Towards the estimates in (1), we first derive the estimates for $\psi_{n}$,
$\tilde{\psi}_{n}$, $r_{n}$, $\Pi_{\geq N_{n}}\mathcal{R}_{n}$ and
$\mathcal{R}_{n}\psi_{n}$.

Estimates for $\psi_{n}$. Applying (10.304) and (10.305) to (10.328), we see
that

$\displaystyle\mathfrak{M}^{\sharp,\gamma}_{\psi_{n}}(0,s)$
$\displaystyle\leq_{\mathtt{pe}}\varepsilon^{2}\gamma^{-2}N_{n}^{2\tau+1}\mathfrak{M}^{\sharp,\gamma}_{\mathcal{R}_{n}}(0,s),$
$\displaystyle\sup_{\vec{b}\in\mathbb{N}^{\nu}_{0},\
|\vec{b}|=\mathtt{b}_{0}}\mathfrak{M}^{\sharp,\gamma}_{\partial_{\varphi}^{\vec{b}}(\psi_{n})}(0,s)$
$\displaystyle\leq_{\mathtt{pe}}\varepsilon^{2}\gamma^{-2}N_{n}^{2\tau+1}\sup_{\vec{b}\in\mathbb{N}^{\nu}_{0},\
|\vec{b}|=\mathtt{b}_{0}}\mathfrak{M}^{\sharp,\gamma}_{\partial_{\varphi}^{\vec{b}}(\mathcal{R}_{n})}(0,s),$
$\displaystyle\mathfrak{M}^{\sharp}_{d_{i}\Psi_{n}(i_{0})[\hat{\textbf{\i}}]}(0,s_{0})$
$\displaystyle\leq_{\mathtt{pe}}\left(\varepsilon^{3}\gamma^{-3}N_{n}^{2\tau+1}\mathfrak{M}^{\sharp}_{\mathcal{R}_{n}}(0,s_{0})\rVert\hat{\textbf{\i}}\rVert_{s_{0}+\mu_{\mathtt{p}}}+\gamma^{-1}N_{n}^{\tau}\mathfrak{M}^{\sharp}_{d_{i}\mathcal{R}_{n}(i_{0})[i\hat{]}}(0,s_{0})\right),$

and

$\displaystyle\sup_{\vec{b}\in\mathbb{N}^{\nu}_{0},\
|\vec{b}|=\mathtt{b}_{0}}\mathfrak{M}^{\sharp}_{\partial_{\varphi}^{\vec{b}}(d_{i}\Psi_{n}(i_{0})[\hat{\textbf{\i}}])}(0,s_{0})$
$\displaystyle\quad\leq_{\mathtt{pe}}\left(\varepsilon^{3}\gamma^{-3}N_{n}^{2\tau+1}\sup_{\vec{b}\in\mathbb{N}^{\nu}_{0},\
|\vec{b}|=\mathtt{b}_{0}}\mathfrak{M}^{\sharp}_{\partial_{\varphi}^{\vec{b}}(\mathcal{R}_{n})}(0,s_{0})\rVert\hat{\textbf{\i}}\rVert_{s_{0}+\mu_{\mathtt{p}}}+\gamma^{-1}N_{n}^{\tau}\sup_{\vec{b}\in\mathbb{N}^{\nu}_{0},\
|\vec{b}|=\mathtt{b}_{0}}\mathfrak{M}^{\sharp}_{\partial_{\varphi}^{\vec{b}}(d_{i}\mathcal{R}_{n}(i_{0})[i\hat{]})}(0,s_{0})\right).$

Using the induction hypotheses for $n$, we can use the estimates in (1) and we
get

$\displaystyle\mathfrak{M}^{\sharp,\gamma}_{\psi_{n}}(0,s)$
$\displaystyle\leq_{\mathtt{pe}}\varepsilon^{2}\gamma^{-2}N_{n}^{2\tau+1}N_{n-1}^{-\sigma_{1}}\mathbb{M}_{0}(s),$
(10.350) $\displaystyle\sup_{\vec{b}\in\mathbb{N}^{\nu}_{0},\
|\vec{b}|=\mathtt{b}_{0}}\mathfrak{M}^{\sharp,\gamma}_{\partial_{\varphi}^{\vec{b}}(\psi_{n})}(0,s)$
$\displaystyle\leq_{\mathtt{pe}}\varepsilon^{2}\gamma^{-2}N_{n}^{2\tau+1}N_{n-1}\mathbb{M}_{0}(s),$
(10.351)
$\displaystyle\mathfrak{M}^{\sharp}_{d_{i}\psi_{n}(i_{0})[\hat{\textbf{\i}}]}(0,s_{0})$
$\displaystyle\leq_{\mathtt{pe}}\left(\varepsilon^{3}\gamma^{-3}N_{n}^{2\tau+1}N_{n-1}^{-\sigma_{1}}\mathbb{M}_{0}(s_{0})+\varepsilon^{3}\gamma^{-2}N_{n}^{\tau}N_{n-1}^{-\sigma_{1}}\right)\rVert\hat{\textbf{\i}}\rVert_{s_{0}+\mu_{\mathtt{p},1}},$
(10.352) $\displaystyle\sup_{\vec{b}\in\mathbb{N}^{\nu}_{0},\
|\vec{b}|=\mathtt{b}_{0}}\mathfrak{M}^{\sharp}_{\partial_{\varphi}^{\vec{b}}(d_{i}\psi_{n}(i_{0})[\hat{\textbf{\i}}])}(0,s_{0})$
$\displaystyle\leq_{\mathtt{pe}}\left(\varepsilon^{3}\gamma^{-3}N_{n}^{2\tau+1}N_{n-1}\mathbb{M}_{0}(s_{0})+\varepsilon^{3}\gamma^{-2}N_{n}^{\tau}N_{n-1}\right)\rVert\hat{\textbf{\i}}\rVert_{s_{0}+\mu_{\mathtt{p},1}}.$
(10.353)

We see from (10.350) that

$\displaystyle\mathfrak{M}^{\sharp,\gamma}_{\psi_{n}}(0,s_{0})$
$\displaystyle\leq_{\mathtt{pe}}\varepsilon^{2}\gamma^{-2}N_{n}^{2\tau+1}N_{n-1}^{-\sigma_{1}}\mathbb{M}_{0}(s_{0}),$

while (10.294) and (10.336) give us that

$\displaystyle\mathbb{M}_{0}(s_{0})\leq_{\mathtt{pe}}\gamma^{-1}\left(\varepsilon^{5}+\varepsilon^{3}\rVert\mathfrak{I}_{\delta}\rVert^{\text{Lip}(\gamma,\Omega_{1})}_{s_{0}+\mu_{\mathtt{p}}}\right)\overset{\eqref{size_assumption_2}}{\leq}_{\mathtt{pe}}\varepsilon^{9-6b}.$
(10.354)

Thus, we have that

$\displaystyle\mathfrak{M}^{\sharp,\gamma}_{\psi_{n}}(0,s_{0})$
$\displaystyle\overset{\eqref{jsdj2sdsd1sd1}}{\leq}_{\mathtt{pe}}\varepsilon^{2}\gamma^{-2}N_{n}^{2\tau+1}N_{n-1}^{-\sigma_{1}}\mathbb{M}_{0}(s_{0})\overset{\eqref{m01sd2est}}{\leq}_{\mathtt{pe}}\varepsilon^{11-10b}N_{n}^{2\tau+1}N_{n-1}^{-\sigma_{1}}\overset{\eqref{para1chs12}}{\leq}_{\mathtt{pe}}\varepsilon^{\frac{1}{22}}$
(10.355)

Furthermore, it follows from (10.352),(10.353) with (10.354) that (using
$\varepsilon^{3}\gamma^{-3}\mathbb{M}_{0}(s_{0})\overset{\eqref{frequency_set2},\eqref{parametersets1231}}{\leq}\varepsilon^{3-4b}=\varepsilon^{3}\gamma^{-2}$)

$\displaystyle\mathfrak{M}^{\sharp}_{d_{i}{\psi}_{n}(i_{0})[\hat{\textbf{\i}}]}(0,s_{0})$
$\displaystyle\leq_{\mathtt{pe}}\varepsilon^{3-4b}N_{n}^{2\tau+1}N_{n-1}^{-\sigma_{1}}\rVert\hat{\textbf{\i}}\rVert_{s_{0}+\mu_{\mathtt{p},1}},$
(10.356) $\displaystyle\sup_{\vec{b}\in\mathbb{N}^{\nu}_{0},\
|\vec{b}|=\mathtt{b}_{0}}\mathfrak{M}^{\sharp}_{\partial_{\varphi}^{\vec{b}}(d_{i}\psi_{n}(i_{0})[\hat{\textbf{\i}}])}(0,s_{0})$
$\displaystyle\leq_{\mathtt{pe}}\varepsilon^{3-4b}N_{n}^{2\tau+1}N_{n-1}\rVert\hat{\textbf{\i}}\rVert_{s_{0}+\mu_{\mathtt{p},1}}.$

Estimates for $\tilde{\psi}_{n}$. For $\tilde{\psi}_{n}$,

Thanks to (10.355), we see from Lemma 2.28 that we can choose $\varepsilon$
sufficiently small, depending on $\mathtt{pe}$ so that $\tilde{\psi}_{n}$ in
(10.328) is well-defined and satisfies

$\displaystyle\mathfrak{M}^{\sharp,\gamma}_{\tilde{\psi}_{n}}(0,s)$
$\displaystyle\leq_{\mathtt{pe}}\mathfrak{M}^{\sharp,\gamma}_{{{\psi}_{n}}}(0,s)\overset{\eqref{jsdj2sdsd1sd1}}{\leq}_{\mathtt{pe}}\varepsilon^{2}\gamma^{-2}N_{n}^{2\tau+1}N_{n-1}^{-\sigma_{1}}\mathbb{M}_{0}(s),$
(10.357) $\displaystyle\sup_{\vec{b}\in\mathbb{N}^{\nu}_{0},\
|\vec{b}|=\mathtt{b}_{0}}\mathfrak{M}^{\sharp,\gamma}_{\partial_{\varphi}^{\vec{b}}(\tilde{\psi}_{n})}(0,s)$
$\displaystyle\leq_{\mathtt{pe}}\sup_{\vec{b}\in\mathbb{N}^{\nu}_{0},\
|\vec{b}|=\mathtt{b}_{0}}\left(\mathfrak{M}^{\sharp,\gamma}_{\partial_{\varphi}^{\vec{b}}(\psi_{n})}(0,s)+\mathfrak{M}^{\sharp,\gamma}_{\partial_{\varphi}^{\vec{b}}(\psi_{n})}(0,s_{0})\mathfrak{M}^{\sharp,\gamma}_{\psi_{n}}(0,s)\right)$
$\displaystyle\overset{\eqref{jsdj2sdsd1sd2}}{\leq}_{\mathtt{pe}}\varepsilon^{2}\gamma^{-2}N_{n}^{2\tau+1}N_{n-1}\mathbb{M}_{0}(s)\left(1+\varepsilon^{2}\gamma^{-2}N_{n}^{2\tau+1}N_{n-1}^{-\sigma_{1}}\mathbb{M}_{0}(s_{0})\right)$
$\displaystyle\overset{\eqref{m01sd2est}}{\leq}_{\mathtt{pe}}\varepsilon^{2}\gamma^{-2}N_{n}^{2\tau+1}N_{n-1}\mathbb{M}_{0}(s)(1+\underbrace{\varepsilon^{11-10b}N_{n}^{2\tau+1}N_{n-1}^{-\sigma_{1}}}_{\leq
1,\ \because\eqref{para1chs12}})$
$\displaystyle\leq_{\mathtt{pe}}\varepsilon^{2}\gamma^{-2}N_{n}^{2\tau+1}N_{n-1}\mathbb{M}_{0}(s).$
(10.358)

Therefore, for $\phi_{n}^{-1}$ in (10.328), we have that

$\displaystyle\mathfrak{M}^{\sharp,\gamma}_{\phi_{n}^{-1}}(0,s)$
$\displaystyle\leq
1+\mathfrak{M}^{\sharp,\gamma}_{\tilde{\psi}_{n}}(0,s)\overset{\eqref{ptildesd1}}{\leq}1+c_{\mathtt{pe}}\varepsilon^{2}\gamma^{-2}N_{n}^{2\tau+1}N_{n-1}^{-\sigma_{1}}\mathbb{M}_{0}(s),$
(10.359) $\displaystyle\sup_{\vec{b}\in\mathbb{N}^{\nu}_{0},\
|\vec{b}|=\mathtt{b}_{0}}\mathfrak{M}^{\sharp,\gamma}_{\phi_{n}^{-1}}(0,s)$
$\displaystyle\leq_{\mathtt{pe}}\sup_{\vec{b}\in\mathbb{N}^{\nu}_{0},\
|\vec{b}|=\mathtt{b}_{0}}\mathfrak{M}^{\sharp,\gamma}_{\partial_{\varphi}^{\vec{b}}(\tilde{\psi}_{n})}(0,s)\overset{\eqref{ptildesd2}}{\leq}_{\mathtt{pe}}\varepsilon^{2}\gamma^{-2}N_{n}^{2\tau+1}N_{n-1}\mathbb{M}_{0}(s).$

Furthermore, using that $\phi_{n}^{-1}\circ\phi_{n}=I$, which implies
$d_{i}(\phi_{n}^{-1})(i_{0})[\hat{\textbf{\i}}]=-\phi_{n}^{-1}\circ
d_{i}(\psi_{n})(i_{0})[\hat{\textbf{\i}}]\circ\phi_{n}^{-1}$ and Lemma 2.23,
we see from (10.359) and (10.356) that

$\displaystyle\mathfrak{M}^{\sharp}_{d_{i}(\phi_{n}^{-1})(i_{0})[\hat{\textbf{\i}}]}(0,s_{0})=\mathfrak{M}^{\sharp}_{d_{i}(\tilde{\psi}_{n})(i_{0})[\hat{\textbf{\i}}]}(0,s_{0})$
$\displaystyle\leq_{\mathtt{pe}}\varepsilon^{3-4b}N_{n}^{2\tau+1}N_{n-1}^{-\sigma_{1}}\rVert\hat{\textbf{\i}}\rVert_{s_{0}+\mu_{\mathtt{p},1}},$
(10.360) $\displaystyle\sup_{\vec{b}\in\mathbb{N}^{\nu}_{0},\
|\vec{b}|=\mathtt{b}_{0}}\mathfrak{M}^{\sharp}_{\partial_{\varphi}^{\vec{b}}(d_{i}(\phi_{n}^{-1})(i_{0})[\hat{\textbf{\i}}])}(0,s_{0})=\sup_{\vec{b}\in\mathbb{N}^{\nu}_{0},\
|\vec{b}|=\mathtt{b}_{0}}\mathfrak{M}^{\sharp}_{\partial_{\varphi}^{\vec{b}}(d_{i}(\tilde{\psi}_{n})(i_{0})[\hat{\textbf{\i}}])}(0,s_{0})$
$\displaystyle\leq_{\mathtt{pe}}\varepsilon^{3-4b}N_{n}^{2\tau+1}N_{n-1}\rVert\hat{\textbf{\i}}\rVert_{s_{0}+\mu_{\mathtt{p},1}}.$

Estimates for $Op^{W}(r_{n})$. Using Lemma 2.25, we have that (in the
applications of the estimates (10.342) and (10.344) below, we use the fact
that $\Omega_{n}^{\gamma}(i_{0})\subset\Omega^{\gamma}_{n-1}(i_{0})$, which
follows from the definition of the set of frequencies in (10.332)),

$\displaystyle\mathfrak{M}_{Op^{W}(r_{n})}^{\sharp,\gamma}(0,s)$
$\displaystyle\leq_{\mathtt{pe},s}|r_{n}|^{\text{Lip}(\gamma,\Omega^{\gamma}_{n}(i_{0}))}_{0,0,0}\overset{\eqref{rsdj1sd11}}{\leq}_{\mathtt{pe},s}\mathbb{M}_{0}(s_{0})N_{n-1}^{-\sigma_{1}},$
(10.361)
$\displaystyle\mathfrak{M}^{\sharp}_{d_{i}(Op^{W}(r_{n}))[\hat{\textbf{\i}}]}(0,s_{0})$
$\displaystyle\leq_{\mathtt{pe},s}|d_{i}r_{n}(i_{0})[\hat{\textbf{\i}}]|^{\sup}_{0,0,0}\overset{\eqref{rsdj1sd12}}{\leq}_{\mathtt{pe},s}\varepsilon^{3}\gamma^{-1}N_{n-1}^{-\sigma_{1}}\rVert\hat{\textbf{\i}}\rVert_{s_{0}+\mu_{\mathtt{p},1}}.$

Estimates for $\Pi_{\geq N_{n}}\mathcal{R}_{n}$. For $\Pi_{\geq
N_{n}}\mathcal{R}_{n}$ in (10.331), it follows from Lemma 2.26 that

$\displaystyle\mathfrak{M}^{\sharp,\gamma}_{\Pi_{\geq
N_{n}}\mathcal{R}_{n}}(0,s)$ $\displaystyle\leq
N_{n}^{-\mathtt{b}_{0}}\sup_{\vec{b}\in\mathbb{N}^{\nu}_{0},\
|\vec{b}|=\mathtt{b}_{0}}\mathfrak{M}^{\sharp,\gamma}_{\partial_{\varphi}^{\vec{b}}(\mathcal{R}_{n})}(0,s)\overset{\eqref{sdjsds11ram2}}{\leq}\mathbb{M}_{0}(s)N_{n}^{-\mathtt{b}_{0}}N_{n-1},$
(10.362) $\displaystyle\sup_{\vec{b}\in\mathbb{N}^{\nu}_{0},\
|\vec{b}|=\mathtt{b}_{0}}\mathfrak{M}^{\sharp,\gamma}_{\partial_{\varphi}^{\vec{b}}(\Pi_{\geq
N_{n}}\mathcal{R}_{n})}(0,s)$
$\displaystyle\leq\sup_{\vec{b}\in\mathbb{N}^{\nu}_{0},\
|\vec{b}|=\mathtt{b}_{0}}\mathfrak{M}^{\sharp,\gamma}_{\partial_{\varphi}^{\vec{b}}(\mathcal{R}_{n})}(0,s)\overset{\eqref{sdjsds11ram2}}{\leq}\mathbb{M}_{0}(s)N_{n-1},$
$\displaystyle\mathfrak{M}^{\sharp}_{d_{i}(\Pi_{\geq
N_{n}}\mathcal{R}_{n})(i_{0})[\hat{\textbf{\i}}]}(0,s_{0})$ $\displaystyle\leq
N_{n}^{-\mathtt{b}_{0}}\sup_{\vec{b}\in\mathbb{N}^{\nu}_{0},\
|\vec{b}|=\mathtt{b}_{0}}\mathfrak{M}^{\sharp}_{\partial_{\varphi}^{\vec{b}}(d_{i}(\mathcal{R}_{n})(i_{0})[\hat{\textbf{\i}}])}(0,s_{0})$
$\displaystyle\overset{\eqref{sdjsds11ram4}}{\leq}_{\mathtt{pe}}\varepsilon^{3}\gamma^{-1}N_{n-1}N_{n}^{-\mathtt{b}_{0}}\rVert\hat{\textbf{\i}}\rVert_{s_{0}+\mu_{\mathtt{p}}},$
$\displaystyle\sup_{\vec{b}\in\mathbb{N}^{\nu}_{0},\
|\vec{b}|=\mathtt{b}_{0}}\mathfrak{M}^{\sharp}_{\partial_{\varphi}^{\vec{b}}(d_{i}(\Pi_{\geq
N_{n}}\mathcal{R}_{n})(i_{0})[\hat{\textbf{\i}}])}(0,s_{0})$
$\displaystyle\leq_{\mathtt{pe}}\sup_{\vec{b}\in\mathbb{N}^{\nu}_{0},\
|\vec{b}|=\mathtt{b}_{0}}\mathfrak{M}^{\sharp}_{\partial_{\varphi}^{\vec{b}}(d_{i}(\mathcal{R}_{n})(i_{0})[\hat{\textbf{\i}}])}(0,s_{0})\overset{\eqref{sdjsds11ram4}}{\leq}_{\mathtt{pe}}\varepsilon^{3}\gamma^{-1}N_{n-1}\rVert\hat{\textbf{\i}}\rVert_{s_{0}+\mu_{\mathtt{p},1}}.$

Estimates for $\mathcal{R}_{n}\psi_{n}$. For $\mathcal{R}_{n}\psi_{n}$ in
(10.331), we use (2.58) to obtain

$\displaystyle\mathfrak{M}^{\sharp,\gamma}_{\mathcal{R}_{n}\psi_{n}}(0,s)$
$\displaystyle\overset{\eqref{jsdj2sdsd1sd1},\eqref{sdjsds11ram1}}{\leq}_{\mathtt{pe}}\varepsilon^{2}\gamma^{-2}N_{n}^{2\tau+1}N_{n-1}^{-2\sigma_{1}}\mathbb{M}_{0}(s_{0})\mathbb{M}_{0}(s)$
$\displaystyle\overset{\eqref{m01sd2est}}{\leq}_{\mathtt{pe}}\varepsilon^{11-10b}N_{n}^{2\tau+1}N_{n-1}^{-2\sigma_{1}}\mathbb{M}_{0}(s),$
(10.363)

and

$\displaystyle\sup_{\vec{b}\in\mathbb{N}^{\nu}_{0},\
|\vec{b}|=\mathtt{b}_{0}}\mathfrak{M}^{\sharp,\gamma}_{\partial_{\varphi}^{\vec{b}}(\mathcal{R}_{n}\psi_{n})}(0,s)$
$\displaystyle\quad\leq_{\mathtt{pe}}\left(\sup_{\vec{b}\in\mathbb{N}^{\nu}_{0},\
|\vec{b}|=\mathtt{b}_{0}}\mathfrak{M}^{\sharp,\gamma}_{\partial_{\varphi}^{\vec{b}}(\mathcal{R}_{n})}(0,s)\mathfrak{M}^{\sharp,\gamma}_{\psi_{n}}(0,s_{0})+\sup_{\vec{b}\in\mathbb{N}^{\nu}_{0},\
|\vec{b}|=\mathtt{b}_{0}}\mathfrak{M}^{\sharp,\gamma}_{\partial_{\varphi}^{\vec{b}}(\mathcal{R}_{n})}(0,s_{0})\mathfrak{M}^{\sharp,\gamma}_{\psi_{n}}(0,s)\right.$
$\displaystyle\left.\quad+\mathfrak{M}^{\sharp,\gamma}_{\mathcal{R}_{n}}(0,s)\sup_{\vec{b}\in\mathbb{N}^{\nu}_{0},\
|\vec{b}|=\mathtt{b}_{0}}\mathfrak{M}^{\sharp,\gamma}_{\partial_{\varphi}^{\vec{b}}(\psi_{n})}(0,s_{0})+\mathfrak{M}^{\sharp,\gamma}_{\mathcal{R}_{n}}(0,s_{0})\sup_{\vec{b}\in\mathbb{N}^{\nu}_{0},\
|\vec{b}|=\mathtt{b}_{0}}\mathfrak{M}^{\sharp,\gamma}_{\partial_{\varphi}^{\vec{b}}(\psi_{n})}(0,s)\right)$
$\displaystyle\quad\overset{\eqref{sdjsds11ram1},\eqref{sdjsds11ram2},\eqref{jsdj2sdsd1sd1},\eqref{jsdj2sdsd1sd2}}{\leq}_{\mathtt{pe}}\varepsilon^{2}\gamma^{-2}N_{n}^{2\tau+1}N_{n-1}^{1-\sigma_{1}}\mathbb{M}_{0}(s_{0})\mathbb{M}_{0}(s)$
$\displaystyle\quad\overset{\eqref{m01sd2est}}{\leq}_{\mathtt{pe}}\varepsilon^{11-10b}N_{n}^{2\tau+1}N_{n-1}^{1-\sigma_{1}}\mathbb{M}_{0}(s).$
(10.364)

Similarly, we have

$\displaystyle\mathfrak{M}^{\sharp}_{d_{i}(\mathcal{R}_{n}\psi_{n})(i_{0})[\hat{\textbf{\i}}]}(0,s_{0})$
$\displaystyle\leq_{\mathtt{pe}}\left(\mathfrak{M}^{\sharp}_{d_{i}(\mathcal{R}_{n})(i_{0})[\hat{\textbf{\i}}]}(0,s_{0})\mathfrak{M}^{\sharp}_{\psi_{n}}(0,s_{0})+\mathfrak{M}^{\sharp}_{d_{i}(\psi_{n})(i_{0})[\hat{\textbf{\i}}]}(0,s_{0})\mathfrak{M}^{\sharp}_{\mathcal{R}_{n}}(0,s_{0})\right)$
$\displaystyle\overset{\eqref{sdjsds11ram3},\eqref{sdjsds11ram1},\eqref{sjdjsddrhcnwkdWlp},\eqref{psiisds2sd}}{\leq_{\mathtt{pe}}}\left(\varepsilon^{3}\gamma^{-1}\varepsilon^{11-10b}+\varepsilon^{3-4b}\mathbb{M}_{0}(s_{0})\right)\varepsilon^{11-10b}N_{n}^{2\tau+1}N_{n-1}^{-2\sigma_{1}}\rVert\hat{\textbf{\i}}\rVert_{s_{0}+\mu_{\mathtt{p},1}}$
$\displaystyle\overset{\eqref{m01sd2est}}{\leq_{\mathtt{pe}}}\varepsilon^{3}\gamma^{-1}\varepsilon^{11-10b}N_{n}^{2\tau+1}N_{n-1}^{-2\sigma_{1}}\rVert\hat{\textbf{\i}}\rVert_{s_{0}+\mu_{\mathtt{p},1}},$
(10.365)

and using (10.338), (10.339), (10.340), (10.341), (10.350), (10.351), (10.356)
and (10.354),

$\displaystyle\sup_{\vec{b}\in\mathbb{N}^{\nu}_{0},\
|\vec{b}|=\mathtt{b}_{0}}\mathfrak{M}^{\sharp}_{\partial_{\varphi}^{\vec{b}}(d_{i}(\mathcal{R}_{n}\psi_{n})(i_{0})[\hat{\textbf{\i}}])}(0,s_{0})$
$\displaystyle\quad\overset{\eqref{chanrul2}}{\leq_{\mathtt{pe}}}\left(\sup_{\vec{b}\in\mathbb{N}^{\nu}_{0},\
|\vec{b}|=\mathtt{b}_{0}}\mathfrak{M}^{\sharp}_{\partial_{\varphi}^{\vec{b}}(d_{i}(\mathcal{R}_{n})(i_{0})[\hat{\textbf{\i}}])}(0,s_{0})\mathfrak{M}^{\sharp}_{\psi_{n}}(0,s_{0})+\sup_{\vec{b}\in\mathbb{N}^{\nu}_{0},\
|\vec{b}|=\mathtt{b}_{0}}\mathfrak{M}^{\sharp}_{\partial_{\varphi}^{\vec{b}}(\mathcal{R}_{n})}(0,s_{0})\mathfrak{M}^{\sharp}_{d_{i}\psi_{n}(i_{0})[\hat{\textbf{\i}}]}(0,s_{0})\right.$
$\displaystyle\quad\left.\ +\sup_{\vec{b}\in\mathbb{N}^{\nu}_{0},\
|\vec{b}|=\mathtt{b}_{0}}\mathfrak{M}^{\sharp}_{\partial_{\varphi}^{\vec{b}}(d_{i}(\psi_{n})(i_{0})[\hat{\textbf{\i}}])}(0,s_{0})\mathfrak{M}^{\sharp}_{\mathcal{R}_{n}}(0,s_{0})+\sup_{\vec{b}\in\mathbb{N}^{\nu}_{0},\
|\vec{b}|=\mathtt{b}_{0}}\mathfrak{M}^{\sharp}_{\partial_{\varphi}^{\vec{b}}(\psi_{n})}(0,s_{0})\mathfrak{M}^{\sharp}_{d_{i}\mathcal{R}_{n}(i_{0})[\hat{\textbf{\i}}]}(0,s_{0})\right)$
$\displaystyle\quad\
\leq_{\mathtt{pe}}\varepsilon^{3}\gamma^{-1}\varepsilon^{11-10b}N_{n}^{2\tau+1}N_{n-1}^{1-\sigma_{1}}\rVert\hat{\textbf{\i}}\rVert_{s_{0}+\mu_{\mathtt{p},1}}.$
(10.366)

Lastly, using Lemma 2.23, (10.354) and the definition of $\mathcal{R}_{n+1}$
in (10.331), we see from the above estimates (10.356), (10.357), (10.358),
(10.360), (10.361), (10.362), (10.363), (10.364) and (10.6.0.3) that

$\displaystyle\mathfrak{M}^{\sharp,\gamma}_{\mathcal{R}_{n+1}}(0,s)$
$\displaystyle\leq_{\mathtt{pe},s}\left(\varepsilon^{11-10b}N_{n}^{2\tau+1}N_{n-1}^{-2\sigma_{1}}+N_{n}^{-\mathtt{b}_{0}}N_{n-1}\right)\mathbb{M}_{0}(s),$
(10.367) $\displaystyle\sup_{\vec{b}\in\mathbb{N}^{\nu}_{0},\
|\vec{b}|=\mathtt{b}_{0}}\mathfrak{M}^{\sharp,\gamma}_{\partial_{\varphi}^{\vec{b}}(\mathcal{R}_{n+1})}(0,s)$
$\displaystyle\leq_{\mathtt{pe},s}\left(\varepsilon^{11-10b}N_{n}^{2\tau+1}N_{n-1}^{1-\sigma_{1}}+N_{n-1}\right)\mathbb{M}_{0}(s),$
$\displaystyle\mathfrak{M}^{\sharp}_{d_{i}(\mathcal{R}_{n+1})(i_{0})[\hat{\textbf{\i}}]}(0,s_{0})$
$\displaystyle\leq_{\mathtt{pe},}(\varepsilon^{11-10b}N_{n}^{2\tau+1}N_{n-1}^{-2\sigma_{1}}+N_{n}^{-\mathtt{b}_{0}}N_{n-1})\varepsilon^{3}\gamma^{-1}\rVert\hat{\textbf{\i}}\rVert_{s_{0}+\mu_{\mathtt{p},1}},$
$\displaystyle\sup_{\vec{b}\in\mathbb{N}^{\nu}_{0},\
|\vec{b}|=\mathtt{b}_{0}}\mathfrak{M}^{\sharp}_{\partial_{\varphi}^{\vec{b}}(d_{i}\mathcal{R}_{n+1}(i_{0})[\hat{\textbf{\i}}])}(0,s_{0})$
$\displaystyle\leq_{\mathtt{pe}}(\varepsilon^{11-10b}N_{n}^{2\tau+1}N_{n-1}^{1-\sigma_{1}}+N_{n-1})\varepsilon^{3}\gamma^{-1}\rVert\hat{\textbf{\i}}\rVert_{s_{0}+\mu_{\mathtt{p},1}}.$

Using (10.322), (10.325) and (10.326), it is clear that

$\displaystyle\varepsilon^{11-10b}N_{n}^{2\tau+1}N_{n-1}^{-2\sigma_{1}}+N_{n}^{-\mathtt{b}_{0}}N_{n-1}$
$\displaystyle\leq_{\mathtt{pe}}(\varepsilon^{\frac{1}{22}}+N_{0}^{1-\sigma_{1}})N_{n}^{-\sigma_{1}},$
$\displaystyle\varepsilon^{11-10b}N_{n}^{2\tau+1}N_{n-1}^{1-\sigma_{1}}+N_{n-1}$
$\displaystyle=\left(\varepsilon^{11-10b}N_{n}^{2\tau+1}N_{n-1}^{-\sigma_{1}}+1\right)\frac{N_{n-1}}{N_{n}}N_{n}$
$\displaystyle\overset{\eqref{para1chs12}}{\leq}\left(\varepsilon^{\frac{1}{22}}+1\right)\frac{N_{-1}}{N_{0}}N_{n}$
$\displaystyle\leq 2N_{0}^{-1}N_{n}.$

Plugging these into (10.367), we have that

$\displaystyle\mathfrak{M}^{\sharp,\gamma}_{\mathcal{R}_{n+1}}(0,s)$
$\displaystyle\leq_{\mathtt{pe},s}(\varepsilon^{\frac{1}{22}}+N_{0}^{1-\sigma_{1}})N_{n}^{-\sigma_{1}}\mathbb{M}_{0}(s),$
$\displaystyle\sup_{\vec{b}\in\mathbb{N}^{\nu}_{0},\
|\vec{b}|=\mathtt{b}_{0}}\mathfrak{M}^{\sharp,\gamma}_{\partial_{\varphi}^{\vec{b}}(\mathcal{R}_{n+1})}(0,s)$
$\displaystyle\leq_{\mathtt{pe},s}N_{0}^{-1}N_{n}\mathbb{M}_{0}(s),$
$\displaystyle\mathfrak{M}^{\sharp}_{d_{i}(\mathcal{R}_{n+1})(i_{0})[\hat{\textbf{\i}}]}(0,s_{0})$
$\displaystyle\leq_{\mathtt{pe}}(\varepsilon^{\frac{1}{22}}+N_{0}^{1-\sigma_{1}})N_{n}^{-\sigma_{1}}\varepsilon^{3}\gamma^{-1}\rVert\hat{\textbf{\i}}\rVert_{s_{0}+\mu_{\mathtt{p},1}},$
$\displaystyle\sup_{\vec{b}\in\mathbb{N}^{\nu}_{0},\
|\vec{b}|=\mathtt{b}_{0}}\mathfrak{M}^{\sharp}_{\partial_{\varphi}^{\vec{b}}(d_{i}\mathcal{R}_{n+1}(i_{0})[\hat{\textbf{\i}}])}(0,s_{0})$
$\displaystyle\leq_{\mathtt{pe}}N_{0}^{-1}N_{n}\varepsilon^{3}\gamma^{-1}\rVert\hat{\textbf{\i}}\rVert_{s_{0}+\mu_{\mathtt{p},1}}.$

Using (10.322), we can choose $\varepsilon$ sufficiently small depending on
$\mathtt{pe}$ and $\mathtt{S}$, so that $N_{0}^{-1}$ and
$N_{0}^{1-\sigma_{1}}$ are as small as we need. This gives the desired
estimates (10.338)-(10.341), for $n+1$.

Inductive step for (2). Assuming (1), (2), and (3) are true for $n$, we aim to
prove (2) for $n\to n+1$. Using (10.329), we see that $Op^{W}(r_{n+2})$ is
also real and reversible, since $\mathcal{R}_{n+1}$ is real and reversible and
$Op^{W}(r_{n+2})$ is nothing but a restriction to the diagonal part of
$\mathcal{R}_{n+1}$. For the estimates, we use Lemma 2.29 and (10.329) to see
that

$\displaystyle|r_{n+2}|^{\text{Lip}(\gamma,\Omega^{\gamma}_{n+1}(i_{0}))}_{0,0,0}$
$\displaystyle\leq_{\mathtt{pe}}\mathfrak{M}^{\sharp,\gamma}_{\mathcal{R}_{n+1}}(0,s_{0}),$
$\displaystyle|d_{i}r_{n+2}(i_{0})[\hat{\textbf{\i}}]|^{\text{Lip}(\gamma,\Omega^{\gamma}_{n+1}(i_{0}))}_{0,0,0}$
$\displaystyle\leq_{\mathtt{pe}}\mathfrak{M}^{\sharp,\gamma}_{d_{i}\mathcal{R}_{n+1}(i_{0})[\hat{\textbf{\i}}]}(0,s_{0}),$

where the second estimate follows from the relation between $r_{n+1}$ and
$\mathcal{R}_{n}$ being linear. Hence, using (10.338), (10.340) for $n\to
n+1$, which have been already proved above, and (10.354), we have

$\displaystyle|r_{n+2}|^{\text{Lip}(\gamma,\Omega^{\gamma}_{n+1}(i_{0}))}_{0,0,0}$
$\displaystyle\leq_{\mathtt{pe}}\varepsilon^{9-6b}N_{n}^{-\sigma_{1}},$
$\displaystyle|d_{i}r_{n+2}(i_{0})[\hat{\textbf{\i}}]|^{\text{Lip}(\gamma,\Omega^{\gamma}_{n+1}(i_{0}))}_{0,0,0}$
$\displaystyle\leq_{\mathtt{pe}}\varepsilon^{3}\gamma^{-1}N_{n}^{-\sigma_{1}}\rVert\hat{\textbf{\i}}\rVert_{s_{0}+\mu_{\mathtt{p},1}},$

and this gives (10.342) and (10.344). Recalling the definitions of $r_{\leq
n+2}$ from (10.330) and noting that $\sum_{n\geq
0}N_{n-1}^{-\sigma_{1}}<\infty$, the estimates for $r_{\leq n+2}$ in (10.343)
and (10.345) follow immediately.

Inductive step for (3). By the induction hypothesis, we already know that
$D_{n+1}$ is real and reversible, and the estimates (10.343) and (10.345) hold
true for $r_{\leq n+1}$. Furthermore, we have already proved that
$\mathcal{R}_{n+1}$ is also real, reversible and satisfying (10.337).
Therefore, we apply Lemma 10.77 to the homological equation in (10.328), which
gives us that $\psi_{n+1}$ is well-defined for
$\omega\in\Omega_{n+2}^{\gamma}(i_{0})$ and it is real, reversibility
preserving and satisfies (10.346). Furthermore, (10.304)with the estimates
(10.338) for $\mathcal{R}_{n+1}$, which we have already proved, gives us the
estimate (10.347) for $\psi_{n+1}$.

Inductive step for (4). Using the induction hypothesis, we have that

$\Omega_{n+1}^{2\gamma}(i_{1})\overset{\eqref{sddpooo3}}{\subset}\Omega^{2\gamma}_{n}(i_{1})\overset{\eqref{tlwoclaosd}}{\subset}\Omega^{2(\gamma-\delta)}_{n}(i_{2})\overset{\because\
\delta<\gamma/2}{\subset}\Omega_{n}^{\gamma}(i_{2})\overset{\eqref{sddpooo3}}{\subset}\Omega_{n-1}^{\gamma}(i_{2}).$

Therefore denoting the dependence of $d_{n}$ on the embedding, we have that
$r_{n}(i_{2})(\omega,j)$ and $r_{n+1}(i_{1})(\omega,j)$ are well-defined for
$\omega\in\Omega_{n+1}^{2\gamma}(i_{1})$ (indeed, $r_{n+1}(i_{1})$ is well-
defined on
$\Omega^{\gamma}_{n}(i_{1})\overset{\eqref{sddpooo3}}{\supset}\Omega^{2\gamma}_{n}(i_{1})$
and $r_{n}(i_{2})$ is well-defined on $\Omega_{n-1}^{\gamma}(i_{2}))$. For
such $\omega$, we have that

$\displaystyle|$
$\displaystyle\left(d_{n}(i_{2})(\omega,j)-d_{n}(i_{2})(\omega,k)\right)-\left(d_{n}(i_{1})(\omega,j)-d_{n}(i_{1})(\omega,k)\right)|$
$\displaystyle\leq|\left(d_{0}(i_{2})(\omega,j)-d_{0}(i_{2})(\omega,k)\right)-\left(d_{0}(i_{1})(\omega,j)-d_{0}(i_{1})(\omega,k)\right)|+2\sup_{j\in
S^{\perp}}|r_{\leq n}(i_{1})(\omega,j)-r_{\leq n}(i_{2})(\omega,j)|.$ (10.368)

From (10.291), we have that

$\displaystyle|$
$\displaystyle\left(d_{0}(i_{2})(\omega,j)-d_{0}(i_{2})(\omega,k)\right)-\left(d_{0}(i_{1})(\omega,j)-d_{0}(i_{1})(\omega,k)\right)|$
$\displaystyle\leq|(\mathtt{m}_{\alpha}(i_{2}(\omega))(\omega)-\mathtt{m}_{\alpha}(i_{1}(\omega))(\omega))(\lambda_{\alpha}(j)-\lambda_{\alpha}(k))|$
$\displaystyle\
+|\left(\mathfrak{m}_{1}(i_{2})(\omega,j)-\mathfrak{m}_{1}(i_{2})(\omega,k)\right)-\left(\mathfrak{m}_{1}(i_{1})(\omega,j)-\mathfrak{m}_{1}(i_{1})(\omega,k)\right)|$
$\displaystyle\overset{\eqref{nusd2sdsd}}{\leq}_{\mathtt{pe}}\varepsilon^{2}\rVert
i_{1}-i_{2}\rVert_{s_{0}+\mu_{\mathtt{p},1}}|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|+\int_{j}^{k}\left|\partial_{\xi}\mathfrak{m}_{1}(i_{1})(\omega,\xi)-\partial_{\xi}\mathfrak{m}_{1}(i_{2})(\omega,\xi)\right|d\xi$
$\displaystyle\overset{\eqref{rejsmsdim1}}{\leq}_{\mathtt{pe}}\left(\varepsilon^{2}\rVert
i_{1}-i_{2}\rVert_{s_{0}+\mu_{\mathtt{p},1}}|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|+\varepsilon^{2}|k-j|\rVert
i_{1}-i_{2}\rVert_{s_{0}+\mu_{\mathtt{p},1}}\right)$
$\displaystyle\overset{\text{
Lemma~{}\ref{melnikove_23s}}}{\leq}_{\mathtt{pe}}\varepsilon^{2}\rVert
i_{1}-i_{2}\rVert_{s_{0}+\mu_{\mathtt{p},1}}|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|.$

Plugging this into (10.368) and using (10.345) for $r_{\leq n}$, we obtain

$\displaystyle|$
$\displaystyle\left(d_{n}(i_{2})(\omega,j)-d_{n}(i_{2})(\omega,k)\right)-\left(d_{n}(i_{1})(\omega,j)-d_{n}(i_{1})(\omega,k)\right)|$
$\displaystyle\leq_{\mathtt{pe}}\varepsilon^{2}\rVert
i_{1}-i_{2}\rVert_{s_{0}+\mu_{\mathtt{p},1}}|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|+\varepsilon^{3}\gamma^{-1}\rVert
i_{1}-i_{2}\rVert_{s_{0}+\mu_{\mathtt{p},1}}$
$\displaystyle\leq_{\mathtt{pe}}\varepsilon^{3}\gamma^{-1}\rVert
i_{1}-i_{2}\rVert_{s_{0}+\mu_{\mathtt{p},1}}|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|$
$\displaystyle\leq C({\mathtt{pe}})\varepsilon^{3}\gamma^{-1}\rVert
i_{1}-i_{2}\rVert_{s_{0}+\mu_{\mathtt{p},1}}|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|,$

for some $C(\mathtt{pe})$. Therefore, if

$C(\mathtt{pe})\varepsilon^{3}\gamma^{-1}N_{n}^{\tau}\rVert
i_{1}-i_{2}\rVert_{s_{0}+\mu_{\mathtt{p},1}}\leq\delta,$

then we have that

$\displaystyle|$
$\displaystyle\left(d_{n}(i_{2})(\omega,j)-d_{n}(i_{2})(\omega,k)\right)-\left(d_{n}(i_{1})(\omega,j)-d_{n}(i_{1})(\omega,k)\right)|\leq\delta
N_{n}^{-\tau}|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|.$ (10.369)

Thus, for $\omega\in\Omega_{n+1}^{2\gamma}(i_{1})$ and $|l|\leq N_{n}$,

$\displaystyle|$
$\displaystyle\text{i}\omega-(d_{n}(i_{2})(\omega,j)-d_{n}(i_{2})(\omega,k))|$
$\displaystyle\geq|\text{i}\omega-(d_{n}(i_{1})(\omega,j)-d_{n}(i_{1})(\omega,k))|-|\left(d_{n}(i_{2})(\omega,j)-d_{n}(i_{2})(\omega,k)\right)-\left(d_{n}(i_{1})(\omega,j)-d_{n}(i_{1})(\omega,k)\right)|$
$\displaystyle\overset{\because\omega\in\Omega_{n+1}^{2\gamma}(i_{1})}{\geq}\gamma|l|^{-\tau}|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|-|\left(d_{n}(i_{2})(\omega,j)-d_{n}(i_{2})(\omega,k)\right)-\left(d_{n}(i_{1})(\omega,j)-d_{n}(i_{1})(\omega,k)\right)|$
$\displaystyle\overset{\eqref{sdskxdnwnfmfcxc}}{\geq}(\gamma-\delta)|l|^{-\tau}|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|.$

which implies $\omega\in\Omega_{n+1}^{2(\gamma-\delta)}(i_{2})$, thus,
$\Omega_{n+1}^{2\gamma}(i_{1})\subset\Omega^{2(\gamma-\delta)}_{n+1}(i_{2})$.

Proof of (10.349). From the above induction steps, it follows that
$D_{n+1},\mathcal{R}_{n},\psi_{n},\tilde{\psi}_{n}$ are all well-defined for
$\omega\in\cap_{n\geq 0}\Omega_{n}^{\gamma}(i_{0})$. Then, (10.349) follows
from (10.6.0.3).

Lastly, since $\phi_{n}$ is real, reversibility and
$\frac{2\pi}{\mathtt{M}}$-translation invariance preserving and $L_{0}$ is
real and reversible and $\frac{2\pi}{\mathtt{M}}$-translation invariant ,
$L_{n}$ is real, reversible and $\frac{2\pi}{\mathtt{M}}$-translation
invariant as well. ∎

######  Lemma 10.80.

For $\omega\in\cap_{n\geq 0}\Omega^{\gamma}_{n}(i_{0})$, let us define (note
that each $\phi_{n}$ depends on $\omega$)

$\displaystyle\Phi_{7,0}:=\phi_{0},\quad\Phi_{7,n+1}:=\Phi_{7,n}\circ\phi_{n+1}.$
(10.370)

Then given $\mathtt{S}\gg s_{0}$, there exists
$\varepsilon_{0}(\mathtt{pe},\mathtt{S})>0$ such that if
$\varepsilon\in(0,\varepsilon_{0})$, then there exists an operator
$\Phi_{7,\infty}$ such that

$\displaystyle\lim_{n\to\infty}\rVert(\Phi_{7,\infty}-\Phi_{7,n})h\rVert_{s}=0,\text{
for all $h\in C^{\infty}_{\varphi,x}$ and $s\in[s_{0},\mathtt{S}]$}.$ (10.371)

Furthermore, $\Phi_{7,\infty}-I$ is Lip-$0$-modulo-tame operator up to
$\mathtt{S}$ with the estimates

$\displaystyle\mathfrak{M}^{\sharp,\gamma}_{\Phi_{7,\infty}-I}(0,s)\leq_{\mathtt{pe},\mathtt{S}}\varepsilon^{2}\gamma^{-2}N_{0}^{2\tau+1}\mathbb{M}_{0}(s).$
(10.372)

Lastly, the same results hold as above for $\phi_{n}{-1}$, that is,
$\Phi_{7,\infty}^{-1}$, the inverse of $\Phi_{7,\infty}$, satisfies

$\lim_{n\to\infty}\rVert(\Phi_{7,\infty}^{-1}-\Phi_{7,n}^{-1})h\rVert_{s}=0,\text{
and
}\mathfrak{M}^{\sharp,\gamma}_{\Phi_{7,\infty}{-1}-I}(0,s)\leq_{\mathtt{S},\mathtt{pe}}\varepsilon^{2}\gamma^{-2}N_{0}^{2\tau+1}\mathbb{M}_{0}(s),$

for $s\in[s_{0},\mathtt{S}]$.

###### Proof.

In the proof, we will denote by $c_{\mathtt{pe}}$ a constant that varies from
line to line, depending on $\mathtt{pe}$ only. We will prove the lemma for
$\Phi_{7,\infty}$ only, since the same argument applies to
$\Phi_{7,\infty}^{-1}$.

First, we derive a tame constant of $\Phi_{7,n}$ We claim that

$\displaystyle\mathfrak{M}^{\sharp,\gamma}_{\Phi_{7,n}}(0,s_{0})\leq
c_{\mathtt{pe}}^{n+1}\text{ for some $c_{\mathtt{pe}}\geq 1$ for all $n\geq
0$}.$ (10.373)

When $n=0$, it follows from (10.347) that

$\displaystyle\mathfrak{M}^{\sharp,\gamma}_{\Phi_{7,0}}(0,s_{0})$
$\displaystyle\leq 1+\mathfrak{M}^{\sharp,\gamma}_{\psi_{0}}(0,s_{0})$
$\displaystyle\leq
1+c_{\mathtt{pe}}\varepsilon^{2}\gamma^{-2}N_{-1}^{-\sigma_{1}}N_{0}^{2\tau+1}\mathbb{M}_{0}(s_{0})$
$\displaystyle\overset{\eqref{m01sd2est},\
N_{-1}=1}{\leq}1+c_{\mathtt{pe}}\varepsilon^{11-10b}N_{-1}^{-2\sigma_{1}}N_{0}^{2\tau+1}$
$\displaystyle\overset{\eqref{para1chs12}}{\leq}c_{\mathtt{pe}}.$

Assuming (10.373) for $n\geq 0$, Lemma 2.23 tells us that

$\displaystyle\mathfrak{M}^{\sharp,\gamma}_{\Phi_{7,n+1}}(0,s_{0})$
$\displaystyle=2\mathfrak{M}^{\sharp,\gamma}_{\Phi_{7,n}}(0,s_{0})\mathfrak{M}^{\sharp,\gamma}_{\phi_{n+1}}(0,s_{0}),$
$\displaystyle\overset{\eqref{claimfors_0023212}}{\leq}2c_{\mathtt{pe}}^{n+1}(1+\mathfrak{M}^{\sharp,\gamma}_{\psi_{n+1}}(0,s_{0}))$
$\displaystyle\overset{\eqref{psdjsds11ram1}}{\leq}2c_{\mathtt{pe}}^{n+1}\left(1+c_{\mathtt{pe}}\varepsilon^{2}\gamma^{-2}N_{n}^{-\sigma_{1}}N_{n+1}^{2\tau+1}\mathbb{M}_{0}(s_{0})\right)$
$\displaystyle\overset{\eqref{m01sd2est}}{\leq}2c_{\mathtt{pe}}^{n+1}(1+c_{\mathtt{pe}}\varepsilon^{11-10b}N_{n}^{-\sigma_{1}}N_{n+1}^{2\tau+1})$
$\displaystyle\overset{\eqref{para1chs12}}{\leq}2c_{\mathtt{pe}}^{n+1}\left(1+\varepsilon^{\frac{1}{22}}c_{\mathtt{pe}}\right)$

Therefore, assuming that $c_{\mathtt{pe}}$ is large enough depending on
$\mathtt{pe}$, we obtain (10.373) for $n+1$.

Now, we claim that

$\displaystyle\mathfrak{M}^{\sharp,\gamma}_{\Phi_{7,n}}(0,s)\leq
c_{\mathtt{pe}}^{n+1}(1+\varepsilon^{2}\gamma^{-2}N^{-\sigma_{1}}_{n-1}N_{n}^{2\tau+1}\mathbb{M}_{0}(s)),$
(10.374)

for some constant $c_{\mathtt{pe}}>0$. When $n=0$, the proof is identical to
the case when $s=s_{0}$ as above. For $n\geq 0$, Lemma 2.23 yields that

$\displaystyle\mathfrak{M}^{\sharp,\gamma}_{\Phi_{7,{n+1}}}(0,s)$
$\displaystyle\leq\mathfrak{M}^{\sharp,\gamma}_{\Phi_{7,n}}(0,s)\mathfrak{M}^{\sharp,\gamma}_{\phi_{n+1}}(0,s_{0})+\mathfrak{M}^{\sharp,\gamma}_{\Phi_{7,n}}(0,s_{0})\mathfrak{M}^{\sharp,\gamma}_{\phi_{n+1}}(0,s)$
$\displaystyle\overset{\eqref{claimfors_0023212}}{\leq}\mathfrak{M}^{\sharp,\gamma}_{\Phi_{7,{n}}}(0,s)(1+\mathfrak{M}^{\sharp,\gamma}_{\psi_{{n+1}}}(0,s_{0}))+c_{\mathtt{pe}}^{n+1}(1+\mathfrak{M}^{\sharp,\gamma}_{\psi_{n+1}}(0,s))$
$\displaystyle\overset{\eqref{psdjsds11ram1},\eqref{sd2sdsdasd}}{\leq}4c_{\mathtt{pe}}^{n+1}(1+\varepsilon^{2}\gamma^{-2}N_{n}^{-\sigma_{1}}N_{n+1}^{2\tau+1}\mathbb{M}_{0}(s)),$

where we used $\mathfrak{M}^{\sharp,\gamma}_{\psi_{{n+1}}}(0,s_{0})\leq 1$ for
sufficiently small $\varepsilon>0$, which follows from (10.355). Assuming that
$c_{\mathtt{pe}}\geq 4$, we obtain (10.374) for $n+1$. Therefore, we have that
for some $c_{\mathtt{pe}}>0$,

$\displaystyle\mathfrak{M}^{\sharp,\gamma}_{\Phi_{7,n}\circ\psi_{n+1}}(0,s)$
$\displaystyle\overset{\text{Lemma~{}\ref{consdsdpcosdldsx}}}{\leq}\mathfrak{M}^{\sharp,\gamma}_{\Phi_{7,n}}(0,s)\mathfrak{M}^{\sharp,\gamma}_{\psi_{n+1}}(0,s_{0})+\mathfrak{M}^{\sharp,\gamma}_{\Phi_{7,n}}(0,s_{0})\mathfrak{M}^{\sharp,\gamma}_{\psi_{n+1}}(0,s)$
$\displaystyle\overset{\eqref{sd2sdsdasd},\eqref{psdjsds11ram1}}{\leq}2c_{\mathtt{pe}}^{n+1}\varepsilon^{2}\gamma^{-2}N_{n}^{-\sigma_{1}}N_{n+1}^{2\tau+1}\left(\mathbb{M}_{0}(s_{0})+(1+\underbrace{\varepsilon^{2}\gamma^{-2}N^{-\sigma_{1}}_{n-1}N_{n}^{2\tau+1}\mathbb{M}_{0}(s_{0})}_{\leq
1,\because\eqref{m01sd2est},\eqref{para1chs12}})\mathbb{M}_{0}(s)\right),$
$\displaystyle\overset{\mathbb{M}_{0}(s_{0})\leq\mathbb{M}_{0}(s)}{\leq}6c_{\mathtt{pe}}^{n+1}\varepsilon^{2}\gamma^{-2}N_{n}^{-\sigma_{1}}N_{n+1}^{2\tau+1}\mathbb{M}_{0}(s)$
$\displaystyle\leq
6c_{\mathtt{pe}}^{n+1}\varepsilon^{2}\gamma^{-2}N_{n}^{-\frac{\sigma_{1}}{2}}\mathbb{M}_{0}(s),$
(10.375)

where the last inequality follows from

$N_{n}^{-\sigma_{1}}N_{n+1}^{2\tau+1}\overset{\eqref{parameterssd1sd}}{=}N_{0}^{\chi^{n}(3/2(2\tau+1)-\sigma_{1})}\leq
N_{0}^{-\chi^{n}\frac{\sigma_{1}}{2}}=N_{n}^{-\frac{\sigma_{1}}{2}}.$

Therefore, for all $n\geq 0$ and $m\geq 1$, noting that

$\displaystyle\Phi_{7,n+m}-\Phi_{7,n}=\sum_{k=n}^{n+m-1}(\Phi_{7,k+1}-\Phi_{7,k})\overset{\eqref{pusjdwsd1ss},\eqref{sddpooo2}}{=}\sum_{k=n}^{n+m-1}\Phi_{7,k}\circ\psi_{k+1},$
(10.376)

we see that

$\displaystyle\mathfrak{M}^{\sharp,\gamma}_{\Phi_{7,n+m}-\Phi_{7,n}}(0,s)\overset{\eqref{rksdopwogodnseos1}}{\leq}\sum_{k=n}^{\infty}\mathfrak{M}^{\sharp,\gamma}_{\Phi_{7,k}\circ\psi_{k+1}}(0,s)\overset{\eqref{ericcartma22n}}{\leq}6\varepsilon^{2}\gamma^{-2}\mathbb{M}_{0}(s)\sum_{k=n}^{\infty}c_{\mathtt{pe}}^{k+1}N_{k}^{-\frac{\sigma_{1}}{2}}.$
(10.377)

For each $\omega\in\omega\in\cap_{n\geq 0}\Omega^{\gamma}_{n}(i_{0})$, for
which the $\phi_{n}$’s are well-defined for all $n\geq 0$ (see (3) of
Proposition 10.79), we define $\Phi_{7,\infty}=\Phi_{7,\infty}(\omega)$ to be

$\displaystyle\Phi_{7,\infty}h:=\lim_{n\to\infty}\Phi_{7,n}h,\text{ for $h\in
C^{\infty}_{\varphi,x}$,}$ (10.378)

in the strong limit in $H^{s}_{\varphi,x}$. Indeed, we need to show that for
each $h\in C^{\infty}_{\varphi,x}$, the sequence $\Phi_{7,\infty}h$ converges
strongly in $H^{s}_{\varphi,x}$. Using the notation in (2.62), it follows from
(10.377) that for $\omega,\omega_{1},\omega_{2}\in\cap_{n\geq
0}\Omega^{\gamma}_{n}(i_{0})$,

$\displaystyle\rVert(\Phi_{7,n+m}(\omega)-\Phi_{7,n}(\omega))h\rVert_{s}$
$\displaystyle\leq\sum_{k=0}^{1}\rVert(\underline{\Delta^{k}_{1,2}\Phi_{7,n+m}}-\underline{\Delta^{k}_{12}\Phi_{7,n}})h\rVert_{s}$
$\displaystyle\leq\mathfrak{M}^{\sharp,\gamma}_{\Phi_{7,n+m}-\Phi_{7,n}}(0,s)\rVert
h\rVert_{s_{0}}+\mathfrak{M}^{\sharp,\gamma}_{\Phi_{7,n+m}-\Phi_{7,n}}(0,s_{0})\rVert
h\rVert_{s}$
$\displaystyle\leq_{\mathtt{pe}}\varepsilon^{2}\gamma^{-2}(\mathbb{M}_{0}(s_{0})\rVert
h\rVert_{s}+\mathbb{M}_{0}(s)\rVert
h\rVert_{s_{0}})\sum_{k=n}^{\infty}c_{\mathtt{pe}}^{k+1}N_{k}^{-\frac{\sigma_{1}}{2}}.$
(10.379)

Using $\sigma_{1}$ and $N_{n}$ in (10.322), one can easily check that
$\sum_{k=0}^{\infty}c_{\mathtt{pe}}^{k+1}N_{k}^{-\frac{\sigma_{1}}{2}}<\infty$
for sufficiently small $\varepsilon>0$, since $N_{n}$ increases to $\infty$
double-exponentially in $n$. Therefore, $n\mapsto\Phi_{7,n}(\omega)h$ is a
Cauchy sequence in $H^{s}_{\varphi,x}$, hence, (10.378) is well-defined.
Furthermore, the above estimates also show that
$n\mapsto\underline{\Delta^{k}_{12}\Phi_{7,n}}h$ is also a Cauchy sequence for
$k=0,1$. Therefore, using the definition of Lip-$0$-modulo tame operators, one
can easily check that
$\lim_{n\to\infty}\underline{\Delta^{k}_{12}\Phi_{n}}h=\underline{\Delta^{k}_{12}\Phi_{\infty}}h$
in $H^{s}_{\varphi,x}$ strongly, therefore, plugging $n=0$ and letting
$m\to\infty$ in (10.379) to obtain that

$\displaystyle\rVert\left(\underline{\Delta^{k}_{12}\Phi_{7,\infty}}-\underline{\Delta^{k}_{12}\Phi_{7,0}}\right)h\rVert_{s}\leq
c_{\mathtt{pe}}\varepsilon^{2}\gamma^{-2}(\mathbb{M}_{0}(s_{0})\rVert
h\rVert_{s}+\mathbb{M}_{0}(s)\rVert h\rVert_{s_{0}}).$

Therefore, taking the supremum in $\omega,\omega_{1},\omega_{2}$ over
$\cap_{n\geq 0}\Omega^{\gamma}_{n}(i_{0})$, we obtain

$\displaystyle\mathfrak{M}^{\sharp,\gamma}_{\Phi_{7,\infty}-\Phi_{7,0}}(0,s)\leq_{\mathtt{pe}}\varepsilon^{2}\gamma^{-2}\mathbb{M}_{0}(s).$
(10.380)

Thus, we get

$\displaystyle\mathfrak{M}^{\sharp,\gamma}_{\Phi_{7,\infty}-I}(0,s)$
$\displaystyle\leq\mathfrak{M}^{\sharp,\gamma}_{\Phi_{7,\infty}-\Phi_{7,0}}(0,s)+\mathfrak{M}^{\sharp,\gamma}_{\Phi_{7,0}-I}(0,s)$
$\displaystyle\overset{\eqref{sddpooo2}}{\leq}\mathfrak{M}^{\sharp,\gamma}_{\Phi_{7,\infty}-\Phi_{7,0}}(0,s)+\mathfrak{M}^{\sharp,\gamma}_{\psi_{0}}(0,s)$
$\displaystyle\overset{\eqref{psdjsds11ram1},\eqref{iirpsdsd1sd}}{\leq}c_{\mathtt{pe}}\varepsilon^{2}\gamma^{-2}\mathbb{M}_{0}(s)+c_{\mathtt{pe}}\varepsilon^{2}\gamma^{-2}N_{0}^{2\tau+1}\mathbb{M}_{0}(s)$
$\displaystyle\leq
c_{\mathtt{pe}}\varepsilon^{2}\gamma^{-2}N_{0}^{2\tau+1}\mathbb{M}_{0}(s),$

which proves (10.372). ∎

Now, we denote

$\displaystyle r_{\infty}(0,j)$ $\displaystyle:=\sum_{n=0}^{\infty}r_{n},\quad
d_{\infty}(i_{0})(\omega,j):=d_{0}(\omega,j)+r_{\infty}(\omega,j),$ (10.381)
$\displaystyle\Omega^{4\gamma}_{\infty}(i_{0})$
$\displaystyle:=\left\\{\omega\in\Omega_{1}:|\text{i}\omega-(d_{\infty}(i_{0})(\omega,j)-d_{\infty}(i_{0})(\omega,k))|\geq
2\gamma|l|^{-\tau}|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|,\quad\forall l\neq
0,\forall j,k\in S_{\mathtt{M}}^{\perp}\cup\left\\{0\right\\}\right\\}.$

######  Lemma 10.81.

We have that

$|r_{\infty}|^{\text{Lip}(\gamma,\cap_{n\geq
0}\Omega^{\gamma}_{n}(i_{0}))}_{0,0,0}\leq_{\mathtt{pe}}\mathbb{M}_{0}(s_{0}),\quad|d_{i}r_{\infty}(i_{0})[\hat{\textbf{\i}}]|^{\sup}_{0,0,0}\leq_{\mathtt{pe}}\varepsilon^{3}\gamma^{-1}\rVert\hat{\textbf{\i}}\rVert_{s_{0}+\mu_{\mathtt{p},1}}.$

We also have that

$\displaystyle\Omega^{4\gamma}_{\infty}(i_{0})\subset\cap_{n\geq
0}\Omega^{\gamma}_{n}(i_{0}),$ (10.382)

therefore $r_{\infty}$ is well-defined for all
$\omega\in\Omega^{4\gamma}_{\infty}(i_{0})$.

###### Proof.

The estimates for $r_{\infty}$ follow immediately from (2) of Proposition
10.79.

For each $\omega\in\Omega^{4\gamma}_{\infty}(i_{0})$, we have
$\omega\in\Omega_{1}\subset\Omega^{\gamma}_{0}(i_{0})$ by definition. For
$n\geq 0$, we have that that (recalling the definition of
$\Omega^{\gamma}_{n+1}(i_{0})$ from (10.332)) for $|l|\leq N_{n}$ and $j\neq
k$,

$\displaystyle|\text{i}\omega\cdot l-(d_{n}(\omega,j)-d_{n}(\omega,k))|$
$\displaystyle\geq|\text{i}\omega\cdot
l-(d_{\infty}(\omega,j)-d_{\infty}(\omega,k))|-2\sum_{k\geq
n+1}|r_{k}|^{\sup}_{0,0,0}$
$\displaystyle\overset{\eqref{sjdjsd2sdsddenoteasd},\eqref{rsdj1sd11},\eqref{m01sd2est}}{\geq}2\gamma|l|^{-\tau}|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|-c_{\mathtt{pe}}\varepsilon^{9-6b}\sum_{k\geq
n+1}N_{k-2}^{-\sigma_{1}}$
$\displaystyle\overset{\text{Lemma~{}\ref{melnikove_23s}}}{\geq}\gamma|l|^{-\tau}|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|\left(2-c_{\mathtt{pe}}\varepsilon^{9-8b}|l|^{\tau}\sum_{k\geq
n}N_{k-1}^{-\sigma_{1}}\right).$ (10.383)

Let us claim

$\displaystyle c_{\mathtt{pe}}\varepsilon^{9-8b}|l|^{\tau}\sum_{k\geq
n}N_{k-1}^{-\sigma_{1}}\leq 1,\text{ for $|l|\leq N_{n},\ n\geq 0$.}$ (10.384)

When $n=0$, we have (using $|l|\leq N_{n}=N_{0}$),

$\displaystyle c_{\mathtt{pe}}\varepsilon^{9-8b}|l|^{\tau}\sum_{k\geq
0}N_{k-1}^{-\sigma_{1}}\leq
c_{\mathtt{pe}}\varepsilon^{9-8b}N_{0}^{\tau}\overset{\eqref{parameterssd1sd}}{\leq}c_{\mathtt{pe}}\varepsilon^{9-8b}(\varepsilon^{1-2b})^{\rho\tau}\leq
c_{\mathtt{pe}}\varepsilon^{(9-8b)/2},$

where the last inequality follows from (10.322), which says
$\rho^{-1}\geq\frac{2\tau(2b-1)}{9-8b}$. This proves (10.384) for $n=0$. For
$n\geq 1$, one can also use (10.322) to see that

$\sum_{k\geq n}N_{k-1}^{-\sigma_{1}}=N_{n-1}^{-\sigma_{1}}\left(1+\sum_{k\geq
n+1}\frac{N_{k-1}^{-\sigma_{1}}}{N_{n-1}^{-\sigma_{1}}}\right)=N_{n-1}^{-\sigma_{1}}\left(1+\sum_{k\geq
n+1}N_{0}^{-\sigma_{1}\chi^{k-1}(1-\chi^{n-k})}\right)\leq
2N_{n-1}^{-\sigma_{1}},$

for sufficiently small $\varepsilon>0$. Therefore,

$c_{\mathtt{pe}}\varepsilon^{9-8b}|l|^{\tau}\sum_{k\geq
n}N_{k-1}^{-\sigma_{1}}\leq
2c_{\mathtt{pe}}\varepsilon^{9-8b}N_{n}^{\tau}N_{n-1}^{-\sigma_{1}}\leq
2c_{\mathtt{pe}}\varepsilon^{9-8b}N_{0}^{\chi^{n-1}(\frac{3}{2}\tau-\sigma_{1})}\leq
c_{\mathtt{pe}}\varepsilon^{9-8b},$

for small $\varepsilon$, which gives (10.384) for $n\geq 1$. Plugging (10.384)
into (10.383), we get

$|\text{i}\omega\cdot
l-(d_{n}(\omega,j)-d_{n}(\omega,k))|\geq\gamma|l|^{-\tau}|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|,\text{
for all $|l|\leq N_{n}$},$

which implies $\omega\in\Omega_{n+1}^{\gamma}(i_{0})$ for all $n\geq 0$. This
proves (10.382). ∎

We summarize the main results of this subsection:

######  Proposition 10.82.

Given $\mathtt{S}\gg s_{0}$, there exists
$\varepsilon_{0}(\mathtt{pe},\mathtt{S})>0$ such that if
$\varepsilon\in(0,\varepsilon_{0})$ and (7.3) holds for
$\mu=\mu_{\mathtt{p},1}$ and for some $\mathtt{C}>0$, there exist a real,
reversibility preserving operator $\Phi_{7}=\Phi_{7}(\omega)$ and a diagonal
operator $D_{\infty}(\omega):=\text{diag}_{j\in
S_{\mathtt{M}}^{\perp}}d_{\infty}(\omega,j)$ that depend on
$\omega\in\Omega_{1}$ satisfying

  1. (1)

(Eigenvalues)
$d_{\infty}(\omega,j)=d_{\infty}(i_{0}(\omega))(\omega,j)=d_{0}(\omega,j)+r_{\infty}(\omega,j)$,
where $r_{\infty}(\omega,j)$ satisfies

$\displaystyle|r_{\infty}|^{\text{Lip}(\gamma,\Omega_{1})}_{0,0,0}\leq_{\mathtt{pe}}\varepsilon^{9-6b},\quad|d_{i}r_{\infty}(i_{0})[\hat{\textbf{\i}}]|^{\sup}_{0,0,0}\leq_{\mathtt{pe}}\varepsilon^{3}\gamma^{-1}\rVert\hat{\textbf{\i}}\rVert_{s_{0}+\mu_{\mathtt{p},1}}.$
(10.385)

  2. (2)

(Conjugation) $\Phi=\Phi_{7},\Phi_{7}^{-1}$ satisfies that (assuming
$\varepsilon\in(0,\varepsilon_{0}(\mathtt{pe},\mathtt{S}))$)

$\displaystyle\rVert\Phi
h\rVert_{s}^{\text{Lip}(\gamma,\Omega_{1})}\leq_{\mathtt{pe},\mathtt{S}}\rVert
h\rVert_{s}^{\text{Lip}(\gamma,\Omega_{1})}+\varepsilon^{7}\gamma^{-4}\rVert\mathfrak{I}_{0}\rVert_{s+\mu_{\mathtt{p},1}}^{\text{Lip}(\gamma,\Omega_{1})}\rVert
h\rVert_{s_{0}},\text{ for $s\in[s_{0},\mathtt{S}]$}.$ (10.386)

  3. (3)

(Diagonalization) For $\omega\in\Omega^{4\gamma}_{\infty}(i_{0})$, where

$\Omega^{4\gamma}_{\infty}(i_{0}):=\left\\{\omega\in\Omega_{1}:|\text{i}\omega-(d_{\infty}(\omega,j)-d_{\infty}(\omega,k))|\geq
2\gamma|l|^{-\tau}|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|,\quad\forall l\neq
0,\forall j,k\in S_{\mathtt{M}}^{\perp}\cup\left\\{0\right\\}\right\\},$

We have that

$\displaystyle\mathcal{L}_{\infty}:=\Phi_{7}(\omega)^{-1}L_{0}(\omega)\Phi_{7}(\omega)=\mathcal{D}_{\omega}-D_{\infty}(\omega).$
(10.387)

Also, $\mathcal{L}_{\infty}$ is real and reversible.

  4. (4)

($\frac{2\pi}{\mathtt{M}}$-translation invariance) $\Phi_{7}$ is
$\frac{2\pi}{\mathtt{M}}$-translation invariance preserving.

###### Proof.

We already have $r_{\infty},\Phi_{7,\infty},\Phi_{7,\infty}^{-1}$, which are
well-defined for $\omega\in\Omega^{4\gamma}_{\infty}(i_{0})$ (Lemma 10.382).
Using Kirszbraun’s Theorem [92], we can extend them to $\omega\in\Omega_{1}$
with the same Lipschitz constant. denoting $\Phi_{7}:=\Phi_{7,\infty}$, where
$\Phi_{7,\infty}$ is as in Lemma 10.80, we see that $\Phi_{7}^{\pm}$ are real
and reversibility preserving, since they are compositions of real,
reversibility preserving operators.

Item (1) follows from Lemma 10.81.

For item (2), we obtain from (10.372), (10.336) and (10.294) that

$\mathfrak{M}^{\sharp,\gamma}_{\Phi_{7}-I}(0,s)\leq_{\mathtt{pe},\mathtt{S}}\varepsilon^{2}\gamma^{-2}N_{0}^{2\tau+1}\gamma^{-1}\left(\varepsilon^{5}+\varepsilon^{3}\rVert\mathfrak{I}_{0}\rVert_{s+\mu_{\mathtt{p},1}}^{\text{Lip}(\gamma,\Omega_{1})}\right).$

Therefore, using (7.3),

$\displaystyle\mathfrak{M}^{\sharp,\gamma}_{\Phi_{7}-I}(0,s_{0})\leq_{\mathtt{pe}}\varepsilon^{11-10b}N_{0}^{2\tau+1}\overset{\eqref{para1chs12},n=0}{\leq}1.$
(10.388)

Also, we have

$\displaystyle\mathfrak{M}^{\sharp,\gamma}_{\Phi_{7}-I}(0,s)$
$\displaystyle\leq_{\mathtt{pe},\mathtt{S}}\varepsilon^{7-4b}N_{0}^{2\tau+1}+\varepsilon^{5}\gamma^{-3}N_{0}^{2\tau+1}\rVert\mathfrak{I}_{0}\rVert^{\text{Lip}(\gamma,\Omega_{1})}_{s+\mu_{\mathtt{p},1}}$
$\displaystyle\overset{\eqref{rkkxcxcppsds11}}{\leq}1+\varepsilon^{5}\gamma^{-3}N_{0}^{2\tau+1}\rVert\mathfrak{I}_{0}\rVert^{\text{Lip}(\gamma,\Omega_{1})}_{s+\mu_{\mathtt{p},1}}$
$\displaystyle\leq
1+\varepsilon^{7-4b}\rVert\mathfrak{I}_{0}\rVert_{s+\mu_{\mathtt{p},1}}^{\text{Lip}(\gamma,\Omega_{1})},$
(10.389)

where the last inequality follows from (10.322), which says
$\rho^{-1}\geq\frac{(2b-1)(2\tau+1)}{2b-2}$, therefore,

$\displaystyle\varepsilon^{5}\gamma^{-3}N_{0}^{2\tau+1}$
$\displaystyle=\varepsilon^{5}\gamma^{-3}\varepsilon^{(1-2b)\rho(2\tau+1)}\leq\varepsilon^{5}\gamma^{-3}\varepsilon^{2-2b}\overset{\eqref{frequency_set2}}{=}\varepsilon^{7}\gamma^{-4}.$

Hence, (10.386) for $\Phi=\Phi_{7}$ follows from (10.388) and (10.6.0.3). The
proof for $\Phi=\Phi_{7}^{-1}$ is identical.

Item (3) follows from (10.349) with the strong convergence of $\Phi_{7,n}$ to
$\Phi_{7}$, which have been proved in Lemma 10.80. $\mathcal{L}_{\infty}$ is
real and reversible, since so is $L_{0}$ (see (10.289) and (4) of Proposition
10.66) and $\Phi_{7}$ is real and reversibility preserving.

Lastly, since $\Phi_{7}$ is composition of
$\frac{2\pi}{\mathtt{M}}$-translation invariance preserving, so is $\Phi_{7}$.
∎

###  10.7 Invertibility of $\mathcal{L}_{\omega}$: Proof of proposition 7.14

  * Proof of Proposition 7.14: Let us fix $\mathtt{S}\gg s_{0}$ and

$\displaystyle\mu_{\mathtt{p},0}:=2\mu_{\mathtt{p},1}+2\tau+1\ (\text{recall
$\mu_{\mathtt{p,1}}$ from
\eqref{parameterssd1sd}}),\quad\Omega_{\infty}(i_{0}):=\Omega^{4\gamma}_{\infty}(i_{0}),\text{
see \eqref{sjdjsd2sdsddenoteasd}}.$ (10.390)

It follows from (4) of Proposition 10.76 and (3) of Proposition 10.82 that
denoting

$\Phi_{1,\infty}:=\Phi_{7}^{-1}\circ\Phi_{1-6,1},\quad\Phi_{2,\infty}:=\Phi_{1-6,2}\circ\Phi_{7}.$

we have

$\displaystyle\Phi_{1,\infty}\mathcal{L}_{\omega}\Phi_{2,\infty}=\mathcal{L}_{\infty},$
(10.391)

Applying (10.297) and (10.386) (we replace $\mathtt{S}$ in the application of
(10.297) and (10.386), by $\mathtt{S}+2\mu_{\mathtt{p},1}+2\tau+1$, where the
later $\mathtt{S}$ was fixed in the beginning of this proof, hence we can
choose $\varepsilon$ sufficiently small depending on only $\mathtt{pe}$ and
$\mathtt{S}$ to obtain the following estimates for $\Phi_{i,\infty}$)

$\displaystyle\rVert(\Phi_{i,\infty})^{\pm}h\rVert^{\text{Lip}(\gamma,\Omega_{\infty})}_{s}\leq_{\mathtt{pe},s}\rVert
h\rVert^{\text{Lip}(\gamma,\Omega_{\infty})}_{s+\mu_{\mathtt{p},1}}+\varepsilon^{7}\gamma^{-4}\rVert\mathfrak{I}_{0}\rVert^{\text{Lip}(\gamma,\Omega_{\infty})}_{s+\mu_{\mathtt{p},1}}\rVert
h\rVert_{s_{0}+\mu_{\mathtt{p},1}}^{\text{Lip}(\gamma,\Omega_{\infty})},\quad
i=1,2,$ (10.392)

for all $s\in[s_{0},\mathtt{S}+2\mu_{\mathtt{p},1}+2\tau+1]$.

Now, it suffices to prove the estimate (7.33), assuming (7.3) holds, for some
$\mathtt{C}>0$ and $\mu=\mu_{\mathtt{p},0}$, which is fixed in (10.390). For
all $\omega\in\Omega_{\infty}(i_{0})$, it follows from (3) of Proposition
10.82 that

$\displaystyle|\text{i}\omega-d_{\infty}(\omega,j)|\geq
2\gamma|l|^{-\tau}|{\lambda}_{\alpha}(j)|,\text{ for all $j\in
S_{\mathtt{M}}^{\perp}$.}$ (10.393)

For such $\omega$, we have that

$\displaystyle\mathcal{L}_{\infty}(\omega)^{-1}g=\sum_{l\in\mathbb{Z}^{\nu},\
j\in\mathbb{Z}\backslash\left\\{0\right\\}}\frac{1}{\text{i}\omega\cdot
l-d_{\infty}(\omega,j)}\widehat{g}_{j}(l)e^{\text{i}(l\cdot\varphi+jx)},\text{
for $g(\varphi,\cdot)\in H_{S^{\perp}}\cap X_{\mathtt{M}}$,}$ (10.394)

hence, (10.393) implies that

$\displaystyle\rVert\mathcal{L}_{\infty}(\omega)^{-1}g\rVert_{s}\leq
2\gamma^{-1}\rVert g\rVert_{s+{\tau}}.$ (10.395)

For $\omega_{1},\omega_{2}\in\Omega_{\infty}(i_{0})$, it follows from
(10.385), (10.290), (10.292) and (10.293) that, denoting
$\delta(\omega,l,j):=\text{i}\omega-d_{\infty}(\omega,j)$,

$\displaystyle\left|\frac{1}{\delta(\omega_{1},l,j)}-\frac{1}{\delta(\omega_{2},l,j)}\right|$
$\displaystyle\leq\frac{|\omega_{1}-\omega_{2}||l|+|d_{\infty}(\omega_{1},j)-d_{\infty}(\omega_{2},j)|}{|\delta(\omega_{1},l,j)||\delta(\omega_{2},l,j)|}$
$\displaystyle\overset{\eqref{firstmelnikov}}{\leq}4\gamma^{-2}|\omega_{1}-\omega_{2}||l|^{2\tau}|\lambda_{\alpha}(j)|^{-2}\left(|l|+\frac{|d_{\infty}(\omega_{1},j)-d_{\infty}(\omega_{2},j)|}{|\omega_{1}-\omega_{2}|}\right)$
$\displaystyle\leq_{\mathtt{pe}}\gamma^{-2}|\omega_{1}-\omega_{2}||\lambda_{\alpha}(j)|^{-2}|l|^{2\tau}\left(|l|+\varepsilon^{2}\lambda_{\alpha}(j)\right)$
$\displaystyle\leq_{\mathtt{pe}}\gamma^{-2}|\omega_{1}-\omega_{2}|\left(|l|^{2\tau+1}+1\right).$

where we used (2.99), which implies
$|\lambda_{\alpha}(j)|^{-2}\leq|j|^{2\alpha}$ for the third inequality (since
$g(\varphi,\cdot)\in H_{S^{\perp}}$, we exclude the case where $j=0$). Hence,
using (10.394), we see that

$\displaystyle\gamma\frac{1}{|\omega_{1}-\omega_{2}|}\rVert\left(\mathcal{L}_{\infty}(\omega_{1})^{-1}-\mathcal{L}_{\infty}(\omega_{2})^{-1}\right)g\rVert_{s}\leq_{\mathtt{pe},s}\gamma^{-1}\rVert
g\rVert_{s+2\tau+1}.$

Combining this with (10.395), we have

$\displaystyle\rVert\mathcal{L}_{\infty}^{-1}g\rVert_{s}^{\text{Lip}(\gamma,\Omega_{\infty}(i_{0}))}\leq_{\mathtt{pe},s}\gamma^{-1}\rVert
g\rVert_{s+2\tau+1}^{\text{Lip}(\gamma,\Omega_{\infty}(i_{0}))},\text{ for all
$s\geq 0$.}$ (10.396)

Clearly, $\mathcal{L}_{\omega}[f]=g$ is equivalent to (from (10.391))

$\displaystyle
f=\Phi_{2,\infty}^{-1}\mathcal{L}_{\infty}^{-1}\Phi_{1,\infty}^{-1}g.$
(10.397)

Since $\Phi_{1,\infty},\Phi_{2,\infty}$ are real, reversibility preserving and
$\frac{2\pi}{\mathtt{M}}$-translation invariance preserving (see Proposition
10.82 and (4), (5) of Proposition 10.76), and $\mathcal{L}_{\infty}$ is real,
reversible and diagonal (see (3) of Proposition 10.82), we have that $g\in
Y_{\perp}\cap X_{\mathtt{M}}$ implies $f\in X_{\perp}\cap X_{\mathtt{M}}$. For
the estimates, it follows from (10.392) and (10.396) that for all
$s\in[s_{0},\mathtt{S}]$,

$\displaystyle\rVert f\rVert_{s}^{\text{Lip}(\gamma,\Omega_{\infty})}$
$\displaystyle\leq_{\mathtt{pe},\mathtt{S}}\gamma^{-1}\left(\rVert
g\rVert^{\text{Lip}(\gamma,\Omega_{\infty})}_{s+2\mu_{\mathtt{p},1}+2\tau+1}+\varepsilon^{7}\gamma^{-4}\rVert\mathfrak{I}_{0}\rVert^{\text{Lip}(\gamma,\Omega_{\infty})}_{s+2\mu_{\mathtt{p},1}+2\tau+1}\rVert
g\rVert_{s_{0}+2\mu_{\mathtt{p},1}+2\tau+1}\right)$
$\displaystyle\overset{\eqref{finally_muisfie}}{=}\gamma^{-1}\left(\rVert
g\rVert^{\text{Lip}(\gamma,\Omega_{\infty})}_{s+\mu_{\mathtt{p},0}}+\varepsilon^{7}\gamma^{-4}\rVert\mathfrak{I}_{0}\rVert^{\text{Lip}(\gamma,\Omega_{\infty})}_{s+\mu_{\mathtt{p},0}}\rVert
g\rVert_{s_{0}+\mu_{\mathtt{p},0}}\right),$

which gives us (7.33). $\Box$

##  11 Nash-Moser iteration

In this section, we aim to prove Theorem 6.5. We denote

$E_{n}:=\left\\{\mathfrak{I}=(\Theta,y,z)\in C^{\infty}_{\varphi}\times
C^{\infty}_{\varphi}\times C^{\infty}_{\varphi,x},\ \Theta=\Pi_{n}\Theta,\
y=\Pi_{n}y,\ \Pi_{n}z=z\right\\},$

where

$\Pi_{n}\Theta(\varphi)=\sum_{|l|\leq
N_{n}}\Theta_{l}e^{\text{i}l\cdot\varphi},\quad\Pi_{n}y(\varphi):=\sum_{|l|\leq
N_{n}}y_{l}e^{\text{i}l\cdot\varphi},\quad\Pi_{n}z(\varphi,x):=\sum_{|l|,|j|\leq
N_{n}}z_{l,j}e^{\text{i}l\cdot\varphi+\text{i}jx},$

where $N_{n}$ is as defined in (10.322). Given
$\mathfrak{I}(\varphi):=(\Theta(\varphi),y(\varphi),z(\varphi))$, we denote
$\Pi_{n}\mathfrak{I}:=(\Pi_{n}\Theta,\Pi_{n}y,\Pi_{n}z)$, so that for
$\Omega\subset\mathbb{R}^{\nu}$,

$\displaystyle\rVert\Pi_{n}\mathfrak{I}\rVert_{s+\mu}^{\text{Lip}(\gamma,\Omega)}\leq_{s,\mu}N_{n}^{\mu}\rVert\mathfrak{I}\rVert^{\text{Lip}(\gamma,\Omega)}_{s},\quad\rVert(I-\Pi_{n})\mathfrak{I}\rVert^{\text{Lip}(\gamma,\Omega)}_{s}\leq_{s,\mu}N_{n}^{-\mu}\rVert\mathfrak{I}\rVert^{\text{Lip}(\gamma,\Omega)}_{s+\mu},\text{
for all $s\geq 0,\mu\geq 0$.}$ (11.1)

We define inductively for $n\geq 0$,

$\displaystyle\mathfrak{I}_{0}:=(0,0,0),\quad\mathfrak{I}_{n+1}:=\mathfrak{I}_{n}-\Pi_{n}T(i_{n})\Pi_{n}\mathcal{F}_{\omega}(i_{n}),\quad
i_{n}:=i_{triv}+\mathfrak{I}_{n},\quad\widehat{\mathfrak{I}}_{n+1}:=\mathfrak{I}_{n+1}-\mathfrak{I}_{n},\quad\widehat{\mathfrak{I}}_{0}:=0,$
(11.2)

where $T(i_{n})$ is the approximate inverse constructed in Proposition 7.3. We
also define

$\displaystyle\mathcal{G}_{0}:=\left\\{\omega\in\Omega_{\varepsilon}:|\omega\cdot
l|\geq 2\gamma|l|^{-\tau},\quad\forall
l\in\mathbb{Z}^{\nu}\backslash\left\\{0\right\\}\right\\}=\Omega_{0},$ (11.3)
$\displaystyle\mathcal{G}_{n+1}:=\left\\{\omega\in\mathcal{G}_{n}:|\text{i}\omega\cdot
l-(d_{\infty}(i_{n}(\omega))(\omega,j)-d_{\infty}(i_{n}(\omega)(\omega,k)))|\geq\frac{2\gamma_{n}|{\lambda}_{\alpha}(j)-{\lambda}_{\alpha}(k)|}{\langle
l\rangle^{\tau}},\quad\forall j,k\in
S_{\mathtt{M}}^{\perp}\cup\left\\{0\right\\},\ l\in\mathbb{Z}^{\nu}\right\\},$
where $\gamma_{n}:=\gamma(1+2^{-n})$.

Let us denote

$\displaystyle A:=\frac{5-4b}{(2b-1)\rho}.$ (11.4)

Recalling $\rho$ from (10.322), which says
$\rho^{-1}\geq\frac{40(\mu_{\mathtt{p,0}}+\mu_{\mathtt{p},2})(2b-1)}{5-4b}$,
we see that

$\displaystyle A\geq
40(\mu_{\mathtt{p},0}+\mu_{\mathtt{p},2})\overset{\eqref{mu0andmu1}}{\geq}40\mu_{\mathtt{p}}.$
(11.5)

Since $\rho$ in (10.322) depends on $\mathtt{p},b$ only (see (6.24) for
$\mathtt{p}$), so does $A$. For such $A$, we fix

$\displaystyle\sigma_{3}=\sigma_{5}:=\frac{9}{16}A,\quad\sigma_{4}:=\frac{3}{4}A,\quad\mathtt{k}:=7A,\quad\mathtt{b}_{1}:=9A,\quad\mathtt{S}:=s_{0}+2\mu_{\mathtt{p}}+\mathtt{b}_{1}.$
(11.6)

######  Proposition 11.1.

There exists
$\varepsilon_{0}=\varepsilon_{0}({\mathtt{p},b,\mathtt{M},S_{0}})>0$ such that
for all $\varepsilon\in(0,\varepsilon_{0})$, and for all $n\geq 0$,

  1. (1)

$Z_{n}(\omega):=\mathcal{F}_{\omega}(i_{n}(\omega))$ is well-defined for
$\omega\in\mathcal{G}_{n}$ and $Z_{n}(\omega)\in Y_{i}\cap X_{i,\mathtt{M}}$.
It also satisfies

$\displaystyle\rVert Z_{n}\rVert_{s_{0}}^{\text{Lip}(\gamma,\mathcal{G}_{n})}$
$\displaystyle\leq_{\mathtt{p},b,\mathtt{M},S_{0}}\varepsilon^{6-2b}N_{n-1}^{-\sigma_{4}},$
(11.7) $\displaystyle\rVert
Z_{n}\rVert_{s_{0}+\mathtt{b}_{1}-\mu_{\mathtt{p}}}^{\text{Lip}(\gamma,\mathcal{G}_{n})}$
$\displaystyle\leq_{\mathtt{p},b,\mathtt{M},S_{0}}\varepsilon^{6-2b}N_{n-1}^{\mathtt{k}},$
(11.8)

Furthermore, we have

$\displaystyle\rVert
Z_{n}\rVert_{s_{0}+\mu_{\mathtt{p}}}^{\text{Lip}(\gamma,\mathcal{G}_{n})}$
$\displaystyle\leq_{\mathtt{p},b,\mathtt{M},S_{0}}\varepsilon^{6-2b}N_{n-1}^{-\sigma_{5}},$
(11.9)

  2. (2)

$\mathfrak{I}_{n+1}=\mathfrak{I}_{n+1}(\omega)$ is well-defined for
$\omega\in\mathcal{G}_{n+1}$ and $\mathfrak{I}_{n+1}(\omega)\in X_{i}\cap
X_{i,\mathtt{M}}$. It also satisfies

$\displaystyle\rVert\widehat{\mathfrak{I}}_{n+1}\rVert^{\text{Lip}(\gamma,\mathcal{G}_{n+1})}_{s_{0}+2\mu_{\mathtt{p}}}$
$\displaystyle\leq_{\mathtt{p},b,\mathtt{M},S_{0}}\varepsilon^{6-4b}N_{n-1}^{-\sigma_{3}},$
(11.10)
$\displaystyle\rVert\widehat{\mathfrak{I}}_{n+1}\rVert^{\text{Lip}(\gamma,\mathcal{G}_{n+1})}_{s_{0}+2\mu_{\mathtt{p}}+\mathtt{b}_{1}}$
$\displaystyle\leq_{\mathtt{p},b,\mathtt{M},S_{0}}\varepsilon^{6-4b}N_{n}^{\mathtt{k}}$
(11.11)

###### Proof.

The proof will be completed by the usual induction argument. For simplicity,
we denote

$\displaystyle|A_{n}|_{s}:=\rVert
A_{n}\rVert_{s}^{\text{Lip}(\gamma,\mathcal{G}_{n})},\quad
A\in\left\\{\mathfrak{I},\widehat{\mathfrak{I}},Z\right\\},\quad\mathtt{p_{e}}:=\left\\{\mathtt{p},b,\mathtt{M},S_{0}\right\\}.$
(11.12)

Proof of (1),(2) for $n=0$. For item (1), we note that $i_{0}=i_{triv}\in
X_{i}\cap X_{i,\mathtt{M}}$, which is well-defined for all
$\omega\in\Omega_{1}=\mathcal{G}_{0}$, hence so is $Z_{0}$. And Lemma 6.3 and
Lemma 6.4 tell us that $Z_{0}\in Y_{i}\cap X_{i,\mathtt{M}}$. The estimates
(11.7), (11.8) and (11.9) follow from (6.16) with $N_{-1}=1$ (see (10.322)).
Indeed, (6.16) tells us that

$\displaystyle|Z_{0}|_{s}\leq_{\mathtt{p_{e}}}\varepsilon^{6-2b},\text{ for
all $s\in[s_{0},\mathtt{S}]$}.$ (11.13)

For (2), we prove (2) for $\mathfrak{I}_{1}$. Trivially, we have
$\mathfrak{I}_{0}(\omega)=0\in X_{i}\cap X_{i,\mathtt{M}}$ well-defined.
Furthermore, by (11.9) for $Z_{0}$, which was just proved, we have that

$\displaystyle|\mathfrak{I}_{0}|_{s_{0}+\mu_{\mathtt{p}}}=0,\quad|Z_{0}|_{s_{0}+\mu_{\mathtt{p}}}\leq_{\mathtt{p_{e}}}\varepsilon^{6-2b}.$
(11.14)

Therefore, (7.3) holds for $\mu=\mu_{\mathtt{p}}$ and for some
$\mathtt{C}=\mathtt{C}_{\mathtt{p_{e}}}$. Then we can apply Proposition 7.3
with $\mathtt{S}$ in (11.6), and $\mathtt{C}=\mathtt{C}_{\mathtt{p_{e}}}$, for
sufficiently $\varepsilon$ small depending only on $\mathtt{p_{e}}$, since
$\mathtt{S}$ in (11.6) and $\mathtt{C}_{\mathtt{p_{e}}}$ are fixed depending
on $\mathtt{p_{e}}$. Note that Proposition 7.3 tells us that we can find an
approximate inverse $T(i_{0})$ for each $\omega\in\Omega_{\infty}(i_{0})$.
From the definition of $\mathcal{G}_{1}$ in (11.3) and
$\Omega_{\infty}(i_{0})$ in (10.390), it is clear that

$\displaystyle\mathcal{G}_{n+1}\subset\Omega_{\infty}(i_{n}).$ (11.15)

Therefore, for all $\omega\in\mathcal{G}_{1}$, $T(i_{0}(\omega)):Y_{i}\cap
X_{i,\mathtt{M}}\mapsto X_{i}\cap X_{i,\mathtt{M}}$ is well-defined. Since
$Z_{0}(\omega)\in Y_{i}\cap X_{i,\mathtt{M}}$, we see that $\mathfrak{I}_{1}$
given in (11.2) is well-defined for $\omega\in\mathcal{G}_{1}$ and
$\mathfrak{I}_{1}\in X_{i}\cap X_{i,\mathtt{M}}$.

Now, we prove (11.10) and (11.11) for $\mathfrak{I}_{1}$. Towards the
estimates (11.10), we use (7.14) to obtain

$\displaystyle|\widehat{\mathfrak{I}}_{1}|_{s_{0}+2\mu_{\mathtt{p}}}$
$\displaystyle=|\Pi_{0}T(i_{0})\Pi_{n}Z_{0}|_{s_{0}+2\mu_{\mathtt{p}}}$
$\displaystyle\leq|T(i_{0})\Pi_{0}Z_{0}|_{s_{0}+2\mu_{\mathtt{p}}}$
$\displaystyle\overset{\eqref{inverse_estimate2}}{\leq_{\mathtt{p_{e}}}}\gamma^{-1}(1+\varepsilon^{7}\gamma^{-4}\underbrace{|\mathfrak{I}_{0}|_{s_{0}+2\mu_{\mathtt{p}}}}_{=0})|\Pi_{0}Z_{0}|_{s_{0}+2\mu_{\mathtt{p}}}$
$\displaystyle\overset{\eqref{sdjj22sdxcxcosd}}{\leq_{\mathtt{p_{e}}}}\varepsilon^{6-2b}\gamma^{-1},$
(11.16)

which gives us (11.10) for $\widehat{\mathfrak{I}}_{1}$. For (11.11), we
compute similarly,

$\displaystyle|\widehat{\mathfrak{I}}_{1}|_{s_{0}+2\mu_{\mathtt{p}}+\mathtt{b}_{1}}$
$\displaystyle=|\Pi_{0}T(i_{0})\Pi_{0}Z_{0}|_{s_{0}+2\mu_{\mathtt{p}}+\mathtt{b}_{1}}$
$\displaystyle\leq|T(i_{0})\Pi_{0}Z_{0}|_{s_{0}+2\mu_{\mathtt{p}}+\mathtt{b}_{1}}$
$\displaystyle\overset{\eqref{inverse_estimate2},\mathfrak{I}_{0}=0}{\leq_{\mathtt{p_{e}}}}\gamma^{-1}|\Pi_{0}Z_{0}|_{s_{0}+2\mu_{\mathtt{p}}+\mathtt{b}_{1}}$
$\displaystyle\overset{\eqref{sdjj22sdxcxcosd}}{\leq_{\mathtt{p_{e}}}}\varepsilon^{6-2b}\gamma^{-1},$
(11.17)

which gives us (11.11) for $\widehat{\mathfrak{I}}_{1}$.

Proof of (1) for $n+1$. We aim to prove (1) for $Z_{n+1}$, assuming (1) and
(2) hold for $Z_{k}$ and $\mathfrak{I}_{k+1}$ for $k=1,\ldots,n$ respectively.
Since we $i_{n+1}(\omega)\in X_{i}\cap X_{i,\mathtt{M}}$, which is well-
defined for $\omega\in\mathcal{G}_{n+1}$, $Z_{n+1}$ is well-defined for
$\omega\in\mathcal{G}_{n+1}$ and Lemma 6.3 and Lemma 6.4 tell us that
$Z_{n+1}\in Y_{i}\cap X_{i,\mathtt{M}}$. Therefore, it suffices to prove
(11.7), (11.8) and (11.9) for $Z_{n+1}$. We first note that the induction
hypotheses on (11.10) and (11.11) tells us that

$\displaystyle|\mathfrak{I}_{k+1}|_{s_{0}+2\mu_{\mathtt{p}}}$
$\displaystyle\overset{\eqref{iteration_step2}}{\leq}\sum_{i=0}^{k}|\widehat{\mathfrak{I}}_{i+1}|_{s_{0}+2\mu_{\mathtt{p}}}\leq_{\mathtt{p_{e}}}\varepsilon^{6-4b}\sum_{i=0}^{\infty}N_{i-1}^{-\sigma_{3}}\leq_{\mathtt{p_{e}}}\varepsilon^{6-4b},\text{
for all $0\leq k\leq n$,}$ (11.18)
$\displaystyle|\mathfrak{I}_{k+1}|_{s_{0}+2\mu_{\mathtt{p}}+\mathtt{b}_{1}}$
$\displaystyle\leq\sum_{i=0}^{k}|\widehat{\mathfrak{I}}_{i+1}|_{s_{0}+2\mu_{\mathtt{p}}+\mathtt{b}_{1}}\leq_{\mathtt{p_{e}}}\varepsilon^{6-4b}\sum_{i=0}^{k}N_{i}^{\mathtt{k}}\overset{\eqref{choices1xs}}{\leq_{\mathtt{p_{e}}}}\varepsilon^{6-4b}N_{k}^{\mathtt{k}},\text{
for all $0\leq k\leq n$.}$ (11.19)

We compute that for all $s\in[s_{0},\mathtt{S}]$ (see $\mathtt{S}$ in (11.6)),

$\displaystyle|Z_{n+1}|_{s}$
$\displaystyle\leq|Z_{n+1}-Z_{n}-d_{i}\mathcal{F}_{\omega}(i_{n})[\widehat{\mathfrak{I}}_{n+1}]|_{s}+|Z_{n}+d_{i}\mathcal{F}_{\omega}(i_{n})[\widehat{\mathfrak{I}}_{n+1}]|_{s}$
$\displaystyle\overset{\eqref{iteration_step2}}{\leq}|Z_{n+1}-Z_{n}-d_{i}\mathcal{F}_{\omega}(i_{n})[\widehat{\mathfrak{I}}_{n+1}]|_{s}+|Z_{n}-d_{i}(\mathcal{F}_{\omega})(i_{n})[\Pi_{n}T(i_{n})\Pi_{n}Z_{n}]|_{s}.$

We further decompose
$|Z_{n}-d_{i}(\mathcal{F}_{\omega})(i_{n})[\Pi_{n}T(i_{n})\Pi_{n}Z_{n}]|_{s}$
into

$\displaystyle|Z_{n}-d_{i}(\mathcal{F}_{\omega})(i_{n})[\Pi_{n}T(i_{n})\Pi_{n}Z_{n}]|_{s}$
$\displaystyle\leq|Z_{n}-d_{i}(\mathcal{F}_{\omega})(i_{n})[T(i_{n})\Pi_{n}Z_{n}]|_{s}+|d_{i}(\mathcal{F}_{\omega})(i_{n})(I-\Pi_{n})T(i_{n})\Pi_{n}Z_{n}|_{s}$
$\displaystyle\leq|(I-\Pi_{n})Z_{n}|_{s}+|(I-d_{i}(\mathcal{F}_{\omega})(i_{n})T(i_{n}))\Pi_{n}Z_{n}|_{s}$
$\displaystyle\
+|d_{i}(\mathcal{F}_{\omega})(i_{n})(I-\Pi_{n})T(i_{n})\Pi_{n}Z_{n}|_{s}.$
$\displaystyle=|(I-\Pi_{n})Z_{n}|_{s}+|\Pi_{n}\left(I-d_{i}(\mathcal{F}_{\omega})(i_{n})T(i_{n}))\Pi_{n}Z_{n}\right)|_{s}$
$\displaystyle\
+|(I-\Pi_{n})d_{i}(\mathcal{F}_{\omega})(i_{n})T(i_{n}))\Pi_{n}Z_{n}|_{s}+|d_{i}(\mathcal{F}_{\omega})(i_{n})(I-\Pi_{n})T(i_{n})\Pi_{n}Z_{n}|_{s}.$

Therefore, we have

$\displaystyle|Z_{n+1}|_{s}$
$\displaystyle\leq|Z_{n+1}-Z_{n}-d_{i}\mathcal{F}_{\omega}(i_{n})[\widehat{\mathfrak{I}}_{n+1}]|_{s}$
(11.20) $\displaystyle\ +|(I-\Pi_{n})Z_{n}|_{s}$ $\displaystyle\
+|\Pi_{n}\left(I-d_{i}(\mathcal{F}_{\omega})(i_{n})T(i_{n}))\Pi_{n}Z_{n}\right)|_{s}$
$\displaystyle\
+|(I-\Pi_{n})d_{i}(\mathcal{F}_{\omega})(i_{n})T(i_{n}))\Pi_{n}Z_{n}|_{s}$
$\displaystyle+|d_{i}(\mathcal{F}_{\omega})(i_{n})(I-\Pi_{n})T(i_{n})\Pi_{n}Z_{n}|_{s}.$

We estimate the low/high norms of each term separately. We recall from Remark
7.4 that

$\displaystyle\mu_{\mathtt{p}}\geq\mu_{1},\mu_{2},\mu_{3}.$ (11.21)

For
$|Z_{n+1}-Z_{n}-d_{i}\mathcal{F}_{\omega}(i_{n})[\widehat{\mathfrak{I}}_{n+1}]|_{s_{0}}$,
we compute

$\displaystyle|Z_{n+1}-Z_{n}-d_{i}\mathcal{F}_{\omega}(i_{n})[\widehat{\mathfrak{I}}_{n+1}]|_{s_{0}}$
$\displaystyle\overset{\eqref{fffs2x},\eqref{muoisduxcs}}{\leq_{\mathtt{p_{e}}}}\varepsilon\left(1+(|\mathfrak{I}_{n+1}|_{s_{0}+\mu_{\mathtt{p}}}+|\mathfrak{I}_{n}|_{s_{0}+\mu_{\mathtt{p}}})\right)|\widehat{\mathfrak{I}}_{n+1}|_{s_{0}+\mu_{\mathtt{p}}}^{2}$
$\displaystyle\overset{\eqref{fortheloveof2xgod}}{\leq}2\varepsilon|\widehat{\mathfrak{I}}_{n+1}|_{s_{0}+\mu_{\mathtt{p}}}^{2}$
$\displaystyle\overset{\eqref{lowernorm_11232}}{\leq_{\mathtt{p_{e}}}}\varepsilon^{13-8b}N_{n-1}^{-2\sigma_{3}}$
$\displaystyle\overset{\eqref{choicess6}}{\leq}\varepsilon^{6-2b}N_{n}^{-\sigma_{4}}.$
(11.22)

Similarly, we have

$\displaystyle|Z_{n+1}-Z_{n}-d_{i}\mathcal{F}_{\omega}(i_{n})[\widehat{\mathfrak{I}}_{n+1}]|_{s_{0}+\mathtt{b}_{1}-\mu_{\mathtt{p}}}$
$\displaystyle\overset{\eqref{fffs2x},\eqref{muoisduxcs}}{\leq_{\mathtt{p_{e}}}}\varepsilon\left(|\widehat{\mathfrak{I}}_{n+1}|_{s_{0}+\mathtt{b}_{1}}|\widehat{\mathfrak{I}}_{n+1}|_{s_{0}+\mu_{\mathtt{p}}}+(|\mathfrak{I}_{n+1}|_{s_{0}+\mathtt{b}_{1}}+|\mathfrak{I}_{n}|_{s_{0}+\mathtt{b}_{1}})|\widehat{\mathfrak{I}}_{n+1}|_{s_{0}+\mu_{\mathtt{p}}}^{2}\right)$
$\displaystyle\overset{\eqref{lowernorm_11232},\eqref{hihsdsd2sdsd},\eqref{forthe2x1loveofgod2}}{\leq_{\mathtt{p_{e}}}}\varepsilon^{13-8b}\left(N_{n}^{\mathtt{k}}N_{n-1}^{-\sigma_{3}}+\varepsilon^{6-4b}N_{n-1}^{-2\sigma_{3}}N_{n}^{\mathtt{k}}\right)$
$\displaystyle\leq_{\mathtt{p_{e}}}\varepsilon^{6-2b}N_{n}^{\mathtt{k}}.$
(11.23)

For $|(I-\Pi_{n})Z_{n}|_{s_{0}}$, we have

$\displaystyle|(I-\Pi_{n})Z_{n}|_{s_{0}}\overset{\eqref{truncation_3212}}{\leq}N_{n}^{-(\mathtt{b}_{1}-\mu_{\mathtt{p}})}|Z_{n}|_{s_{0}+\mathtt{b}_{1}-\mu_{\mathtt{p}}}\overset{\eqref{jjsd2asdsa2}}{\leq_{\mathtt{p_{e}}}}\varepsilon^{6-2b}N_{n-1}^{\mathtt{k}}N_{n}^{-(\mathtt{b}_{1}-\mu_{\mathtt{p}})}\overset{\eqref{choicess2}}{\leq}\varepsilon^{6-2b}N_{n}^{-\sigma_{4}}.$
(11.24)

For $|(I-\Pi_{n})Z_{n}|_{s_{0}+\mathtt{b}_{1}-\mu_{\mathtt{p}}}$, we simply
estimate it as

$\displaystyle|(I-\Pi_{n})Z_{n}|_{s_{0}+\mathtt{b}_{1}-\mu_{\mathtt{p}}}\leq|Z_{n}|_{s_{0}+\mathtt{b}_{1}-\mu_{\mathtt{p}}}\overset{\eqref{jjsd2asdsa2}}{\leq_{\mathtt{p_{e}}}}\varepsilon^{6-2b}N_{n}^{\mathtt{k}}.$
(11.25)

For
$|\Pi_{n}\left(I-d_{i}(\mathcal{F}_{\omega})(i_{n})T(i_{n}))\Pi_{n}Z_{n}\right)|_{s_{0}}$,
we have that

$\displaystyle|\Pi_{n}\left(I-d_{i}(\mathcal{F}_{\omega})(i_{n})T(i_{n}))\Pi_{n}Z_{n}\right)|_{s_{0}}$
$\displaystyle\leq|\left(I-d_{i}(\mathcal{F}_{\omega})(i_{n})T(i_{n}))\Pi_{n}Z_{n}\right)|_{s_{0}}$
$\displaystyle\overset{\eqref{inverse_estimate1}}{\leq_{\mathtt{p_{e}}}}\varepsilon^{2b-1}\gamma^{-2}\left(1+\varepsilon^{7}\gamma^{-4}|\mathfrak{I}_{n}|_{s_{0}+\mu_{\mathtt{p}}}\right)|Z_{n}|_{s_{0}+\mu_{\mathtt{p}}}|\Pi_{n}Z_{n}|_{s_{0}+\mu_{\mathtt{p}}}$
$\displaystyle\overset{\eqref{fortheloveof2xgod},\gamma=\varepsilon^{2b}}{\leq_{\mathtt{p_{e}}}}\varepsilon^{2b-1}\gamma^{-2}(1+\varepsilon^{13-12b})|Z_{n}|_{s_{0}+\mu_{\mathtt{p}}}^{2}$
(11.26)
$\displaystyle\overset{\eqref{jjsd2asdsa3},\eqref{parametersets1231}}{\leq_{\mathtt{p_{e}}}}\varepsilon^{11-6b}N_{n-1}^{-2\sigma_{5}}$
$\displaystyle\overset{\eqref{choicess3}}{\leq}\varepsilon^{6-2b}N_{n}^{-\sigma_{4}}.$
(11.27)

For
$|\Pi_{n}\left(I-d_{i}(\mathcal{F}_{\omega})(i_{n})T(i_{n}))\Pi_{n}Z_{n}\right)|_{s_{0}+\mathtt{b}_{1}-\mu_{\mathtt{p}_{1}}}$,
we have

$\displaystyle|\Pi_{n}$
$\displaystyle\left(I-d_{i}(\mathcal{F}_{\omega})(i_{n})T(i_{n}))\Pi_{n}Z_{n}\right)|_{s_{0}+\mathtt{b}_{1}-\mu_{\mathtt{p}_{1}}}$
$\displaystyle\overset{\eqref{truncation_3212}}{\leq}N_{n}^{\mu_{\mathtt{p}}}|\left(I-d_{i}(\mathcal{F}_{\omega})(i_{n})T(i_{n}))\Pi_{n}Z_{n}\right)|_{s_{0}+\mathtt{b}_{1}-2\mu_{\mathtt{p}}}$
$\displaystyle\overset{\eqref{inverse_estimate1}}{\leq_{\mathtt{p_{e}}}}N_{n}^{\mu_{\mathtt{p}}}\varepsilon^{2b-1}\gamma^{-2}\left(|Z_{n}|_{s_{0}+\mu_{\mathtt{p}}}|\Pi_{n}Z_{n}|_{s_{0}+\mathtt{b}_{1}-\mu_{\mathtt{p}}}+(|Z_{n}|_{s_{0}+\mathtt{b}_{1}-\mu_{\mathtt{p}}}+\varepsilon^{7}\gamma^{-4}|Z_{n}|_{s_{0}+\mu_{\mathtt{p}}}|\mathfrak{I}_{n}|_{s_{0}+\mathtt{b}_{1}-\mu_{\mathtt{p}}})|\Pi_{n}Z_{n}|_{s_{0}+\mu_{\mathtt{p}}}\right)$
$\displaystyle\overset{\gamma=\varepsilon^{2b}}{\leq_{\mathtt{p_{e}}}}N_{n}^{\mu_{\mathtt{p}}}\varepsilon^{-2b-1}|Z_{n}|_{s_{0}+\mu_{\mathtt{p}}}\left(|Z_{n}|_{s_{0}+\mathtt{b}_{1}-\mu_{\mathtt{p}}}+\varepsilon^{7}\gamma^{-4}|Z_{n}|_{s_{0}+\mu_{\mathtt{p}}}|\mathfrak{I}_{n}|_{s_{0}+\mathtt{b}_{1}-\mu_{\mathtt{p}}}\right)$
$\displaystyle\overset{\eqref{jjsd2asdsa3},\eqref{forthe2x1loveofgod2},\eqref{jjsd2asdsa2}}{\leq_{\mathtt{p_{e}}}}\varepsilon^{5-4b}N_{n}^{\mu_{\mathtt{p}}}N_{n-1}^{-\sigma_{5}}\left(\varepsilon^{6-2b}N_{n-1}^{\mathtt{k}}+\varepsilon^{6-2b}\underbrace{\varepsilon^{13-12b}}_{\leq
1,\because\eqref{parametersets1231}}N_{n-1}^{-\sigma_{5}}N_{n}^{\mathtt{k}}\right)$
$\displaystyle\overset{\eqref{choicess4}}{\leq_{\mathtt{p_{e}}}}\varepsilon^{6-2b}N_{n}^{\mathtt{k}}.$
(11.28)

For
$|(I-\Pi_{n})d_{i}(\mathcal{F}_{\omega})(i_{n})T(i_{n}))\Pi_{n}Z_{n}|_{s_{0}}$,
We see from (7.1), (6.14) and (11.21) that

$\displaystyle|d_{i}\mathcal{F}_{\omega}(i_{n})[\hat{\textbf{\i}}]|_{s}\leq_{\mathtt{p_{e}}}|\hat{\textbf{\i}}|_{s+\mu_{\mathtt{p}}}+|\mathfrak{I}_{n}|_{s+\mu_{\mathtt{p}}}|\hat{\textbf{\i}}|_{s_{0}+\mu_{\mathtt{p}}}.$
(11.29)

Then, we compute

$\displaystyle|(I$
$\displaystyle-\Pi_{n})d_{i}(\mathcal{F}_{\omega})(i_{n})T(i_{n}))\Pi_{n}Z_{n}|_{s_{0}}$
$\displaystyle\overset{\eqref{truncation_3212}}{\leq}N_{n}^{-(\mathtt{b}_{1}-\mu_{\mathtt{p}})}|d_{i}(\mathcal{F}_{\omega})(i_{n})T(i_{n}))\Pi_{n}Z_{n}|_{s_{0}+\mathtt{b}_{1}-\mu_{\mathtt{p}}}$
$\displaystyle\overset{\eqref{kkksdsxcxx222xc11}}{\leq_{\mathtt{p_{e}}}}N_{n}^{-(\mathtt{b}_{1}-\mu_{\mathtt{p}})}\left(|T(i_{n})\Pi_{n}Z_{n}|_{s_{0}+\mathtt{b}_{1}}+|\mathfrak{I}_{n}|_{s_{0}+\mathtt{b}_{1}}|T(i_{n})\Pi_{n}Z_{n}|_{s_{0}+\mu_{\mathtt{p}}}\right).$
(11.30)

From (7.14), it follows that

$\displaystyle|T(i_{n})\Pi_{n}Z_{n}|_{s_{0}+\mathtt{b}_{1}}$
$\displaystyle\leq_{\mathtt{p_{e}}}\gamma^{-1}\left(|\Pi_{n}Z_{n}|_{s_{0}+\mathtt{b}_{1}+\mu_{\mathtt{p}}}+\varepsilon^{7}\gamma^{-4}|\mathfrak{I}_{n}|_{s_{0}+\mathtt{b}_{1}+\mu_{\mathtt{p}}}|\Pi_{n}Z_{n}|_{s_{0}+\mu_{\mathtt{p}}}\right)$
$\displaystyle\leq\gamma^{-1}\left(N_{n}^{2\mu_{\mathtt{p}}}|Z_{n}|_{s_{0}+\mathtt{b}_{1}-\mu_{\mathtt{p}}}+\varepsilon^{7}\gamma^{-4}|\mathfrak{I}_{n}|_{s_{0}+\mathtt{b}_{1}+\mu_{\mathtt{p}}}|\Pi_{n}Z_{n}|_{s_{0}+\mu_{\mathtt{p}}}\right)$
$\displaystyle\overset{\eqref{jjsd2asdsa2},\eqref{jjsd2asdsa3},\eqref{forthe2x1loveofgod2}}{\leq_{\mathtt{p_{e}}}}\varepsilon^{6-2b}\gamma^{-1}\left(N_{n}^{2\mu_{\mathtt{p}}}N_{n-1}^{\mathtt{k}}+\varepsilon^{13-12b}N_{n-1}^{\mathtt{k}}N_{n-1}^{-\sigma_{5}}\right)$
$\displaystyle\overset{\eqref{parametersets1231}}{\leq_{\mathtt{p_{e}}}}\varepsilon^{6-2b}\gamma^{-1}(N_{n}^{2\mu_{\mathtt{p}}}N_{n-1}^{\mathtt{k}}+N_{n-1}^{\mathtt{k}})$
$\displaystyle\overset{\eqref{choicess5}}{\leq}2\varepsilon^{6-2b}N_{n}^{\mathtt{k}}.$
(11.31)

Similarly,

$\displaystyle|\mathfrak{I}_{n}|_{s_{0}+\mathtt{b}_{1}}|T(i_{n})\Pi_{n}Z_{n}|_{s_{0}+\mu_{\mathtt{p}}}$
$\displaystyle\overset{\eqref{inverse_estimate2}}{\leq_{\mathtt{p_{e}}}}|\mathfrak{I}_{n}|_{s_{0}+\mathtt{b}_{1}+\mu_{\mathtt{p}}}\gamma^{-1}\left(|\Pi_{n}Z_{n}|_{s_{0}+2\mu_{\mathtt{p}}}+\varepsilon^{7}\gamma^{-4}|\mathfrak{I}_{n}|_{s_{0}+2\mu_{\mathtt{p}}}|\Pi_{n}Z_{n}|_{s_{0}+\mu_{\mathtt{p}}}\right)$
$\displaystyle\overset{\eqref{truncation_3212},\eqref{forthe2x1loveofgod2}}{\leq_{\mathtt{p_{e}}}}|\mathfrak{I}_{n}|_{s_{0}+\mathtt{b}_{1}+\mu_{\mathtt{p}}}\gamma^{-1}\left(1+\varepsilon^{13-12b}\right)N_{n}^{2\mu_{\mathtt{p}}}|Z_{n}|_{s_{0}}$
$\displaystyle\overset{\eqref{jjsd2asdsa1},\eqref{forthe2x1loveofgod2}}{\leq_{\mathtt{p_{e}}}}\varepsilon^{6-4b}N_{n-1}^{\mathtt{k}}\gamma^{-1}\varepsilon^{6-2b}N_{n}^{2\mu_{\mathtt{p}}}N_{n-1}^{-\sigma_{4}}$
$\displaystyle\overset{\eqref{choicess5}}{\leq}\varepsilon^{6-2b}N_{n}^{\mathtt{k}}.$

Therefore, together with (11.31), we obtain

$\displaystyle\left(|T(i_{n})\Pi_{n}Z_{n}|_{s_{0}+\mathtt{b}_{1}}+|\mathfrak{I}_{n}|_{s_{0}+\mathtt{b}_{1}}|T(i_{n})\Pi_{n}Z_{n}|_{s_{0}+\mu_{\mathtt{p}}}\right)\leq_{\mathtt{p_{e}}}\varepsilon^{6-2b}N_{n}^{\mathtt{k}}.$
(11.32)

Plugging this into (11.30), we have

$\displaystyle|(I$
$\displaystyle-\Pi_{n})d_{i}(\mathcal{F}_{\omega})(i_{n})T(i_{n}))\Pi_{n}Z_{n}|_{s_{0}}\leq_{\mathtt{p_{e}}}N_{n}^{-\mathtt{b}_{1}+\mu_{\mathtt{p}}}\varepsilon^{6-2b}N_{n}^{\mathtt{k}}\overset{\eqref{choicess2}}{\leq}\varepsilon^{6-2b}N_{n}^{-\sigma_{4}}.$
(11.33)

For
$|(I-\Pi_{n})d_{i}(\mathcal{F}_{\omega})(i_{n})T(i_{n}))\Pi_{n}Z_{n}|_{s_{0}+\mathtt{b}_{1}-\mu_{\mathtt{p}}}$,
we have

$\displaystyle|(I-\Pi_{n})d_{i}(\mathcal{F}_{\omega})(i_{n})T(i_{n}))\Pi_{n}Z_{n}|_{s_{0}+\mathtt{b}_{1}-\mu_{\mathtt{p}}}$
$\displaystyle\leq|d_{i}(\mathcal{F}_{\omega})(i_{n})T(i_{n}))\Pi_{n}Z_{n}|_{s_{0}+\mathtt{b}_{1}-\mu_{\mathtt{p}}}$
$\displaystyle\overset{\eqref{kkksdsxcxx222xc11}}{\leq_{\mathtt{p_{e}}}}\left(|T(i_{n})\Pi_{n}Z_{n}|_{s_{0}+\mathtt{b}_{1}}+|\mathfrak{I}_{n}|_{s_{0}+\mathtt{b}_{1}}|T(i_{n})\Pi_{n}Z_{n}|_{s_{0}+\mu_{\mathtt{p}}}\right)$
$\displaystyle\overset{\eqref{jjsd11009sd2}}{\leq_{\mathtt{p_{e}}}}\varepsilon^{6-2b}N_{n}^{\mathtt{k}}.$
(11.34)

For
$|d_{i}(\mathcal{F}_{\omega})(i_{n})(I-\Pi_{n})T(i_{n})\Pi_{n}Z_{n}|_{s_{0}}$,
we have

$\displaystyle|$ $\displaystyle
d_{i}(\mathcal{F}_{\omega})(i_{n})(I-\Pi_{n})T(i_{n})\Pi_{n}Z_{n}|_{s_{0}}$
$\displaystyle\overset{\eqref{kkksdsxcxx222xc11}}{\leq_{\mathtt{p_{e}}}}\left(|(I-\Pi_{n})T(i_{n})\Pi_{n}Z_{n}|_{s_{0}+\mu_{\mathtt{p}}}+|\mathfrak{I}_{n}|_{s_{0}+\mu_{\mathtt{p}}}|(I-\Pi_{n})T(i_{n})\Pi_{n}Z_{n}|_{s_{0}+\mu_{\mathtt{p}}}\right)$
$\displaystyle\overset{\eqref{truncation_3212}}{\leq_{\mathtt{p_{e}}}}N_{n}^{-(\mathtt{b}_{1}-\mu_{\mathtt{p}})}\left(|T(i_{n})\Pi_{n}Z_{n}|_{s_{0}+\mathtt{b}_{1}}+|\mathfrak{I}_{n}|_{s_{0}+\mu_{\mathtt{p}}}|T(i_{n})\Pi_{n}Z_{n}|_{s_{0}+\mathtt{b}_{1}}\right)$
$\displaystyle\overset{\eqref{inverse_estimate2}}{\leq_{\mathtt{p_{e}}}}N_{n}^{-\mathtt{b}_{1}+\mu_{\mathtt{p}}}\left(|T(i_{n})\Pi_{n}Z_{n}|_{s_{0}+\mathtt{b}_{1}}|+|\mathfrak{I}_{n}|_{s_{0}+\mu_{\mathtt{p}}}\gamma^{-1}(|\Pi_{n}Z_{n}|_{s_{0}+\mathtt{b}_{1}+\mu_{\mathtt{p}}}+\varepsilon^{7}\gamma^{-4}|\mathfrak{I}_{n}|_{s_{0}+\mathtt{b}_{1}+\mu_{\mathtt{p}}}|\Pi_{n}Z_{n}|_{s_{0}+\mu_{\mathtt{p}}})\right)$
(11.35)

We see each term in
$|\mathfrak{I}_{n}|_{s_{0}+\mu_{\mathtt{p}}}\gamma^{-1}(|\Pi_{n}Z_{n}|_{s_{0}+\mathtt{b}_{1}+\mu_{\mathtt{p}}}+\varepsilon^{7}\gamma^{-4}|\mathfrak{I}_{n}|_{s_{0}+\mathtt{b}_{1}+\mu_{\mathtt{p}}}|\Pi_{n}Z_{n}|_{s_{0}+\mu_{\mathtt{p}}})$
separately. For the first term, we have that

$\displaystyle|\mathfrak{I}_{n}|_{s_{0}+\mu_{\mathtt{p}}}\gamma^{-1}|\Pi_{n}Z_{n}|_{s_{0}+\mathtt{b}_{1}+\mu_{\mathtt{p}}}$
$\displaystyle\overset{\eqref{truncation_3212},\eqref{fortheloveof2xgod}}{\leq_{\mathtt{p_{e}}}}\gamma^{-1}\varepsilon^{6-4b}N_{n}^{2\mu_{\mathtt{p}}}|Z_{n}|_{s_{0}+\mathtt{b}_{1}-\mu_{\mathtt{p}}}$
$\displaystyle\overset{\eqref{jjsd2asdsa2}}{\leq_{\mathtt{p_{e}}}}\varepsilon^{6-2b}\varepsilon^{6-4b}\gamma^{-1}N_{n}^{2\mu_{\mathtt{p}}}N_{n-1}^{\mathtt{k}}$
$\displaystyle\overset{\eqref{choicess5}}{\leq}\varepsilon^{6-2b}N_{n}^{\mathtt{k}}.$
(11.36)

For the second term, we have

$\displaystyle\gamma^{-1}|\mathfrak{I}_{n}|_{s_{0}+\mu_{\mathtt{p}}}\varepsilon^{7}\gamma^{-4}|\mathfrak{I}_{n}|_{s_{0}+\mathtt{b}_{1}+\mu_{\mathtt{p}}}|\Pi_{n}Z_{n}|_{s_{0}+\mu_{\mathtt{p}}}$
$\displaystyle\overset{\eqref{fortheloveof2xgod},\eqref{forthe2x1loveofgod2},\gamma=\varepsilon^{2b}}{\leq_{\mathtt{p_{e}}}}\gamma^{-1}\underbrace{\varepsilon^{19-16b}}_{\leq
1,\
\because\eqref{parametersets1231}}N_{n-1}^{\mathtt{k}}|\Pi_{n}Z_{n}|_{s_{0}+\mu_{\mathtt{p}}}$
$\displaystyle\overset{\eqref{jjsd2asdsa3}}{\leq_{\mathtt{p_{e}}}}\varepsilon^{6-2b}\gamma^{-1}N_{n-1}^{\mathtt{k}}$
$\displaystyle\overset{\eqref{choicess5}}{\leq}\varepsilon^{6-2b}N_{n}^{\mathtt{k}}.$
(11.37)

Plugging (11), (11) and (11.31) into (11.35), we obtain

$\displaystyle|$ $\displaystyle
d_{i}(\mathcal{F}_{\omega})(i_{n})(I-\Pi_{n})T(i_{n})\Pi_{n}Z_{n}|_{s_{0}}\leq_{\mathtt{p_{e}}}\varepsilon^{6-2b}N_{n}^{\mathtt{k}+\mu_{\mathtt{p}}-\mathtt{b}_{1}}\overset{\eqref{choicess2}}{\leq}\varepsilon^{6-2b}N_{n}^{-\sigma_{4}}.$
(11.38)

For
$|d_{i}(\mathcal{F}_{\omega})(i_{n})(I-\Pi_{n})T(i_{n})\Pi_{n}Z_{n}|_{s_{0}+\mathtt{b}_{1}-\mu_{\mathtt{p}}}$,
we have

$\displaystyle|$ $\displaystyle
d_{i}(\mathcal{F}_{\omega})(i_{n})(I-\Pi_{n})T(i_{n})\Pi_{n}Z_{n}|_{s_{0}+\mathtt{b}_{1}-\mu_{\mathtt{p}}}$
$\displaystyle\overset{\eqref{kkksdsxcxx222xc11}}{\leq_{\mathtt{p_{e}}}}\left(|(I-\Pi_{n})T(i_{n})\Pi_{n}Z_{n}|_{s_{0}+\mathtt{b}_{1}}+|\mathfrak{I}_{n}|_{s_{0}+\mathtt{b}_{1}}|(I-\Pi_{n})T(i_{n})\Pi_{n}Z_{n}|_{s_{0}+\mu_{\mathtt{p}}}\right)$
$\displaystyle\leq|T(i_{n})\Pi_{n}Z_{n}|_{s_{0}+\mathtt{b}_{1}}+|\mathfrak{I}_{n}|_{s_{0}+\mathtt{b}_{1}}|T(i_{n})\Pi_{n}Z_{n}|_{s_{0}+\mu_{\mathtt{p}}}$
$\displaystyle\overset{\eqref{jjsd11009sd2}}{\leq}\varepsilon^{6-2b}N_{n}^{\mathtt{k}}.$
(11.39)

Then, plugging (11.22), (11.24), (11.27), (11.33) and (11.38) into (11.20), we
obtain

$\displaystyle|Z_{n+1}|_{s_{0}}\leq_{\mathtt{p_{e}}}\varepsilon^{6-2b}N_{n}^{-\sigma_{4}}.$
(11.40)

Similarly, plugging (11.23), (11.25), (11), (11.34), and (11.39) into (11.20),
we obtain

$\displaystyle|Z_{n+1}|_{s_{0}+\mathtt{b}_{1}-\mu_{\mathtt{p}}}\leq_{\mathtt{p_{e}}}\varepsilon^{6-2b}N_{n}^{\mathtt{k}}.$
(11.41)

Therefore, we prove (11.7) and (11.8) for $Z_{n+1}$.

Now we aim to prove (11.9) for $Z_{n+1}$. We simply apply Lemma 2.3 and obtain

$\displaystyle|Z_{n+1}|_{s_{0}+\mu_{\mathtt{p}}}\leq|Z_{n+1}|_{s_{0}}^{1-\frac{\mu_{\mathtt{p}}}{\mathtt{b}_{1}-\mu_{\mathtt{p}}}}|Z_{n+1}|_{s_{0}+\mathtt{b}_{1}-\mu_{\mathtt{p}}}^{\frac{\mu_{\mathtt{p}}}{\mathtt{b}_{1}-\mu_{\mathtt{p}}}}\overset{\eqref{znplus1l},\eqref{znplus1l2}}{\leq_{\mathtt{p_{e}}}}\varepsilon^{6-2b}N_{n}^{-\sigma_{4}\left(1-\frac{\mu_{\mathtt{p}}}{\mathtt{b}_{1}-\mu_{\mathtt{p}}}\right)+\frac{\mathtt{k}\mu_{\mathtt{p}}}{\mathtt{b}_{1}-\mu_{\mathtt{p}}}}\overset{\eqref{choicess71}}{\leq}\varepsilon^{6-2b}N_{n}^{-\sigma_{5}},$

which proves (11.9) for $Z_{n+1}$.

Proof of (2) for $n+1$. We assume (1), (2) hold true for $Z_{n+1}$, which was
just proved above, and $\widehat{\mathfrak{I}}_{k}$ for $k\leq n+1$ and will
aim to prove (2) for $\widehat{\mathfrak{I}}_{n+2}$.

Since $\mathfrak{I}_{n+1}(\omega)\in X_{i}\cap X_{i,\mathtt{M}}$ is already
well-defined by the induction hypotheses for $\omega\in\mathcal{G}_{n+1}$,
using (11.18), (11.19), and (11.9) for $Z_{n+1}$, we have

$\displaystyle|\mathfrak{I}_{n+1}|_{s_{0}+2\mu_{\mathtt{p}}}\leq_{\mathtt{p_{e}}}\varepsilon^{6-4b},\quad|\mathfrak{I}_{n+1}|_{s_{0}+2\mu_{\mathtt{p}}+\mathtt{b}_{1}}\leq\varepsilon^{6-4b}N_{n-1}^{\mathtt{k}},\quad|Z_{n+1}|_{s_{0}+\mu_{\mathtt{p}}}\leq_{\mathtt{p_{e}}}\varepsilon^{6-2b}.$
(11.42)

therefore, (7.3) holds for $\mu=\mu_{\mathtt{p}}$ and for some
$\mathtt{C}=\mathtt{C}_{\mathtt{p_{e}}}$. Then we can apply Proposition 7.3
with $\mathtt{S}$ in (11.6), and $\mathtt{C}=\mathtt{C}_{\mathtt{p_{e}}}$, for
sufficiently small depending on $\mathtt{p_{e}}$, since $\mathtt{S}$ in (11.6)
and $\mathtt{C}_{\mathtt{p_{e}}}$ are fixed depending on $\mathtt{p_{e}}$.
Note that Proposition 7.3 tells us that we can find an approximate inverse
$T(i_{n+1})$ for all $\omega\in\Omega_{\infty}(i_{n+1})$. From definition of
$\mathcal{G}_{n+2}$ in (11.3) and $\Omega_{\infty}(i_{n+1})$ in (10.390), it
is clear that

$\displaystyle\mathcal{G}_{n+2}\subset\Omega_{\infty}(i_{n+1}).$ (11.43)

Therefore, for all $\omega\in\mathcal{G}_{n+2}$, $T(i_{n+1}(\omega)):Y_{i}\cap
X_{i,\mathtt{M}}\mapsto X_{i}\cap X_{i,\mathtt{M}}$ is well-defined. Since
$Z_{n+1}(\omega)\in Y_{i}\cap X_{i,\mathtt{M}}$ is well-defined for
$\omega\in\mathcal{G}_{n+1}\subset\mathcal{G}_{n+2}$, we see that
$\mathfrak{I}_{n+2}$ in (11.2) is well-defined for
$\omega\in\mathcal{G}_{n+2}$ and $\mathfrak{I}_{n+2}\in X_{i}\cap
X_{i,\mathtt{M}}$.

Now, it suffices to show the estimates (11.10) and (11.11) for
$\widehat{\mathfrak{I}}_{n+2}$. Towards the estimates (11.10), we use (7.14)
to obtain

$\displaystyle|\widehat{\mathfrak{I}}_{n+2}|_{s_{0}+2\mu_{\mathtt{p}}}$
$\displaystyle=|\Pi_{n+1}T(i_{n+1})\Pi_{n+1}Z_{n+1}|_{s_{0}+2\mu_{\mathtt{p}}}$
$\displaystyle\overset{\eqref{truncation_3212}}{\leq}N_{n+1}^{\mu_{\mathtt{p}}}|T(i_{n+1})\Pi_{n+1}Z_{n+1}|_{s_{0}+\mu_{\mathtt{p}}}$
$\displaystyle\overset{\eqref{inverse_estimate2}}{\leq}_{\mathtt{p_{e}}}N_{n+1}^{\mu_{\mathtt{p}}}\gamma^{-1}(1+\varepsilon^{7}\gamma^{-4}|\mathfrak{I}_{n+1}|_{s_{0}+2\mu_{\mathtt{p}}})|\Pi_{n+1}Z_{n+1}|_{s_{0}+2\mu_{\mathtt{p}}}$
$\displaystyle\overset{\eqref{fortheloveof2xgod},\eqref{parametersets1231},\eqref{truncation_3212}}{\leq_{\mathtt{p_{e}}}}N_{n+1}^{3\mu_{\mathtt{p}}}\gamma^{-1}(1+\underbrace{\varepsilon^{7}\gamma^{-4}\varepsilon^{6-4b}}_{\overset{\gamma=\varepsilon^{2b}}{=}\varepsilon^{13-12b}\overset{\eqref{parametersets1231}}{\leq}1})|Z_{n+1}|_{s_{0}}$
$\displaystyle\overset{\eqref{znplus1l}}{\leq}_{\mathtt{p_{e}}}\varepsilon^{6-2b}\gamma^{-1}N_{n+1}^{3\mu_{\mathtt{p}}}N_{n}^{-\sigma_{4}}$
$\displaystyle\overset{\eqref{choicess31}}{\leq}\varepsilon^{6-2b}\gamma^{-1}N_{n}^{-\sigma_{3}},$
(11.44)

which gives us (11.10) for $\widehat{\mathfrak{I}}_{n+2}$. For (11.11), we
compute similarly,

$\displaystyle|\widehat{\mathfrak{I}}_{n+2}|_{s_{0}+2\mu_{\mathtt{p}}+\mathtt{b}_{1}}$
$\displaystyle=|\Pi_{n+1}T(i_{n+1})\Pi_{n+1}Z_{n+1}|_{s_{0}+2\mu_{\mathtt{p}}+\mathtt{b}_{1}}$
$\displaystyle\overset{\eqref{truncation_3212}}{\leq}N_{n+1}^{4\mu_{\mathtt{p}}}|T(i_{n+1})\Pi_{n+1}Z_{n+1}|_{s_{0}-2\mu_{\mathtt{p}}+\mathtt{b}_{1}}$
$\displaystyle\overset{\eqref{inverse_estimate2}}{\leq_{\mathtt{p_{e}}}}N_{n+1}^{4\mu_{\mathtt{p}}}\gamma^{-1}(|\Pi_{n+1}Z_{n+1}|_{s_{0}-\mu_{\mathtt{p}}+\mathtt{b}_{1}}+\varepsilon^{7}\gamma^{-4}|\mathfrak{I}_{n+1}|_{s_{0}-\mu_{\mathtt{p}}+\mathtt{b}_{1}}|\Pi_{n+1}Z_{n+1}|_{s_{0}+\mu_{\mathtt{p}}})$
$\displaystyle\overset{\eqref{truncation_3212}}{\leq}N_{n+1}^{5\mu_{\mathtt{p}}}\gamma^{-1}(|Z_{n+1}|_{s_{0}-\mu_{\mathtt{p}}+\mathtt{b}_{1}}+\varepsilon^{7}\gamma^{-4}|\mathfrak{I}_{n+1}|_{s_{0}-\mu_{\mathtt{p}}+\mathtt{b}_{1}}|Z_{n+1}|_{s_{0}})$
$\displaystyle\overset{\eqref{forthe2x1loveofgod2},\eqref{znplus1l},\eqref{znplus1l2}}{\leq_{\mathtt{b}}}\gamma^{-1}\varepsilon^{6-2b}N_{n+1}^{5\mu_{\mathtt{p}}}\left(N_{n}^{\mathtt{k}}+\underbrace{\varepsilon^{13-12b}}_{\leq
1,\because\eqref{parametersets1231}}N_{n}^{\mathtt{k}}N_{n}^{-\sigma_{4}}\right)$
$\displaystyle\leq\gamma^{-1}\varepsilon^{6-2b}N_{n+1}^{5\mu_{\mathtt{p}}}N_{n}^{\mathtt{k}}$
$\displaystyle\overset{\eqref{choicess09}}{\leq}\gamma^{-1}\varepsilon^{6-2b}N_{n+1}^{\mathtt{k}},$
(11.45)

which gives us (11.11) for $\widehat{\mathfrak{I}}_{n+2}$. ∎

######  Lemma 11.2.

With the choice of
$\mathtt{b}_{1},\sigma_{3},\sigma_{4},\sigma_{5},\mathtt{k}$ in (11.6) and the
sequence $N_{n}$ in (10.322), we have the following: For all sufficiently
small $\varepsilon$ depending on $\mathtt{p},b$,

$\displaystyle\gamma^{-1}N_{n}^{2\mu_{\mathtt{p}}}N_{n-1}^{\mathtt{k}}$
$\displaystyle\leq N_{n}^{\mathtt{k}},$ (11.46)
$\displaystyle\varepsilon^{7-6b}N_{n-1}^{-2\sigma_{3}}$ $\displaystyle\leq
N_{n}^{-\sigma_{4}},$ (11.47) $\displaystyle
N_{n}^{\mathtt{k}}N_{n}^{-(\mathtt{b}_{1}-\mu_{\mathtt{p}})}$
$\displaystyle\leq N_{n}^{-\sigma_{4}},$ (11.48)
$\displaystyle\varepsilon^{5-4b}N_{n-1}^{-2\sigma_{5}}$ $\displaystyle\leq
N_{n}^{-\sigma_{4}},$ (11.49)
$\displaystyle\varepsilon^{5-4b}N_{n}^{\mu_{\mathtt{p}}}N_{n-1}^{-\sigma_{5}}$
$\displaystyle\leq 1,$ (11.50) $\displaystyle
N_{n+1}^{3\mu_{\mathtt{p}}}N_{n}^{-\sigma_{4}}$ $\displaystyle\leq
N_{n}^{-\sigma_{3}},$ (11.51) $\displaystyle
N_{n+1}^{5\mu_{\mathtt{p}}}N_{n}^{\mathtt{k}}$ $\displaystyle\leq
N_{n+1}^{\mathtt{k}},$ (11.52)
$\displaystyle-\sigma_{4}\left(1-\frac{\mu_{\mathtt{p}}}{\mathtt{b}_{1}-\mu_{\mathtt{p}}}\right)+\frac{\mathtt{k}\mu_{\mathtt{p}}}{\mathtt{b}_{1}-\mu_{\mathtt{p}}}$
$\displaystyle\leq-\sigma_{5},$ (11.53)
$\displaystyle\sum_{k=0}^{n}N_{k}^{\mathtt{k}}$ $\displaystyle\leq
2N_{n}^{\mathtt{k}},$ (11.54)

for all $n\geq 0$.

###### Proof.

Recalling $\gamma$, $N_{-1}$, $N_{0}$ and $N_{n}$ from (6.3) and (10.322), we
have

$\displaystyle\gamma^{-1}=N_{0}^{\frac{2b}{(2b-1)\rho}},\quad N_{-1}=1,\quad
N_{0}=\varepsilon^{(1-2b)\rho},\quad\chi=3/2,\quad
N_{n}=N_{0}^{(\chi)^{n}},\text{ for $n\geq 0$.}$ (11.55)

Proof of (11.46). When $n=0$, we see that

$\displaystyle\gamma^{-1}N_{n}^{2\mu_{\mathtt{p}}}N_{n-1}^{\mathtt{k}}\leq
N_{n}^{\mathtt{k}}\overset{n=0}{\iff}\gamma^{-1}N_{0}^{2\mu_{\mathtt{p}}}\leq
N_{0}^{\mathtt{k}}\overset{\eqref{N_0andjr}}{\iff}N_{0}^{\frac{2b}{(2b-1)\rho}+2\mu_{\mathtt{p}}}\leq
N_{0}^{\mathtt{k}}\iff\mathtt{k}\geq\frac{2b}{(2b-1)\rho}+2\mu_{\mathtt{p}}.$
(11.56)

When $n\geq 1$, we see that

$\gamma^{-1}N_{n}^{2\mu_{\mathtt{p}}}N_{n-1}^{\mathtt{k}}\leq
N_{n}^{\mathtt{k}}\iff\frac{2b}{(2b-1)\rho}+2\mu_{\mathtt{p}}\chi^{n}+\mathtt{k}\chi^{n-1}\leq\mathtt{k}\chi^{n}\overset{\chi=3/2}{\Longleftarrow}\mathtt{k}\geq\frac{4b}{(2b-1)\rho}+6\mu_{\mathtt{p}}.$

Therefore, for any $n\geq 0$, it suffices to show that

$\displaystyle\mathtt{k}\geq\frac{4b}{(2b-1)\rho}+6\mu_{\mathtt{p}}.$ (11.57)

Since
$\frac{4b}{(2b-1)\rho}\overset{\eqref{rhoreplacesds}}{=}A\frac{4b}{5-4b}\overset{b\in(1,1+1/12)}{\leq}\frac{13}{2}A,$
it suffices to show that

$\displaystyle\mathtt{k}\geq\frac{13}{2}A+6\mu_{\mathtt{p}}.$ (11.58)

From (11.6), we have $\mathtt{k}=7A$, therefore

$\mathtt{k}-\left(\frac{13}{2}A+6\mu_{\mathtt{p}}\right)=\frac{1}{2}A-6\mu_{\mathtt{p}}\overset{\eqref{sdsdxc2sdAAA}}{\geq}0.$

Therefore, (11.58) holds true.

Proof of (11.47). When $n=0$, we see that

$\displaystyle\varepsilon^{7-6b}N_{-1}^{-2\sigma_{3}}$ $\displaystyle\leq
N_{0}^{-\sigma_{4}}\iff\varepsilon^{7-6b}\leq
N_{0}^{-\sigma_{4}}\overset{N_{0}=(\varepsilon\gamma^{-1})^{\rho}=\varepsilon^{\rho(1-2b)}}{\iff}\varepsilon^{7-6b}\leq\varepsilon^{(2b-1)\rho\sigma_{4}}\iff\sigma_{4}\leq\frac{7-6b}{(2b-1)\rho}$
$\displaystyle\iff\sigma_{4}\leq\frac{5-4b}{(2b-1)\rho}\frac{7-6b}{5-4b}$
$\displaystyle\overset{\eqref{rhoreplacesds}}{\iff}\sigma_{4}\leq
A\frac{7-6b}{5-4b}$
$\displaystyle\overset{b\in(1,1+1/12)}{\Longleftarrow}\sigma_{4}\leq\frac{3}{4}A$

which holds true thanks to $\sigma_{4}$ in (11.6). When $n\geq 1$, we see that

$\varepsilon^{7-6b}N_{n-1}^{-2\sigma_{3}}\leq
N_{n}^{-\sigma_{4}}\Longleftarrow N_{n-1}^{-2\sigma_{3}}\leq
N_{n}^{-\sigma_{4}}\iff N_{0}^{-2\sigma_{3}\chi^{n-1}}\leq
N_{0}^{-\sigma_{4}\chi^{n}}\iff\frac{3}{2}\sigma_{4}\leq 2\sigma_{3},$

which holds true thanks to $\sigma_{4},\sigma_{3}$ in (11.6).

Proof of (11.48). We see that

$N_{n}^{\mathtt{k}}N_{n}^{-(\mathtt{b}_{1}-\mu_{\mathtt{p}})}\leq
N_{n}^{-\sigma_{4}}\iff\sigma_{4}\leq\mathtt{b}_{1}-\mathtt{k}-\mu_{\mathtt{p}}.$

From (11.6), we see that

$\mathtt{b}_{1}-\mathtt{k}-\mu_{\mathtt{p}}-\sigma_{4}=\frac{5}{4}A-\mu_{\mathtt{p}}\overset{\eqref{sdsdxc2sdAAA}}{\geq}0.$

Therefore, (11.48) holds true.

Proof of (11.49). When $n=0$, we see that

$\varepsilon^{5-4b}\leq
N_{0}^{-\sigma_{4}}\overset{\eqref{N_0andjr}}{\iff}\varepsilon^{5-4b}\leq\varepsilon^{(2b-1)\rho\sigma_{4}}\iff\sigma_{4}\leq\frac{5-4b}{(2b-1)\rho}\overset{\eqref{rhoreplacesds}}{\iff}\sigma_{4}\leq
A,$

which holds true thanks to $\sigma_{4}$ in (11.6). When $n\geq 1$, we see that

$\varepsilon^{5-4b}N_{n-1}^{-2\sigma_{5}}\leq
N_{n}^{-\sigma_{4}}\Longleftarrow N_{n-1}^{-2\sigma_{5}}\leq
N_{n}^{-\sigma_{4}}\iff N_{0}^{\sigma_{4}\chi^{n}}\leq
N_{0}^{2\sigma_{5}\chi^{n-1}}\iff\frac{3}{2}\sigma_{4}\leq 2\sigma_{5},$

which holds true, thanks to $\sigma_{4},\sigma_{5}$ in (11.6).

Proof of (11.50). When $n=0$, we see that

$\displaystyle\varepsilon^{5-4b}N_{0}^{\mu_{\mathtt{p}}}$ $\displaystyle\leq
1\overset{\eqref{N_0andjr}}{\iff}\varepsilon^{5-4b+\mu_{\mathtt{p}}\rho(1-2b)}\leq
1\iff\rho\mu_{\mathtt{p}}(2b-1)\leq
5-4b\iff\frac{1}{\rho}\geq\frac{(2b-1)\mu_{\mathtt{p}}}{5-4b}$
$\displaystyle\overset{\eqref{rhoreplacesds}}{\iff}A\geq\mu_{\mathtt{p}},$

which holds true thanks to (11.5). When $n\geq 1$, we see that

$\varepsilon^{5-4b}N_{n}^{\mu_{\mathtt{p}}}N_{n-1}^{-\sigma_{5}}\leq
1\Longleftarrow N_{n}^{\mu_{\mathtt{p}}}N_{n-1}^{-\sigma_{5}}\leq 1\iff
N_{0}^{\mu_{\mathtt{p}}\chi^{n}-\sigma_{5}\chi^{n-1}}\leq
1\iff\sigma_{5}\geq\frac{3}{2}\mu_{\mathtt{p}},$

which holds also true, thanks to the choice of $\sigma_{5}$ in (11.6) and the
estimate for $A$ in (11.5).

Proof of (11.51). We see that

$N_{n+1}^{3\mu_{\mathtt{p}}}N_{n}^{-\sigma_{4}}\leq N_{n}^{-\sigma_{3}}\iff
N_{0}^{3\mu_{\mathtt{p}}\chi^{n+1}-\sigma_{4}\chi^{n}}\leq
N_{0}^{-\sigma_{3}\chi^{n}}\overset{\chi=3/2}{\iff}\frac{9}{2}\mu_{\mathtt{p}}-\sigma_{4}\leq-\sigma_{3}\iff\sigma_{4}-\sigma_{3}\geq\frac{9}{2}\mu_{\mathtt{p}}.$

From $\sigma_{4},\sigma_{3}$ in (11.6), we have

$\sigma_{4}-\sigma_{3}-\frac{9}{2}\mu_{\mathtt{p}}=\frac{3}{16}A-\frac{9}{2}{\mu_{\mathtt{p}}}\overset{\eqref{sdsdxc2sdAAA}}{\geq}\frac{15}{2}\mu_{\mathtt{p}}-\frac{9}{2}\mu_{\mathtt{p}}\geq
0.$

Therefore, (11.51) holds true.

Proof of (11.52). We see that

$N_{n+1}^{5\mu_{\mathtt{p}}}N_{n}^{\mathtt{k}}\leq N_{n+1}^{\mathtt{k}}\iff
N_{0}^{5\mu_{\mathtt{p}}\chi^{n+1}+\mathtt{k}\chi^{n}}\leq
N_{0}^{\mathtt{k}\chi^{n+1}}\iff
5\mu_{\mathtt{p}}\chi^{n+1}+\mathtt{k}\chi^{n}\leq\mathtt{k}\chi^{n+1}\overset{\chi=3/2}{\Longleftarrow}15\mu_{\mathtt{p}}\leq\mathtt{k}.$

From (11.6), we have

$\mathtt{k}-15\mu_{\mathtt{p}}=7A-15\mu_{\mathtt{p}}\overset{\eqref{sdsdxc2sdAAA}}{\geq}0.$

Therefore, (11.52) holds true.

Proof of (11.53). Plugging $\sigma_{4},\sigma_{5}$ and $\mathtt{k}$ from
(11.6) into (11.53), it suffices to show that

$-(1-\frac{\mu_{\mathtt{p}}}{\mathtt{b}_{1}-\mu_{\mathtt{p}}})+\frac{7\mu_{\mathtt{p}}}{\mathtt{b}_{1}-\mu_{\mathtt{p}}}\leq-\frac{3}{4},$

which is equivalent to

$\frac{8\mu_{\mathtt{p}}}{\mathtt{b}_{1}-\mu_{\mathtt{p}}}\leq\frac{1}{4}.$

This is equivalent to $33\mu_{\mathtt{p}}\leq\mathtt{b}_{1}$, while we have
$\mathtt{b}_{1}\overset{\eqref{parambesd2sdjj22}}{=}9A\overset{\eqref{sdsdxc2sdAAA}}{\geq}360\mu_{\mathtt{p}}.$
Therefore, (11.53) holds true.

Proof of (11.54). Recalling $N_{n}$ from (11.55), we see that it suffices to
show that

$\sum_{k=0}^{n-1}N_{k}^{\mathtt{k}}\leq N_{n}^{\mathtt{k}},\text{ for all
$n\geq 1$.}$

This can be proved by the usual induction argument. If $n=1$, the result
follows immediately since $n\mapsto N_{n}$ is an increasing sequence. For
$n+1$, we have that

$\sum_{k=0}^{n}N_{k}^{\mathtt{k}}=\sum_{k=0}^{n-1}N_{k}^{\mathtt{k}}+N_{n}^{\mathtt{k}}\leq
2N_{n}^{\mathtt{k}},$

where the last inequality follows from the induction hypothesis. Therefore, it
suffices to show that $2N_{n}^{\mathtt{k}}\leq N_{n+1}^{\mathtt{k}}$, in other
words (recalling $N_{n}$ from (11.55))

$2N_{0}^{\mathtt{k}\chi^{n}}\leq N_{0}^{\mathtt{k}\chi^{n+1}},\text{ for all
$n\geq 0$.}$

Since $\mathtt{k}>0$, $\chi=3/2$ and $n\geq 0$, this certainly holds for
sufficiently large $N_{0}$, that is, for sufficiently small $\varepsilon$ (see
$N_{0}$ in (11.55)). ∎

###  11.1 Measure of the frequency set

In this subsection, we estimate the measure of the set $\mathcal{G}_{n}$ in
(11.3). Recall from (11.2) and (2) of Proposition 11.1 that $i_{n}(\omega)$ is
a well-defined reversible, $\frac{2\pi}{\mathtt{M}}$-translation invariant
embedding for each $\omega\in\mathcal{G}_{n}$ for all $n\geq 0$. It follows
from (11.10) and (11.9) that (using the sequence
$\sum_{n=-1}^{\infty}N_{n}^{-\sigma}$ is summable, and arguing as in (11.18)),
each $i_{n}$ for all $n\geq 0$ satisfies

$\displaystyle\rVert\mathfrak{I}_{n}\rVert^{\text{Lip}(\gamma,\mathcal{G}_{n})}_{s_{0}+2\mu_{\mathtt{p}}}\leq_{\mathtt{p_{e}}}\varepsilon^{6-4b},\quad\rVert
Z_{n}\rVert^{\text{Lip}(\gamma,\mathcal{G}_{n})}_{s_{0}+\mu_{\mathtt{p}}}\leq_{\mathtt{p_{e}}}\varepsilon^{6-2b},\text{
where $\mathtt{p_{e}}=\left\\{\mathtt{p},b,\mathtt{M},S_{0}\right\\}$ (recall
from \eqref{nzzotzzations1zzdddd}).}$ (11.59)

Espeically, $i_{n}$ satisfies (7.3) with $\Omega_{1}=\mathcal{G}_{n}$,
$\mu=\mu_{\mathtt{p}}$ and for some $\mathtt{C}=\mathtt{C}(\mathtt{p_{e}})$.
Since $\mu_{\mathtt{p}}>\mu_{\mathtt{p},1}$ (see (10.390) and (7.65)), we
apply Proposition 10.82 and (1) of Proposition 10.76 with $\mathtt{S}$ (fixed
in (11.6)) to see that the eigenvalues

$\displaystyle d_{\infty}(i_{n}(\omega))(\omega,j)$
$\displaystyle=d_{0}(i_{n}(\omega))(\omega,j)+r_{\infty}(i_{n}(\omega))(\omega,j),$
(11.60) $\displaystyle d_{0}(i_{n}(\omega))(\omega,j)$
$\displaystyle=\text{i}\left(\mathtt{m}_{\alpha}(i_{n}(\omega))\lambda_{\alpha}(j)+j\frac{T_{\alpha}}{4}+\mathfrak{m}_{1}(i_{n}(\omega))(\omega,j)\right),$

are well-defined and satisfy the decompositions:

$\displaystyle\mathtt{m}_{\alpha}(i_{n}(\omega))$
$\displaystyle=-\frac{1}{2}+\varepsilon^{2}\mathtt{m}_{\alpha,1}(\omega)+\mathtt{m}_{\alpha,2}(i_{n}(\omega)),$
(11.61) $\displaystyle\mathfrak{m}_{1}(\omega,\xi)$
$\displaystyle:=\xi\left(\mathfrak{m}_{\leq
0}(i_{n}(\omega))(\omega,\xi)+\varepsilon^{2}\mathfrak{m}_{\mathfrak{b}}(\omega,\xi)\right)$
$\displaystyle\overset{\eqref{docsd2ksdsdsd}}{=}\xi\left(\varepsilon^{2}(\mathfrak{m}_{\leq
0,1}(\omega,\xi)+\mathfrak{m}_{\mathfrak{b}}(\omega,\xi))+\mathfrak{m}_{\leq
0,2}(i_{n}(\omega))(\omega,\xi)\right),$

with the estimates (see (10.3), (10.293), (10.122) and (10.385) and recall
$\mathtt{p_{e}}$ from (11.12)),

$\displaystyle|\mathtt{m}_{\alpha,1}|^{\text{Lip}(\gamma,\mathcal{G}_{n})}$
$\displaystyle\leq_{\mathtt{p_{e}}}1,\quad|\mathtt{m}_{\alpha,2}|^{\text{Lip}(\gamma,\mathcal{G}_{n})}\leq_{\mathtt{p_{e}}}\varepsilon^{7-4b},\quad|\mathtt{m}_{\alpha,2}(i_{n})-\mathtt{m}_{\alpha,2}(i_{m})|\leq_{\mathtt{p_{e}}}\varepsilon\rVert
i_{n}-i_{m}\rVert_{s_{0}+\mu_{\mathtt{p}}},$ (11.62)
$\displaystyle|\mathfrak{m}_{1}|^{\text{Lip}(\gamma,\mathcal{G}_{n})}_{1,0,2}$
$\displaystyle\leq_{\mathtt{p_{e}}}\varepsilon^{2},\quad|\mathfrak{m}_{1}(i_{n})-\mathfrak{m}_{1}(i_{m})|_{1,0,2}\leq_{\mathtt{p_{e}}}\varepsilon\rVert
i_{n}-i_{m}\rVert_{s_{0}+\mu_{\mathtt{p}}},$ (11.63)
$\displaystyle|\mathfrak{m}_{\leq
0,1}|^{\text{Lip}(\gamma,\Omega_{1})}_{0,0,2}$
$\displaystyle\leq_{d}1,\quad|\mathfrak{m}_{\leq
0,2}|^{\text{Lip}(\gamma,\Omega_{1})}_{0,0,2}\leq_{\mathtt{p_{e}}}\varepsilon^{7-4b},\quad|\mathfrak{m}_{\leq
0,2}(i_{n})-\mathfrak{m}_{\leq
0,2}(i_{m})|_{0,0,2}\leq_{\mathtt{p_{e}}}\varepsilon\rVert
i_{n}-i_{m}\rVert_{s_{0}+\mu_{\mathtt{p}}},$ (11.64)
$\displaystyle|r_{\infty}|^{\text{Lip}(\gamma,\mathcal{G}_{n})}_{0,0,0}$
$\displaystyle\leq_{\mathtt{p_{e}}}\varepsilon^{9-6b},\quad|r_{\infty}(i_{n})-r_{\infty}(i_{m})|_{0,0,0}\leq_{\mathtt{p_{e}}}\varepsilon^{3}\gamma^{-1}\rVert
i_{n}-i_{m}\rVert_{s_{0}+\mu_{\mathtt{p}}},,$ (11.65)

for all $n,m\in\mathbb{N}_{0}$, where we used the fact that the inequalities
in the estimates depend on $\mathtt{pe}$ (see (7.15)), while $\mathtt{C}$
depends on $\mathtt{p_{e}}$, which is defined in (11.12), hence the
inequalities depend on only $\mathtt{p_{e}}$. Furthermore, (10.194) tells us
that

$\displaystyle\mathfrak{m}_{\mathfrak{b}}(i_{n}(\omega))(\omega,\xi)+\mathtt{m}_{\alpha,1}(\omega)m_{1,\alpha}(\xi)+\mathfrak{m}_{\leq
0,1}(i_{n}(\omega))(\omega,\xi)=\mathfrak{F}_{\alpha-1}(\omega,\xi).$ (11.66)

Hence, plugging (11.66) and (11.61) into (11.60), we have (rearranging terms
according to the power of $\varepsilon$)

$\displaystyle d_{\infty}(i_{n}(\omega))(\omega,j)$
$\displaystyle=\text{i}\left(-\frac{1}{2}\lambda_{\alpha}(j)+j\frac{T_{\alpha}}{4}+\varepsilon^{2}j\mathfrak{F}_{\alpha-1}(\omega,j)\right.$
(11.67) $\displaystyle\left.\
+\left(\mathtt{m}_{\alpha,2}(i_{n}(\omega))\lambda_{\alpha}(j)+j\mathfrak{m}_{\leq
0,2}(i_{n}(\omega))(\omega,j)\right)\right)+r_{\infty}(i_{n}(\omega))(\omega,j),\text{
if $j\in S^{\perp}$}$

while,
$r_{\infty}(i_{n}(\omega))(\omega,0)\overset{\eqref{sddpooo4},\eqref{sjdjsd2sdsddenoteasd}}{=}0$
and
$d_{0}(i_{n}(\omega))(\omega,0)\overset{\eqref{d0defsd2},\eqref{d0defsd22}}{=}0$,
therefore, (11.60) yields that

$\displaystyle d_{\infty}(i_{n}(\omega))(\omega,0)=0.$ (11.68)

In view of the definition of $\mathcal{G}_{n}$, let us define

$\displaystyle
R_{ljk}(i_{n}):=\left\\{\omega\in\mathcal{G}_{n}:|\text{i}\omega\cdot
l-(d_{\infty}(i_{n}(\omega))(\omega,j)-d_{\infty}(i_{n}(\omega))(\omega,k))|<2\gamma_{n}\langle
l\rangle^{-\tau}|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|\right\\}.$ (11.69)

From (11.3), it is clear that

$\displaystyle\mathcal{G}_{n}\backslash\mathcal{G}_{n+1}=\cup_{l\in\mathbb{Z}^{\nu},\
j,k\in S_{\mathtt{M}}^{\perp}\cup\left\\{0\right\\}}R_{ljk}(i_{n}),\text{ for
all $n\geq 0$.}$ (11.70)

In view of (11.70), we will assume in the rest of the proof that

$\displaystyle j,k\subset S_{\mathtt{M}}^{\perp}\cup\left\\{0\right\\},\text{
and }j\neq k.$ (11.71)

######  Lemma 11.3.

For small enough $\varepsilon>0$ depending on $\mathtt{p_{e}}$, it holds that
$R_{ljk}(i_{n+1})\subset R_{ljk}(i_{n})$, if $|l|<N_{n}$ for $n\geq 0$.

###### Proof.

Let $\omega\subset R_{ljk}(i_{n+1})$, that is,

$\displaystyle|\text{i}\omega\cdot
l-(d_{\infty}(i_{n+1}(\omega))(\omega,j)-d_{\infty}(i_{n+1}(\omega))(\omega,k))|<2\gamma_{n+1}\langle
l\rangle^{-\tau}|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|.$ (11.72)

Hence, for $|l|<N_{n}$, we have

$\displaystyle|\text{i}\omega\cdot l$
$\displaystyle-(d_{\infty}(i_{n}(\omega))(\omega,j)-d_{\infty}(i_{n}(\omega))(\omega,k)))|\leq|\text{i}\omega\cdot
l-(d_{\infty}(i_{n+1}(\omega))(\omega,j)-d_{\infty}(i_{n+1}(\omega))(\omega,k))|$
(11.73) $\displaystyle\
+|(d_{\infty}(i_{n+1}(\omega))(\omega,j)-d_{\infty}(i_{n+1}(\omega))(\omega,k))-(d_{\infty}(i_{n}(\omega))(\omega,j)-d_{\infty}(i_{n}(\omega))(\omega,k))|$
$\displaystyle\overset{\eqref{omega11xxxsd}}{<}2\gamma_{n+1}\langle
l\rangle^{-\tau}|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|$ $\displaystyle\
+|(d_{\infty}(i_{n+1}(\omega))(\omega,j)-d_{\infty}(i_{n+1}(\omega))(\omega,k))-(d_{\infty}(i_{n}(\omega))(\omega,j)-d_{\infty}(i_{n}(\omega))(\omega,k))|.$
(11.74)

We claim that

$\displaystyle|(d_{\infty}(i_{n+1}(\omega))(\omega,j)-d_{\infty}(i_{n+1}(\omega))(\omega,k))-(d_{\infty}(i_{n}(\omega))(\omega,j)-d_{\infty}(i_{n}(\omega))(\omega,k))|\leq
2^{-n}\gamma\langle
l\rangle^{-\tau}|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|,$ (11.75)

for $|l|<N_{n}$. Assuming (11.75) holds, plugging this into (11.74), we see
that

$\displaystyle|\text{i}\omega\cdot l$
$\displaystyle-(d_{\infty}(i_{n}(\omega))(\omega,j)-d_{\infty}(i_{n}(\omega))(\omega,k)))|<2\langle
l\rangle^{-\tau}|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|(\gamma_{n+1}+2^{-n-1}\gamma)$
$\displaystyle=2\langle
l\rangle^{-\tau}|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|\gamma_{n},$ (11.76)

where the last inequality follows from $\gamma_{n}=\gamma(1+2^{-n})$ (see
(11.3)). Clearly, (11.1) and (11.69) imply that $\omega\in R_{ljk}(i_{n})$.

To finish the proof, let us prove (11.75). From (11.60), we have

$\displaystyle|($ $\displaystyle
d_{\infty}(i_{n+1}(\omega))(\omega,j)-d_{\infty}(i_{n+1}(\omega))(\omega,k))-(d_{\infty}(i_{n}(\omega))(\omega,j)-d_{\infty}(i_{n}(\omega))(\omega,k))|$
$\displaystyle\leq|(\mathtt{m}_{\alpha}(i_{n+1}(\omega))-\mathtt{m}_{\alpha}(i_{n}(\omega)))(\lambda_{\alpha}(j)-\lambda_{\alpha}(k))|$
$\displaystyle\
+|(\mathfrak{m}_{1}(i_{n+1}(\omega))(\omega,j)-\mathfrak{m}_{1}(i_{n+1}(\omega))(\omega,k))-(\mathfrak{m}_{1}(i_{n}(\omega))(\omega,j)-\mathfrak{m}_{1}(i_{n}(\omega))(\omega,k))|$
$\displaystyle\ +2\sup_{j\in
S^{\perp}\left\\{0\right\\}}|r_{\infty}(i_{n}(\omega))(\omega,j)-r_{\infty}(i_{n+1}(\omega))(\omega,j)|$
$\displaystyle=\left|(\mathtt{m}_{\alpha}(i_{n+1}(\omega))-\mathtt{m}_{\alpha}(i_{n}(\omega)))(\lambda_{\alpha}(j)-\lambda_{\alpha}(k))\right|+\left|\int_{k}^{j}\partial_{\xi}\mathfrak{m}_{1}(i_{n+1}(\omega))(\omega,\xi)-\partial_{\xi}\mathfrak{m}_{1}(i_{n}(\omega))(\omega,\xi)d\xi\right|$
$\displaystyle\ +2\sup_{j\in
S^{\perp}\left\\{0\right\\}}|r_{\infty}(i_{n}(\omega))(\omega,j)-r_{\infty}(i_{n+1}(\omega))(\omega,j)|$
$\displaystyle\overset{\eqref{decoprjdmlRmxdlqhdlsekxxx}}{=}|(\mathtt{m}_{\alpha,2}(i_{n+1}(\omega))-\mathtt{m}_{\alpha,2}(i_{n}(\omega)))(\lambda_{\alpha}(j)-\lambda_{\alpha}(k))|$
$\displaystyle\ +\left|\int_{k}^{j}\partial_{\xi}\left(\xi\mathfrak{m}_{\leq
0,2}(i_{n+1}(\omega))(\omega,\xi)-\xi\mathfrak{m}_{\leq
0,2}(i_{n}(\omega))(\omega,\xi)\right)d\xi\right|+2\sup_{j\in
S^{\perp}\left\\{0\right\\}}|r_{\infty}(i_{n}(\omega))(\omega,j)-r_{\infty}(i_{n+1}(\omega))(\omega,j)|$
$\displaystyle\overset{\eqref{measure_est_11s1},\eqref{measure_est_11s3},\eqref{measure_est_11s4}}{\leq_{\mathtt{p_{e}}}}\varepsilon^{3}\gamma^{-1}\rVert
i_{n+1}(\omega)-i_{n}(\omega)\rVert_{s_{0}+\mu_{\mathtt{p}}}\left(|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|+|j-k|+1\right)$
$\displaystyle\overset{\eqref{pheobe1},\eqref{jkasxcmfoldandnonjexc}}{\leq_{\mathtt{p_{e}}}}\varepsilon^{3}\gamma^{-1}\rVert
i_{n+1}(\omega)-i_{n}(\omega)\rVert_{s_{0}+\mu_{\mathtt{p}}}|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|$
$\displaystyle\overset{\eqref{lowernorm_11232}}{\leq}C_{\mathtt{p_{e}}}\varepsilon^{9-6b}N_{n-1}^{-\sigma_{3}}|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|,$

for some constant $C_{\mathtt{p_{e}}}$, where the the third equality follows
from the fact that $\mathtt{m}_{\alpha,1},\mathfrak{m}_{\leq 0,1}$ and
$\mathfrak{m}_{\mathfrak{b}}$ are independent of embeddings. Comparing this to
the claim (11.75), it suffices to show that

$C_{\mathtt{p_{e}}}\varepsilon^{9-6b}N_{n-1}^{-\sigma_{3}}\leq
2^{-n}\gamma\langle l\rangle^{-\tau},\text{ for $|l|\leq N_{n}$ for all $n\geq
0$.}$

From $\gamma=\varepsilon^{2b}$ (see (6.3)) and $\langle l\rangle=1+|l|$ (see
(2.4)), it suffices to show that

$\displaystyle C_{\mathtt{p_{e}}}\varepsilon^{9-8b}N_{n-1}^{-\sigma_{3}}\leq
2^{-n}(1+N_{n})^{-\tau},\text{ for all $n\geq 0$.}$ (11.77)

We consider two cases: $n=0$ and $n\geq 1$. If $n=0$, then it follows from the
definition of $N_{n}$ in (10.322) that we need to show

$C_{\mathtt{p_{e}}}\varepsilon^{9-8b}\leq\varepsilon^{(2b-1)\rho\tau},\text{
for some $C_{\mathtt{p_{e}}}$, possibly smaller than the one in
\eqref{parameterssd1sd}}.$

Since (10.322) tells us that $\rho^{-1}\geq\frac{2\tau(2b-1)}{9-8b}$, which
imples $2(b-1)\rho\tau\leq\frac{9-8b}{2}$. Therefore, it is enough to show
$C_{\mathtt{p_{e}}}\varepsilon^{9-8b}\leq\varepsilon^{\frac{9-8b}{2}}$, which
is equivalent to $C_{\mathtt{p_{e}}}\leq\varepsilon^{-\frac{9-8b}{2}}$. Since
$0<b-1<\frac{1}{2}$ (see (6.24)), this holds true for sufficiently small
$\varepsilon$ depending on $\mathtt{p_{e}}$.

Now we turn to $n\geq 1$. Again from (10.322) for $N_{n}$, we see that (11.77)
can be implied by showing

$C_{\mathtt{p_{e}}}\varepsilon^{9-8b}N_{0}^{-\sigma_{3}\chi^{n-1}}\leq
2^{-n}N_{0}^{-\tau\chi^{n}},$

which is equivalent to (using $\chi=\frac{3}{2}$ in (10.322))
$C_{\mathtt{p_{e}}}\varepsilon^{9-8b}2^{n}\leq
N_{0}^{\chi^{n-1}(\sigma_{3}-\frac{3}{2}\tau)}$. Note that

$\sigma_{3}\overset{\eqref{parambesd2sdjj22}}{=}\frac{3}{4}A\overset{\eqref{sdsdxc2sdAAA}}{\geq}15\mu_{\mathtt{p},0}\overset{\eqref{parameterssd1sd}}{\geq}15(2\tau+1),$

therefore, $\sigma_{3}-\frac{3}{2}\tau\geq 15$, and we are led to show
$C_{\mathtt{p_{e}}}\varepsilon^{9-8b}2^{n}\leq
N_{0}^{15\left(3/2\right)^{n-1}}$ for all $n\geq 1$. This inequality obviously
true for sufficiently small $\varepsilon$ since the right-hand side increases
double-exponentially. Therefore, (11.77) is proved. ∎

######  Lemma 11.4.

If $|l|<N_{n}$, then $R_{ljk}(i_{n+1})=\emptyset$, for $n\geq 0$.

###### Proof.

By definition in (11.69), we have that
$R_{ljk}(i_{n+1})\subset\mathcal{G}_{n+1}$, while (11.70) tells us that
$\mathcal{G}_{n+1}\cap R_{ljk}(i_{n})=\emptyset$. Therefore, for $|l|<N_{n}$,
Lemma 11.3 implies that $R_{ljk}(i_{n+1})\subset\mathcal{G}_{n+1}\cap
R_{ljk}(i_{n})=\emptyset$, which proves the lemma. ∎

######  Lemma 11.5.

There exists $C_{\mathtt{p_{e}}}>0$ such that if
$\max\left\\{|j|,|k|\right\\}\geq
C_{\mathtt{p_{e}}}(|l|^{\frac{1}{\alpha-1}}+1)$, then
$R_{ljk}(i_{n})=\emptyset$ for $n\geq 0$.

###### Proof.

Let us assume that $R_{ljk}(i_{n})\neq\emptyset$ for some

$\displaystyle\max\left\\{|j|,|k|\right\\}\geq N(|l|^{\frac{1}{\alpha-1}}+1),$
(11.78)

for some $N$. Assuming $N$ is sufficiently large, depending on
$\mathtt{p_{e}}$, we will derive a contradiction, which yields that
$R_{ljk}(i_{n})$ has to be empty.

For $\omega\in R_{ljk}(i_{n})$, (11.69) tells us that

$|\text{i}\omega\cdot
l-(d_{\infty}(i_{n}(\omega))(\omega,j)-d_{\infty}(i_{n}(\omega))(\omega,k))|<2\gamma_{n}\langle
l\rangle^{-\tau}|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|,$

that is,

$\displaystyle|d_{\infty}(i_{n}(\omega))(\omega,j)-d_{\infty}(i_{n}(\omega))(\omega,k)|-2\gamma_{n}\langle
l\rangle^{-\tau}|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|<|\omega\cdot l|.$
(11.79)

From (11.60) and (11.61), we have

$\displaystyle|d_{\infty}(i_{n}(\omega))(\omega,j)-d_{\infty}(i_{n}(\omega))(\omega,k)|$
$\displaystyle\geq\left|-\frac{1}{2}+\varepsilon^{2}\mathtt{m}_{\alpha,1}(\omega)+\mathtt{m}_{\alpha,2}(i_{n}(\omega))\right||\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|-\frac{T_{\alpha}}{4}|j-k|$
$\displaystyle\
-|\mathfrak{m}_{1}(i_{n}(\omega))(\omega,j)-\mathfrak{m}_{1}(i_{n}(\omega))(\omega,k)|-|r_{\infty}(i_{n}(\omega))(\omega,j)-r_{\infty}(i_{n}(\omega))(\omega,k)|$
$\displaystyle\overset{\eqref{measure_est_11s1},\eqref{measure_est_11s2},\eqref{measure_est_11s3},\eqref{measure_est_11s4}}{\geq_{\mathtt{p_{e}}}}\left(\frac{1}{2}-\varepsilon^{2}\right)|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|-c_{\mathtt{p_{e}}}|j-k|$
$\displaystyle\geq_{\mathtt{p_{e}}}|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|\left(\frac{1}{2}-c_{\mathtt{p_{e}}}\frac{|j-k|}{|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|}\right)$
$\displaystyle\overset{\eqref{pheobe1}}{\geq_{\mathtt{p_{e}}}}|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|\left(\frac{1}{2}-\frac{c_{\mathtt{p_{e}}}}{|j|^{\alpha-1}+|k|^{\alpha-1}}\right).$
(11.80)

Therefore, for sufficiently large $N$ in (11.78) depending on
$\mathtt{p_{e}}$, we have

$|d_{\infty}(i_{n}(\omega))(\omega,j)-d_{\infty}(i_{n}(\omega))(\omega,k)|\geq
C_{\mathtt{p_{e}}}|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|.$

Plugging this into (11.79), we have

$|\omega\cdot
l|\geq|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|\left(C_{\mathtt{p_{e}}}-2\gamma_{n}\langle
l\rangle^{-\tau}\right),$

Recalling $\gamma_{n}$ from (11.3) and using (2.99), we can choose
$\varepsilon$ small enough depending on $\mathtt{p_{e}}$ so that

$\displaystyle|\omega\cdot
l|\geq_{\mathtt{p_{e}}}|j-k|\left(|j|^{\alpha-1}+|k|^{\alpha-1}\right)\overset{\eqref{rlkxcxcjqqsd2xx1},\eqref{jkasxcmfoldandnonjexc}}{\geq}C_{\mathtt{p_{e}}}N^{\alpha-1}|l|.$
(11.81)

On the other hand, (5.27) and (5.25) tells us

$\displaystyle|\omega|\leq_{\mathtt{p_{e}}}1,\text{ for all
$\omega\in\Omega_{\varepsilon}$.}$ (11.82)

Noting that $R_{ljk}(i_{n})\subset\Omega_{\varepsilon}$, we have $|\omega\cdot
l|\leq_{\mathtt{p_{e}}}|l|$. Combining this with (11.81), we get

$C_{\mathtt{p_{e}}}N^{\alpha-1}|l|\leq|l|,$

which is a contradiction if $N$ is sufficiently large depending on
$\mathtt{p_{e}}$. ∎

By Lemma 11.4 and Lemma 11.5, we are led to estimate the measure of
$R_{ljk}(i_{n})$, only for $(l,j,k)$ such that $l\geq N_{n}$ and
$|j|,|k|\leq_{\mathtt{p_{e}}}\langle l\rangle^{\frac{1}{\alpha-1}}$, since
otherwise, $R_{ljk}(i_{n})=\emptyset$. For such $(l,j,k)$, we now aim to
estimate the measure of $R_{ljk}(i_{n})$.

######  Lemma 11.6.

Let the tangential sites $S$ in (4.6) be chosen so that it satisfies the
hypotheses $(\mathtt{H}1)$, $(\mathtt{H}2)$ and $(\mathtt{H}3)$ in Subsection
5.1. Then,

$\displaystyle|R_{ljk}(i_{n})|\leq_{\mathtt{p_{e}}}\varepsilon^{2(\nu-1)}\gamma\langle
l\rangle^{-\tau},\text{ for all $n\geq 0$ and $j,k\in
S_{\mathtt{M}}^{\perp}\cup\left\\{0\right\\}$ such that $j\neq k$.}$ (11.83)

We will postpone the proof of Lemma 11.6 to the end of this section.

######  Proposition 11.7.

Let the tangential sites $S$ in (4.6) be chosen so that they satisfy the
hypotheses $(\mathtt{H}1)$, $(\mathtt{H}2)$ and $(\mathtt{H}3)$ in Subsection
5.1. Then, for sufficiently small $\varepsilon$, depending on
$\mathtt{p_{e}}$, it holds that

$\displaystyle|\cap_{n\geq
0}\mathcal{G}_{n}|\geq|\Omega_{\varepsilon}|-C_{\mathtt{p_{e}}}\varepsilon^{2(\nu-1)}\gamma.$
(11.84)

###### Proof.

We have

$\displaystyle|\mathcal{G}_{0}\backslash\mathcal{G}_{1}|$
$\displaystyle\overset{\eqref{rkaksxcxskfk}}{\leq}\sum_{l\in\mathbb{Z}^{\nu},\
j,k\in S^{\perp}_{\mathtt{M}}\cup\left\\{0\right\\},j\neq k}|R_{ljk}(i_{0})|$
$\displaystyle\overset{Lemma~{}\ref{rkawkskfk182s2}}{\leq}\sum_{l\in\mathbb{Z}^{\nu},|j|,|k|\leq
C_{\mathtt{p_{e}}}\left(|l|^{\frac{1}{\alpha-1}}+1\right)}|R_{ljk}(i_{0})|$
$\displaystyle\overset{Lemma~{}\ref{rkhxcxc11}}{\leq_{\mathtt{p_{e}}}}\sum_{l\in\mathbb{Z}^{\nu}}\varepsilon^{2(\nu-1)}\gamma\langle
l\rangle^{-\tau}\langle l\rangle^{\frac{2}{\alpha-1}}$
$\displaystyle\overset{\eqref{frequency_set2}}{\leq_{\mathtt{p_{e}}}}\varepsilon^{2(\nu-1)}\gamma\sum_{l\in\mathbb{Z}^{\nu}}\langle
l\rangle^{-\nu-2}$
$\displaystyle\leq_{\mathtt{p_{e}}}\varepsilon^{2(\nu-1)}\gamma.$ (11.85)

For all $n\geq 0$,

$\displaystyle|\mathcal{G}_{n+1}\backslash\mathcal{G}_{n+2}$
$\displaystyle|\overset{\eqref{rkaksxcxskfk}}{\leq}\sum_{l\in\mathbb{Z}^{\nu},j,k\in
S_{\mathtt{M}}^{\perp}\cup\left\\{0\right\\},j\neq k}|R_{ljk}(i_{n+1})|$
$\displaystyle\overset{Lemma~{}\ref{measurelemma1}}{\leq}\sum_{|l|\geq N_{n},\
j,k\in S_{\mathtt{M}}^{\perp}\cup\left\\{0\right\\},j\neq
k}|R_{ljk}(i_{n+1})|$
$\displaystyle\overset{Lemma~{}\ref{rkawkskfk182s2}}{\leq}\sum_{|l|\geq
N_{n},\ |j|,|k|\leq
C_{\mathtt{p_{e}}}\left(|l|^{\frac{1}{\alpha-1}}+1\right)}|R_{ljk}(i_{n+1})|$
$\displaystyle\overset{Lemma~{}\ref{rkhxcxc11}}{\leq_{\mathtt{p_{e}}}}\sum_{|l|\geq
N_{n}}\varepsilon^{2(\nu-1)}\gamma\langle l\rangle^{-\tau+\frac{2}{\alpha-1}}$
$\displaystyle\overset{\eqref{frequency_set2}}{\leq}_{\mathtt{p_{e}}}\varepsilon^{2(\nu-1)}\gamma\sum_{l\in\mathbb{Z}^{\nu},\
|l|\geq N_{n}}\langle l\rangle^{-\nu-2}$
$\displaystyle\leq_{\mathtt{p_{e}}}\varepsilon^{2(\nu-1)}\gamma N_{n}^{-1}.$

Therefore, recalling $\mathcal{G}_{n}$ from (11.3), we have that

$|\cap_{n\geq
0}\mathcal{G}_{n}|=|\mathcal{G}_{0}|-|\mathcal{G}_{0}\backslash\mathcal{G}_{1}|-\sum_{n\geq
1}|\mathcal{G}_{n}\backslash\mathcal{G}_{n+1}|\geq|\Omega_{0}|-C_{\mathtt{p_{e}}}\varepsilon^{2(\nu-1)}\gamma\left(1+\sum_{n\geq
0}N_{n}^{-1}\right)\geq|\Omega_{0}|-C_{\mathtt{p_{e}}}\varepsilon^{2(\nu-1)}\gamma,$

where $N_{n}$ is given in (10.322) and it is a double-exponentially increasing
sequence depending on $\mathtt{p_{e}}$. Recalling
$\Omega_{\varepsilon},\Omega_{0}$ from (5.25) and (6.3), and using (5.27) it
is classical that

$|\Omega_{\varepsilon}\backslash\Omega_{0}|\leq_{\mathtt{p_{e}}}\varepsilon^{2(\nu-1)}\gamma,$

therefore,

$|\cap_{n\geq
0}\mathcal{G}_{n}|\geq|\Omega_{0}|-C_{\mathtt{p_{e}}}\varepsilon^{2(\nu-1)}\gamma\geq|\Omega_{\varepsilon}|-|\Omega_{\varepsilon}\backslash\Omega_{0}|-C_{\mathtt{p_{e}}}\varepsilon^{2(\nu-1)}\gamma\geq|\Omega_{\varepsilon}|-C_{\mathtt{p_{e}}}\varepsilon^{2(\nu-1)}\gamma,$

which proves (11.84). ∎

#####  11.1.0.1 Proof of Lemma 11.6.

We notice from (10.209) that $\zeta\mapsto\mathfrak{F}_{\alpha-1}(\omega,\xi)$
is linear. Indeed, it follows from (10.209) and (5.1.0.2) that

$\displaystyle\mathfrak{F}_{\alpha-1}$
$\displaystyle(\omega,\xi)=\vec{D}(\xi)\cdot\vec{\zeta}\overset{\eqref{xi_omega_dependent}}{=}\frac{\pi}{6\varepsilon^{2}}\vec{D}(\xi)\cdot\mathbb{A}^{-1}(\omega-\overline{\omega}),\text{
for $\xi\in S^{\perp}$.}$ (11.86)

Then we see from (11.86), (5.29), (11.67) and (11.68) that

$\displaystyle d_{\infty}(i_{n}(\omega))(\omega,j)$
$\displaystyle=\text{i}\left(-\frac{1}{2}\lambda_{\alpha}(j)+j\frac{T_{\alpha}}{4}+\frac{j\pi}{6}\vec{D}(j)\cdot\mathbb{A}^{-1}(\omega-\overline{\omega})\right)$
$\displaystyle\
+\text{i}\left(\mathtt{m}_{\alpha,2}(i_{n}(\omega))\lambda_{\alpha}(j)+j\mathfrak{m}_{\leq
0,2}(i_{n}(\omega))(\omega,j)\right)+r_{\infty}(i_{n}(\omega))(\omega,j).$
(11.87)

Therefore, we can decompose $\omega\cdot
l-\frac{1}{\text{i}}(d_{\infty}(i_{n}(\omega))(\omega,j)-d_{\infty}(i_{n}(\omega))(\omega,k))$
as

$\displaystyle\phi(\omega)$ $\displaystyle:=\omega\cdot
l-\frac{1}{\text{i}}(d_{\infty}(i_{n}(\omega))(\omega,j)-d_{\infty}(i_{n}(\omega))(\omega,k))=a_{jk}+\vec{b}_{ljk}\cdot\omega+q_{jk}(\omega)$
(11.88) $\displaystyle a_{jk}$
$\displaystyle:=-\left(-\frac{1}{2}(\lambda_{\alpha}(j)-\lambda_{\alpha}(k))+\frac{T_{\alpha}}{4}(j-k))\right)+\frac{\pi}{6}\left((j\vec{D}(j)-k\vec{D}(k))\cdot\mathbb{A}^{-1}\overline{\omega}\right)$
(11.89) $\displaystyle\vec{b}_{ljk}$
$\displaystyle:=l-\frac{\pi}{6}\left(j\left(\mathbb{A}^{-1}\right)^{T}\vec{D}(j)-k\left(\mathbb{A}^{-1}\right)^{T}\vec{D}(k)\right),$
(11.90) $\displaystyle q_{jk}(\omega)$
$\displaystyle:=-\mathtt{m}_{\alpha,2}(i_{n}(\omega))(\lambda_{\alpha}(j)-\lambda_{\alpha}(k))-(j\mathfrak{m}_{\leq
0,2}(i_{n}(\omega))(\omega,j)-k\mathfrak{m}_{\leq
0,2}(i_{n}(\omega))(\omega,k))$ $\displaystyle\
-\frac{1}{\text{i}}\left(r_{\infty}(i_{n}(\omega))(\omega,j)-r_{\infty}(i_{n}(\omega))(\omega,k)\right).$
(11.91)

From (11.89) and our hypothesis on the tangential sites in $(\mathtt{H}2)$
(see (5.32)), we have that

  1. $(\mathtt{H}2)$

There exists a constant $C_{\mathtt{H}2}(\mathtt{p_{e}})>0$ such that if
$j,k\in S_{\mathtt{M}}^{\perp}\left\\{0\right\\}$, $j\neq k$ and
$\max\left\\{|j|,|k|\right\\}\geq C_{\mathtt{H}2}$, then

$\displaystyle|j-k||(|j|^{\alpha-1}+|k|^{\alpha-1})$
$\displaystyle\leq_{\mathtt{p_{e}}}|a_{jk}|.$ (11.92)

Lemma 11.6 will be proved by using several auxiliary lemmas that will be
proved below. If $\max\left\\{|j|,|k|\right\\}\geq C_{\mathtt{H}2}$, then
(11.83) follows from Lemma 11.8. Assuming
$\max\left\\{|j|,|k|\right\\}<C_{\mathtt{H}2}$, then Lemma 11.9 tells us that
(11.83) holds for $|l|\geq L$ for some $L>0$. If
$\max\left\\{|j|,|k|\right\\}<C_{\mathtt{H}2}$ and $|l|\leq L$, then the
estimate (11.83) follows from Lemma 11.11.

######  Lemma 11.8.

For $j,k\in S_{\mathtt{M}}^{\perp}\left\\{0\right\\}$, $j\neq k$ and
$\max\left\\{|j|,|k|\right\\}\geq C_{\mathtt{H}2}$, it holds that

$\displaystyle|R_{ljk}(i_{n})|\leq_{\mathtt{p_{e}}}\varepsilon^{2(\nu-1)}\gamma\langle
l\rangle^{-\tau},$ (11.93)

###### Proof.

Let $l,j,k$ be fixed so that $j\neq k$ and $\max\left\\{|j|,|k|\right\\}\geq
C_{\mathtt{H}2}$. For $\omega\in R_{ljk}(i_{n})$, we see from (11.88) and
(11.69) that $|\phi(\omega)|<2\gamma_{n}\langle
l\rangle^{-\tau}|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|$, therefore,

$|a_{j,k}|-|\vec{b}_{ljk}\cdot\omega|-|q_{jk}(\omega)|<2\gamma_{n}\langle
l\rangle^{-\tau}|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|<2\gamma_{n}|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|\overset{\eqref{setoffreqences}}{\leq}4\gamma|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|,$

which implies

$\displaystyle|\vec{b}_{ljk}\cdot\omega|\geq|a_{j,k}|-|q_{jk}(\omega)|-4\gamma|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|.$
(11.94)

For $|q_{jk}(\omega)|$, it follows from its definition in (11.91) and the
estimates in (11.62), (11.64) and (11.65) that

$\displaystyle|q_{jk}|^{\text{Lip}(\gamma,\mathcal{G}_{n})}\leq_{\mathtt{p_{e}}}\varepsilon^{9-6b}|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|,$
(11.95)

therefore , plugging this and (11.92) into (11.94) and using (2.99), we obtain

$|\vec{b}_{ljk}\cdot\omega|\geq(C_{\mathtt{p_{e}}}-(4\gamma+\varepsilon^{9-6b}))|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|\overset{\gamma=\varepsilon^{2b}}{\geq}\frac{C_{\mathtt{p_{e}}}}{2}|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|,$

for sufficiently small $\varepsilon>0$. Meanwhile, we see from (11.82) that
$|\vec{b}_{ljk}\cdot\omega|\leq_{\mathtt{p_{e}}}|\vec{b}_{ljk}|.$ Hence, we
have that

$\displaystyle|\vec{b}_{ljk}|\geq
C_{\mathtt{p_{e}}}|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|.$ (11.96)

Towards the estimate of $|R_{ljk}(i_{n})|$, we define

$\displaystyle b^{\perp}$
$\displaystyle:=\left\\{\vec{w}\in\mathbb{R}^{\nu}:\vec{w}\cdot\vec{b}_{ljk}=0,\text{
and $s\frac{\vec{b}_{ljk}}{|\vec{b}_{ljk}|}+\vec{w}\in R_{ljk}(i_{n})$ for
some $s\in\mathbb{R}$ }\right\\},$ (11.97) $\displaystyle I_{b}(\vec{w})$
$\displaystyle:=\left\\{s\in\mathbb{R}:s\frac{\vec{b}_{ljk}}{|\vec{b}_{ljk}|}+\vec{w}\in
R_{ljk}(i_{n})\right\\},$

so that each $\omega\in R_{ljk}(i_{n})$ can be uniquely decomposed as

$\displaystyle\omega=s\frac{\vec{b}_{ljk}}{|\vec{b}_{ljk}|}+\vec{w}=:\omega(s,\vec{w}),\text{
for some $\vec{w}\in b^{\perp}$ and $s\in I_{b}(\vec{w})$}.$ (11.98)

Denoting $\Psi(s):=\phi(s\frac{\vec{b}_{ljk}}{|\vec{b}_{ljk}|}+\vec{w})$ for
each fixed $\vec{w}\in b^{\perp}$, we have that for each $\vec{w}\in
b^{\perp}$ and $s_{1},s_{2}\in I_{b}(\vec{w})$,

$\displaystyle|\Psi(s_{1})-\Psi(s_{2})|$
$\displaystyle=|\phi(\omega(s_{1},\vec{w}))-\phi(\omega(s_{2},\vec{w}))|$
$\displaystyle\overset{\eqref{ab11xxcx1}}{=}|\vec{b}_{ljk}\cdot(\omega(s_{1},\vec{w})-\omega(s_{2},\vec{w}))+q_{jk}(\omega(s_{1},\vec{w}))-q_{jk}(\omega(s_{2},\vec{w}))|$
$\displaystyle\overset{\eqref{decopomegs}}{=}\left|(s_{1}-s_{2})|\vec{b}_{ljk}|+\gamma\frac{q_{jk}(\omega(s_{1},\vec{w}))-q_{jk}(\omega(s_{2},\vec{w}))}{|\omega(s_{1},\vec{w})-\omega(s_{2},\vec{w})|}\cdot\frac{|\omega(s_{1},\vec{w})-\omega(s_{2},\vec{w})|}{\gamma}\right|$
$\displaystyle\overset{\eqref{qdsxxcxc2ekdtlsgksxp},\eqref{constnas1}}{\geq_{\mathtt{p_{e}}}}|s_{1}-s_{2}||\vec{b}_{ljk}|-\varepsilon^{9-6b}|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|\gamma^{-1}|s_{1}-s_{2}|$
$\displaystyle\overset{\eqref{vecbislargess},\gamma=\varepsilon^{2b}}{\geq_{\mathtt{p_{e}}}}|s_{1}-s_{2}||\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|-\varepsilon^{9-8b}|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)||s_{1}-s_{2}|$
$\displaystyle\geq_{\mathtt{p_{e}}}|s_{1}-s_{2}||\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|,$

for all sufficiently small $\varepsilon$, depending on $\mathtt{p_{e}}$. On
the other hand, we also have (using
$\omega(s_{1},\vec{w}),\omega(s_{2},\vec{w})\in R_{ljk}(i_{n})$, by (11.97))

$\displaystyle|\Psi(s_{1})-\Psi(s_{2})|$
$\displaystyle=|\phi(\omega(s_{1},\vec{w}))-\phi(\omega(s_{2},\vec{w}))|$
$\displaystyle\overset{\eqref{ab11xxcx1},\eqref{define_rlkj}}{<}4\gamma_{n}\langle
l\rangle^{-\tau}|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|$
$\displaystyle\overset{\eqref{setoffreqences}}{<}8\gamma\langle
l\rangle^{-\tau}|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|.$

Therefore, we have that

$|s_{1}-s_{2}|\leq_{\mathtt{p_{e}}}\gamma\langle l\rangle^{-\tau},\text{ for
all $\vec{w}\in b^{\perp}$, and $s_{1},s_{2}\in I_{b}(\vec{w})$,}$

in other words,

$\displaystyle|I_{b}(\vec{w})|\leq_{\mathtt{p_{e}}}\gamma\langle
l\rangle^{-\tau},\text{ for all $\vec{w}\in b^{\perp}$.}$ (11.99)

For the set $b^{\perp}$ in (11.97), it follows from
$R_{ljk}(i_{n})\subset\mathcal{G}_{n}\subset\Omega_{\varepsilon}$ (see
(11.70), (11.3)) and the definition of $b^{\perp}$ in (11.97) that

$b^{\perp}\subset\left\\{\vec{w}\in\mathbb{R}^{\nu}:\vec{w}\cdot\vec{b}_{ljk}=0,\text{
and $s\frac{\vec{b}_{ljk}}{|\vec{b}_{ljk}|}+\vec{w}\in\Omega_{\varepsilon}$
for some $s\in\mathbb{R}$ }\right\\}.$

Since $\Omega_{\varepsilon}$ in (5.25) is a $\nu$-dimensional cube with side-
length $O(\varepsilon^{2})$ (see (5.27)), we have that
$|b^{\perp}|\leq_{\mathtt{p_{e}}}\varepsilon^{2(\nu-1)}$. Lastly, it follows
from (11.97) that

$|R_{ljk}(i_{n})|=\int_{b^{\perp}}\int_{I_{b}(\vec{w})}1dsd\vec{w}\overset{\eqref{setofIestimate}}{\leq_{\mathtt{p_{e}}}}\gamma\langle
l\rangle^{-\tau}|b^{\perp}|\leq_{\mathtt{p_{e}}}\varepsilon^{2(\nu-1)}\gamma\langle
l\rangle^{-\tau},$

which proves (11.93). ∎

######  Lemma 11.9.

There exists $L=L(\mathtt{p_{e}})>0$ such that if $|l|\geq L$ and
$|j|,|k|<C_{\mathtt{H}2}$, then

$\displaystyle|R_{ljk}(i_{n})|\leq_{\mathtt{p_{e}}}\varepsilon^{2(\nu-1)}\gamma\langle
l\rangle^{-\tau}.$ (11.100)

###### Proof.

First we see from (11.86) and (11.66) that $\xi\mapsto(\vec{D}(\xi))_{k}$ for
each $k=1,\ldots,\nu$ is a Fourier multiplier in $\mathcal{S}^{\alpha-1}$,
therefore,

$\displaystyle|j\vec{D}(j)-k\vec{D}(k)|\leq_{\mathtt{p_{e}}}|j-k|(|j|^{\alpha-1}+|k|^{\alpha-1}),\text{
for all $j,k\in S_{\mathtt{M}}^{\perp}\cup\left\\{0\right\\}$}.$ (11.101)

Then, it follows from (5.27) that

$\displaystyle|(|j\vec{D}(j)-k\vec{D}(k))\cdot\mathbb{A}^{-1}\vec{w}|$
$\displaystyle\leq_{\mathtt{p_{e}}}|\vec{w}||j-k|(|j|^{\alpha-1}+|k|^{\alpha-1}),\text{
for all $\vec{w}\in\mathbb{R}^{\nu}$, $j,k\in S^{\perp}\left\\{0\right\\}$.}$
(11.102)

Given $\omega_{1},\omega_{2}\in R_{ljk}(i_{n})$, we have

$\displaystyle\left|\left(d_{\infty}(i_{n}(\omega_{1}))(\omega_{1},j)-d_{\infty}(i_{n}(\omega_{1}))(\omega_{1},j)\right)-\left(d_{\infty}(i_{n}(\omega_{1}))(\omega_{1},j)-d_{\infty}(i_{n}(\omega_{1}))(\omega_{1},j)\right)\right|$
$\displaystyle\overset{\eqref{rlajjxcxcx2yysssd}}{\leq}\left|\frac{\pi}{6}\left(j\vec{D}(j)-k\vec{D}(k)\right)\cdot\mathbb{A}^{-1}(\omega_{1}-\omega_{2})\right|+\left|(\mathtt{m}_{\alpha,2}(i_{n}(\omega_{1}))-\mathtt{m}_{\alpha,2}(i_{n}(\omega_{2})))(\lambda_{\alpha}(j)-\lambda_{\alpha}(k))\right|$
$\displaystyle\ +\left|\int_{k}^{j}\partial_{\xi}\left(\xi\mathfrak{m}_{\leq
0,2}(i_{n}(\omega_{1}))(\omega_{1},\xi)-\xi\mathfrak{m}_{\leq
0,2}(i_{n}(\omega_{2}))(\omega_{2},\xi)\right)d\xi\right|+2|r_{\infty}(i_{n}(\omega_{1}))-r_{\infty}(i_{n}(\omega_{2}))|_{0,0,0}$
$\displaystyle\overset{\eqref{measure_est_11s1},\eqref{measure_est_11s3},\eqref{measure_est_11s4}}{\leq_{\mathtt{p_{e}}}}\left|\frac{\pi}{6}\left(j\vec{D}(j)-k\vec{D}(k)\right)\cdot\mathbb{A}^{-1}(\omega_{1}-\omega_{2})\right|+\varepsilon^{9-6b}\gamma^{-1}|\omega_{1}-\omega_{2}|$
$\displaystyle\overset{\eqref{ajsj1estimax2}}{\leq_{\mathtt{p_{e}}}}|j-k|(|j|^{\alpha-1}+|k|^{\alpha-1})|\omega_{1}-\omega_{2}|+\varepsilon^{9-8b}|\omega_{1}-\omega_{2}|$
$\displaystyle\leq_{\mathtt{p_{e}}}|j-k|(|j|^{\alpha-1}+|k|^{\alpha-1})|\omega_{1}-\omega_{2}|,$
(11.103)

for sufficiently small $\varepsilon$, depending on $\mathtt{p_{e}}$.

For a fixed $0\neq l\in\mathbb{R}^{\nu}$, we define

$\displaystyle l^{\perp}$
$\displaystyle:=\left\\{\vec{w}\in\mathbb{R}^{\nu}:\vec{w}\cdot l=0,\text{ and
}s\widehat{l}+\vec{w}\in R_{ljk}(i_{n}),\text{ for some $s\in\mathbb{R}$,
where $\widehat{l}:=\frac{l}{|l|}$}\right\\},$ $\displaystyle I_{l}(\vec{w})$
$\displaystyle:=\left\\{s\in\mathbb{R}:s\widehat{l}+\vec{w}\in
R_{ljk}(i_{n})\right\\},$

so that each $\omega\in R_{ljk}(i_{n})$ can be uniquely written as

$\displaystyle\omega=\omega(s,\vec{w}):=s\widehat{l}+\vec{w},\text{ for some
$s\in I_{l}(\vec{w})$ and $\vec{w}\in l^{\perp}$.}$

For a fixed $\vec{w}\in l^{\perp}$, let us denote

$\Psi(s):=\phi(s\widehat{l}+\vec{w}),\text{ for $s\in I_{l}(\vec{w})$}.$

Denoting $\omega_{1}=\omega(s_{1},\vec{w}),\omega_{2}=\omega(s_{2},\vec{w})$,
we have that

$\displaystyle|\Psi(s_{1})-\Psi(s_{2})|$
$\displaystyle=|\phi(\omega_{1})-\phi(\omega_{2})|$
$\displaystyle\overset{\eqref{ab11xxcx1}}{\geq}|(\omega_{1}-\omega_{2})\cdot
l|$ $\displaystyle\
-\left|\left(d_{\infty}(i_{n}(\omega_{1}))(\omega_{1},j)-d_{\infty}(i_{n}(\omega_{1}))(\omega_{1},j)\right)-\left(d_{\infty}(i_{n}(\omega_{1}))(\omega_{1},j)-d_{\infty}(i_{n}(\omega_{1}))(\omega_{1},j)\right)\right|$
$\displaystyle\overset{\eqref{rkshxxxcxc1}}{\geq}|l||s_{1}-s_{2}|-c_{\mathtt{p_{e}}}|j-k|(|j|^{\alpha-1}+|k|^{\alpha-1})|s_{1}-s_{2}|$
$\displaystyle=||l|-c_{\mathtt{p_{e}}}|j-k|(|j|^{\alpha-1}+|k|^{\alpha-1})||s_{1}-s_{2}|$
$\displaystyle\geq(L-\mathtt{c}_{d})|s_{1}-s_{2}|,$

where the last inequality follows from $|l|\geq L$ and
$|j|,|k|<C_{\mathtt{H}2}=C_{\mathtt{H}2}(\mathtt{p_{e}})$. Therefore we can
choose $L$ large enough depending on $\mathtt{p_{e}}$ so that

$\displaystyle|\Psi(s_{1})-\Psi(s_{2})|\geq\frac{L}{2}|s_{1}-s_{2}|,\text{ for
all $s_{1},s_{2}\in I_{l}(\vec{w})$, for all $\vec{w}\in l^{\perp}$.}$
(11.104)

On the other hand, it follows from $\omega_{1},\omega_{2}\in R_{ljk}(i_{n})$
and (11.69) that

$|\Psi(s_{1})-\Psi(s_{2})|=|\phi(\omega_{1})-\phi(\omega_{2})|\overset{\gamma_{n}=(1+2^{-n}\gamma)}{<}2\gamma\langle
l\rangle^{-\tau}|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|\leq_{\mathtt{p_{e}}}\gamma\langle
l\rangle^{-\tau}.$

Combining this with (11.104), we have

$|I_{l}(\vec{w})|\leq_{\mathtt{p_{e}}}\gamma\langle l\rangle^{-\tau},\text{
for all $\vec{w}\in l^{\perp}$.}$

As in the proof of Lemma 11.8, we have
$R_{ljk}(i_{n})\subset\Omega_{\varepsilon}$, hence,
$|l^{\perp}|\leq_{\mathtt{p_{e}}}\varepsilon^{2(\nu-1)},$ therefore

$|R_{ljk}(i_{n})|=\int_{l^{\perp}}\int_{I_{l}(\vec{w})}1dsd\vec{w}\leq_{\mathtt{p_{e}}}\gamma\langle
l\rangle^{-\tau}|l^{\perp}|\leq_{\mathtt{p_{e}}}\varepsilon^{2(\nu-1)}\gamma\langle
l\rangle^{-\tau},$

which proves (11.100). ∎

######  Lemma 11.10.

Let $L$ be fixed as in Lemma 11.9 and we recall the definition of $W$ from
(5.31):

$W(j)=-\frac{1}{2}\lambda_{\alpha}(j)+\frac{T_{\alpha}}{4}j.$

If $|l|\leq L$, $|j|,|k|<C_{\mathtt{H}2}$, and

$\delta:=|\overline{\omega}\cdot l-(W(j)-W(k))|\neq 0,$

then $R_{ljk}(i_{n})=\emptyset$ for all sufficiently small $\varepsilon$
depending on $\mathtt{p_{e}}$ and $\delta$.

###### Proof.

For each $\omega\in\Omega_{\varepsilon}$, it follows from (5.25) and (5.27)
that

$\displaystyle|\omega-\overline{\omega}|\leq_{\mathtt{p_{e}}}\varepsilon^{2}.$
(11.105)

Furthermore, using (11.60), (11.61), we see that

$\displaystyle|d_{\infty}(i_{n}(\omega))(\omega,j)-\text{i}W(j)|$
$\displaystyle=|(\varepsilon^{2}\mathtt{m}_{\alpha,1}(\omega)+\mathtt{m}_{\alpha,2}(i_{n}(\omega)))\lambda_{\alpha}(j)+\mathfrak{m}_{1}(i_{n}(\omega))(\omega,j)+r_{\infty}(i_{n}(\omega))(\omega,j)|$
$\displaystyle\overset{\eqref{measure_est_11s1},\eqref{measure_est_11s2},\eqref{measure_est_11s4}}{\leq_{\mathtt{p_{e}}}}\varepsilon^{2}|\lambda_{\alpha}(j)|\leq_{\mathtt{p_{e}}}\varepsilon^{2},$

where the last inequality follows from $|j|<C_{\mathtt{H}2}$. Therefore, if
$\omega\in R_{ljk}(i_{n})$, then

$\displaystyle 2\gamma_{n}\langle
l\rangle^{-\tau}|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|$
$\displaystyle\overset{\eqref{define_rlkj}}{>}|\text{i}\omega\cdot
l-(d_{\infty}(i_{n}(\omega))(\omega,j)-d_{\infty}(i_{n}(\omega))(\omega,k))|$
$\displaystyle\geq|\overline{\omega}\cdot
l-(W(j)-W(k))|-|(\omega-\overline{\omega})\cdot l|$ $\displaystyle\
-|d_{\infty}(i_{n}(\omega))(\omega,j)-\text{i}W(j)|-|d_{\infty}(i_{n}(\omega))(\omega,k)-\text{i}W(k)|$
$\displaystyle\geq\delta-c_{\mathtt{p_{e}}}\varepsilon^{2}(|l|+1)$
$\displaystyle\geq\delta-c_{\mathtt{p_{e}}}\varepsilon^{2},$

where the last inequality follows from that $|l|\leq L$ and $L$ in Lemma 11.9
depends on only $\mathtt{p_{e}}$. Hence, for small enough $\varepsilon$
depending on $\delta$ and $\mathtt{p_{e}}$, we have

$\frac{\delta}{2}<2\gamma_{n}\langle
l\rangle^{-\tau}|\lambda_{\alpha}(j)-\lambda_{\alpha}(k)|\overset{\gamma_{n}=(1+2^{-n})\gamma,\
\eqref{pheobe1}}{\leq_{\mathtt{p_{e}}}}\gamma(|j|^{\alpha}+|k|^{\alpha})\leq_{\mathtt{p_{e}}}\gamma\leq\varepsilon^{2b},$

which is a contradiction for sufficiently small $\varepsilon$, therefore
$R_{ljk}(i_{n})$ is empty. ∎

######  Lemma 11.11.

Let $L$ be fixed as in Lemma 11.9. If $|l|\leq L$, $j,k<C_{\mathtt{H}2}$ then,
$|R_{ljk}(i_{n})|\leq_{\mathtt{p_{e}}}\varepsilon^{2(\nu-1)}\gamma\langle
l\rangle^{-\tau}$ for all sufficiently small $\varepsilon$ depending on
$\mathtt{p_{e}}$.

###### Proof.

Let us suppose $l,j,k$ are chosen according to the assumptions of the lemma.
If $|\overline{\omega}\cdot l-(W(j)-W(k))|\neq 0$, then
$R_{ljk}(i_{n})=\emptyset$ which follows from Lemma 11.10. Therefore, let us
assume that

$\displaystyle|\overline{\omega}\cdot l-(W(j)-W(k))|=0.$ (11.106)

We first write $d_{\infty}(i_{n})$ in (11.1.0.1) using $W(j)$ in (5.31) as

$\displaystyle d_{\infty}(i_{n}(\omega))(\omega,j)$
$\displaystyle=\text{i}\left(W(j)+\frac{j\pi}{6}\vec{D}(j)\cdot\mathbb{A}^{-1}(\omega-\overline{\omega})\right)$
$\displaystyle\
+\text{i}\left(\mathtt{m}_{\alpha,2}(i_{n}(\omega))\lambda_{\alpha}(j)+j\mathfrak{m}_{\leq
0,2}(i_{n}(\omega))(\omega,j)\right)+r_{\infty}(i_{n}(\omega))(\omega,j)$
$\displaystyle=:\text{i}(W(j)+\frac{j\pi}{6}\vec{D}(j)\cdot\mathbb{A}^{-1}(\omega-\overline{\omega}))+\tilde{r}_{\infty}(\omega,j).$

Using (11.62), (11.64), (11.65), we have that for each
$\omega_{1},\omega_{2}\in R_{ljk}(i_{n})$,

$\displaystyle|\tilde{r}_{\infty}(\omega_{1},j)-\tilde{r}_{\infty}(\omega_{2},j)|\leq_{\mathtt{p_{e}}}\varepsilon^{9-6b}\gamma^{-1}|\omega_{1}-\omega_{2}|=\varepsilon^{9-8b}|\omega_{1}-\omega_{2}|,$
(11.107)

for all $j\in S_{\mathtt{M}}^{\perp}$ such that $j<C_{\mathtt{H}2}\max_{j\in
S^{+}}|j|$. Therefore, the expression of $\phi$ in (11.88) gives us that for
$\omega\in R_{ljk}(i_{n})$,

$\displaystyle\phi(\omega)$ $\displaystyle=\omega\cdot
l-\frac{1}{\text{i}}(d_{\infty}(i_{n}(\omega))(\omega,j)-d_{\infty}(i_{n}(\omega))(\omega,k))$
$\displaystyle=\omega\cdot
l-(W(j)-W(k)+\frac{\pi}{6}(j\vec{D}(j)-k\vec{D}(k))\cdot\mathbb{A}^{-1}(\omega-\overline{\omega}))-\frac{1}{\text{i}}(\tilde{r}_{\infty}(\omega,j)-\tilde{r}_{\infty}(\omega,k))$
$\displaystyle=\omega\cdot
l-(W(j)-W(k)+\frac{\pi}{6}\frac{(j\vec{D}(j)-k\vec{D}(k))}{W(j)-W(k)}\cdot\mathbb{A}^{-1}(\omega-\overline{\omega})(W(j)-W(k)))$
$\displaystyle\
-\frac{1}{\text{i}}(\tilde{r}_{\infty}(\omega,j)-\tilde{r}_{\infty}(\omega,k))$
$\displaystyle\overset{\eqref{degenerxxx11}}{=}\left(\omega-\overline{\omega}\right)\cdot
l-\frac{\pi}{6}\frac{(j\vec{D}(j)-k\vec{D}(k))}{W(j)-W(k)}\cdot\mathbb{A}^{-1}(\omega-\overline{\omega})\overline{\omega}\cdot
l-\frac{1}{\text{i}}(\tilde{r}_{\infty}(\omega,j)-\tilde{r}_{\infty}(\omega,k))$
$\displaystyle\overset{\eqref{mathbsk2sdx}}{=}\left((\mathbb{A}^{-1})^{T}\mathbb{A}^{T}(\omega-\overline{\omega})\right)\cdot
l-\left(\left(\mathbb{A}^{-1}\right)^{T}\frac{\pi}{6}\mathbb{B}_{j,k}(\omega-\overline{\omega})\right)\cdot
l-\frac{1}{\text{i}}(\tilde{r}_{\infty}(\omega,j)-\tilde{r}_{\infty}(\omega,k))$
$\displaystyle=\left(\mathbb{A}^{-1}\right)^{T}(\mathbb{A}^{T}-\frac{\pi}{6}\mathbb{B}_{j,k})(\omega-\overline{\omega})\cdot
l-\frac{1}{\text{i}}(\tilde{r}_{\infty}(\omega,j)-\tilde{r}_{\infty}(\omega,k))$
$\displaystyle\overset{\eqref{mathbsk2sdx}}{=}\left(\mathbb{A}^{-1}\right)^{T}\mathbb{C}_{j,k}(\omega-\overline{\omega})\cdot
l-\frac{1}{\text{i}}(\tilde{r}_{\infty}(\omega,j)-\tilde{r}_{\infty}(\omega,k))$
$\displaystyle=\mathbb{C}_{j,k}^{T}\mathbb{A}^{-1}l\cdot(\omega-\overline{\omega})-\frac{1}{\text{i}}(\tilde{r}_{\infty}(\omega,j)-\tilde{r}_{\infty}(\omega,k)).$

Therefore, for $\omega_{1},\omega_{2}\in R_{ljk}(i_{n})$, denoting

$\displaystyle
l_{ac}:=\frac{\mathbb{C}_{j,k}^{T}\mathbb{A}^{-1}l}{|\mathbb{C}_{j,k}^{T}\mathbb{A}^{-1}l|},$
(11.108)

we have