An Abstract Birkhoff Normal Form Theorem and Exponential Type Stability of the 1d NLS

Abstract

We study stability times for a family of parameter dependent nonlinear Schrödinger equations on the circle, close to the origin. Imposing a suitable Diophantine condition (first introduced by Bourgain), we prove a rather flexible Birkhoff Normal Form theorem, which implies, e.g., exponential and sub-exponential time estimates in the Sobolev and Gevrey class respectively.

Appendices

Part 3. Appendices

Appendix A. Constants

In this subsection are listed all the constants appearing along the paper. We first introduce some auxiliary constants. Given $t,{\sigma },\zeta >0,$$p>1/2,$$0<\theta <1,$$s,{q}\ge 0,$ we set²⁶

$$\begin{aligned} {C_{\mathtt {alg}}(p)}:= & {} 2^p\Big (\sum _{i\in \mathbb {Z}} \langle i \rangle ^{-2p}\Big )^{1/2} , \qquad \\ {C_{\mathtt {alg},\mathtt {M}}(p)}:= & {} \sqrt{2}\sqrt{2+\frac{2p+1}{2p-1}} , \\ {C_{\mathtt {Nem}}}(p,s,t):= & {} {C_{\mathtt {alg}}(p)}\big (e^s+\sup _{x\ge 1} x^p e^{- t x+s x^\theta } \big ), \quad \\ {C_{\mathtt {mon}}}(t, {\sigma },p):= & {} 2^{p+1} p^{p} \max \left\{ {\left( 2t\right) }^{p} , {\sigma }^{-p}, 1\right\} , \\ {{\mathcal {C}}_1}:= & {} 28 \,\theta ^{-1}({q}+3) \Big (\frac{2^3\cdot 13 ({q}+3)}{\theta (1-\theta )}\Big )^{\frac{2}{\theta }} \\&{\left( \ln \big (\frac{2^3\cdot 13 ({q}+3)}{\theta (1-\theta )}\big )\right) }^{\frac{2}{\theta }+1} , \quad C_* = 13/(1-\theta ), \\ {{\mathcal {C}}_2}(t,{\sigma },\zeta ):= & {} e^{27(2+{q})}{C_{\mathtt {mon}}}(t,{\sigma },3\zeta ), \\ \tau:= & {} \tau _0(2+{q}),\qquad \tau _0:=15/2, \\ \tau _1:= & {} 2{\left( \tau _0+\frac{3}{2\ln 2}\right) }(2+{q}). \end{aligned}$$

Here are the constants appearing in Theorem 1.1:

$$\begin{aligned} {{\varvec{\delta }}_\mathtt {G}}:= & {} \min \left\{ \frac{\sqrt{R}}{4{C_{\mathtt {alg}}(p)}} ,\ \delta _\mathtt {G}e^{ - \left( \max \left\{ 16(4{{\mathcal {C}}_1})^\theta \eta _\mathtt {G}^{-3}, 2^{\frac{2\theta +4}{4-\theta }} \right\} \right) ^{4/\theta }} \right\} , \quad \\ \mathtt {T}_\mathtt {G}:= & {} \frac{2^4 e {{\varvec{\delta }}_\mathtt {G}}^2}{\gamma },\qquad \text{ where }\\ \delta _\mathtt {G}:= & {} \frac{\sqrt{\gamma R}}{{C_{\mathtt {alg}}(p)}\sqrt{2^{11}e {C_{\mathtt {Nem}}}(p,s-\eta _\mathtt {G},{\mathtt {a}}- a-\eta _\mathtt {G})|f|_{{\mathtt {a}},R}}}, \qquad \\ \eta _\mathtt {G}:= & {} \min \left\{ \frac{{\mathtt {a}}-a}{2},s\right\} . \end{aligned}$$

Here are the constants appearing in Proposition 1.1

$$\begin{aligned} {\tau _\mathtt {S}}:= & {} \tau =\frac{15}{2}(2+q),\qquad \\ {{\varvec{\delta }}_\mathtt {S}}= & {} \min \Big \{\frac{\sqrt{3^\tau \tau \gamma R}}{ 2^{10} e^{2\tau } (\mathtt {k}_\mathtt {S}\tau )^{4\tau } \sqrt{ {C_{\mathtt {Nem}}}(p-{\mathtt {N}}\tau _\mathtt {S},0,{\mathtt {a}}/2)|f|_{{\mathtt {a}},R}} },\frac{\sqrt{R}}{ 2^{\tau +5} }\Big \}\\ {\mathtt {k}}_\mathtt {S}:= & {} \sqrt{\frac{12}{\tau }}\max \left\{ 2 , {\mathtt {a}}^{-1/2}\right\} ,\qquad \\ \mathtt {T}_\mathtt {S}= & {} \frac{ { 2^6 e^{2\tau } 3^{3\tau } (4 \max \left\{ 4 , (1/{\mathtt {a}})\right\} )^{4\tau }}}{ \tau \gamma }. \end{aligned}$$

Here are the constants in Theorem 1.2

$$\begin{aligned} {\tau _\mathtt {M}}:= & {} \tau _1={\left( 15+\frac{3}{\ln 2}\right) }(2+{q}), \qquad {{\varvec{\delta }}_\mathtt {M}}:= \min \Big \{\frac{\sqrt{\tau _1}\delta _\mathtt {M}}{4},\sqrt{\frac{{R}}{ 2^{\tau _1 +8} }} \Big \}, \qquad \\ \mathtt {T}_\mathtt {M}:= & {} \frac{R}{2^{\tau _1+13}\delta _\mathtt {M}^2|f|_{R}}= \frac{2^{3} e 6^{\tau _1} (4^6 e^{27})^{2+{q}}}{ \gamma } \,; \end{aligned}$$

where, recalling 8.2,

$$\begin{aligned} \delta _\mathtt {M}= \frac{\sqrt{ \gamma R }}{\sqrt{2^{17} e 12^{\tau _1} (4^6 e^{27})^{2+{q}}|f|_{R}}}. \end{aligned}$$

Here are the constants appearing in Corollary 1.1:

$$\begin{aligned} \bar{\delta }_\mathtt {S}:= & {} {{\varvec{\delta }}_\mathtt {S}}e^{-\max \{\mathtt {k}_\mathtt {S},\,24 \tau ^2\}}, \qquad {\bar{\delta }}_\mathtt {M}:= \frac{{{\varvec{\delta }}_\mathtt {M}}}{4\tau _1} . \end{aligned}$$

Appendix B. Proofs of the Main Properties of the Norms

Lemma B.1

Let $0<r_1<r.$ Let E be a Banach space endowed with the norm $|\cdot |_E$. Let $X:B_r \rightarrow E$ a vector field satisfying

$$\begin{aligned} \sup _{B_r}|X|_E\le \delta _0. \end{aligned}$$

Then the flow $\Phi (u,t)$ of the vector field²⁷ is well defined for every

$$\begin{aligned} |t|\le T:=\frac{r-r_1}{\delta _0} \end{aligned}$$

and $u\in B_{r_1}$ with estimate

$$\begin{aligned} |\Phi (u,t)-u|_E\le \delta _0 |t|,\qquad \forall \, |t|\le T . \end{aligned}$$

Proof

Fix $u\in B_{r_1}$. Let us first prove that $\Phi (u,t)$ exists $\forall \, |t|\le T.$ Otherwise there exists a time²⁸$0<t_0<T$ such that $|\Phi (u,t)|_E<r$ for every $0\le t<t_0$ but $|\Phi (u,t_0)|_E=r.$ Then, by the fundamental theorem of calculus

$$\begin{aligned} \Phi (u,t_0)-u=\int _0^{t_0}X(\Phi (u,\tau ))d\tau . \end{aligned}$$

(B.1)

Therefore

$$\begin{aligned} r-r_1\le & {} |\Phi (u,t_0)|_E-|u|_E\le |\Phi (u,t_0)-u|_E\le \int _0^{t_0}|X(\Phi (u,\tau ))|_Ed\tau \le \delta _0 t_0 \\< & {} \delta _0 T= r-r_1, \end{aligned}$$

which is a contradiction. Finally, for every $|t|\le T,$

$$\begin{aligned} |\Phi (u,t)-u|_E\le \left| \int _0^{t}|X(\Phi (u,\tau ))|_Ed\tau \right| \le \delta _0 |t|. \end{aligned}$$

$\square$

Proof of Lemma 2.1

For brevity we set, for every $r'>0$

$$\begin{aligned} |\cdot |_{r'}:=|\cdot |_{r',\eta ,{\mathtt {w}}}. \end{aligned}$$

We use Lemma B.1, with $E\rightarrow {\mathtt {h}}_{\mathtt {w}}$, $X\rightarrow X_S$, $\delta _0\rightarrow (r+\rho ) |S|_{r+\rho },$$r\rightarrow r+\rho ,$$r_1\rightarrow r,$$T\rightarrow 8e.$ Then the fact that the time 1-Hamiltonian flow $\Phi ^1_S: B_r({\mathtt {h}}_{\mathtt {w}}) \rightarrow B_{r + \rho }({\mathtt {h}}_{\mathtt {w}})$ is well defined, analytic, symplectic follows, since for any $\eta \ge 0$

$$\begin{aligned} \sup _{u\in B_{r+\rho }({\mathtt {h}}_{\mathtt {w}})} |X_S|_{{\mathtt {h}}_{\mathtt {w}}} \le (r+\rho ) |S|_{r+\rho }<\frac{\rho }{8 e}. \end{aligned}$$

Regarding the estimate (2.3), again by Lemma B.1 (choosing $t=1$), we get

$$\begin{aligned} \sup _{u\in B_{r}({\mathtt {h}}_{\mathtt {w}})} {\left| \Phi ^1_S(u)-u\right| } _{{\mathtt {h}}_{\mathtt {w}}} \le (r+\rho ) |S|_{r+\rho } <\frac{\rho }{8 e} . \end{aligned}$$

Estimates (2.4), (2.5), (2.6) directly follow by (2.7) with $h=0,1,2,$ respectively and $c_k=1/k!$, recalling that by Lie series

$$\begin{aligned} H \circ \Phi ^1_S = e^\mathrm{ad_S} H = \sum _{k=0}^\infty \frac{ \mathrm{ad}_S ^k H}{k!} = \sum _{k=0}^\infty \frac{ H^{(k)}}{k!}, \end{aligned}$$

where $H^{(i)} := \mathrm{ad}_S^i (H)= \mathrm{ad}_S ( H^{(i-1)})$, $H^{(0)}:=H$.

Let us prove (2.7). Fix $k\in {\mathbb {N}},$$k>0$ and set

$$\begin{aligned} r_i := r +\rho (1 - \frac{i}{k}) \, \, , \qquad i = 0,\ldots ,k \, . \end{aligned}$$

Note that, by the immersion properties of the norm (recall Remark 2.1)

$$\begin{aligned} |S|_{r_i}\le |S|_{r+\rho },\qquad \forall \, i = 0,\ldots ,k. \end{aligned}$$

(B.2)

Noting that

$$\begin{aligned} 1+\frac{k r_i}{\rho } \, \le k {\left( 1+\frac{r}{\rho }\right) }, \qquad \forall \, i=0,\ldots ,k, \end{aligned}$$

(B.3)

by using k times (2.1) we have

$$\begin{aligned} | {H^{(k)}}|_r= & {} | \{S, {H^{(k-1)}}\} |_r \le 4 (1+\frac{ k r}{\rho }) |{H^{(k-1)}}|_{r_{k-1}}| S|_{r_{k-1}}\\&{\mathop {\le }\limits ^{(B.2)}}|H|_{r+\rho } |S|_{r+\rho }^k 4^k \prod _{i=1}^k (1+\frac{ k r_i}{\rho }) {\mathop {\le }\limits ^{(B.3)}}|H|_{r+\rho } \left( 4k {\left( 1+\frac{r}{\rho }\right) }|S|_{r+\rho } \right) ^k . \end{aligned}$$

Then, using $k^k\le e^k k!,$ we get

$$\begin{aligned} \left| \sum _{k\ge h} c_k {H^{(k)}}\right| _{r}\le & {} \sum _{k\ge h} |c_k| |{H^{(k)}}|_{r} \le |H|_{r+\rho } \sum _{k\ge h} \left( 4e {\left( 1+\frac{r}{\rho }\right) }|S|_{r+\rho } \right) ^k\\= & {} | H|_{r+\rho } \sum _{k\ge h} (|S|_{r+\rho }/2\delta )^k {\mathop {\le }\limits ^{(2.2)}}2 |H|_{r+\rho } (|S|_{r+\rho }/2\delta )^h. \end{aligned}$$

Finally, if S and H satisfy mass conservation so does each $\mathrm{ad}_S^k H$, $k \ge 1$, hence $H \circ \Phi ^1_S$ too. $\square$

Proof of Lemma 3.1

We first prove (i). It is easily seen that:

$$\begin{aligned} X_{\underline{H}_\eta }^{(j)}(u) = \mathrm{i}\sum _{{\varvec{{\alpha }}},{\varvec{{\beta }}}\in {\mathbb {N}}^\mathbb {Z}} {\left| H_{{\varvec{{\alpha }}},{\varvec{{\beta }}}}\right| }{\varvec{{\beta }}}_j e^{\eta |\pi ({\varvec{{\alpha }}}-{\varvec{{\beta }}})|}u^{\varvec{{\alpha }}}{\bar{u}}^{{\varvec{{\beta }}}-e_j}. \end{aligned}$$

Now

$$\begin{aligned} |X_{\underline{H}_\eta }(u) |_{\mathtt {w}}\le |X_{\underline{H}_\eta }(\underline{u}) |_{\mathtt {w}},\quad \underline{u}={\left( |u_j|\right) }_{j\in \mathbb {Z}} \end{aligned}$$

hence, in evaluating the supremum of $|X_{\underline{H}_\eta }|_{\mathtt {w}}$ over $|u|_{\mathtt {w}}\le r$ we ca restrict to the case in which $u=(u_j)_{j\in \mathbb {Z}}$ has all real positive components. Hence

$$\begin{aligned} {\left| H\right| }_{r,\eta ,{\mathtt {w}}} = r^{-1}\sup _{{\left| u\right| }_{\mathtt {w}}\le r} {\left| {\left( \sum ^*{\left| H_{{\varvec{{\alpha }}},{\varvec{{\beta }}}}\right| }{\varvec{{\beta }}}_j e^{\eta {\left| \pi ({\varvec{{\alpha }}}-{\varvec{{\beta }}})\right| }} {\left| u\right| }^{{\varvec{{\alpha }}}+ {\varvec{{\beta }}}- e_j} \right) }_{j\in \mathbb {Z}}\right| }_{\mathtt {w}}. \end{aligned}$$

Then

$$\begin{aligned} |H|_{r,\eta ,{\mathtt {w}}} = \frac{1}{2 r} \sup _{|u|_{\mathtt {w}}\le r} {\left| {\left( W^{(j)}_{\eta }(u)\right) }_{j\in \mathbb {Z}}\right| }_{{\mathtt {w}}} \end{aligned}$$

(B.4)

where

$$\begin{aligned} W^{(j)}_{\eta }(u)= \sum ^*{\left| H_{{\varvec{{\alpha }}},{\varvec{{\beta }}}}\right| }{\left( {\varvec{{\alpha }}}_j + {\varvec{{\beta }}}_j\right) } e^{\eta {\left| \pi ({\varvec{{\alpha }}}-{\varvec{{\beta }}})\right| }}{u}^{{\varvec{{\alpha }}}+ {\varvec{{\beta }}}- e_j}, \end{aligned}$$

since, by the reality condition 1.23, we have

$$\begin{aligned} \sum ^*{\left| H_{{\varvec{{\alpha }}},{\varvec{{\beta }}}}\right| }{\varvec{{\beta }}}_j e^{\eta {\left| \pi ({\varvec{{\alpha }}}-{\varvec{{\beta }}})\right| }} {u}^{{\varvec{{\alpha }}}+ {\varvec{{\beta }}}- e_j} = \sum ^*{\left| H_{{\varvec{{\alpha }}},{\varvec{{\beta }}}}\right| }{\varvec{{\alpha }}}_j e^{\eta {\left| \pi ({\varvec{{\alpha }}}-{\varvec{{\beta }}})\right| }} {u}^{{\varvec{{\alpha }}}+ {\varvec{{\beta }}}- e_j} = \frac{1}{2} W^{(j)}_{\eta }(u). \end{aligned}$$

By the linear map

$$\begin{aligned} L_{r,{\mathtt {w}}}:\ell ^2\rightarrow {\mathtt {h}}_{\mathtt {w}},\qquad y_j\mapsto \frac{r}{{\mathtt {w}}_j}y_j = u_j, \end{aligned}$$

the ball of radius 1 in $\ell ^2$ is isomorphic to the the ball of radius r in ${\mathtt {h}}_{\mathtt {w}}$, namely $L_{r,{\mathtt {w}}}(B_1(\ell ^2))=B_r({\mathtt {h}}_{\mathtt {w}}).$ We have

$$\begin{aligned} Y^{(j)}_{H}(y;r,\eta ,{\mathtt {w}}) = \frac{1}{2} W^{(j)}_{\eta }(L_{r,{\mathtt {w}}} y). \end{aligned}$$

Then (i) follows.

In order to prove item (ii) we rely on the fact that, since we are using the $\eta$-majorant norm, the supremum over y in the norm is achieved on the real positive cone. Moreover, given $u,v\in \ell ^2$, if

$$\begin{aligned} |u_j|\le |v_j|,\quad \forall j\in \mathbb {Z}\end{aligned}$$

then $|u|_{\ell ^2}\le |v|_{\ell ^2}$. $\square$

Proof of Lemma 5.4

Let us look at the time evolution of $|v(t)|_{\mathtt {w}}^2$. By construction and Cauchy-Schwarz inequality

$$\begin{aligned} 2|v(t)|_{\mathtt {w}}\left| \frac{d}{dt} |v(t)|_{\mathtt {w}}\right|= & {} \left| \frac{d}{dt} |v(t)|_{\mathtt {w}}^2 \right| = 2| {\text {Re}}(v,\dot{v})_{{\mathtt {h}}_{\mathtt {w}}}| = 2| {\text {Re}}(v,X_R)_{{\mathtt {h}}_{\mathtt {w}}}|\\\le & {} 2 |v(t)|_{\mathtt {w}}|X_{\underline{R}}|_{\mathtt {w}}\le 2 r |v(t)|_{\mathtt {w}}|R|_{r,\eta ,{\mathtt {w}}}\ \end{aligned}$$

as long as $|v(t)|_{\mathtt {w}}\le r$; namely

$$\begin{aligned} \left| \frac{d}{dt} |v(t)|_{\mathtt {w}}\right| \le r |R|_{r,\eta ,{\mathtt {w}}} \end{aligned}$$

(B.5)

as long as $|v(t)|_{\mathtt {w}}\le r.$

Assume by contradiction that there exists a time²⁹

$$\begin{aligned} 0<T_0<\frac{1}{8|R|_{r,\eta ,{\mathtt {w}}}} \end{aligned}$$

such that

$$\begin{aligned} \Big | |v(t)|_{\mathtt {w}}-|v_0|_{\mathtt {w}}\Big |< \frac{r}{8}, \quad \forall \, 0\le t<T_0,\qquad \mathrm{but}\ \ \Big | |v(T_0)|_{\mathtt {w}}-|v_0|_{\mathtt {w}}\Big |= \frac{r}{8} . \end{aligned}$$

(B.6)

Then

$$\begin{aligned} |v(t)|_{\mathtt {w}}\le |v_0|_{\mathtt {w}}+ \frac{r}{8} < r\, \qquad \quad \forall \, 0\le t\le T_0 . \end{aligned}$$

By (B.5) we get

$$\begin{aligned} \Big | |v(T_0)|_{\mathtt {w}}-|v_0|_{\mathtt {w}}\Big | \le r |R|_{r,\eta ,{\mathtt {w}}} T_0 <\frac{r}{8}, \end{aligned}$$

which contradicts (B.6), proving (5.26). $\square$

Proof of Lemma 5.5

We first note that (see, e.g. Lemma 17 of [BDG10]) for $p >1/2$ and every sequence $\{x_i\}_{i\in \mathbb {Z}}$, $x_i\ge 0,$

$$\begin{aligned} \left( \sum _{i\in \mathbb {Z}} x_i \right) ^2 \le c\sum _{i\in \mathbb {Z}} \left( \frac{\langle i \rangle ^p\langle j-i \rangle ^p}{\langle j \rangle ^p} x_i\right) ^2, \end{aligned}$$

with $c:=4^p\sum _{i\in \mathbb {Z}} \langle i \rangle ^{-2p}=({C_{\mathtt {alg}}(p)})^2.$ Then

$$\begin{aligned} {\left| f\star g\right| }_{p,s,a}^{2}\le & {} \sum _{j}e^{2s\langle j \rangle ^\theta } e^{2a{\left| j\right| }}\langle j \rangle ^{2p}\Big (\sum _{i}|f_{i}||g_{j-i}|\Big )^2\\\le & {} c \sum _{j}e^{2s\langle j \rangle ^\theta } e^{2a{\left| j\right| }}\sum _{i} \langle i \rangle ^{2p}\langle j-i \rangle ^{2p}|f_{i}|^2 |g_{j-i}|^2\\= & {} c \sum _{i} e^{2s\langle i \rangle ^\theta } e^{2a{\left| i\right| }} \langle i \rangle ^{2p} |f_{i}|^2 \sum _{j} \langle j-i \rangle ^{2p} e^{2s\langle j-i \rangle ^\theta } e^{2a{\left| j-i\right| }} |g_{j-i}|^2\\= & {} c {\left| f\right| }_{p,s,a}^{2} {\left| g\right| }_{p,s,a}^{2}. \end{aligned}$$

Regarding the second estimate, we set

$$\begin{aligned} \phi (i,j):=\frac{\lfloor j \rfloor }{\lfloor i \rfloor \lfloor j-i \rfloor },\qquad \qquad \forall \, i,j\in \mathbb Z . \end{aligned}$$

Note that

$$\begin{aligned} \phi (i,j)=\phi (j,i)=\phi (-i,-j). \end{aligned}$$

(B.7)

We claim that

$$\begin{aligned} \phi (i,j)\le 1. \end{aligned}$$

(B.8)

Indeed by (B.7) we can consider only the case $j\ge 0.$ Since $\phi (-|i|,j)\le \phi (|i|,j)$ we can consider only the case $i\ge 0$. Again by (B.7) we can assume $j\ge i.$ In particular we can take $j>i>0,$ (B.8) being trivial in the cases $j=i,$$i=0$. We have

$$\begin{aligned} \phi (i+1,i)=\frac{i+1}{2\lfloor i \rfloor }\le \frac{3}{4}, \qquad \phi (j,1)=\frac{j}{2(j-1)}\le 1. \end{aligned}$$

Then it remains also to discuss the case $j-2\ge i\ge 2;$ we have

$$\begin{aligned} \phi (i,j)=\frac{j}{i(j-i)}=\frac{1}{i}+\frac{1}{j-i}\le 1, \end{aligned}$$

proving (B.8).

For $q\ge 0$ set

$$\begin{aligned} c_q:=\sup _{j\in \mathtt {Z}}\sum _{i\in \mathtt {Z}}(\phi (i,j))^q = \sup _{j\ge 0}\sum _{i\in \mathtt {Z}}(\phi (i,j))^q. \end{aligned}$$

(B.9)

We claim that

$$\begin{aligned} c_q\le 4+2\frac{q+1}{q-1}<\infty ,\qquad \forall \, q>1. \end{aligned}$$

(B.10)

Indeed, since $\lfloor j \rfloor /\lfloor j+1 \rfloor \le 1$ and $\lfloor j \rfloor /\lfloor j-1 \rfloor \le 3/2$ for $j\ge 0$, we have³⁰

$$\begin{aligned} c_q= & {} \sup _{j\ge 0} \left( \frac{\lfloor j \rfloor ^q}{2^{q-1}\lfloor j+1 \rfloor ^q} +\frac{1}{2^{q-1}} +\frac{\lfloor j \rfloor ^q}{2^{q-1}\lfloor j-1 \rfloor ^q} + \sum _{i\le -2,\, 2\le i\le j-2,\, i\ge j+2} (\phi (i,j))^q \right) \\\le & {} 2^{3-q} + \sup _{j\ge 0} \left( \sum _{i\ge 2} \frac{\lfloor j \rfloor ^q}{i^q(j+i)^q} +\sum _{2\le i\le j-2}\left( \frac{1}{i}+\frac{1}{(j-i)}\right) ^q +\sum _{i\ge j+2} \frac{\lfloor j \rfloor ^q}{i^q(i-j)^q} \right) \\\le & {} 2^{3-q} +\sum _{i\ge 2} \frac{1}{i^q} +2^{q-1}\sum _{2\le i\le j-2}\left( \frac{1}{i^q}+\frac{1}{(j-i)^q}\right) +\sum _{i\ge j+2} \frac{1}{(i-j)^q}\\\le & {} 4+2\frac{q+1}{q-1}, \end{aligned}$$

using that $(x+y)^q\le 2^{q-1}(x^q+y^q)$ for $x,y\ge 0$ and that³¹

$$\begin{aligned} \sum _{i\ge 2} i^{-q}\le \frac{q+1}{2^q(q-1)}. \end{aligned}$$

Note that for every $q,q_0\ge 0$ we have

$$\begin{aligned} c_{q_0+q}\le c_{q_0} \end{aligned}$$

(B.11)

since

$$\begin{aligned} c_{q_0+q} := \sup _{j\in \mathtt {Z}}\sum _{i\in \mathtt {Z}}(\phi (i,j))^{q_0} (\phi (i,j))^q {\mathop {\le }\limits ^{(B.8)}}\sup _{j\in \mathtt {Z}}\sum _{i\in \mathtt {Z}}(\phi (i,j))^{q_0} =c_{q_0}. \end{aligned}$$

We now note that for $p >1/2$, $j\in \mathbb Z$ and every sequence $\{x_i\}_{i\in \mathbb {Z}}$, $x_i\ge 0,$ we have by Cauchy-Schwarz inequality

$$\begin{aligned} \left( \sum _{i\in \mathbb {Z}} x_i \right) ^2 = \left( \sum _{i\in \mathbb {Z}} (\phi (i,j))^p (\phi (i,j))^{-p} x_i \right) ^2 \le c_{2p}\sum _{i\in \mathbb {Z}} \left( (\phi (i,j))^{-p} x_i\right) ^2, \end{aligned}$$

with $c_{2p}$ defined in (B.9). Using the above inequality we get

$$\begin{aligned} \Vert f\star g\Vert _p^{2}\le & {} \sum _{j}\lfloor j \rfloor ^{2p}\Big (\sum _{i}|f_{i}||g_{j-i}|\Big )^2\\\le & {} c_{2p} \sum _{j}\sum _{i} \lfloor i \rfloor ^{2p}\lfloor j-i \rfloor ^{2p}|f_{i}|^2 |g_{j-i}|^2\\= & {} c_{2p} \sum _{i} \lfloor i \rfloor ^{2p} |f_{i}|^2 \sum _{j} \lfloor j-i \rfloor ^{2p} |g_{j-i}|^2\\= & {} c_{2p} \Vert f\Vert _p^2 \Vert g\Vert _p^2. \end{aligned}$$

The proof ends recalling (B.10). $\square$

Lemma B.2

(Nemitskii operators). Let $p> 1/2.$ (i) Fix $s\ge 0, a_0\ge 0$. Consider a sequence $F^{(d)}={\left( F^{(d)}_j\right) }_{j\in \mathbb {Z}}\in {\mathtt {h}}_{p,s,a_0}$, $d\ge 1,$ such that

$$\begin{aligned} \sum _{d=1}^\infty d |F^{(d)}|_{p,s,a_0} R^{d} < \infty \end{aligned}$$

(B.12)

for some $R>0$.

For $u={\left( u_j\right) }_{j\in \mathbb {Z}}$ let ${\bar{u}}= {\left( \overline{u_{-j}}\right) }_{j\in \mathbb {Z}}$ and consider the Hamiltonian

$$\begin{aligned} H(u)= \sum _{d=1}^\infty {\left( F^{(d)}\star \underbrace{ u \star \cdots \star u}_{d \,\text{ times }} \star \underbrace{{\bar{u}} \star \cdots \star {\bar{u}}}_{d \,\text{ times }}\right) }_0. \end{aligned}$$

For all $(\eta ,a,r)$ such that $\eta +a \le a_0$ and $({C_{\mathtt {alg}}(p)}r)^2 \le R$, we have that $H\in {{\mathcal {H}}}_{r,\eta }({\mathtt {h}}_{p,s,a})$ and

$$\begin{aligned} |H|_{r,\eta ,{\mathtt {w}}(p,s,a)} \le r^{-1}\sum _{d=1}^\infty d |F^{(d)}|_{p,s,a_0} ({C_{\mathtt {alg}}(p)}r)^{2d-1} <\infty . \end{aligned}$$

(ii) Analogously if $F^{(d)}$ are constants satisfying

$$\begin{aligned} \sum _{d=1}^\infty d |F^{(d)}| R^{d} < \infty \end{aligned}$$

(B.13)

and $({C_{\mathtt {alg},\mathtt {M}}(p)}r)^2\le R,$ then $H\in {{\mathcal {H}}}^{r,p}$ with

$$\begin{aligned} \Vert H\Vert _{r,p} \le 2^p r^{-1}\sum _{d=1}^\infty d |F^{(d)}| ({C_{\mathtt {alg},\mathtt {M}}(p)}r)^{2d-1}<\infty . \end{aligned}$$

(B.14)

Proof

(i) By definition the $\eta$-majorant Hamiltonian is

$$\begin{aligned} \underline{H}_\eta = \sum _d \sum _{\begin{array}{c} j_0,j_1\dots , j_{2d}\\ j_0+\sum _{i=1}^{2d} (-1)^{i} j_i = 0 \end{array}} e^{\eta |\pi _{j_1,\dots ,j_{2d}}|}|F^{(d)}_{j_0}| \overline{u_{j_1}} u_{j_2}\overline{u_{j_3}}\dots u_{j_{2d}} \end{aligned}$$

where

$$\begin{aligned} \pi _{j_1,\dots ,j_{2d}} = \sum _{i=1}^{2d} (-1)^{i} j_i= -j_0, \end{aligned}$$

hence

$$\begin{aligned} \underline{H}_\eta = \sum _d {\left( \underline{F}_\eta ^{(d)}\star \underbrace{ u \star \cdots \star u}_{d \,\text{ times }} \star \underbrace{{\bar{u}} \star \cdots \star {\bar{u}}}_{d \,\text{ times }}\right) }_0,\quad \underline{F}^{(d)}_\eta := {\left( e^{\eta |j|}|F^{(d)}_j|\right) }_{j\in \mathbb {Z}}\, \end{aligned}$$

consequently

$$\begin{aligned} X^{(j)}_{\underline{H}_\eta }= \sum _d d {\left( \underline{F}_\eta ^{(d)}\star \underbrace{ u \star \cdots \star u}_{d \,\text{ times }} \star \underbrace{{\bar{u}} \star \cdots \star {\bar{u}}}_{d-1 \,\text{ times }}\right) }_j\, . \end{aligned}$$

Moreover

$$\begin{aligned} |X_{\underline{H}_\eta }|_{p,s,a} \le \sum _d d ({C_{\mathtt {alg}}(p)})^{2d-1}| \underline{F}_\eta ^{(d)}|_{p,s,a} (|u|_{p,s,a} )^{2d-1}. \end{aligned}$$

Since

$$\begin{aligned} |\underline{F}_\eta ^{(d)}|_{p,s,a} = |F^{(d)}|_{s,a+\eta ,p} \le |F^{(d)}|_{ p,s,a_0} \end{aligned}$$

we get

$$\begin{aligned} |X_{\underline{H}_\eta }|_{p,s,a} \le \sum _d d ({C_{\mathtt {alg}}(p)})^{2d-1} |F^{(d)}|_{ p,s,a_0} (|u|_{p,s,a} )^{2d-1}. \end{aligned}$$

Therefore

$$\begin{aligned} |H|_{r,p,\eta }^{(p,s,a,0)} = r^{-1} {\left( \sup _{{\left| u\right| }_{p,s,a}< r} {\left| {X}_{{{\underline{H}}}_\eta }\right| }_{p,s,a} \right) } \le r^{-1}\sum _d d |F^{(d)}|_{p,s,a_0} ({C_{\mathtt {alg}}(p)}r)^{2d-1}<\infty . \end{aligned}$$

(ii) The proof is analogous to point (i). $\square$

Proof of Proposition 6.2

We start by Taylor expanding H in homogeneous components. The majorant analiticity implies that for a homogeneous component of degree d one has

$$\begin{aligned} |H^{(d)}|_{r,\eta ,{\mathtt {w}}(p,s,a)}^{}\le |H|_{r,\eta ,,{\mathtt {w}}(p,s,a)}^{}. \end{aligned}$$

Now let us consider the polinomial map (homogeneous of degree $d-1$) $X_{H^{(d)}}: {\mathtt {h}}_{p,s,a} \rightarrow {\mathtt {h}}_{p,s,a}$; as is habitual we identify the polynomial map with the corresponding symmetric multilinear operator $M^{(d-1)}: {\mathtt {h}}_{p,s,a}^{d-1} \rightarrow {\mathtt {h}}_{p,s,a}$. Since we are in a Hilbert space, one has that

$$\begin{aligned} |\underline{M}|^\mathrm{op}_{p,s,a}&:= \sup _{\begin{array}{c} u_1,\dots u_{d-1}\in {\mathtt {h}}_{p,s,a}\\ |u_i|_{p,s,a}\le 1 \end{array}} | \underline{M}^{(d-1)}(u_1,\dots ,u_{d-1})|_{p,s,a}= \sup _{|u|_{p,s,a}\le 1} | \underline{M}^{(d-1)}(u,\dots ,u)|_{p,s,a} \\&= \sup _{|u|_{p,s,a}\le 1} |X_{\underline{H}^{(d)}}|_{p,s,a} \le r^{-d+2} |H|_{r,\eta ,{\mathtt {w}}(p,s,a)}^{}\end{aligned}$$

for all $\eta \ge 0$. Now let us compute the tame norm on a homogeneous component, i.e.

$$\begin{aligned} \sup _{|u|_{p_0,s,a}\le r-\rho } \frac{| \underline{M}^{(d-1)}(u^{d-1})|_{p,s,a}}{|u|_{p,s,a}} = \sup _{|u|_{ p_0,s,a}\le r-\rho } \frac{| \underline{N_p}^{(d-1)}(u^{d-1})|_{ p_0,s,a}}{|u|_{p,s,a}} \end{aligned}$$

where

$$\begin{aligned} \underline{N_p}^{(d-1,j)}(u^{d-1})={ \langle j \rangle ^{p-p_0} \sum _{j_1,\dots ,j_{d-1}} |M_{j_1,\dots j_{d-1}}^{(d-1,j)}| u_{j_1}\dots u_{j_{d-1}}} \end{aligned}$$

now setting $\pi = \sum _i j_i- j$ we have

$$\begin{aligned}&{\underline{N_p}^{(d-1)}(u_1,\dots ,u_{d-1})}\\&\quad \le (d-1) \langle j \rangle ^{p-p_0} \sum _{\begin{array}{c} j_1,\dots ,j_{d-1}: \\ |j_1|\ge |j_i| \end{array}} |M_{j_1,\dots j_{d-1} }^{(d-1,j)}| u_{j_1}\dots u_{j_{d-1}}\\&\quad \le (d-1) \sum _{\begin{array}{c} j_1,\dots ,j_{d-1}: \\ |j_1|\ge |j_i| \end{array}}{\left( \sum _i\langle j_i \rangle +|\pi | \right) }^{p-p_0} |M_{j_1,\dots j_{d-1}}^{(d-1,j)}| u_{j_1}\dots u_{j_{d-1}}\\&\quad \le (d-1)2^{p-p_0} C(\eta ,p)\sum _{\begin{array}{c} j_1,\dots ,j_{d-1} \end{array}} e^{\eta |\pi |} |M_{j_1,\dots j_{d-1}}^{(d-1,j)}| u_{j_1}\dots u_{j_{d-1}} \\&\qquad + (d-1)2^{p-p_0} (d-1)^{p-p_0} \sum _{{j_1,\dots ,j_{d-1}}} |M_{j_1,\dots j_{d-1}}^{(d-1,j)}| \langle j_1 \rangle ^{p-p_0}u_{j_1}\dots u_{j_{d-1}} \end{aligned}$$

which means that for any $|u|_{ p_0,s,a}\le r-\rho$ one has

$$\begin{aligned}&| \underline{N_p}^{(d-1)}(u^{d-1})|_{ p_0,s,a}\\&\quad \le (d-1)2^{p-p_0} C(\eta ,p) |H^{(d)}|_{r-\rho ,\eta ,{\mathtt {w}}(p_0,s,a)}|u|_{p_0,s,a}\\&\qquad + 2^{p-p_0} (d-1)^{p-p_0+1} |\underline{M}|^\mathrm{op}_{ p_0,s,a} (r-\rho )^{d-2} |u|_{p,s,a}\\&\quad \le (d-1)2^{p-p_0}( C(\eta ,p) + (d-1)^{p-p_0})(1-\frac{\rho }{r})^{d-2} |H|_{r,\eta ,{\mathtt {w}}(p_0,s,a)} |u|_{p,s,a}. \end{aligned}$$

We conclude that

$$\begin{aligned} \sup _{|u|_{ p_0,s,a}\le r }\frac{|X_{\underline{H}}|_{p,s,a}}{|u|_{p,s,a}} \le 2^{p-p_0}|H|_{r,\eta ,{\mathtt {w}}(p_0,s,a)} \sum _{d\ge 2} (d-1){\left( C(\eta ,p) + (d-1)^{p-p_0}\right) }(1-\frac{\rho }{r})^{d-2}\, \end{aligned}$$

and the thesis follows since the right hand side is convergent. $\square$

Appendix C. Small Divisor Estimates

Let us start with two techincal lemmata.

Lemma C.1

For $p,\beta >0$ and $x_0\ge 0$ we have that

$$\begin{aligned} \max _{x\ge x_0} x^p e^{-\beta x}= {\left\{ \begin{array}{ll} &{}(p/\beta )^p e^{-p} \quad \text{ if } \, \ \ x_0\le p/\beta ,\\ &{}x_0^p e^{-\beta x_0} \quad \text{ if } \, \ \ x_0> p/\beta . \end{array}\right. } \end{aligned}$$

Lemma C.2

Let $0<a<1$ and $x_1\ge x_2\ge \cdots \ge x_N\ge 2.$ Then

$$\begin{aligned} \frac{\sum _{1\le \ell \le N} x_\ell }{\prod _{1\le \ell \le N} x_\ell ^a} \le x_1^{1-a}+\frac{2}{a x_1^a}. \end{aligned}$$

Proof

By induction over N. It is obviously true for $N=1.$ Assume that it hols for N and prove it for $N+1.$$\square$

Proof of Lemma 6.1

The fact that this (6.5) holds true when $\pi =0$ is proven in [Bou96b] and [CLSY]. The bound (6.5) is equivalent to proving

$$\begin{aligned} \sum _{l\ge 1} \widehat{n}_l^\theta - 2 \widehat{n}^\theta _1+ \theta {\left| \pi \right| } - {\left( 2 - 2^\theta \right) }\sum _{l\ge 3} \widehat{n}_l^\theta \ge 0. \end{aligned}$$

(C.1)

i.e.

$$\begin{aligned} \sum _{l\ge 2} \widehat{n}_l^\theta - \widehat{n}^\theta _1+ \theta {\left| \pi \right| } - {\left( 2 - 2^\theta \right) }\sum _{l\ge 3} \widehat{n}_l^\theta \ge 0. \end{aligned}$$

(C.2)

Inequality (C.2) then follows from

$$\begin{aligned} f{\left( {\left| \pi \right| }\right) }:= \sum _{l\ge 2} \widehat{n}_l^\theta - {\left( {\left| \pi \right| } + \sum _{l\ge 2}\widehat{n}_l\right) }^\theta + \theta {\left| \pi \right| } - {\left( 2 - 2^\theta \right) }\sum _{l\ge 3} \widehat{n}_l^\theta \ge 0, \end{aligned}$$

(C.3)

which we are now going to prove. We shall show that the function f(x) is increasing in $x\ge 0$; then the result follows by showing $f(0) \ge 0$, which is what was proven by Yuan and Bourgain.

We now verify that $f'(x)\ge 0$. By direct computation we see that

$$\begin{aligned} f'(x) = - \theta {\left( x + \sum _{l\ge 2}\widehat{n}_l\right) }^{\theta - 1} +\theta , \end{aligned}$$

so it suffices to prove that

$$\begin{aligned} 1 \le {\left( x + \sum _{l\ge 2}\widehat{n}_l\right) }^{1 - \theta }, \end{aligned}$$

(C.4)

which is indeed true, since $\sum _{i\ge 2}\widehat{n}_i\ge \widehat{n}_2\ge 1$ holds, by mass conservation. $\square$

Proof of Lemma 7.1

In this subsection we will take

$$\begin{aligned} {\varvec{{\alpha }}},{\varvec{{\beta }}}\in {\mathbb {N}}^\mathbb {Z}\quad \mathrm{with} \quad 1\le |{\varvec{{\alpha }}}|=|{\varvec{{\beta }}}|<\infty . \end{aligned}$$

(C.5)

Given $u\in \mathbb {Z}^\mathbb {Z}$, with $|u|<\infty ,$ consider the set

$$\begin{aligned} {\left\{ j\ne 0 ,\quad \text{ repeated }\quad {\left| u_j\right| } \,\text{ times }\right\} }, \end{aligned}$$

where $D<\infty$ is its cardinality. Define the vector $m=m(u)$ as the reordering of the elements of the set above such that $|m_1|\ge |m_2|\ge \dots \ge |m_D|\ge 1.$ Given ${\varvec{{\alpha }}}\ne {\varvec{{\beta }}}\in {\mathbb {N}}^\mathbb {Z},$ with $|{\varvec{{\alpha }}}|=|{\varvec{{\beta }}}|<\infty$ we consider $m=m({\varvec{{\alpha }}}-{\varvec{{\beta }}})$ and $\widehat{n}=\widehat{n}({\varvec{{\alpha }}}+{\varvec{{\beta }}}).$ If we denote by D the cardinality of m and N the one of $\widehat{n}$ we have

$$\begin{aligned} D+{\varvec{{\alpha }}}_0+{\varvec{{\beta }}}_0\le N \end{aligned}$$

(C.6)

and

$$\begin{aligned} (|m_1|,\dots ,|m_D|,\underbrace{1,\,\dots \,,1}_{N-D\,\mathrm {times}} )\, \le \, {\left( \widehat{n}_1,\dots \widehat{n}_N\right) }. \end{aligned}$$

(C.7)

Set

$$\begin{aligned} {\sigma }_l= \mathrm{sign}({\varvec{{\alpha }}}_{m_l}-{\varvec{{\beta }}}_{m_l}). \end{aligned}$$

For every function g defined on $\mathbb {Z}$ we have that

$$\begin{aligned} \sum _{i\in \mathbb {Z}} g(i) |{\varvec{{\alpha }}}_i-{\varvec{{\beta }}}_i|= & {} g(0)|{\varvec{{\alpha }}}_0-{\varvec{{\beta }}}_0|+ \sum _{l\ge 1} g(m_l), \nonumber \\ \sum _{i\in \mathbb {Z}} g(i) ({\varvec{{\alpha }}}_i-{\varvec{{\beta }}}_i)= & {} g(0)({\varvec{{\alpha }}}_0-{\varvec{{\beta }}}_0)+ \sum _{l\ge 1} {\sigma }_l g(m_l). \end{aligned}$$

(C.8)

Lemma C.3

Assume that g defined on $\mathbb {Z}$ is non negative, even and not decreasing on ${\mathbb {N}}.$ Then, if ${\varvec{{\alpha }}}\ne {\varvec{{\beta }}}$,

$$\begin{aligned} \sum _{i\in \mathbb {Z}} g(i) |{\varvec{{\alpha }}}_i-{\varvec{{\beta }}}_i| \le 2g(m_1)+ \sum _{l\ge 3} g(\widehat{n}_l). \end{aligned}$$

(C.9)

Proof

By (C.8)

$$\begin{aligned} \sum _{i\in \mathbb {Z}} g(i) |{\varvec{{\alpha }}}_i-{\varvec{{\beta }}}_i|= & {} g(0)|{\varvec{{\alpha }}}_0-{\varvec{{\beta }}}_0|+ \sum _{l\ge 1} g(m_l)\\\le & {} g(1)({\varvec{{\alpha }}}_0+{\varvec{{\beta }}}_0)+ 2g(m_1)+ \sum _{l\ge 3} g(m_l) \end{aligned}$$

and (C.9) follows by (C.6) and (C.7). $\square$

We denote as before the momentum by $\pi$ so by (C.8)

$$\begin{aligned} \pi = \sum _{i\in \mathbb {Z}}{\left( {\varvec{{\alpha }}}_i-{\varvec{{\beta }}}_i\right) }i = \sum _l {\sigma }_l m_l \end{aligned}$$

(C.10)

and

$$\begin{aligned} {\sum _i{{\left( {\varvec{{\alpha }}}_i-{\varvec{{\beta }}}_i\right) }i^2}}= \sum _l {\sigma }_l m^2_l. \end{aligned}$$

(C.11)

Analogously

$$\begin{aligned} {\sum _i{{\left| {\varvec{{\alpha }}}_i-{\varvec{{\beta }}}_i\right| }}} = D+|{\varvec{{\alpha }}}_0-{\varvec{{\beta }}}_0| {\mathop {\le }\limits ^{(C.6)}}N. \end{aligned}$$

(C.12)

Finally note that

$$\begin{aligned} \sigma _l\sigma _{l'} =-1\qquad \Longrightarrow \qquad m_l \ne m_{l'}. \end{aligned}$$

(C.13)

Note that

$$\begin{aligned} {\varvec{{\alpha }}}\ne {\varvec{{\beta }}}\quad \Longrightarrow \quad N\ge 3 \ \ \mathrm{or}\ \ \pi \ne 0, \end{aligned}$$

(C.14)

indeed, by mass conservation, $|{\varvec{{\alpha }}}|=|{\varvec{{\beta }}}|=1$ therefore if $N=2$ we get ${\varvec{{\alpha }}}-{\varvec{{\beta }}}= e_{j_1}-e_{j_2}$ so if $\pi =0$ we have ${\varvec{{\alpha }}}={\varvec{{\beta }}}$. Note also that

$$\begin{aligned} {\varvec{{\alpha }}}\ne {\varvec{{\beta }}}\qquad \Longrightarrow \qquad D \ge 1, \end{aligned}$$

(C.15)

indeed, if $D=0$ then ${\varvec{{\alpha }}}_l-{\varvec{{\beta }}}_l=0$ for every $|l|\ge 1$ and, by mass conservation ${\varvec{{\alpha }}}_0={\varvec{{\beta }}}_0$, contradicting ${\varvec{{\alpha }}}\ne {\varvec{{\beta }}}$ .

Lemma C.4

Given ${\varvec{{\alpha }}}\ne {\varvec{{\beta }}}\in {\mathbb {N}}^\mathbb {Z},$ with $1\le |{\varvec{{\alpha }}}|=|{\varvec{{\beta }}}|<\infty$ and satisfying (7.1), we have³²

$$\begin{aligned} {\left| m_1\right| }\le 20 |\pi |+ 31\sum _{l\ge 3}\widehat{n}_l^2. \end{aligned}$$

(C.16)

Proof

In the case $D=1$ by (C.10) $|\pi |=|m_1|$ and (C.16) follows. Let us now consider the case $D=2$, i.e.

$$\begin{aligned} {\varvec{{\alpha }}}-{\varvec{{\beta }}}={\sigma }_1 e_{m_1}+{\sigma }_2 e_{m_2} +({\varvec{{\alpha }}}_0-{\varvec{{\beta }}}_0)e_0. \end{aligned}$$

Let us start with the case ${\sigma }_1{\sigma }_2=1.$ By mass conservation $|{\sigma }_1+{\sigma }_2|=|{\varvec{{\beta }}}_0-{\varvec{{\alpha }}}_0|=2.$ By (C.12) $N\ge 4.$ Then conditions (7.1) and (C.12) imply that

$$\begin{aligned} m_1^2 + m_2^2 \le 20+10|{\varvec{{\alpha }}}_0-{\varvec{{\beta }}}_0|=40. \end{aligned}$$

Then

$$\begin{aligned} |m_1|\le \sqrt{40} \le \frac{\sqrt{40}}{2} \sum _{\ell =3}^N \widehat{n}_\ell ^2 \end{aligned}$$

since $N\ge 4$ and $\widehat{n}_\ell \ge 1.$ When ${\sigma }_1{\sigma }_2=-1$ we have $m_1\ne m_2$, $|\pi |=|m_1-m_2|\ge 1$ and by mass conservation ${\varvec{{\alpha }}}_0-{\varvec{{\beta }}}_0=0.$ Then

$$\begin{aligned} (|m_1|+|m_2|)(|m_1|-|m_2|)= m_1^2 - m_2^2 \le 20. \end{aligned}$$

If $|m_1|>|m_2|$ then

$$\begin{aligned} |m_1|\le 20\le 20 |\pi |. \end{aligned}$$

(C.17)

Otherwise $m_1=-m_2$ and, therefore, $|\pi |=2|m_1|,$ completing the proof in the case $D=2.$

Let us now consider the case $D \ge 3$. By (7.1), (C.11) and (C.12)

$$\begin{aligned} m_1^2 +{\sigma }_1{\sigma }_2 m_2^2\le & {} 10 N + \sum _{l=3}^Dm_l^2 \le 10 N + \sum _{l=3}^N\widehat{n}_l^2\\= & {} 20 +\sum _{l=3}^N (10 + \widehat{n}_l^2) {\le } 20+ 11 \sum _{l=3}^N \widehat{n}_l^2 {\mathop {\le }\limits ^{N\ge 3}} 31\sum _{l=3}^N \widehat{n}_l^2. \end{aligned}$$

If $\sigma _1\sigma _2 = 1$ then

$$\begin{aligned} {\left| m_1\right| }, {\left| m_2\right| } \le \sqrt{31\sum _{l\ge 3} \widehat{n}_l^2}. \end{aligned}$$

If ${\sigma }_1{\sigma }_2 = -1$

$$\begin{aligned} (|m_1|+|m_2|)(|m_1|-|m_2|)= m_1^2 - m_2^2 \le 31\sum _{l\ge 3} \widehat{n}_l^2. \end{aligned}$$

Now, if ${\left| m_1\right| }\ne {\left| m_2\right| }$ then

$$\begin{aligned} {\left| m_1\right| } + {\left| m_2\right| } \le 31\sum _{l\ge 3} \widehat{n}_l^2. \end{aligned}$$

Conversely, if ${\left| m_1\right| } = {\left| m_2\right| }$, by (C.13), $m_1\ne m_2$, hence $m_1 = - m_2$. By substituting this relation into (C.10), we have

$$\begin{aligned} 2{\left| m_1\right| } \le {\left| \pi \right| } + \sum _{l\ge 3}{\left| m_l\right| } \le {\left| \pi \right| } + \sum _{l\ge 3}\widehat{n}_l^2, \end{aligned}$$

concluding the proof. $\square$

Conclusion of the proof of Lemma 7.1

As above, given ${\varvec{{\alpha }}},{\varvec{{\beta }}}\in {\mathbb {N}}^\mathbb {Z},$ with $1\le |{\varvec{{\alpha }}}|=|{\varvec{{\beta }}}|<\infty$ we consider $m=m({\varvec{{\alpha }}}-{\varvec{{\beta }}})$ and $\widehat{n}=\widehat{n}({\varvec{{\alpha }}}+{\varvec{{\beta }}}).$ Note that $N:=|{\varvec{{\alpha }}}+{\varvec{{\beta }}}|\ge 2.$

We have³³

$$\begin{aligned}&\sum _i{\left| {\varvec{{\alpha }}}_i-{\varvec{{\beta }}}_i\right| }\langle i \rangle ^{\theta /2} {\mathop {\le }\limits ^{(C.9)}}2{\left| m_1\right| }^{\frac{\theta }{2}} + \sum _{l\ge 3} \widehat{n}_l^{\frac{\theta }{2}} \nonumber \\&\quad {\mathop {\le }\limits ^{(C.16)}}2{\left( 20|\pi | + 31 \sum _{l\ge 3} \widehat{n}_l^2 \right) }^{\frac{\theta }{2}} + \sum _{l\ge 3} \widehat{n}_l^{\frac{\theta }{2}} \nonumber \\&\quad \le 2 {\left( 20|\pi | \right) }^{\frac{\theta }{2}} + 2(31)^{\frac{\theta }{2}}\sum _{l\ge 3}\widehat{n}_l^\theta + \sum _{l\ge 3} \widehat{n}_l^{\frac{\theta }{2}} \nonumber \\&\quad \le \frac{13 }{1-\theta }{\left( (1-\theta ) {\left| \pi \right| }+ (2-2^\theta ){\left( \sum _{l\ge 3}\widehat{n}_l^\theta \right) }\right) }, \end{aligned}$$

(C.18)

using that $1-\theta \le 2-2^\theta$ for $0\le \theta \le 1.$ Then by Lemma 6.1 and (C.18) we get

$$\begin{aligned} \sum _i{\left| {\varvec{{\alpha }}}_i-{\varvec{{\beta }}}_i\right| }\langle i \rangle ^{\theta /2}\le & {} \frac{13 }{1-\theta }{\left( (1-\theta ) {\left| \pi \right| }+ \sum _i \langle i \rangle ^\theta {\left( {\varvec{{\alpha }}}_i +{\varvec{{\beta }}}_i\right) } + \theta {\left| \pi \right| } - 2\widehat{n}_1^\theta \right) }\\\le & {} \frac{13 }{1-\theta } {\left[ \sum _i \langle i \rangle ^\theta {\left( {\varvec{{\alpha }}}_i +{\varvec{{\beta }}}_i\right) } + {\left| \pi \right| } - 2\langle j \rangle ^\theta \right] }, \end{aligned}$$

proving (7.2).

Let us now prove (7.3) passing to the logarithm. We have

$$\begin{aligned} \begin{aligned}&\sum _i\ln (1+{\left| {\varvec{{\alpha }}}_i-{\varvec{{\beta }}}_i\right| }{\langle i \rangle })\\&\quad = \sum _{|i|\le 1}\ln (1+{\left| {\varvec{{\alpha }}}_i-{\varvec{{\beta }}}_i\right| }) + \sum _{|i|\ge 2}\ln (1+{\left| {\varvec{{\alpha }}}_i-{\varvec{{\beta }}}_i\right| }|i|)\\&\quad \le 3\ln (1+N ) + \sum _{|i|\ge 2}\ln (1+{\left| {\varvec{{\alpha }}}_i-{\varvec{{\beta }}}_i\right| }|i|)\\&\quad \le 3\ln 2+3\ln N + \frac{3}{2} \sum _{|i|\ge 2}{\left| {\varvec{{\alpha }}}_i-{\varvec{{\beta }}}_i\right| }\ln |i| , \end{aligned} \end{aligned}$$

(C.19)

using that $1+cx \le \frac{3}{2} x^c$ for $c\ge 1,$$x\ge 2.$ If ${\varvec{{\alpha }}}_i-{\varvec{{\beta }}}_i=0$ for every $|i|\ge 2$ then (7.3) follows. Assume now that ${\varvec{{\alpha }}}_i-{\varvec{{\beta }}}_i\ne 0$ for some $|i|\ge 2.$ By (C.14) we have

$$\begin{aligned} N\ge 3\quad \mathrm{or}\quad \pi \ne 0. \end{aligned}$$

(C.20)

We claim that, when $N\ge 3,$

$$\begin{aligned} \ln {\left( \sum _{l= 3}^N \widehat{n}_l^2\right) } \le \ln N +\sum _{l= 3}^N \ln \widehat{n}_l^2. \end{aligned}$$

(C.21)

Let ${\mathcal {S}}:=\{3\le l\le N,\ \mathrm{s.t.}\ \widehat{n}_l\ge 2\}.$ If ${\mathcal {S}}=\emptyset$ we have the equality in (C.21). Otherwise $\sum _{l\in {\mathcal {S}}}\widehat{n}_l^2\ge 4$ and³⁴

$$\begin{aligned} \ln {\left( \sum _{l= 3}^N \widehat{n}_l^2\right) } \le \ln {\left( N+\sum _{l\in {\mathcal {S}}}\widehat{n}_l^2\right) } \le \ln N +\sum _{l\in {\mathcal {S}}} \ln \widehat{n}_l^2, \end{aligned}$$

proving (C.21).

We claim that

$$\begin{aligned} \ln {\left( 20 {\left| \pi \right| }+ 31 \sum _{l= 3}^N \widehat{n}_l^2 \right) } \le \ln (1+|\pi |)+\ln N+\sum _{l= 3}^N \ln \widehat{n}_l^2 +\ln 20+\ln 31. \end{aligned}$$

(C.22)

Indeed consider first the case $\pi =0,$ then $N\ge 3$ by (C.20) and (C.22) follows by (C.21). Consider now the case $|\pi |\ge 1.$ If $N<3$ (C.22) follows (there is no sum). If $N\ge 3$ we have³⁵

$$\begin{aligned}&\ln {\left( 20 {\left| \pi \right| }+ 31 \sum _{l= 3}^N \widehat{n}_l^2 \right) } \le \ln {\left( 20 {\left| \pi \right| }\right) }+\ln {\left( 31 \sum _{l= 3}^N \widehat{n}_l^2 \right) }\\&\quad \le \ln (|\pi |)+\ln {\left( \sum _{l= 3}^N \widehat{n}_l^2 \right) } +\ln 20+\ln 31. \end{aligned}$$

Recalling (C.21) this complete the proof of (C.22).

Let us continue the proof of (7.3). Set $g(i):=0$ if $|i|\le 1$ and $g(i):=\ln |i|$ if $|i|\ge 2$ and apply (C.9) to (C.19); we get

$$\begin{aligned}&\sum _{|i|\ge 2}{\left| {\varvec{{\alpha }}}_i-{\varvec{{\beta }}}_i\right| }\ln |i| \le 2 \ln |m_1|+\sum _{l\ge 3} \ln |\widehat{n}_l|\\&\quad {\mathop {\le }\limits ^{(C.16)}}2\ln {\left( 20 {\left| \pi \right| }+ 31 \sum _{l\ge 3} \widehat{n}_l^2 \right) } +\sum _{l\ge 3} \ln \widehat{n}_l \\&\quad {\mathop {\le }\limits ^{(C.22)}}2\ln (1+|\pi |)+2\ln N+5\sum _{l= 3}^N \ln \widehat{n}_l +16. \end{aligned}$$

Inserting in (C.19) we obtain

$$\begin{aligned} \sum _i\ln (1+{\left| {\varvec{{\alpha }}}_i-{\varvec{{\beta }}}_i\right| }{\langle i \rangle }) \le 3\ln (1+|\pi |)+6\ln N+\frac{15}{2}\sum _{l= 3}^N \ln \widehat{n}_l +27\, \end{aligned}$$

concluding the proof of (7.3). $\square$

Proof of Lemma 7.2

First of all we note that

$$\begin{aligned} \sum _i f_i(|\ell _i|)= \sum _{i\ \text {s.t.} \ \ell _i\ne 0} f_i(|\ell _i|) \end{aligned}$$

since $f_i(0)=0.$ We have that³⁶

$$\begin{aligned} f_i(x) \le -\frac{{\sigma }}{C_* }\langle i \rangle ^{\frac{\theta }{2}}x + {2} \ln (x)+ {\left( {2}+{q}\right) }\ln \langle i \rangle +1,\qquad \forall \, x\ge 1. \end{aligned}$$

We have that

$$\begin{aligned} \max _{x\ge 1} \left( -\frac{{\sigma }}{C_* }\langle i \rangle ^{\frac{\theta }{2}}x + 2 \ln (x)\right) = \left\{ \begin{array}{l} \displaystyle -\frac{{\sigma }}{C_* }\langle i \rangle ^{\frac{\theta }{2}} \qquad \qquad \qquad \ \,\qquad \text {if}\quad \langle i \rangle \ge i_0, \\ \\ \displaystyle -{2}+{2}\ln \frac{2C_* }{{\sigma }}-\theta \ln \langle i \rangle \qquad \text {if}\quad \langle i \rangle < i_0, \end{array} \right. \end{aligned}$$

where

$$\begin{aligned} i_0:=\left( \frac{2C_* }{{\sigma }}\right) ^{\frac{2}{\theta }}, \end{aligned}$$

since the maximum is achieved for $x=1$ if $\langle i \rangle \ge i_0$ and $x=\frac{2C_* }{{\sigma }\langle i \rangle ^{\theta /2}}$ if $\langle i \rangle < i_0$. Note that $i_0\ge e.$ Then we get

$$\begin{aligned}&\sum _i f_i(|\ell _i|) = \sum _{i\ \text {s.t.} \ \ell _i\ne 0} f_i(|\ell _i|) \le \\&\quad \sum _{\langle i \rangle \ge i_0\ \text {s.t.} \ \ell _i\ne 0} \left( {\left( {2}+{q}\right) }\ln \langle i \rangle +1 -\frac{{\sigma }}{C_* }\langle i \rangle ^{\frac{\theta }{2}} \right) + \sum _{\langle i \rangle < i_0} \left( 2\ln \frac{2C_* }{{\sigma }}+\Big ({2}+{q}-\theta \Big ) \ln \langle i \rangle \right) . \end{aligned}$$

We immediately have that

$$\begin{aligned}&\sum _{\langle i \rangle < i_0} \left( 2\ln \frac{2C_* }{{\sigma }}+\Big ({2}+{q}-\theta \Big ) \ln \langle i \rangle \right) \le 3 i_0 \left( 2\ln \frac{2C_* }{{\sigma }}+({2}+{q}) \ln i_0 \right) \\&\quad = 3\left( 2+ \frac{2}{\theta }({2}+{q})\right) \left( \frac{2C_* }{{\sigma }}\right) ^{\frac{2}{\theta }} \ln \frac{2C_* }{{\sigma }}. \end{aligned}$$

Moreover, in the case $\langle i \rangle \ge i_0\ge e,$

$$\begin{aligned} {\left( {2}+{q}\right) }\ln \langle i \rangle +1 -\frac{{\sigma }}{C_* }\langle i \rangle ^{\frac{\theta }{2}}\le & {} {\left( {2}+{q}+1\right) }\ln \langle i \rangle -\frac{{\sigma }}{C_* }\langle i \rangle ^{\frac{\theta }{2}} \\= & {} \frac{2}{\theta }{\left( {2}+{q}+1\right) }\Big ( \ln \langle i \rangle ^{\frac{\theta }{2}}-2{\mathfrak {C} } \langle i \rangle ^{\frac{\theta }{2}} \Big ) \end{aligned}$$

where

$$\begin{aligned} {\mathfrak {C} }:=\frac{{\sigma }\theta }{4C_* {\left( {2}+{q}+1\right) }}<1 . \end{aligned}$$

Therefore

$$\begin{aligned} S_*:=\sum _{\langle i \rangle \ge i_0\ \text {s.t.} \ \ell _i\ne 0} \left( {\left( {2}+{q}\right) }\ln \langle i \rangle +1 -\frac{{\sigma }}{C_* }\langle i \rangle ^{\frac{\theta }{2}} \right) \end{aligned}$$

satisfies

$$\begin{aligned} S_* \le \sum _{\langle i \rangle \ge i_0\ \text {s.t.} \ \ell _i\ne 0} \frac{2}{\theta }{\left( {2}+{q}+1\right) }\Big ( \ln \langle i \rangle ^{\frac{\theta }{2}}-2{\mathfrak {C}} \langle i \rangle ^{\frac{\theta }{2}} \Big ). \end{aligned}$$

We have that³⁷

$$\begin{aligned} \ln \langle i \rangle ^{\frac{\theta }{2}}-2{\mathfrak {C}} \langle i \rangle ^{\frac{\theta }{2}} \le -{\mathfrak {C}} \langle i \rangle ^{\frac{\theta }{2}}, \qquad \text {when}\qquad \langle i \rangle \ge i_*:= \left( \frac{2}{{\mathfrak {C}}}\ln \frac{1}{{\mathfrak {C}}}\right) ^{\frac{2}{\theta }} . \end{aligned}$$

Note that

$$\begin{aligned} i_\sharp \ge \max \{ i_0, i_*\}. \end{aligned}$$

Therefore

$$\begin{aligned} S_*\le & {} \frac{2}{\theta }{\left( {2}+{q}+1\right) } \left( \sum _{\langle i \rangle <i_\sharp } \ln \langle i \rangle ^{\frac{\theta }{2}} - \sum _{\langle i \rangle \ge i_\sharp \ \text {s.t.} \ \ell _i\ne 0} \Big ( {\mathfrak {C}} \langle i \rangle ^{\frac{\theta }{2}} \Big ) \right) \\\le & {} {\left( {2}+{q}+1\right) } \left( 3 i_\sharp \ln i_\sharp -\frac{2{\mathfrak {C}}}{\theta } M_\ell ^{\frac{\theta }{2}} \right) \, \end{aligned}$$

where

$$\begin{aligned} M_\ell :=\max \{ |i|\ge i_\sharp ,\ \ \text {s.t.}\ \ \ell _i\ne 0 \} \end{aligned}$$

and $M_\ell :=0$ if $|\ell _i|=0$ for every $|i|\ge i_\sharp .$ In conclusion we get

$$\begin{aligned} \sum _i f_i(|\ell _i|)\le & {} 3\left( {2}+ \frac{2}{\theta }({2}+{q})\right) \left( \frac{2C_* }{{\sigma }}\right) ^{\frac{2}{\theta }} \ln \frac{2C_* }{{\sigma }} + {\left( {2}+{q}+1\right) } \left( 3 i_\sharp \ln i_\sharp -\frac{2{\mathfrak {C}}}{\theta } M_\ell ^{\frac{\theta }{2}} \right) \\\le & {} 6({q}+3) i_\sharp \ln i_\sharp - \frac{{\sigma }}{2C_* } M_\ell ^{\frac{\theta }{2}}\\\le & {} 7({q}+3) i_\sharp \ln i_\sharp - \frac{{\sigma }}{2C_* } \big (\widehat{n}_1(\ell )\big )^{\frac{\theta }{2}}, \end{aligned}$$

noting that $\widehat{n}_1(\ell )=M_\ell$ if $M_\ell \ne 0,$ otherwise $\widehat{n}_1(\ell )< i_\sharp ,$ and, therefore,

$$\begin{aligned} \frac{{\sigma }}{2C_* } \big (\widehat{n}_1(\ell )\big )^{\frac{\theta }{2}} < \frac{{\sigma }}{2C_* } i_\sharp ^{\frac{\theta }{2}} \le ({q}+3) i_\sharp \ln i_\sharp . \end{aligned}$$

$\square$

Proof of Lemma  4.1

For $\ell \in \mathbb {Z}^\mathbb {Z}$ with $0<|\ell |<\infty$ we define

$$\begin{aligned} {\mathcal {R}}_\ell := {\left\{ \omega \in \Omega _{q}\,:\, |\omega \cdot \ell |\le \frac{\gamma }{1+|\ell _0|^{\mu _1}} \prod _{n\ne 0}\frac{1}{(1+|\ell _n|^{\mu _1} | n|^{{\mu _2}+{q}})}\right\} } \end{aligned}$$

• if $\ell$ is such that $\ell _n=0$$\forall n\ne 0$ then

  $$\begin{aligned} \mu ({\mathcal {R}}_\ell ) = \frac{\gamma }{1+|\ell _0|^{\mu _1} }. \end{aligned}$$

• Otherwise: let $s=s(\ell )>0$ be the smallest positive index i such that $|\ell _i |+|\ell _{-i}|\ne 0$ and $S=S(\ell )$ be the biggest. Then we have³⁸

  $$\begin{aligned} \mu ({\mathcal {R}}_\ell ) \le \frac{\gamma s^{q}}{{\left( 1+|\ell _0|^{\mu _1}\right) } }\prod _{n\ne 0}\frac{1}{(1+|\ell _n|^{\mu _1} |n|^{{\mu _2}+{q}})}. \end{aligned}$$

Let us write

$$\begin{aligned}&\frac{1}{1 + |\ell _0|^{\mu _1}}\prod _{n\ne 0}\frac{1}{(1+|\ell _n|^{\mu _1} |n|^{{\mu _2}+{q}})} \\&\quad = \frac{1}{1 + |\ell _0|^{\mu _1}}\prod _{n>0}\frac{1}{(1+|\ell _n|^{\mu _1} |n|^{{\mu _2}+{q}})} \frac{1}{(1+|\ell _{-n}|^{\mu _1} |n|^{{\mu _2}+{q}})}\\&\quad = \frac{1}{1 + |\ell _0|^{\mu _1}}\prod _{s(\ell )\le n\le S(\ell )}\frac{1}{(1+|\ell _n|^{\mu _1} |n|^{{\mu _2}+{q}})} \frac{1}{(1+|\ell _{-n}|^{\mu _1} |n|^{{\mu _2}+{q}})}. \end{aligned}$$

Now

$$\begin{aligned}&\mu (\Omega _{q}\setminus {\mathtt {D}_{\gamma ,{q}}})\le \sum _{\ell } \mu ({{\mathcal {R}}}_\ell )= \sum _{ \ell _0}\frac{\gamma }{1+|\ell _0|^{\mu _1} } \\&\quad + \sum _{s>0} \sum _{\begin{array}{c} \ell : s(\ell )= S(\ell )=s \end{array}}\frac{1}{1 + |\ell _0|^{\mu _1}}\frac{\gamma s^{q}}{|\ell _s|(1+|\ell _s|^{\mu _1} |s|^{{\mu _2}+{q}})} \frac{1}{(1+|\ell _{-s}|^{\mu _1} |s|^{{\mu _2}+{q}})} \end{aligned}$$

(C.23)

$$\begin{aligned}&\quad + \sum _{0<s< S} \sum _{\begin{array}{c} \ell : s(\ell )=s,\\ S(\ell )=S \end{array}}\frac{\gamma s^{q}}{1 + |\ell _0|^{\mu _1}}\prod _{s\le n\le S }\frac{1}{(1+|\ell _n|^{\mu _1} |n|^{{\mu _2}+{q}})} \frac{1}{(1+|\ell _{-n}|^{\mu _1} |n|^{{\mu _2}+{q}})}. \end{aligned}$$

(C.24)

Let us estimate (C.24)

$$\begin{aligned}&\sum _{s>0}\sum _{\ell _0\in \mathbb {Z}}\frac{1}{1 + |\ell _0|^{\mu _1}}\sum _{ \begin{array}{c} \ell _s,\ell _{-s}\in \mathbb {Z}\\ |\ell _s|+|\ell _{-s}|>0 \end{array}}\frac{\gamma s^{q}}{(1+|\ell _s|^{\mu _1} |s|^{{\mu _2}+{q}})} \frac{1}{(1+|\ell _{-s}|^{\mu _1} |s|^{{\mu _2}+{q}})}\\&\le c(\mu _1)\gamma \sum _{s>0} s^{q}\sum _{ \begin{array}{c} \ell _s,\ell _{-s}\in \mathbb {Z}\\ |\ell _s|+|\ell _{-s}|>0 \end{array}}\frac{1}{(1+|\ell _s|^{\mu _1} |s|^{{\mu _2}+{q}})} \frac{1}{(1+|\ell _{-s}|^{\mu _1} |s|^{{\mu _2}+{q}})}. \end{aligned}$$

Now since

$$\begin{aligned} \sum _{h=1}^\infty \frac{1}{(1+h^{\mu _1} |n|^{{\mu _2}+{q}})} \le \sum _{h=1}^\infty \frac{1}{h^{\mu _1} |n|^{{\mu _2}+{q}}} \le \frac{c(\mu _1)}{|n|^{{\mu _2}+{q}}} \end{aligned}$$

we have

$$\begin{aligned} \sum _{h\in \mathbb {Z}} \frac{1}{(1+|h|^{\mu _1} |n|^{{\mu _2}+{q}})} \le 1+ \frac{2c(\mu _1)}{|n|^{{\mu _2}+{q}}}. \end{aligned}$$

Then we have

$$\begin{aligned} \sum _{ \begin{array}{c} \ell _s,\ell _{-s}\in \mathbb {Z}\\ |\ell _s|+|\ell _{-s}|>0 \end{array}}\frac{1}{(1+|\ell _s|^{\mu _1} |s|^{{\mu _2}+{q}})} \frac{1}{(1+|\ell _{-s}|^{\mu _1} |s|^{{\mu _2}+{q}})} \le \frac{c_1(\mu _1)}{|s|^{{\mu _2}+{q}}} \end{aligned}$$

and consequently (C.24) is bounded by

$$\begin{aligned} c_2(\mu _1)\gamma \sum _{s>0} |s|^b \le c_3(\mu _1,\mu _2)\gamma . \end{aligned}$$

Regarding the third line in (C.23), we note that for all n we have

$$\begin{aligned} \sum _{ \ell _n,\ell _{-n}\in \mathbb {Z}} \frac{1}{(1+|\ell _n|^{\mu _1} |n|^{{\mu _2}+{q}})} \frac{1}{(1+|\ell _{-n}|^{\mu _1} |n|^{{\mu _2}+{q}})}\le {\left( 1 + 2 \frac{c(\mu _1)}{|n|^{{\mu _2}+{q}}}\right) }^2. \end{aligned}$$

Hence

$$\begin{aligned}&\sum _{\begin{array}{c} \ell : s(\ell )=s,\\ S(\ell )=S \end{array}}\frac{1}{1 + |\ell _0|^{\mu _1}}\prod _{s\le n\le S }\frac{1}{(1+|\ell _n|^{\mu _1} |n|^{{\mu _2}+{q}})} \frac{1}{(1+|\ell _{-n}|^{\mu _1} |n|^{{\mu _2}+{q}})}\\&\quad = \sum _{ \ell _0\in \mathbb {Z}}\frac{1}{1 + |\ell _0|^{\mu _1}}\times \sum _{ \begin{array}{c} \ell _s,\ell _{-s}\in \mathbb {Z}\\ |\ell _s|+|\ell _{-s}|>0 \end{array}}\frac{1}{(1+|\ell _s|^{\mu _1} |s|^{{\mu _2}+{q}})} \frac{1}{(1+|\ell _{-s}|^{\mu _1} |s|^{{\mu _2}+{q}})}\\&\qquad \times \sum _{ \begin{array}{c} \ell _S,\ell _{-S}\in \mathbb {Z}\\ |\ell _S|+|\ell _{-S}|>0 \end{array}}\frac{1}{(1+|\ell _S|^{\mu _1} |S|^{{\mu _2}+{q}})} \frac{1}{(1+|\ell _{-S}|^{\mu _1} |S|^{{\mu _2}+{q}})}\\&\qquad \times \prod _{s< n< S }\sum _{ \ell _n,\ell _{-n}\in \mathbb {Z}}\frac{1}{(1+|\ell _n|^{\mu _1} |n|^{{\mu _2}+{q}})} \frac{1}{(1+|\ell _{-n}|^{\mu _1} |n|^{{\mu _2}+{q}})}\\&\quad \le \frac{c_4(\mu _1)}{s^{{\mu _2}+{q}}S^{{\mu _2}+{q}}} \prod _{s< n< S }{\left( 1 + 2 \frac{c(\mu _1)}{|n|^{{\mu _2}+{q}}}\right) }^2\\&\quad \le \frac{c_4(\mu _1)}{s^{{\mu _2}+{q}}S^{{\mu _2}+{q}}} \exp \left( \sum _{n\ge 1 } \ln {\left( 1 + 2 \frac{c(\mu _1)}{|n|^{{\mu _2}+{q}}}\right) }^2 \right) \\&\quad \le \frac{c_5(\mu _1,\mu _2)}{s^{{\mu _2}+{q}}S^{{\mu _2}+{q}}}. \end{aligned}$$

Then, multiplying by $\gamma s^{q}$ and taking the $\sum _{0<s<S},$ we have that also (C.25) is bounded by some constant ${C_{\mathtt {meas}}}(\mu _1,\mu _2)\gamma$. $\square$