol {\omega }\in \Omega _{N_2}\subset \Omega _{N_1}$ .

Then we have

Lemma 2.20. There are some $0<c_2=c_2(b,d)\leq \frac {1}{10^4(b+d)^2}$ and some

$$ \begin{align*}\delta_3=\delta_3(b, d, K, V, \max\limits_{1\leq \ell\leq b}|\boldsymbol{n}_\ell|, \kappa_1, \kappa_2, \rho, c_1, C_2)>0\end{align*} $$

so that the following holds true for $0<\varepsilon +\delta \leq \delta _3$ , $\varepsilon \leq \delta \leq \log ^{-1}\frac {1}{\varepsilon }$ and $0<\rho \leq c_2\kappa _1$ with $\kappa _1$ given by Lemma 2.5. Let $(\boldsymbol {\alpha }, \boldsymbol {\theta })\in \mathcal W$ (cf. (2.11)) and N satisfy

$$ \begin{align*} (\varepsilon+\delta)^{-c_1}<N<\infty. \end{align*} $$

Then there is a semi-algebraic set $\widetilde \Omega _N\subset \Omega $ (cf. Lemma 2.26) with $\mathrm { meas}(\Omega \setminus \widetilde \Omega _N)\leq e^{-\frac 15N^{\frac {3\kappa _1\rho ^2}{4}}}$ and $\deg \ \widetilde \Omega _N\leq N^{5(b+d)}$ so that, for $\boldsymbol {\omega }\in \mathrm {DC}_{\boldsymbol {\omega }}(N)\cap \widetilde \Omega _N\cap {\Omega _{N_2}}$ (with $\mathrm { DC}_{\boldsymbol {\omega }}(N)$ given by (2.12)), there is some $\Sigma _N\subset \mathbb R$ with $\mathrm {meas}(\Sigma _N)\leq e^{-N^{\rho }}$ , so that, if $\sigma \notin \Sigma _N$ , then for all $\Lambda \in (\boldsymbol {0}, \boldsymbol {n})+\mathcal {ER}_{\boldsymbol {0}}(N)$ with $|\boldsymbol {n}|\leq 10N,$

$$ \begin{align*} \|G_{\Lambda}(\sigma)\|&\leq e^{N^{\frac{3}{4}}},\\ |G_{\Lambda}(\sigma)((\boldsymbol{x},\xi);(\boldsymbol{x}', \xi'))|&\leq e^{-\gamma_N|\boldsymbol{x}-\boldsymbol{x}'|}\ \mathrm{for}\ |\boldsymbol{x}-\boldsymbol{x}'|\geq N^{\frac{8}{9}}, \end{align*} $$

where $[\frac \gamma 2, \gamma ]\ni \gamma _N\geq \gamma _{N_1}-N^{-\kappa }$ (cf. (2.35)).

We begin with a useful definition.

Definition 2.21 ( $\sigma $ -good)

For $\Lambda \in \mathcal {ER}(L)$ , we say $\Lambda $ is $\sigma $ -good if

$$ \begin{align*} \|G_{\Lambda}(\sigma)\|&\leq e^{L^{\frac{3}{4}}},\\ |G_{\Lambda}(\sigma)(\boldsymbol{x},\xi);(\boldsymbol{x}', \xi'))|&\leq e^{-\gamma_L |\boldsymbol{x}-\boldsymbol{x}'|}\ \mathrm{for}\ |\boldsymbol{x}-\boldsymbol{x}'|\geq L^{\frac{8}{9}}, \end{align*} $$

where $\gamma _L\in [\frac \gamma 2, \gamma ]$ , and $\sigma $ -bad, if one of the inequalities is violated.

The proof of Lemma 2.20 is based on matrix-valued Cartan’s lemma (Lemma 2.25) and iterations of resolvent identities. To apply Cartan’s lemma, one needs to establish the sublinear bound on $\sigma $ -bad $\Lambda \in (\boldsymbol {k}, \boldsymbol {n})+\mathcal {ER}_{\boldsymbol {0}}(N_1)$ contained in $\Lambda _N$ : If $|\boldsymbol {n}|\leq 10N_1$ , the sublinear bound can be obtained via the Töplitz property of H in the $\boldsymbol {k}$ -direction together with the semi-algebraic sets theory (cf. Lemma 2.22); for $|\boldsymbol {n}|>10N_1$ , we will use the short-range property of S and Bourgain’s geometric lemma (Lemma 2.5) to prove the sublinear bound. Once the sublinear bound is derived, one can use the Cartan’s lemma and resolvent identities to finish the proof of large scales LDE.

We need the following ingredients to prove Lemma 2.20.

Lemma 2.22. There is some $0<c_2=c_2(b,d)\leq \frac {1}{10^4(b+d)^2}$ so that the following holds true. Assume that LDE hold true at scale $N_1$ with

$$ \begin{align*}N_1\geq N_0(b,d,\rho,K,V).\end{align*} $$

If $0<\rho \leq c_2$ , then for all $\sigma \in \mathbb R$ , the number of $\sigma $ -bad $\Lambda \in (\boldsymbol {k},\boldsymbol {n})+\mathcal {ER}_{\boldsymbol {0}}(N_1)$ satisfying $|\boldsymbol {k}|\leq N^2$ and $|\boldsymbol {n}|\leq 10N_1$ is at most $N^{\frac {1}{10^3(b+d)^2}}.$

Proof. First, we define $\widetilde \Sigma _{N_1}$ to be the set of $\sigma \in \mathbb R$ so that, there is $\Lambda \in (\boldsymbol {0}, \boldsymbol {n})+\mathcal {ER}_{\boldsymbol {0}}(N_1)$ satifying $|\boldsymbol {n}|\leq 10N_1$ , which is $\sigma $ -bad. We then obtain using LDE at scale $N_1,$

$$ \begin{align*} \mathrm{meas}(\widetilde \Sigma_{N_1})\leq C(b,d)N_1^{C(b,d)}e^{-N_1^{\rho}}<e^{-\frac12 N_1^{\rho}} \end{align*} $$

for $N_1\geq N_0(b,d,\rho ).$ Using the Hilbert-Schmidt norm and Cramer’s rule, we can identify $\widetilde \Sigma _{N_1}$ with a semi-algebraic set of $\deg \widetilde \Sigma _{N_1}\leq N_1^{C(b,d)}$ . Then using Basu-Pollack-Roy Theorem [Reference Basu, Pollack and RoyBPR96] on Betti numbers of a semi-algebraic set (cf. also Proposition 9.2 of [Reference BourgainBou05a]), we have a decomposition of

$$ \begin{align*}\widetilde \Sigma_{N_1}\subset\bigcup_{1\leq \ell\leq N_1 ^{C(b,d)}}I_\ell,\end{align*} $$

where each $I_\ell $ is an interval of length $|I_\ell |\leq e^{-\frac 12 N_1^{\rho }}.$ Now assume that $\Lambda \in (\boldsymbol {k},\boldsymbol {n})+\mathcal {ER}_{\boldsymbol {0}}(N_1)$ and $\Lambda '\in (\boldsymbol {k}',\boldsymbol {n})+\mathcal {ER}_{\boldsymbol {0}}(N_1)$ , which are all $\sigma $ -bad for some $|\boldsymbol {n}|\leq 10N_1, \boldsymbol {k}\neq \boldsymbol {k}'$ and $|\boldsymbol {k}|, |\boldsymbol {k}'|\leq N^2$ . Then from the Töplitz property of ${H}(\sigma )$ in the $\boldsymbol {k}$ -direction, we obtain

$$ \begin{align*}\sigma+\boldsymbol{k}\cdot\boldsymbol{\omega},\ \sigma+\boldsymbol{k}'\cdot\boldsymbol{\omega}\in\widetilde \Sigma_{N_1}.\end{align*} $$

As a result, by $\boldsymbol {k}\neq \boldsymbol {k}'$ with $|\boldsymbol {k}|, |\boldsymbol {k}'|\leq N^2$ , $\boldsymbol {\omega }\in \mathrm {DC}_{\boldsymbol {\omega }}(N)$ (cf. (2.12)) and $N\sim N_1^{\frac {1}{\rho ^2}}$ , we have for $N_1\geq N_0(b,d, \rho , K,V),$

$$ \begin{align*} {e^{-\frac12 N_1^{\rho}}}=e^{-\frac12 N^{\rho^3}}< e^{-N^{\rho^4}}\leq |(\boldsymbol{k}-\boldsymbol{k}')\cdot\boldsymbol{\omega}|, \end{align*} $$

which implies $\boldsymbol {k}$ and $\boldsymbol {k}'$ cannot stay in the same interval $I_\ell $ for $1\leq \ell \leq N_1^{C(b,d)}.$ Thus we have shown that the number of $\sigma $ -bad $\Lambda \in (\boldsymbol {k},\boldsymbol {n})+\mathcal {ER}_{\boldsymbol {0}}(N_1)$ with $|\boldsymbol {n}|\leq 10N_1$ and $|\boldsymbol {k}|\leq N^2$ is at most

$$ \begin{align*} N_1^{C(b,d)}\leq N^{\rho^2 \cdot C(b,d)} \leq N^{\frac{1}{10^3(b+d)^2}}, \end{align*} $$

assuming $0<\rho \leq c_2(b,d)\leq \frac {1}{10^4(b+d)^2}. $ This proves Lemma 2.22.

Next, we will deal with $|\boldsymbol {n}|>10N_1$ by combining the short-range property of S and Bourgain’s geometric lemma (cf. Lemma 2.5). For $\Lambda \subset \mathbb Z^{b+d}$ , denote by $\Pi _b\Lambda $ (resp. $\Pi _d\Lambda $ ) the projection of $\Lambda $ on $\mathbb Z^b$ (resp. $\mathbb Z^d$ ). For each $\boldsymbol {k}\in \Pi _b\Lambda , $ define

$$ \begin{align*}\Lambda(\boldsymbol{k})=\{\boldsymbol{n}\in\mathbb Z^d:\ (\boldsymbol{k},\boldsymbol{n})\in\Lambda\}.\end{align*} $$

We have

Lemma 2.23. Assume that $(\boldsymbol {\alpha }, \boldsymbol {\theta })\in \mathcal {\mathcal W'}$ (cf. Lemma 2.5) and $L\geq 10^{-\frac {32}{\kappa _1^2}}\log ^{\frac {4}{\kappa _1}}(\log \frac 1\varepsilon )$ . We have

1. (1) Fix $\Lambda \in (\boldsymbol {k}', \boldsymbol {n}')+\mathcal {ER}_{\boldsymbol {0}}(L) $ with $|\boldsymbol {n}'|>10L$ . There is a collection of $\{\lambda _\ell =\lambda _\ell (\boldsymbol {\alpha }, \boldsymbol {\theta })\}_{1\leq \ell \leq L^{b+2d}}$ depending only on $\Lambda , \boldsymbol {\alpha }, \boldsymbol {\theta }, V$ so that, if

   $$ \begin{align*}\min_{\boldsymbol{k}\in\Pi_b\Lambda, 1\leq \ell\leq L^{b+2d}, \xi=\pm1}|\sigma+\boldsymbol{k}\cdot\boldsymbol{\omega}+\xi{\lambda_\ell}|> e^{-\frac14 L^{\frac{3\kappa_1}{4}}},\end{align*} $$
   then $\Lambda $ is $\sigma $ -good with $\gamma _L=\gamma $ , where $\kappa _1$ is given in Lemma 2.5.

2. (2) Let $L\leq N\leq e^{\log ^2 L}$ . There exists some $\tilde \Sigma _{L}\subset \mathbb R$ with $\mathrm { meas}(\tilde \Sigma _{L})\leq e^{-\frac 15 L^{\frac {3\kappa _1}{4}}}$ so that, if $\sigma \notin \tilde \Sigma _{L}$ , then each $\Lambda \in (\boldsymbol {k}, \boldsymbol {n})+\mathcal {ER}_{\boldsymbol {0}}(L) $ satisfying $|(\boldsymbol {k},\boldsymbol {n})|\leq 100N^2$ and $|\boldsymbol {n}|>10L$ is $\sigma $ -good with $\gamma _L=\gamma $ .

Proof. (1) Let $\boldsymbol {k}\in \Pi _b\Lambda $ . It is important that either $\Lambda ({\boldsymbol {k}})\in \mathcal {ER}_{\mathbb Z^d}(L),$ or $\Lambda ({\boldsymbol {k}})$ is a rectangle of width at least L (cf. Lemma B.8). In each of the two cases, we have $\Lambda ({\boldsymbol {k}})\in \mathcal E_{2L}^{L}$ .

For $Q\subset \Lambda ({\boldsymbol {k}}), $ write

$$ \begin{align*} A_{\boldsymbol{k}, Q}=A_{\boldsymbol{k}, Q}(\sigma)&=R_{Q}({D}(\sigma)+\varepsilon(\Delta\oplus\Delta))R_{Q}\\ &=(-\sigma-\boldsymbol{k}\cdot\boldsymbol{\omega}+\mathcal{L}_{Q}(\boldsymbol{\theta})) \oplus (\sigma+\boldsymbol{k}\cdot\boldsymbol{\omega}+\mathcal{L}_{Q}(\boldsymbol{\theta})), \end{align*} $$

where $\mathcal {L}_{Q}(\boldsymbol {\theta }):=\mathcal H_Q(\boldsymbol {\alpha }, \boldsymbol {\theta })$ is defined in Lemma 2.5. Then the set of all eigenvalues of $A_{\boldsymbol {k}, Q}$ is given by

$$ \begin{align*}\{\pm (\sigma+\boldsymbol{k}\cdot\boldsymbol{\omega})+\lambda_{\ell, Q}(\boldsymbol{\alpha}, \boldsymbol{\theta})\}_{\ell=1}^{\#Q},\end{align*} $$

where $\{\lambda _\ell :=\lambda _{\ell , Q}(\boldsymbol {\alpha }, \boldsymbol {\theta })\}_{\ell =1}^{\#Q}$ denotes the spectrum of $\mathcal {L}_{Q}(\boldsymbol {\theta }).$ So we have

$$ \begin{align*} \|A_{\boldsymbol{k}, Q}^{-1}\|&\leq \max_{\xi=\pm1, 1\leq \ell\leq \#Q}|\xi (\sigma+\boldsymbol{k}\cdot\boldsymbol{\omega})+\lambda_\ell|^{-1}. \end{align*} $$

We now can use Lemma 2.5 and resolvent identities. Since $\boldsymbol {\alpha }\in \mathcal {A}\subset \mathcal {A}_{L}$ , applying (1) of Lemma 2.5 with $E_{\boldsymbol {k}}=\sigma +\boldsymbol {k}\cdot \boldsymbol {\omega }$ similar to [Reference Jitomirskaya, Liu and ShiJLS20] (cf. pages 471–472, the proof of Theorem 3.7) gives the following: For each $\boldsymbol {m}\in \Lambda (\boldsymbol {k})$ , there exist $\frac {1}{4}L^{\kappa _1}\leq \tilde {L} \leq L^{\kappa _2}$ , $Q_{\boldsymbol {m}} \in \mathcal {ER}_{\mathbb Z^d}(\tilde L)$ and $\tilde {Q}_{\boldsymbol {m}}$ , such that

$$ \begin{align*} &Q_{\boldsymbol{m}} \subset \boldsymbol{m}+[-\tilde{L}, \tilde{L}]^d, \ \boldsymbol{m}+[-\tilde{L}^{\frac{1}{10d}}, \tilde{L}^{\frac{1}{10d}}]^d\subset\tilde{Q}_{\boldsymbol{m}},\\ & \boldsymbol{m}\in Q_{\boldsymbol{m}} \subset \Lambda(\boldsymbol{k}), \mathrm{dist}(\boldsymbol{m}, \Lambda({\boldsymbol{k}})\setminus Q_{\boldsymbol{m}})\geq \frac{ \tilde{L}}{2},\ \mathrm{ diam}\ \tilde Q_{\boldsymbol{m}} \leq 4 \tilde{L}^{\frac{1}{10d}}. \end{align*} $$

Also for any $ \boldsymbol {n}\in Q_{\boldsymbol {m}}\setminus \tilde {Q}_{\boldsymbol {m}}$ , there exist $L_1\sim (\log L)^{\frac {4}{\kappa _1}}$ and some $\mathcal {ER}_{\mathbb Z^d}(L_1)\ni W \subset Q_{\boldsymbol {m}}\setminus \tilde {Q}_{\boldsymbol {m}}$ such that

$$ \begin{align*} \boldsymbol{n}\in W,\ \mathrm{dist}(\boldsymbol{n}, Q_{\boldsymbol{m}}\setminus \tilde{Q}_{\boldsymbol{m}}\setminus W)\geq \frac{L_1}{2}, \end{align*} $$

and

$$ \begin{align*} \|\mathcal{H}^{-1}_{W}(E_{\boldsymbol{k}};\boldsymbol{\theta})\|&\leq e^{\sqrt{L_1}},\\ |\mathcal{H}^{-1}_{W}(E_{\boldsymbol{k}};\boldsymbol{\theta})(\boldsymbol{n}; \boldsymbol{n}')|&\leq e^{-\frac12|\log\varepsilon|\cdot|\boldsymbol{n}-\boldsymbol{n}'|}\ \mathrm{for}\ |\boldsymbol{n}-\boldsymbol{n}'|\geq {L_1^{\frac89}}. \end{align*} $$

Next, assume that (set $Q=Q_{\boldsymbol {m}}$ )

$$ \begin{align*} \min_{\boldsymbol{m}\in\Lambda(\boldsymbol{k}), \xi=\pm1, 1\leq \ell\leq \#Q_{\boldsymbol{m}}}|\sigma+\boldsymbol{k}\cdot\boldsymbol{\omega}+\xi{\lambda_\ell}|>e^{-\frac 14 \tilde L^{\frac{3}{4}}}. \end{align*} $$

Then applying Lemma B.6 of [Reference Jitomirskaya, Liu and ShiJLS20] yields for

$$ \begin{align*}A_{\boldsymbol{k}}:=A_{\boldsymbol{k}}(\sigma)=A_{\boldsymbol{k}, \Lambda(\boldsymbol{k})}(\sigma),\end{align*} $$

the following estimates

$$ \begin{align*} \|A^{-1}_{\boldsymbol{k}}\|&\leq e^{\frac 12 L^{\frac{3}{4}}},\\ |A_{\boldsymbol{k}}^{-1}(\boldsymbol{n};\boldsymbol{n}')|&\leq e^{-\frac14|\log\varepsilon|\cdot|\boldsymbol{n}-\boldsymbol{n}'|}\ \mathrm{for}\ |\boldsymbol{n}-\boldsymbol{n}'|\geq {L}^{\frac{8}{9}}. \end{align*} $$

Note that

$$ \begin{align*} \tilde G_{\Lambda}(\sigma)=\bigoplus_{\boldsymbol{k}\in\Pi_b\Lambda}A_{\boldsymbol{k}}^{-1}, \end{align*} $$

where

$$ \begin{align*} \tilde G_{\Lambda}(\sigma)=\left(R_{\Lambda} ({D}(\sigma)+\varepsilon(\Delta\oplus\Delta))R_{\Lambda}\right)^{-1}. \end{align*} $$

By taking into account all $\boldsymbol {k}\in \Pi _b\Lambda ,$ we obtain that if

(2.29) $$ \begin{align} \min_{\boldsymbol{k}\in\Pi_b\Lambda, \boldsymbol{m}\in\Lambda(\boldsymbol{k}), \xi=\pm1, 1\leq \ell\leq \#Q_{\boldsymbol{m}}}|\sigma+\boldsymbol{k}\cdot\boldsymbol{\omega}+\xi{\lambda_\ell}|> e^{-\frac 14 L^{\frac{3\kappa_1}{4}}}, \end{align} $$

then

$$ \begin{align*} \|\tilde G_{\Lambda}(\sigma)\|&\leq e^{\frac12L^{\frac{3}{4}}},\\ |\tilde G_{\Lambda}(\sigma)((\boldsymbol{k}, \boldsymbol{n}, \xi);(\boldsymbol{k}', \boldsymbol{n}', \xi'))|&\leq \delta_{\boldsymbol{k}, \boldsymbol{k}'}e^{-\frac14|\log\varepsilon| \cdot|\boldsymbol{n}-\boldsymbol{n}'|}\ \mathrm{for}\ |\boldsymbol{n}-\boldsymbol{n}'|\geq L^{\frac{8}{9}}. \end{align*} $$

Note that the total number of $\lambda _\ell $ in (2.29) is at most

$$ \begin{align*}C(b, d)L^b L^dL^{\kappa_2}L^{\kappa_2d}\ll L^{b+2d}.\end{align*} $$

Finally, since $|\boldsymbol {n}|>10L$ , we know that S has the decay estimate

$$ \begin{align*} |S((\boldsymbol{x}, \xi);(\boldsymbol{x}', \xi'))|\leq C_2(1+|\boldsymbol{x}-\boldsymbol{x}'|)^{C_2}e^{-6\gamma L}e^{-2\gamma |\boldsymbol{x}-\boldsymbol{x}'|}. \end{align*} $$

Then using the perturbation Lemma B.1 implies that $\Lambda $ is $\sigma $ -good assuming (2.29) holds true. In fact, we can apply Lemma B.1 (we hide the dependence on $\xi , \xi '$ ) with $B=\delta S_\Lambda (\boldsymbol {x}, \boldsymbol {x}')$ , $A^{-1}(\boldsymbol {x}, \boldsymbol {x}')=\tilde G_{\Lambda }(\boldsymbol {x}, \boldsymbol {x}')$ and $c=2\gamma , \epsilon _1=e^{-\frac 12 L^{\frac 34}}, M=L^{\frac 89}, \epsilon _2=C\delta e^{-6\gamma L}$ . Since $\gamma \in (\frac 12, 10)$ , we obtain for $L\geq N_0(b,d, C_2)\gg~1,$

$$ \begin{align*}C(b,d, C_2)L^{2(b+d)+C_2}\delta e^{-6\gamma L}e^{2\gamma L^{\frac89}}e^{\frac 12 L^{\frac34}} \leq \frac12.\end{align*} $$

This verifies all assumptions of Lemma B.1. So using Lemma B.1, we get

$$ \begin{align*} \|G_{\Lambda}(\sigma)\|&\leq 2 e^{\frac12L^{\frac{3}{4}}}<e^{L^{\frac34}}, \end{align*} $$

and for $|\boldsymbol {x}-\boldsymbol {x}'|\geq L^{\frac {8}{9}}$ (since $\frac 14 |\log \varepsilon |>2\gamma >1$ ),

$$ \begin{align*} |G_{\Lambda}(\sigma)((\boldsymbol{x}, \xi);(\boldsymbol{x}', \xi'))|&\leq |\tilde G_{\Lambda}(\sigma)((\boldsymbol{x}, \xi);(\boldsymbol{x}', \xi'))|+ e^{\frac 12 L^{\frac34}}e^{-2\gamma |\boldsymbol{x}-\boldsymbol{x}'|} \\ &\leq e^{L^{\frac34}-L^{\frac89}}e^{-\gamma |\boldsymbol{x}-\boldsymbol{x}'|}\\ &\leq e^{-\gamma |\boldsymbol{x}-\boldsymbol{x}'|}. \end{align*} $$

This proves (1) of Lemma 2.23.

Remark 2.24.

• ○ Let $L\leq N\leq e^{(\log L)^2}$ . We would also like to remark that if we take into account all these $\Lambda \in (\boldsymbol {k}, \boldsymbol {n})+\mathcal {ER}_{\boldsymbol {0}}(L)$ with $|(\boldsymbol {k}, \boldsymbol {n})|\leq 100N^2$ and $|\boldsymbol {n}|>10L$ (the total number of these $\Lambda $ is at most $C(b,d)N^{2(b+d)}$ ), then we get a sequence of real numbers $\{{\lambda _\ell (\boldsymbol {\alpha }, \boldsymbol {\theta })}\}_{1\leq \ell \leq N^{4(b+d)}}$ . As a result, there is a set $\tilde \Sigma _{L}\subset \mathbb R$ satisfying

  $$ \begin{align*}\mathrm{ meas}(\tilde\Sigma_{L})\leq e^{C(b,d)\log^2 L}e^{-\frac14 L^{\frac{3}{4}\kappa_1}}\leq e^{-\frac15 L^{\frac{3}{4}\kappa_1}}\end{align*} $$
  so that, for $\sigma \not \in \tilde \Sigma _{L}$ , all $\Lambda \in (\boldsymbol {k}, \boldsymbol {n})+\mathcal {ER}_{\boldsymbol {0}}(L)$ satisfying $|(\boldsymbol {k}, \boldsymbol {n})|\leq 100N^2$ and $|\boldsymbol {n}|>10L$ are $\sigma $ -good.

• ○ In the application, we typically choose $L=N_1\geq 10^{-\frac {32}{\kappa _1^2}}\log ^{\frac {4}{\kappa _1}}(\log \frac 1\varepsilon )$ . It then follows from $ N_1\sim N^{\rho ^2}\geq (\varepsilon +\delta )^{-c_1\rho ^2}$ that a restriction like $\delta \leq \log ^{-1}\frac 1\varepsilon $ is needed.

(2) The conclusion is just that in Remark 2.24.

This completes the proof of Lemma 2.23.

By combining Lemma 2.22 and Lemma 2.23, we are ready to prove LDE at the scale N (indeed all scales in $[N, N^2]$ ). We first recall an important lemma.

Lemma 2.25 (Matrix-valued Cartan’s lemma, cf. Proposition 14.1 in [Reference BourgainBou05a])

Let $T(\sigma )$ be a self-adjoint $N\times N$ matrix-valued function of a parameter $\sigma \in [-\beta , \beta ]$ satisfying the following conditions:

1. (i) $T(\sigma )$ is real analytic in $\sigma $ and has a holomorphic extension to

   $$ \begin{align*} \mathbb{D}=\left\{z\in\mathbb{C}: \ |\Re z|\leq \beta,\ |\Im z|\leq \beta_1\right\} \end{align*} $$
   satisfying $\sup \limits _{z\in \mathbb {D}}\|T(z)\|\leq K_1,\ K_1\geq 1.$

2. (ii) For each $\sigma \in [-\beta , \beta ]$ , there is a subset $Y\subset [1,N]$ with $\#Y\leq M$ such that

   $$ \begin{align*} \|(R_{[1,N]\setminus Y}T(\sigma)R_{[1,N]\setminus Y})^{-1}\|\leq K_2, \ K_2\geq 1. \end{align*} $$

3. (iii) Assume

   $$ \begin{align*} \mathrm{meas}\ \left(\{\sigma\in[-{\beta}, {\beta}]: \ \|T^{-1}(\sigma)\|\geq K_3\}\right)\leq 10^{-3}\beta_1(1+K_1)^{-1}(1+K_2)^{-1}. \end{align*} $$

Let $0<\epsilon \leq (1+K_1+K_2)^{-10 M}.$ Then we have

(2.30) $$ \begin{align} \mathrm{meas}\ \left(\left\{\sigma\in\left[-{\beta}/{2}, {\beta}/{2}\right]:\ \|T^{-1}(\sigma)\|\geq \frac1\epsilon\right\}\right)\leq Ce^{\frac{-c\log \frac1\epsilon}{M\log( M+K_1+K_2+K_3)}}, \end{align} $$

where $C, c>0$ are some absolute constants.

Now, we can prove Lemma 2.20.

Proof of Lemma 2.20

Recalling

$$ \begin{align*} N_2=N_1^{\frac 2\rho},\ N=N_1^{\frac 1{\rho^2}}, \end{align*} $$

we assume that for $L=N_1, N_2$ and $\Lambda \in (\boldsymbol {0}, \boldsymbol {n})+\mathcal {ER}_{\boldsymbol {0}}(L)$ with $|\boldsymbol {n}|\leq 10L$ , $G_{\Lambda }(\sigma )$ satisfies the LDE.

Without loss of generality, we only establish LDE for the Green’s function on $\Lambda _{\mathrm {pm}, N}=\{(\boldsymbol {x}, \xi )\in \mathbb Z_{\mathrm {pm}, *}^{b+d}:\ |\boldsymbol {x}|\leq N \}= (\Lambda _{N}\times \{+, -\})\cap \mathbb Z_{\mathrm {pm}, *}^{b+d}$ , the other cases (i.e., $\Lambda \in (\boldsymbol {0}, \boldsymbol {n})+{\mathcal {ER}_{\boldsymbol {0}}(N_3)}$ for $|\boldsymbol {n}|\leq 10N_3, N_3\in [N, N^2]$ ) can be dealt with similarly. We define three intermediate scales between $[N_2, N]$ :

(2.31) $$ \begin{align} L_1=N^{\frac{1}{100(b+d)^2}},\ L_2=N^{\frac{1}{50(b+d)^2}},\ L_3= N^{\frac{1}{10(b+d)}}. \end{align} $$

Note that from $0<\rho \leq c_2(b,d)\leq {\frac {1}{10^4(b+d)^2}}$ (cf. Lemma 2.22), we have

$$ \begin{align*}\log N_1\ll\log N_2\ll \log L_1= \frac{1}{100(b+d)^2}\log N.\end{align*} $$

We first cover the $\Lambda _{\mathrm {pm}, N}$ with two overlapped subregions, $\Lambda _{\mathrm {pm}, N}=\mathcal {R}_1\cup \mathcal {R}_2$ satisfying

$$ \begin{align*} \mathcal{R}_1=\{(\boldsymbol{k}, \boldsymbol{n}, \xi )\in\Lambda_{\mathrm{pm}, N}:\ |\boldsymbol{n}|\leq L_2\},\\ \mathcal{R}_2=\{(\boldsymbol{k}, \boldsymbol{n}, \xi )\in\Lambda_{\mathrm{pm}, N}:\ |\boldsymbol{n}|\geq L_1\}. \end{align*} $$

We will apply matrix-valued Cartan’s lemma (cf. Lemma 2.25) and Bourgain’s geometric lemma [Reference BourgainBou07] (cf. Lemma 2.5) to obtain the sub-exponential growth estimates on Green’s functions. Then we apply the coupling lemmas of [Reference Bourgain, Goldstein and SchlagBGS02, Reference Jitomirskaya, Liu and ShiJLS20, Reference LiuLiu22] (cf., e.g., Lemmas B.3, B.5, B.6) based on the iteration of resolvent identities to prove the off-diagonal exponential decay of $G_{\Lambda _{\mathrm {pm}, N}}(\sigma )$ if desired $\sigma $ are removed.

Before we enter the details, let us first explain why we need to divide $\Lambda _{\mathrm {pm}, N}$ into the above two regions. This is quite different from that in [Reference Shi and WangSW23]. Based on Lemma 2.23, we can impose the so-called weak second Melnikov’s condition on $\boldsymbol {\omega }$ so that all $\sigma $ -bad (cf. Definition 2.21) $\Lambda \in (\boldsymbol {k}, \boldsymbol {n})+\mathcal {ER}_{\boldsymbol {0}}(N_1)\subset \Lambda _{\mathrm {pm}, N}$ can stay only in a narrow strip region (cf. Figure 1), namely, there is some $\boldsymbol {k}_*\in [-N, N]^b$ so that

$$ \begin{align*}|\boldsymbol{k}-\boldsymbol{k}_*|\leq 10N_1\leq N^{0+}.\end{align*} $$

This gives the desired control on $\sigma $ -bad $\Lambda $ in the $\boldsymbol {k}$ -direction. To establish the sublinear bound, it remains to handle the $\boldsymbol {n}$ -direction. In [Reference Shi and WangSW23], the $\boldsymbol {n}$ -direction estimates can be obtained directly using the fact that $\theta \in \mathbb R$ , Diophantine $\boldsymbol {\omega }$ and LDT for the corresponding quasi-periodic Schrödinger operators. Such arguments inevitably fail for $\boldsymbol {\theta }\in \mathbb R^d$ : There is simply no one-dimensional interval decomposition of d-dimensional semi-algebraic sets, and more essentially, the Diophantine property of $\boldsymbol {\omega }$ cannot ensure the sublinear estimate in the $\boldsymbol {n}$ -direction. This motivates us to restrict $|\boldsymbol {n}|\leq L_2$ (i.e., the region $\mathcal {R}_1$ ). We can use the Cartan’s lemma and resolvent identities to deal with $\mathcal {R}_1$ . For $\mathcal {R}_2$ , we take advantage of Bourgain’s geometric lemma and the short-range property of H using Lemma 2.23. Finally, it suffices to cover $\Lambda _{\mathrm {pm}, N}$ with four types of regions of sizes $N_1, L_1^{\frac {9}{10}}, L_1, L_3$ (will be specified below) contained in $\mathcal {R}_1$ , $\mathcal {R}_2$ , and apply the resolvent identities again. See Figure 1.

We first deal with $\mathcal R_2$ , and then $\mathcal R_1$ .

Analysis in $\mathcal {R}_2$

In this region, we can apply directly the conclusion of Lemma 2.23. In this case, we make no restriction on $\boldsymbol {\omega }.$ We let

$$ \begin{align*} L_0 =L_1^{\frac {9}{10}}= N^{\frac{9}{1000(b+d)^2}}. \end{align*} $$

For any $\Lambda \subset \mathcal R_2$ satisfying $\Lambda =(\boldsymbol {k}, \boldsymbol {n})+\mathcal {ER}(L_0)$ , we have $|\boldsymbol {n}|>L_1>10L_0$ . Then applying Lemma 2.23 with $L=L_0$ and since $0<\rho <c_2(b,d)\kappa _1\leq \frac {\kappa _1}{10^4(b+d)^2},$ we get a set $\tilde \Sigma _{N, 1}\subset \mathbb R$ satisfying

$$ \begin{align*}\mathrm{meas}(\tilde\Sigma_{N,1})\leq e^{-\frac15 L_0^{\frac{3\kappa_1}{4}}}\leq e^{-\frac15N^{\frac{27\kappa_1}{4000(b+d)^2}}}<e^{-N^{3\rho}},\end{align*} $$

so that, if $\sigma \notin \tilde \Sigma _{N,1}$ , all $\Lambda \in \mathcal {ER}(L_0)$ contained in $ \mathcal R_2$ are $\sigma $ -good (cf. Definition 2.21).

We remark that in this case, we do not use Cartan’s lemma.

Analysis in $\mathbf {\mathcal {R}_1}$

In this region, we need to make further restriction on $\boldsymbol {\omega }$ called the weak second Melnikov’s condition. More precisely, we have

Lemma 2.26. There is a sequence of real numbers $\{\lambda _{\ell }=\lambda _\ell (\boldsymbol {\alpha }, \boldsymbol {\theta })\}_{1\leq \ell \leq N^{4(b+d)}}$ depending only on $V, \boldsymbol {\alpha }, \boldsymbol {\theta }$ so that the following holds true. If $\boldsymbol {\omega }\in \widetilde \Omega _{N}$ with

(2.32) $$ \begin{align} \nonumber&\ \ \ \widetilde\Omega_N\\ &:=\bigcap_{1\leq \ell,\ell'\leq N^{4(b+d)}, \xi=\pm1, \xi'=\pm1, 0<|\boldsymbol{k}|\leq 2N^2}\left\{\boldsymbol{\omega}\in\Omega:\ |\boldsymbol{k}\cdot\boldsymbol{\omega}+\xi {\lambda_\ell}-\xi'{\lambda_{\ell'}}|>10 e^{-\frac 14 N^{\frac{3\kappa_1\rho^2}{4}}}\right\}, \end{align} $$

then for any $\sigma \in \mathbb R,$ there is some $\boldsymbol {k}_*\in [-N^2, N^2]^b$ so that, all $\sigma $ -bad $\Lambda \in (\boldsymbol {k}, \boldsymbol {n})+\mathcal {ER}_{\boldsymbol {0}}(N_1)$ satisfying $|\boldsymbol {k}|\leq N^2$ and $10N_1\leq |\boldsymbol {n}|\leq 100N^2$ must obey

(2.33) $$ \begin{align} \Pi_b \Lambda\subset [\boldsymbol{k}_*-10N_1, \boldsymbol{k}_*+10N_1]^b. \end{align} $$

In particular, we have

(2.34) $$ \begin{align} \mathrm{meas}(\Omega\setminus\widetilde\Omega_N) \leq e^{-\frac 15 N^{\frac{3\kappa_1\rho^2}{4}}}. \end{align} $$

Remark 2.27. Indeed, in this lemma, $\lambda _\ell (\boldsymbol {\alpha }, \boldsymbol {\theta })$ ( ${1\leq \ell \leq N^{4(b+d)}})$ are eigenvalues of the operator $\mathcal H_{Q}(\boldsymbol {\alpha }, \boldsymbol {\theta })$ for all $Q\subset [-100N^2, 100N^2]^d$ satisfying $Q\in \mathcal {ER}_{\mathbb Z^d}(\tilde N_1)$ for some $\tilde N_1\in [\frac 14N_1^{\kappa _1}, N_1^{\kappa _2}]$ .

Proof of Lemma 2.26

The proof follows from Lemma 2.23. Applying (1) of Lemma 2.23 (cf. also Remark 2.24) with $L=N_1$ and taking into account of all $\Lambda \in (\boldsymbol {k}, \boldsymbol {n})+\mathcal {ER}_{\boldsymbol {0}}(N_1)$ with $|(\boldsymbol {k},\boldsymbol {n})|\leq 100N^2$ and $|\boldsymbol {n}|>10N_1$ yield a sequence $\{{\lambda _\ell }\}_{\leq \ell \leq N^{4(b+d)}}\subset \mathbb R$ and each $\lambda _\ell $ depending only on $\boldsymbol {\alpha }, \boldsymbol {\theta }$ (but not on $\sigma ,\boldsymbol {\omega }$ ), so that if $\Lambda =(\boldsymbol {k}, \boldsymbol {n})+\mathcal {ER}_{\boldsymbol {0}}(N_1)$ satisfies $|(\boldsymbol {k}, \boldsymbol {n})|\leq 100N^2$ , $|\boldsymbol {n}|>10N_1$ and

$$ \begin{align*}\min_{\boldsymbol{k}\in\Pi_b\Lambda, \xi=\pm1, 1\leq \ell\leq N^{4(b+d)}}|\sigma+\boldsymbol{k}\cdot\boldsymbol{\omega}+\xi {\lambda_\ell}|>e^{-\frac 14 N_1^{\frac{3\kappa_1}{4}}}= e^{-\frac 14 N^{\frac{3\kappa_1\rho^2}{4}}},\end{align*} $$

then $\Lambda $ is $\sigma $ -good. So recalling (2.32), we have

$$ \begin{align*}\mathrm{meas}(\Omega\setminus \widetilde\Omega_N)\leq N^{C(b,d)}e^{-\frac 14 N^{\frac{3\kappa_1\rho^2}{4}}}\leq e^{-\frac 15 N^{\frac{3\kappa_1\rho^2}{4}}},\end{align*} $$

which implies (2.34).

In the following, we assume $\boldsymbol {\omega }\in \widetilde \Omega _N.$

Suppose now $\Lambda '$ with $\Lambda '\in (\boldsymbol {k}', \boldsymbol {n}')+\mathcal {ER}_{\boldsymbol {0}}(N_1)$ satisfying $|\boldsymbol {k}'|\leq N^2$ and $10N_1<|\boldsymbol {n}'|\leq 100N^2$ is $\sigma $ -bad. Then there are some $\boldsymbol {k}_*\in \Pi _b\Lambda '\subset [-N^2, N^2]^b$ , $\xi _*=\pm 1$ and some $\ell _*\in [1, N^{4(b+d)}]$ so that

$$ \begin{align*} |\sigma+\boldsymbol{k}_*\cdot\boldsymbol{\omega}+\xi_*{\lambda_{\ell_*}}|\leq e^{-\frac14 N_1^{\frac{3\kappa_1}{4}}}. \end{align*} $$

Now let $\Lambda ''$ (satisfying $\Lambda ''\in (\boldsymbol {k}'', \boldsymbol {n}'')+\mathcal {ER}_{\boldsymbol {0}}(N_1)$ with $|\boldsymbol {k}''|\leq N^2$ and $10N_1<|\boldsymbol {n}''|\leq 100N^2$ ) be another $\sigma $ -bad region. From $\boldsymbol {\omega }\in \widetilde \Omega _N,$ we must have

$$ \begin{align*} \boldsymbol{k}_*\in \Pi_b\Lambda". \end{align*} $$

Otherwise, there must be some $\boldsymbol {k}'''\in \Pi _b\Lambda '', \xi '\in \{\pm 1\}, \ell '$ so that, $\boldsymbol {k}_*\neq \boldsymbol {k}'''$ and

$$ \begin{align*}|\sigma+\boldsymbol{k}"'\cdot\boldsymbol{\omega}+\xi'{\lambda_{\ell'}}|\leq e^{-\frac14 N_1^{\frac{3\kappa_1}{4}}},\end{align*} $$

which contradicts $\boldsymbol {\omega }\in \widetilde \Omega _N$ . We have established that all $\sigma $ -bad elementary $N_1$ -regions $\Lambda $ with centers $(\boldsymbol {k}, \boldsymbol {n})$ satisfying $|\boldsymbol {k}|\leq N^2$ and $10N_1<|\boldsymbol {n}|\leq 100N^2$ must satisfy (2.33).

In the following, we always assume $\boldsymbol {\omega }\in \widetilde \Omega _N$ . Under this condition, we first give a geometric construction of the a region $\Lambda ^\dagger $ , on which the worst resonances appear. It turns out that this region has the annulus structure similar to [Reference BourgainBou07, Reference Jitomirskaya, Liu and ShiJLS20].

Lemma 2.28. Let $\boldsymbol {\omega }\in \widetilde \Omega _N$ and $\sigma \in \mathbb R$ . Let $\boldsymbol {k}_*\in [-N, N]^b$ (if it exists) be defined by Lemma 2.26. Then there are some $\Lambda ^{\dagger }_1\subset \Lambda ^{\dagger }\subset \mathcal R_1$ satisfying $\mathrm {diam} \ \Lambda _1^\dagger \leq 4L_1, \boldsymbol {k}_*\in \Lambda _1^\dagger $ and

$$ \begin{align*} \Lambda^{\dagger}&=((\boldsymbol{k}_*+[-L_3+O(L_1), L_3+O(L_1)]^b)\times [-L_2, L_2]^d)\cap \mathcal R_1, \end{align*} $$

so that the following holds true (recalling $L_1\ll L_2\ll L_3$ given by (2.31)). For any $\boldsymbol {x}\in \Lambda ^{\dagger }\setminus \Lambda ^{\dagger }_1$ , there are some $L\in \{N_1, L_1, L_0\}$ and some $\mathcal {ER}(L)\ni W\subset \Lambda ^{\dagger }\setminus \Lambda ^{\dagger }_1$ so that

$$ \begin{align*}\boldsymbol{x}\in W,\ \mathrm{dist}(\boldsymbol{x}, \Lambda^{\dagger}\setminus \Lambda^{\dagger}_1\setminus W)\geq \frac{L}{2}.\end{align*} $$

Proof. The proof is similar to that of Theorem 3.7 of [Reference Jitomirskaya, Liu and ShiJLS20] (cf. pages 471–472). We initially set $Q^\dagger =(\boldsymbol {k}_*+[-L_3, L_3]^b)\times [-L_2, L_2]^d$ and $Q_1=(\boldsymbol {k}_*+[-L_1, L_1]^b)\times [-L_1, L_1]^d$ . Then we slightly change $Q^\dagger , Q_1^\dagger $ to $\Lambda ^\dagger , \Lambda _1^\dagger $ with the diameter modulations of length $2L_1$ if the boundary of $Q^\dagger , Q_1^\dagger $ approaches the boundary of $\mathcal R_1$ with the distance of $0<L\leq 2L_1$ .

Next, let $\boldsymbol {x}=(\boldsymbol {k}, \boldsymbol {n})\in \Lambda ^{\dagger }\setminus \Lambda _1^{\dagger }$ . For $|\boldsymbol {n}|\leq 10N_1$ , we use $L_1$ size regions to cover $\boldsymbol {x}$ ; for $10N_1<|\boldsymbol {n}|\leq L_1$ , we use the $N_1$ size regions to cover $\boldsymbol {x}$ ; for $L_1< |\boldsymbol {n}|\leq L_2$ , we use the $L_0$ size regions to cover $\boldsymbol {x}.$

Next, we show that all $Q\in (\boldsymbol {k}, \boldsymbol {0})+\mathcal {ER}_{\boldsymbol {0}}(L_1)$ satisfying $|\boldsymbol {k}|\leq N$ and $|\boldsymbol {k}-\boldsymbol {k}_*|>10N_1$ are $\sigma $ -good for “most” $\sigma $ , by using Cartan’s lemma and coupling lemma of [Reference LiuLiu22]. More precisely, we have

Lemma 2.29. Let $\boldsymbol {\omega }\in \widetilde \Omega _N\cap \Omega _{N_2}$ . There is some $\tilde \Sigma _{N,2}\subset \mathbb R$ satisfying

$$ \begin{align*}\mathrm{meas}(\tilde\Sigma_{N,2})\leq e^{-N^{3\rho}},\end{align*} $$

so that for $\sigma \notin \tilde \Sigma _{N, 2}$ , the following holds true. If $Q\subset \mathcal R_1$ satisfies

$$ \begin{align*}Q\in \mathcal Q:=\bigcup_{|\boldsymbol{k}-\boldsymbol{k}_*|>10N_1, |\boldsymbol{k}|\leq N}\left\{(\boldsymbol{k},\boldsymbol{0})+Q:\ Q\in\mathcal{ER}_{\boldsymbol{0}}(L_1)\right\},\end{align*} $$

then Q is $\sigma $ -good with $\gamma _{L_1}=\gamma _{N_1}-N_1^{-\vartheta }$ , where $\vartheta>0$ is an absolute constant defined in Lemma B.3.

Proof of Lemma 2.29

We choose any $Q\in \mathcal {ER}(L_1)$ with $Q\in \mathcal Q$ . Then we take any $Q'\in \mathcal R_{L_*}^{{\sqrt {L_1}}}$ satisfying $Q'\subset Q$ and $\sqrt {{L_1}}\leq L_*\leq 2L_1$ . From Lemma 2.22 and $\boldsymbol {\omega }\in \widetilde \Omega _N$ , we have for any $\sigma \in \mathbb R,$

$$ \begin{align*}\#\{\Lambda\in \mathcal{ER}(N_1):\ \Lambda\ \mathrm{is}\ \sigma{-\mathrm{bad}}, \ \Lambda\subset Q'\}\leq N^{\frac{1}{10^3(b+d)^2}}.\end{align*} $$

Denote by $Y_1(\sigma ):=\bigcup \limits _{1\leq \ell \leq k} Q_{\boldsymbol {x}_\ell }, k\leq N^{\frac {1}{10^3(b+d)^2}}$ all those $\sigma $ -bad $N_1$ size elementary regions. We claim that, there is some $Y=Y(\sigma )\subset Q'$ satisfying $\#Y\leq C(b,d)N_1^{b+d} N^{\frac {1}{10^3(b+d)^2}}$ , so that if $\boldsymbol {x}\in Q'\setminus Y$ , then there is some $\sigma $ -good $W\in \mathcal {ER}(N_1), W\subset Q'\setminus Y$ satisfying

$$ \begin{align*}\boldsymbol{x}\in W, \ \mathrm{dist}(\boldsymbol{x}, Q'\setminus Y\setminus W)\geq \frac{N_1}{2}.\end{align*} $$

Indeed, we pave $Q'$ with $\Lambda _\alpha $ for $\Lambda _\alpha =Q'\cap \Lambda _{10N_1}(\boldsymbol {x}), \boldsymbol {x}\in 10N_1\mathbb Z^{b+d}$ . Then we can enlarge $\Lambda _\alpha $ to $\tilde \Lambda _\alpha \subset Q'$ so that $\tilde \Lambda _\alpha $ has width at least $N_1$ and diameter at most $C(b,d)N_1$ , and $\bigcup \limits _{\alpha }\tilde \Lambda _\alpha $ remains a tiling of $Q'$ . It suffices to take

$$ \begin{align*}Y(\sigma)=\bigcup\limits_{\alpha:\ \tilde \Lambda_\alpha\cap Y_1(\sigma)\neq\emptyset}\tilde \Lambda_\alpha.\end{align*} $$

Since each $Q_{\boldsymbol {x}_\ell }$ can only intersect with at most $C(b,d)$ many $\tilde \Lambda _\alpha $ , we have

$$ \begin{align*}\#Y(\sigma)\leq C(b,d)N_1^{b+d}N^{\frac{1}{10^3(b+d)^2}}.\end{align*} $$

We are ready to apply Cartan’s lemma (cf. Lemma 2.25). From Lemma B.1, any $\sigma $ -good $N_1$ region remains essentially $\sigma '$ -good if $|\sigma '-\sigma |\leq e^{-10\gamma N_1}$ . So on an interval I of length at most $e^{-10\gamma N_1}$ , we can fix Y to associate with the midpoint of I. In addition, we can assume $|\sigma |\leq C(b,d)N$ since if $|\sigma |>CN$ , then $\|D_{Q'}^{-1}(\sigma )\|\leq 1$ . In this case, $Q'$ becomes $\sigma $ -good via the Neumann series argument. We apply Lemma 2.25 with

$$ \begin{align*} T(\sigma)={{H}}_{Q'}(\sigma), \ \beta=\beta_1=e^{- 10\gamma N_1}. \end{align*} $$

It remains to verify the assumptions of Lemma 2.25. First, one has by the definition of the operator $H(\sigma ),$

$$ \begin{align*} K_1=CN, C=C(V,b,d,C_2)>0. \end{align*} $$

We let Y be defined in the above claim. Then using sublinear bound conclusion and Lemma B.5, we have

$$ \begin{align*} &M=\#Y\leq C(b,d)N^{\rho^2(b+d)}N^{\frac{1}{10^3(b+d)^2}}<N^{\frac{1}{999(b+d)^2}}\leq L_*^{ \frac14}\\ & (\mathrm{since}\ \rho\leq c_2\leq \frac{1}{10^4(b+d)^2},\ L_*\geq N^{\frac{1}{200(b+d)^2}}), \\ &K_2=e^{2{N_1}^{\frac{3}{4}}}. \end{align*} $$

To apply the matrix-valued Cartan’s lemma, we need the scale

$$ \begin{align*}N_2=N_1^{\frac2\rho}.\end{align*} $$

Recall that the LDE hold at scale $N_2$ . Applying the resolvent identity Lemma B.5 and using (2) of Lemma 2.23 with $L=N_2$ yield

$$ \begin{align*} \|T^{-1}(\sigma)\| \leq e^{2{N_2}^{\frac{3}{4}}}=K_3 \end{align*} $$

for $\sigma $ away from a set of measure at most (since $0<\rho <c_2\kappa _1$ )

$$ \begin{align*}CN^{C(b,d)}e^{-{N_2^{\rho}}}+e^{-\frac15{N_2^{\frac{3\kappa_1}{4}}}}\leq e^{-{N_2^{\rho}}/{2}}.\end{align*} $$

It follows from $N_1=N_2^{\frac \rho 2}$ that

$$ \begin{align*} 10^{-3}\beta_1(1+K_1)^{-1}(1+K_2)^{-1}&\geq e^{-10\gamma N_1}N^{-2} e^{-2N_1^{\frac34}}\\ &> e^{-{N_2^{\rho}}/{2}}. \end{align*} $$

This verifies (iii) of Lemma 2.25. If $\epsilon =e^{- {L_*}^{\frac {1}{2}}}$ , then one has $\epsilon <(1+K_1+K_2)^{-10M}.$ Cover $[-CN, CN]$ with disjoint intervals of length $e^{- 10\gamma N_1}$ . Define

$$ \begin{align*} \tilde \Sigma_{Q'}=\{\sigma\in\mathbb R:\ \|G_{Q'}(\sigma)\|\geq e^{L_*^{\frac{3}{4}}}\}. \end{align*} $$

Then by (2.30) of Lemma 2.25, one obtains

$$ \begin{align*} \mathrm{meas}(\tilde \Sigma_{Q'})\leq CN^Ce^{ 10\gamma N_1}e^{-\frac{c{{L_*}^{\frac{3}{4}}}}{CN_1N_2 L_*^{\frac{1}{4}}\log N}}\leq e^{-{L_*}^{\frac13} }\ll e^{-N^{3\rho}}. \end{align*} $$

Note that the number of $Q'\subset Q$ with $Q'\in \mathcal {R}_{L_*}^{\sqrt {L_1}}, \sqrt {{L_1}}\leq L_*\leq 2L_1$ is at most $L^{C(b,d)}$ and the number of $Q\in \mathcal Q$ is at most $N^{(b,d)}$ . We take the union over all these $Q'\subset Q, Q\in \mathcal Q$ leading to the desired $\tilde \Sigma _{N,2}\subset \mathbb R$ with (since $0<\rho <c_2\leq \frac {1}{10^4(b+d)^2}$ )

$$ \begin{align*} \mathrm{meas} (\tilde \Sigma_{N,2})\leq N^{C(b,d)}e^{-L_1^{\frac{1}{6}}}<e^{-N^{\frac{1}{600(b+d)^2}}}\ll e^{-N^{3\rho}}. \end{align*} $$

To finish the proof, it suffices to prove the off-diagonal exponential decay of $G_Q(\sigma )$ for $Q\in \mathcal Q$ and $\sigma \notin \tilde \Sigma _{N,2}$ . This follows from using the coupling lemma of [Reference LiuLiu22]. Indeed, fix $Q\in \mathcal Q$ (then $Q\in \mathcal {ER}(L_1)$ ). Let $\mathcal F$ be any family of pairwise disjoint $\sqrt {L_1}$ size elementary regions in Q. Then using Lemma 2.22 and $\boldsymbol {\omega }\in \widetilde \Omega _N,$ we get

$$ \begin{align*}\# \{\Lambda\in\mathcal F:\ \Lambda \ \mathrm{is }\ \sigma{-\mathrm{bad}}\}\leq N^{\frac{1}{10^3(b+d)^2}}\leq L_1^{\frac19}.\\[-24pt]\end{align*} $$

So applying Lemma B.3 leads to the off-diagonal exponential decay of $G_Q(\sigma )$ with the decay rate

$$ \begin{align*}\gamma_{L_1}=\gamma_{N_1}-N_1^{-\vartheta}.\\[-34pt]\end{align*} $$

We then deal with $G_{\Lambda ^{\dagger }}(\sigma )$ with $\Lambda ^{\dagger }$ given by Lemma 2.28. We have

Lemma 2.30. Let $\boldsymbol {\omega }\in \widetilde \Omega _N\cap \Omega _{N_2}$ . Then there is some $\widetilde \Sigma _{N, 3}\subset \mathbb R$ satisfying

$$ \begin{align*}\mathrm{meas}(\tilde\Sigma_{N,3})<e^{-N^{3\rho}}\end{align*} $$

so that, if $\sigma \notin \bigcup \limits _{\ell =1, 2,3}\tilde \Sigma _{N,\ell },$ then $G_{\Lambda ^{\dagger }}(\sigma )$ is $\sigma $ -good in the sense of Definition 2.21 (replacing $\Lambda $ with $\Lambda ^\dagger $ ) with $\gamma _{L_2}=\gamma _{L_1}-N_1^{-\frac {1}{10}}$ .

Proof of Lemma 2.30

The proof is similar to that of Lemma 2.29, but is more in the spirit of [Reference Jitomirskaya, Liu and ShiJLS20] when dealing with off-diagonal exponential decay estimate, as $\Lambda ^{\dagger }$ has the special annulus structure.

To get the sub-exponential growth estimate of $\|G_{\Lambda ^\dagger }(\sigma )\|$ , the key point is to apply Cartan’s lemma only involving $N_1, N_2$ sizes regions similar to the proof of Lemma 2.29. This means precisely that we do not choose $Y=\Lambda _1^\dagger $ when applying Lemma 2.25.

From $\boldsymbol {\omega }\in \widetilde \Omega _N$ , the definition of $\Lambda ^\dagger $ (cf. Lemma 2.28) and Lemma 2.22, we have for any $\sigma \in \mathbb R,$

$$ \begin{align*} &\ \ \ \#\{\Lambda\in \mathcal{ER}(N_1):\ \Lambda\ \mathrm{is}\ \sigma{-\mathrm{bad}}, \ \Lambda\subset \Lambda^\dagger\}\\ &\leq N^{\frac{1}{10^3(b+d)^2}}+ C(b,d)N_1^{b}L_2^{d}\\ &\leq N^{\frac{1}{10^3(b+d)^2}}+C(b,d)N^{\rho^2b+\frac{1}{50(b+d)}}\\ &\leq 2N^{\rho^2b+\frac{1}{50(b+d)}}. \end{align*} $$

Similar to the proof of Lemma 2.29, we can find some $Y=Y(\sigma )\subset \Lambda ^\dagger $ satisfying $\#Y\leq C(b,d)N^{2\rho ^2(b+d)} N^{\frac {1}{50(b+d)}}<N^{\frac {1}{49(b+d)}}$ so that, if $\boldsymbol {x}\in \Lambda ^\dagger \setminus Y$ , then there is some $\sigma $ -good $W\in \mathcal {ER}(N_1), W\subset \Lambda ^\dagger \setminus Y$ satisfying

$$ \begin{align*}\boldsymbol{x}\in W,\ \mathrm{dist}(\boldsymbol{x}, \Lambda^\dagger\setminus Y\setminus W)\geq \frac{N_1}{2}.\end{align*} $$

Let $\boldsymbol {\omega }\in \widetilde \Omega _N$ . Again, both the site $\boldsymbol {k}_*\in [-N, N]^b$ (given in Lemma 2.26) and $Y(\sigma )$ depend on $\sigma $ . From Lemma B.1, any $\sigma $ -good $N_1$ size region remains essentially $\sigma '$ -good if $|\sigma '-\sigma |\leq e^{-10\gamma N_1}$ . So on an interval I of length at most $e^{-10\gamma N_1}$ , we can choose $Y, \boldsymbol {k}_*$ to associate with the midpoint of I. Also, we can assume $|\sigma |\leq C(b,d)N$ since if $|\sigma |>CN$ , then $\|D_{\Lambda ^\dagger }^{-1}(\sigma )\|\leq 1$ , and $\Lambda ^\dagger $ becomes $\sigma $ -good via the Neumann series argument.

We now apply Lemma 2.25 with

$$ \begin{align*} T(\sigma)={{H}}_{\Lambda^\dagger}(\sigma), \ \beta=\beta_1=e^{- 10\gamma N_1}. \end{align*} $$

We have $ K_1=C N. $ Using the sublinear bound conclusion and the resolvent identity Lemma B.5, we have

$$ \begin{align*} M=\#Y&\leq N^{\frac{1}{49(b+d)}}< L_3^{\frac14},\ K_2=e^{2{N_1}^{\frac{3}{4}}}. \end{align*} $$

We again use the scale $N_2=N_1^{\frac 2\rho }$ , at which the LDE hold. Similar to the proof of Lemma 2.29, we can verify all assumptions of Lemma 2.25. Define

$$ \begin{align*} \tilde \Sigma_{N,3}=\{\sigma\in\mathbb R:\ \|G_{\Lambda^\dagger}(\sigma)\|\geq e^{L_3^{\frac{3}{4}}}\}. \end{align*} $$

Then by (2.30) of Lemma 2.25, one obtains

$$ \begin{align*} \mathrm{meas}(\tilde \Sigma_{N,3})\leq CN^Ce^{ 10\gamma N_1}e^{-\frac{c{{L_3}^{\frac{3}{4}}}}{CN_1N_2 L_3^{\frac{1}{4}}\log N}}\leq e^{-{L_3}^{\frac13} }\ll e^{-N^{3\rho}}. \end{align*} $$

Finally, it suffices to prove the off-diagonal exponential decay of $G_{\Lambda ^\dagger }(\sigma ).$ This follows directly from the resolvent identity of [Reference Jitomirskaya, Liu and ShiJLS20], that is, Lemma B.6. In fact, by the analysis in $\mathcal R_2$ , Lemma 2.29 and Lemma 2.28, all elementary regions contained in $\Lambda ^\dagger \setminus \Lambda _1^\dagger $ with sizes $N_1, L_0, L_1$ are $\sigma $ -good. Then applying Lemma B.6 with $\Lambda =\Lambda ^\dagger , \Lambda _1=\Lambda _1^\dagger $ (since $\mathrm {diam}\ \Lambda ^\dagger \sim L_3>100^{2(b+d)} L_1^{2(b+d)}\geq (\mathrm {diam}\ \Lambda _1^\dagger )^{2(b+d)}$ ) leads to the desired off-diagonal exponential decay estimate of $G_{\Lambda ^\dagger }(\sigma )$ with

$$ \begin{align*}\gamma_{L_2}=\gamma_{L_1}-N_1^{-\frac{1}{10}}=\gamma_{N_1}-N_1^{-\vartheta}-N_1^{\frac {1}{10}}.\\[-24pt]\end{align*} $$

Application of the resolvent identity again

Figure 1

The distribution of resonant blocks.

Figure 1 Long description

At the center is a blue hatched square intersected by horizontal and vertical bands of small red squares labeled S. Surrounding this are four nested rectangles: the innermost green, the next orange, then a larger blue, and the largest black. Each rectangle is annotated with a scale: L sub 0 is approximately L sub 1 to the 9 all over 10 power, L sub 1 is approximately N to the 1 all over 100 times open parenthesis b plus d close parenthesis squared, L sub 2 is approximately N to the 50 all over open parenthesis b plus d close parenthesis squared, and L sub 3 is approximately N to the 1 all over 10 times open parenthesis b plus d close parenthesis. The axes at top left are labeled k (vertical) and n (horizontal). The horizontal axis is marked with log N sub 1 much less than log N and extends to greater than 10 N sub 1. The right side shows an enlarged orange square with a dot, labeled L sub 0. Dashed lines and colored boxes indicate the spatial distribution and scaling of resonant blocks.

Finally, we define $\Sigma _{N}=\bigcup \limits _{\ell =1,2,3}\tilde \Sigma _{N, \ell }$ with $\tilde \Sigma _{N, 1}, \tilde \Sigma _{N, 2}, \tilde \Sigma _{N, 3}$ given by the analysis in $\mathcal R_2$ , Lemma 2.29, Lemma 2.30, respectively. Then

$$ \begin{align*}\mathrm{meas}(\Sigma_N)\leq 3e^{-2N^{3\rho}}\ll e^{-N^{2\rho}},\end{align*} $$

and for $\sigma \notin \Sigma _N$ , we can cover $\Lambda _{\mathrm {pm}, N}$ with $\sigma $ -good regions of sizes $N_1, L_0, L_1, L_3$ . More precisely, we have (cf. Figure 1)

• ○ For $\boldsymbol {x}\in \mathcal R_2$ , cover $\boldsymbol {x}$ with $\Lambda \in \mathcal {ER}(L_0)$ with $L_0\sim N^{\frac {9}{1000(b+d)^2}}$ .

• ○ For $\boldsymbol {x}\in \Lambda _1^\dagger $ , cover $\boldsymbol {x}$ with $\Lambda ^\dagger $ .

• ○ For $\boldsymbol {x}\in \mathcal R_1\setminus \Lambda _1^\dagger $ , cover $\boldsymbol {x}$ with regions in

  $$ \begin{align*}\bigcup_{L=N_1, L_0, L_1}\mathcal{ER}(L).\end{align*} $$

As a result, for $\sigma \notin \Sigma _N$ , we can apply Lemma B.5 and Lemma B.6 to show that $\Lambda _{\mathrm {pm}, N}$ is $\sigma $ -good. Indeed, we first apply Lemma B.5 with

$$ \begin{align*}\Lambda=\Lambda_{\mathrm{pm}, N}, W\in \bigcup_{L=N_1, L_0, L_1}(\mathcal{ER}(L)\cup\{\Lambda^\dagger\})\end{align*} $$

to obtain

$$ \begin{align*}\|G_{\Lambda_{\mathrm{pm}, N}}(\sigma)\|\leq e^{N^{\frac34}}.\end{align*} $$

Next, for the off-diagonal exponential decay, we use Lemma B.6 with $\Lambda =\Lambda _{\mathrm {pm}, N},\ \Lambda _1=\emptyset $ , $M_0=N_1$ to obtain (recalling the decay rate estimates in Lemmas 2.29, 2.30) for $|\boldsymbol {x}-\boldsymbol {x}'|>N^{\frac 89},$

$$ \begin{align*} |G_{\Lambda_{\mathrm{pm}, N}}(\sigma)((\boldsymbol{x},\xi);(\boldsymbol{x}',\xi'))|&\leq e^{-(\gamma_{L_2}-N_1^{-\frac{1}{10}})|\boldsymbol{x}-\boldsymbol{x}'|}\\ &\leq e^{-(\gamma_{N_1}-N_1^{-\vartheta}-2N_1^{-\frac{1}{10}}) |\boldsymbol{x}-\boldsymbol{x}'|}\\ &=e^{-\gamma_{N} |\boldsymbol{x}-\boldsymbol{x}'|}, \end{align*} $$

where

(2.35) $$ \begin{align} \gamma_N\geq \gamma_{N_1}-N^{-\kappa},\ \kappa={\frac12}\min\{\vartheta, \frac{1}{10}\}\leq \frac{1}{20}. \end{align} $$

Since $N\sim N_1^{\frac {1}{\rho ^2}}$ , we choose $\ell _*$ so that $N^{\rho ^{2\ell _*}}:=N_*\sim \log ^{\frac 1\kappa }\frac {1}{\varepsilon +\delta }$ . It follows from applying Lemma 2.15 (i.e., $\gamma _{N^{\rho ^{2\ell _*}}}=\gamma _0=\gamma -\log ^{-2}\frac {1}{\varepsilon +\delta }$ ) and iterating (2.35) that

(2.36) $$ \begin{align} \nonumber\gamma_N&\geq \gamma_{N_*}-\sum_{\ell=1}^{\ell_*}N_*^{-\frac{1}{\rho^{2\ell}}\kappa}\\ \nonumber&\geq \gamma_{N^{\rho^{2\ell_*}}}-N_*^{-\frac{\kappa}{\rho^2}}-\sum_{\ell\geq 2}N_*^{-\frac{1}{\rho^{2\ell}}\kappa}\\\nonumber&\geq \gamma-\log^{-2}{\frac{1}{\varepsilon+\delta}}-2N_*^{-\frac{\kappa}{\rho^2}}\\&\geq \gamma-3\log^{-2}{\frac{1}{\varepsilon+\delta}}\geq\frac\gamma 2 \end{align} $$

provided $0<\varepsilon +\delta \leq c(\rho , c_1, \kappa ).$

The above arguments hold indeed for all scales in $[N, N^2]$ . Then we can propagate the LDE from small scales interval $[N_1, N_2]$ to large scales one $[N, N^2]$ similar to the proof of Theorem 4.1 in [Reference Jitomirskaya, Liu and ShiJLS20]. This completes the proof of Lemma 2.20.

Proof of Theorem 2.11

The proof is just a combination of Lemmas 2.15, 2.17 and 2.20 by taking

$$ \begin{align*}0<\varepsilon+\delta\leq\delta_0=\min\{\delta_1, \delta_2, \delta_3\}.\\[-24pt]\end{align*} $$

3.1 The Lyapunov-Schmidt decomposition

We look for quasi-periodic solutions to (1.1) near

$$ \begin{align*}u^{(0)}=u^{(0)}(t,\boldsymbol{n})=\sum_{\ell=1}^ba_\ell e^{i\omega_\ell^{(0)}t}\delta_{\boldsymbol{n}_{\ell}, \boldsymbol{n}},\end{align*} $$

using the Lyapunov-Schmidt decomposition. The Fourier support of $u^{(0)}$ is $\mathcal {S}_+=\{(\boldsymbol {e}_\ell , \boldsymbol {n}_\ell )_{\ell =1}^b\}\subset \mathbb Z^{b+d}$ (we identify $ \mathcal {S}_+$ with $\{(\boldsymbol {e}_\ell , \boldsymbol {n}_\ell )_{\ell =1}^b\}\times \{+\}\subset \mathbb Z^{b+d}_{\mathrm {pm}}$ ). Similarly, let $\mathcal {S}_-=\{(-\boldsymbol {e}_\ell , \boldsymbol {n}_\ell )_{\ell =1}^b\}\subset \mathbb Z^{b+d}$ be the Fourier support of $v^{(0)}:=\overline {u^{(0)}}$ , which is then identified with $\{(-\boldsymbol {e}_\ell , \boldsymbol {n}_\ell )_{\ell =1}^b\}\times \{-\}\subset \mathbb Z^{b+d}_{\mathrm {pm}}$ . Let $\mathcal {S}=\mathcal {S}_+\cup \mathcal {S}_-$ and write $\mathbb Z^{b+d}_{\mathrm {pm}, *}=\mathbb Z^{b+d}_{\mathrm {pm}}\setminus \mathcal {S}:=\mathcal S^c.$

We divide the equation (3.2) into the P-equations

(3.5) $$ \begin{align} {F}\left(\vec u\right)\big |_{\mathcal S^c}=0,\ \end{align} $$

and the Q-equations

(3.6) $$ \begin{align} {F}(\vec u)\big |_{\mathcal{S}}=0. \end{align} $$

We iteratively solve the Q-equations and then the P-equations.

3.2 The Q-equations and extraction of parameters

The Q-equations are $2b$ -dimensional.

On $\mathcal S$ , $\vec u$ is held fixed, and the Q-equations (3.6) are viewed instead as equations for $\boldsymbol {\omega }$ , and will be solved using the implicit function theorem. The Q-equations are used to relate the frequencies $\boldsymbol {\omega }$ to the amplitudes $\boldsymbol {a}$ , permitting amplitude-frequency modulation to solve the P-equations.

Note that

(3.7) $$ \begin{align} u^{(0)}(t, \boldsymbol{n})=\sum_{(\boldsymbol{k}, \boldsymbol{n})\in\mathcal{S}_{+}}\hat u^{(0)}(\boldsymbol{k}, \boldsymbol{n})e^{i\boldsymbol{k}\cdot\boldsymbol{\omega}^{(0)}t}, \end{align} $$

where

$$ \begin{align*} \hat u^{(0)}(\boldsymbol{e}_\ell, \boldsymbol{n}_\ell)=\hat v^{(0)}(-\boldsymbol{e}_\ell,\boldsymbol{n} _\ell)=a_\ell\ \mathrm{for}\ 1\leq \ell\leq b. \end{align*} $$

The solutions $\vec u=(\hat u, \hat v)^t$ (with $(\cdot )^t$ denoting the transpose) on $\mathcal S$ are held fixed: $\vec u \equiv (\hat u^{(0)}, \hat v^{(0)})^t=\vec u^{(0)}$ on $\mathcal S$ , the Q-equations are used instead to solve for the frequencies. Due to symmetry, the Q-equations for $\hat v$ are the same as those for $\hat u$ . So we only need to consider those for $\hat u$ . Solving the equations at the initial step

$$ \begin{align*} {F}(\vec u^{(0)})\big|_{\mathcal{S_+}}=0 \end{align*} $$

leads to, for $1\leq \ell \leq b$ , the initial modulated frequencies

(3.8) $$ \begin{align} \omega_\ell=\omega_\ell^{(1)}&=\omega_\ell^{(0)}+\delta a_\ell^{2p} \end{align} $$

satisfying

$$ \begin{align*}\det \left(\frac{\partial \boldsymbol{\omega}^{(1)}}{\partial \boldsymbol{a}}\right)=\delta^b\prod_{\ell=1}^b(2p a_\ell^{2p-1}).\end{align*} $$

Along the way, we will show that $\boldsymbol {\omega }^{(r)}=\boldsymbol {\omega }^{(0)}+O(\delta )$ , for all $r=0, 1, 2, \cdots $ . So $\boldsymbol {\omega }^{(r)}\in \Omega $ , the frequency set in Theorem 2.11, hence Theorem 2.11 is at our disposal to solve the P-equations. To construct approximate solutions to the P-equations, it is convenient to view $\boldsymbol {\omega }\in \Omega $ as an independent parameter, and work in the $(\boldsymbol {\omega }, \boldsymbol {a})$ -variable, $(\boldsymbol {\omega }, \boldsymbol {a})\in \Omega \times [1,2]^b:=(\boldsymbol {\omega }^{(0)}+[\delta , 2^{2p}\delta ]^b)\times [1,2]^b$ . The approximate solutions to the nonlinear matrix equation (3.1) will then be obtained by taking into account the solutions to the Q-equations as well, which restricts $(\boldsymbol {\omega }, \boldsymbol {a})$ to the b-dimensional hypersurface $\boldsymbol {\omega }=\boldsymbol {\omega }^{(r)} (\boldsymbol {a})$ , at the r-th iteration.

3.3 The P-equations

The P-equations are infinite-dimensional.

We use (3.5) to solve for $\vec u |_{\mathcal S^c}$ . Solving the P-equations requires analyzing the invertibility of the linearized operator $H=H_0+\delta T_{\vec u}$ restricted to $\mathbb Z^{b+d}_{\mathrm {pm}, *}$ , which we do in the previous section. Then the resolution of the P-equations combines the Newton scheme and the linear analysis in Section 2. The large deviation estimates in $\sigma $ in Theorem 2.11 will be converted into estimates in the amplitudes $\boldsymbol {a}$ by using a semi-algebraic projection lemma (cf. Lemma 3.7 below), and amplitude-frequency modulation, $\boldsymbol {\omega }=\boldsymbol {\omega }(\boldsymbol {a})$ . This is an established scheme, which has recently been extensively elaborated on with detailed proofs in sects. IV-VI, [Reference Kachkovskiy, Liu and WangKLW24] and [Reference Liu and WangLW24]. They will serve as the basic reference point for the present section.

Recall first the formal Newton scheme. Let $\vec u$ be an approximate solution to the P-equations (3.5). For the next approximation ${\vec u}'$ , write

$$ \begin{align*}{\vec u}'=\vec u+\Delta_{\mathrm{cor}}\vec u'.\end{align*} $$

Then $\Delta _{\mathrm {cor}}\vec u'$ is set to be

$$ \begin{align*} \Delta_{\mathrm{cor}}\vec u'=-(H(\vec u))^{-1}F( \vec u), \end{align*} $$

where the linearized operator $H(\vec u)=H_0+\delta T_{\vec u}(\boldsymbol {\omega }, \boldsymbol {a})$ (cf. (3.3) and (3.4)). This is, however, only indicative since we assume that $H(\vec u)$ is invertible. Due to the small divisor difficulties in the inversion process, we need to regularize the linearized operator $H(\vec u)$ and do a multi-scale Newton iteration instead.

Toward that purpose, let M be a large integer and consider the geometric sequence of scales $M^\ell $ , $\ell =1, 2, \cdots $ . Denote by $\vec u^{(r)}$ , the r-th approximate solution to the P-equations (3.5). For the $(r+1)$ -th approximation, write

$$ \begin{align*}\vec u^{(r+1)}=\vec u^{(r)}+\Delta_{\mathrm{cor}} \vec u^{(r+1)}.\end{align*} $$

We define (if it exists)

$$ \begin{align*}\Delta_{\mathrm{cor}} \vec u^{(r+1)}=-(R_{N}{{H(\vec u^{(r)})}}R_{N})^{-1}F( \vec u^{(r)}),\end{align*} $$

where $N=M^{r+1}$ and $R_{N}$ is the restriction operator to the cube $\Lambda _{N}\cap \mathbb Z^{b+d}_{\mathrm {pm, *}}.$ This leads to estimate the inverse (i.e., the Green’s function),

$$ \begin{align*} {{G}}_{N}(\vec u ):= \left(R_{{N}}{{H(\vec u^{(r)})}} R_{N}\right)^{-1}, \ r=0, 1, 2, \cdots. \end{align*} $$

The following estimates are crucial for the convergence of the approximate solutions with exponential decay:

$$ \begin{align*} \| {{G}}_{N}(\vec u^{(r)})\|&\leq e^{(\log N)^C}, \\ | {{G}}_{N}(\vec u^{(r)})((\boldsymbol{x}, \xi);(\boldsymbol{x}',\xi'))|&\leq e^{-c|\boldsymbol{x}-\boldsymbol{x}'|}\ \mathrm{for} \ |\boldsymbol{x}-\boldsymbol{x}'|\geq (\log N)^C, \end{align*} $$

for $\boldsymbol {x}=(\boldsymbol {k}, \boldsymbol {n})$ and some $c, C>0$ , see sect. V, [Reference Kachkovskiy, Liu and WangKLW24].

The LDT, Theorem 2.11, in the previous section plays an essential role in the above Green’s function estimates. The operator $T_{\vec u}$ plays the role of S in (2.1) (cf. Remark 2.13 for details). The following lemma implies that $T_{\vec u}$ satisfies the decay assumption of S (cf. also Remark 2.13). This will then enable us to use Theorem 2.11.

3.4 The induction hypothesis and the proof

Let M be a large integer and recall that $\Lambda _{R}\subset \mathbb Z^{b+d}$ denotes the $\ell ^\infty $ -norm-induced cube of radius R and centered at the origin.

In the following, $C>0$ is a large constant and $c>0$ is a small one.

We begin with the analysis of initial induction steps, which relies on the Neumann series argument and Diophantine estimates (2.5) of Theorem 2.2.

3.4.1 The initial $2r_\star -r_0$ steps

We choose $1<r_0<r_\star $ such that

$$ \begin{align*} M^{r_0}\leq \log \frac{1}{\varepsilon+\delta}\leq M^{r_0+1} \end{align*} $$

and

(3.9) $$ \begin{align} M^{2r_\star}\leq (\varepsilon+\delta)^{-c_1}<M^{2r_\star+1}, \end{align} $$

where $c_1>0$ is given in Theorem 2.11.

Let $\vec u^{(0)}(\boldsymbol {\omega },\boldsymbol {a})=\vec u^{(0)}(\boldsymbol {\omega }^{(0)},\boldsymbol {a})$ with $\boldsymbol {\omega }=\boldsymbol {\omega }^{(1)}$ given by (3.8).

We start by constructing $\vec u_{\mathrm {in}}^{(1)}(\boldsymbol {\omega }, \boldsymbol {a})$ . Obviously,

$$ \begin{align*}{F}(\vec u^{(0)})=O(\varepsilon+\delta)\end{align*} $$

and (the support) $\mathrm {supp}\ {F}(\vec u^{(0)})\subset \Lambda _{M_0}$ for some $M_0=M_0(p, \mathcal {S})>0.$ Let $L_1=M^{r_0+1}>M_0.$ It suffices to estimate

$$ \begin{align*} H_{L_1}^{-1}=(R_{\Lambda_{L_1}}({D}(0)+\varepsilon(\Delta\oplus\Delta)+\delta{T}_{\vec u^{(0)}})R_{\Lambda_{L_1}})^{-1}. \end{align*} $$

It follows from (2.5) of Theorem 2.2, $\boldsymbol {\omega }=\boldsymbol {\omega }^{(0)}+O(\delta )$ and the Neumann series argument (cf. Lemma B.1) that

$$ \begin{align*} \|H_{L_1}^{-1}\|&\leq 2L_1^{C_1}, \\ |H_{L_1}^{-1}((\boldsymbol{x}, \xi); (\boldsymbol{x}', \xi'))|&\leq L_1^{C_1}e^{-\frac{\log \frac{1}{\varepsilon+\delta}}{2}|\boldsymbol{x}-\boldsymbol{x}'|}\ \mathrm{ for}\ \boldsymbol{x}\neq \boldsymbol{x}'\in \Lambda_{L_1}, \end{align*} $$

since $L_1=M^{r_0+1}\ll (\varepsilon +\delta )^{-c_1}$ . Then we have

$$ \begin{align*} \Delta_{\mathrm{cor}}\vec u_{\mathrm{in}}^{(1)}=-H_{L_1}^{-1}{F}(\vec u^{(0)}), \end{align*} $$

and consequently

$$ \begin{align*} \vec u_{\mathrm{in}}^{(1)}=\vec u^{(0)}+\Delta_{\mathrm{cor}}\vec u_{\mathrm{in}}^{(1)}=\vec u^{(0)}-H_{L_1}^{-1}{F}(\vec u^{(0)}). \end{align*} $$

Substituting $\vec u_{\mathrm {in}}^{(1)}$ into the Q-equations using

$$ \begin{align*}\hat u_{\mathrm{in}}^{(1)}\big |_{\mathcal{S}+}=\hat v_{\mathrm{in}}^{(1)}\big |_{\mathcal{S}-}=\boldsymbol{a},\end{align*} $$

we obtain

$$ \begin{align*} \omega_\ell&=\omega_\ell^{(0)}+\varepsilon\left(\frac{1}{a_\ell}(\Delta \hat u_{\mathrm{in}}^{(1)})(\boldsymbol{e}_\ell, \boldsymbol{n}_\ell)\right)(\boldsymbol{\omega}, \boldsymbol{a})\\ &\ \ +\delta\left(\frac{1}{a_\ell}(\hat u_{\mathrm{in}}^{(1)}*\hat v_{\mathrm{in}}^{(1)})^{*p} *\hat u_{\mathrm{in}}^{(1)}(\boldsymbol{e}_\ell, \boldsymbol{n}_\ell)\right)(\boldsymbol{\omega}, \boldsymbol{a}), \, \ell= 1, 2, \cdots, b. \end{align*} $$

Since the second and the third terms are smooth in $\boldsymbol {\omega }$ and $\boldsymbol {a}$ , using the implicit function theorem yields $\boldsymbol {\omega }_{\mathrm {in}}^{(2)}=\boldsymbol {\omega }_{\mathrm {in}}^{(2)}(\boldsymbol {a})=(\omega _{\mathrm {in}, \ell }^{(2)}(\boldsymbol {a}))_{\ell =1}^b$ , written in the form:

$$ \begin{align*} \omega_{\mathrm{in}, \ell}^{(2)}(\boldsymbol{a})=\omega_\ell^{(0)}+(\varepsilon+\delta) \varphi_{\mathrm{in}, \ell}^{(2)}(\boldsymbol{a}), \, \ell= 1, 2, \cdots, b, \end{align*} $$

with a smooth function $\varphi _{\mathrm {in}, \ell }^{(2)}$ (comparing with (3.8)).

The above constructions can be inductively performed for $2r_\star -r_0$ steps, with $r_\star $ satisfying (3.9). So we have obtained $\vec u_{\mathrm {in}}^{(r)}=\vec u_{\mathrm {in}}^{(r)}(\boldsymbol {\omega }, \boldsymbol {a})$ ( $1\leq r\leq 2r_\star -r_0$ ) for all $(\boldsymbol {\omega }, \boldsymbol {a})\in \Omega \times [1,2]^b$ .

3.4.2 The Inductive Theorem

Recall that

$$ \begin{align*}\Omega=\boldsymbol{\omega}^{(0)}+[\delta, 2^{2p}\delta]^b\subset\mathbb R^b.\end{align*} $$

We first state the induction hypothesis, which can be verified if $0<\varepsilon \leq \delta \leq \log ^{-1}\frac {1}{\varepsilon }\leq \delta _0\ll 1$ and $(\boldsymbol {\alpha }, \boldsymbol {\theta })\in \mathcal W$ (cf. (2.11)). Given its significance, we refer to it as the induction theorem.

Theorem 3.2. For $r\geq r_\star ,$ we have

1. (Hi) $\mathrm {supp}\ \vec u^{(r)}\subset \Lambda _{M^{\tilde r}}$ , $\tilde r=r+r_\star .$

2. (Hii) $\|\Delta _{\mathrm {cor}} \vec u^{(r)}\|<\delta _r,\ \|\partial \Delta _{\mathrm {cor}} \vec u^{(r)}\|<{\bar \delta }_{r}$ , where $\partial $ refers to derivation in $\boldsymbol {\omega }$ or $\boldsymbol {a}$ ,

   $$ \begin{align*}\vec u^{(r)}=\vec u^{(r-1)}+\Delta_{\mathrm{cor}} \vec u^{(r)},\end{align*} $$
   and $\|\cdot \|=\sup _{\boldsymbol {\omega }, \boldsymbol {a}}\|\cdot \|_{\ell ^2(\mathbb Z_{\mathrm {pm}}^{b+d})}.$

   Remark 3.3. The size of $\delta _r, \bar \delta _r$ will satisfy $\log \log \frac {1}{\delta _r+\bar \delta _r}\sim r.$

3. (Hiii) $|\vec u^{(r)}(\boldsymbol {k}, \boldsymbol {n})|\leq e^{-c(|\boldsymbol {k}|+|\boldsymbol {n}|)}$ for some $c>0.$

   Remark 3.4. The constant $c>0$ will decrease slightly along the iterations but remain bounded away from $0$ . This will become clear in the proof, cf. sect. V, G. Step 4, [Reference Kachkovskiy, Liu and WangKLW24]. We also remark that $\vec u ^{(r)}$ can be defined as a $C^1$ function on the entire parameter space $(\boldsymbol {\omega }, \boldsymbol {a})\in \Omega \times [1,2]^b$ by using an extension argument, cf., [Reference BourgainBou98, Reference Bourgain and WangBW08] and sect. V, L. Step 9, [Reference Kachkovskiy, Liu and WangKLW24]. So one can apply the implicit function theorem to solve the Q-equations leading to

   (3.10) $$ \begin{align} \omega_\ell^{(r)}(\boldsymbol{a})=\omega_\ell^{(0)}+(\varepsilon+\delta)\varphi_\ell^{(r)}(\boldsymbol{a}) \ (1\leq \ell\leq b), \end{align} $$

   where $\|\partial \boldsymbol {\varphi }^{(r)}\|=\sup _{1\leq \ell \leq b}\|\partial \varphi _\ell ^{(r)}\|\lesssim 1$ and $ \boldsymbol {\varphi }^{(r)}=( \varphi ^{(r)}_\ell )_{\ell =1}^b.$ By (Hii), we have

   $$ \begin{align*} |\boldsymbol{\varphi}^{(r)}-\boldsymbol{\varphi}^{(r-1)}|\lesssim (\varepsilon+\delta)\|\vec u^{(r)}-\vec u^{(r-1)}\|\lesssim {(\varepsilon+\delta)}\delta_r. \end{align*} $$

   Denote by $\Gamma _r$ the graph of $\boldsymbol {\omega }^{(r)}=\boldsymbol {\omega }^{(r)}(\boldsymbol {a})$ . We have $\|\Gamma _r-\Gamma _{r-1}\|\lesssim {(\varepsilon +\delta )}\delta _r.$ Recall that $\boldsymbol {\omega }$ is given by (3.8) and $\boldsymbol {\varphi }^{(0)}=\boldsymbol {0}$ . Thus we have a diffeomorphism from $\boldsymbol {\omega }^{(r)}\in \Omega $ to $\boldsymbol {a}\in [1,2]^b$ .

4. (Hiv) There is a collection $\mathcal {I}_r$ of intervals $I\subset \Omega \times [1,2]^b$ of size $M^{-{\tilde r}^{10C}},$ so that

   1. (a) On each $I\in \mathcal {I}_r,$ both $\hat u^{(r)}(\boldsymbol {\omega }, \boldsymbol {a})$ and $ \hat v^{(r)}(\boldsymbol {\omega }, \boldsymbol {a})$ are given by rational functions in $(\boldsymbol {\omega }, \boldsymbol {a})$ of degree at most $M^{{\tilde r}^3}$ .

   2. (b) For $(\boldsymbol {\omega }, \boldsymbol {a})\in \bigcup _{I\in \mathcal {I}_r}I$ ,

      $$ \begin{align*} \|F(\vec u^{(r)})\|\leq \kappa_r,\ \|\partial F(\vec u^{(r)})\|\leq \bar\kappa_r, \end{align*} $$
      where $\partial $ refers to derivation in $\boldsymbol {\omega }$ or $\boldsymbol {a}$ , and $\log \log \frac {1}{\kappa _r+\bar \kappa _r}\sim r$ .

   3. (c) For $(\boldsymbol {\omega }, \boldsymbol {a})\in \bigcup _{I\in \mathcal {I}_r}I$ and $H=H(\vec u^{(r-1)})$ , one has

      (3.11) $$ \begin{align} \|H_{M^{\tilde r}}^{-1}\|&\leq M^{{\tilde r}^C}, \end{align} $$
      (3.12) $$ \begin{align} |H_{M^{\tilde r}}^{-1}((\boldsymbol{x},\xi); (\boldsymbol{x}',\xi'))|&\leq e^{-c|\boldsymbol{x}-\boldsymbol{x}'|}\ \mathrm{for}\ |\boldsymbol{x}-\boldsymbol{x}'|>{\tilde r}^C, \end{align} $$
      where $\boldsymbol {x}=(\boldsymbol {k}, \boldsymbol {n}), \boldsymbol {x}'=(\boldsymbol {k}', \boldsymbol {n}')$ and $H_{M^{\tilde r}}$ refers to the restriction of H to $\Lambda _{M^{\tilde r}}$ .

   4. (d) Each $I\in \mathcal {I}_r$ is contained in some $I'\in \mathcal {I}_{r-1}$ and

      $$ \begin{align*} \mathrm{meas}\ \left(\Pi_{\boldsymbol{a}}(\Gamma_{r-1}\cap(\bigcup_{I'\in\mathcal{I}_{r-1}}I'\setminus\bigcup_{I\in\mathcal{I}_r}I))\right)\leq M^{-\frac{\tilde r}{C(b)}}, \end{align*} $$
      where $C(b)>0$ depends only on b, and $\Pi _{\boldsymbol {a}}$ denotes the projection of the set on the $\boldsymbol {a}$ -variable.

5. (Hv) The following precise relations hold true

   $$ \begin{align*} \delta_r= {(\varepsilon+\delta)}^{\frac{1}{2}}M^{-(\frac{4}{3})^r}, \, \bar\delta_r= {(\varepsilon+\delta)}^{\frac{1}{8}} M^{-\frac{1}{2}(\frac{4}{3})^r};\\ \kappa_r= {(\varepsilon+\delta)}^{\frac{3}{4}} M^{-(\frac{4}{3})^{r+2}}, \, \bar\kappa_r= {(\varepsilon+\delta)}^{\frac{3}{8}} M^{-\frac{1}{2}(\frac{4}{3})^{r+2}}. \end{align*} $$