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\datereceived{2017-07-06}
\dateaccepted{2018-10-04}
\dateepreuves{2018-10-15}

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\begin{document}
\frontmatter

\title[Quasi-invariant measures for the cubic fourth order NLS]
{An optimal regularity result on the~quasi-invariant Gaussian measures for the~cubic fourth order nonlinear Schrödinger~equation}

\author[\initial{T.} \lastname{Oh}]{\firstname{Tadahiro} \lastname{Oh}}
\address{School of Mathematics, The University of Edinburgh and The Maxwell Institute for the Mathematical Sciences\\
James Clerk Maxwell Building, The King's Buildings, Peter Guthrie Tait Road, Edinburgh EH9\,3FD, United Kingdom}
\email{hiro.oh@ed.ac.uk}
\urladdr{https://www.maths.ed.ac.uk/~toh/}

\author[\initial{P.} \lastname{Sosoe}]{\firstname{Philippe} \lastname{Sosoe}}
\address{Department of Mathematics, Cornell University\\
584 Malott Hall, Ithaca, New York 14853, USA}
\email{psosoe@math.cornell.edu}
\urladdr{http://www1.math.cornell.edu/m/People/bynetid/ps934}

\author[\initial{N.} \lastname{Tzvetkov}]{\firstname{Nikolay} \lastname{Tzvetkov}}
\address{Université de Cergy-Pontoise\\
2, av.~Adolphe Chauvin, 95302 Cergy-Pontoise Cedex, France}
\email{nikolay.tzvetkov@u-cergy.fr}
\urladdr{http://tzvetkov.perso.math.cnrs.fr/}

\thanks{T.O.~was supported by the European Research Council (grant no.~637995 ``ProbDynDispEq'').
N.T.~was supported by the European Research Council (grant no.~257293 ``DISPEQ'').}

\subjclass{35Q55}

\keywords{Fourth order nonlinear Schrödinger equation,
biharmonic nonlinear Schrödinger equation,
quasi-invariant measure, normal form method}

\begin{abstract}
We study the transport properties of the Gaussian measures on Sobolev spaces under the dynamics of the cubic fourth order nonlinear Schrödinger equation on the circle. In particular, we establish an optimal regularity result for quasi-invariance of the mean-zero Gaussian measures on Sobolev spaces. The main new ingredient is an improved energy estimate established by performing an infinite iteration of normal form reductions on the energy functional. Furthermore, we show that the dispersion is essential for such a quasi-invariance result by proving non quasi-invariance of the Gaussian measures under the dynamics of the dispersionless model.
\end{abstract}

\alttitle{Régularité optimale pour la quasi-invariance de mesures gaussiennes par le flot de l'équation de Schrödinger non linéaire d'ordre~$4$}

\altkeywords{Équation de Schrödinger non linéaire, mesures quasi-invariantes, méthode de forme normale}

\begin{altabstract}
Nous étudions le transport de mesures gaussiennes par le flot de l'équation de Schrödinger non linéaire d'ordre~$4$. La nouveauté principale est une estimation d'énergie améliorée faisant appel à un nombre infini de transformations de forme normale sur la fonctionnelle d'énergie. De plus, nous démontrons que la dispersion est essentielle dans cette problématique en prouvant qu'en son absence le même résultat de quasi-invariance ne peut être vrai.
\end{altabstract}

\maketitle
\tableofcontents
\mainmatter

\section{Introduction}
\label{SEC:1}

In this paper, we complete the study of the transport properties of Gaussian measures on Sobolev spaces for the cubic nonlinear Schrödinger equation (NLS) with quartic dispersion,
initiated by the first and third authors in \cite{OTz1}.

The question addressed in this work is motivated by a number of perspectives. In probability theory, absolute continuity properties for the pushforward of Gaussian measures under linear and nonlinear
transformations have been studied extensively, starting with the classical work of Cameron-Martin; see \cite{CM, KU, RA}. More generally, questions of absolute continuity of the distribution of solutions
to differential and stochastic differential equations
with respect to a given initial distribution or some chosen reference measure are also central to stochastic analysis. For example, close to the topic
of the current paper,
see the work of Cruzeiro \cite{Cru1,Cru2}.
We also note a recent work
\cite{NRBSS}
establishing absolute continuity
of the Gaussian measure associated to the complex Brownian bridge
on the circle
under certain gauge transformations.

On the other hand, in the analysis of partial differential equations (PDEs),
Hamiltonian PDE dynamics with initial data distributed according to measures of Gibbs type have been studied intensively over the last two decades, starting with the work of Bourgain \cite{BO94, BO96}.
See \cite{OTz1} for the references therein.
These Gibbs-type measures are constructed as weighted Gaussian measures
and are usually supported on Sobolev spaces of low regularity
with the exception of completely integrable Hamiltonian PDEs
such as the cubic NLS on the circle.
In the approach initiated by Bourgain and successfully
applied to many equations since then,
invariance of such Gibbs-type measures
under the flow of the equation
has been established
by combining the Hamiltonian structure of suitable finite dimensional approximations,
in particular invariance of the finite dimensional Gibbs-type measures,
with PDE approximation arguments.
Invariance of such weighted Gaussian measures
implies
absolute continuity of the pushforward of
the base Gaussian measures.
If we substitute the underlying measure with a different Gaussian measure, however,
the question of absolute continuity becomes non-trivial.
See also \cite{Bo-Gelfand}
for a related question by Gel'fand on building a direct method to prove absolute continuity properties
without relying on invariant measures.

In \cite{TzBBM}, the third author initiated the study of transport properties of Gaussian measures
under the flow of a Hamiltonian PDE, combining probabilistic and PDE techniques.
The result proved there
for a specific Hamiltonian equation (the generalized BBM equation) went beyond general results
on the pushforwards of Gaussian measures by nonlinear transformations such as Ramer's \cite{RA}.
It was shown in \cite{TzBBM} that a key step to showing absolute continuity is to establish a smoothing effect on the nonlinear part.
In \cite{OTz1}, the first and third authors studied the transport of Gaussian measures
for the cubic NLS with quartic dispersion.
An additional difficulty compared to \cite{TzBBM} is the absence of explicit smoothing coming from the nonlinearity,
thus requiring the use of {\it dispersion} in an explicit manner.
In~\cite{OTz1}, such dispersion was manifested
through the normal form method.
In this paper, we improve the result in \cite{OTz1} to the optimal range of Sobolev exponents
by pushing the normal form method to the limit.
Furthermore, we present a result showing that, in the absence of dispersion, the distribution of the solution of the resulting dispersionless equation is {\it not}
absolutely continuous with respect to the Gaussian initial data for any non-zero time.
This in particular establishes the necessity of dispersion
for an absolute continuity property.
Since the linear equation is easily seen to leave the distribution of the Gaussian initial data invariant, this highlights that the question of transport properties for a Hamiltonian PDE is a probabilistic manifestation of the competition between the dispersion and the nonlinear part, familiar for the study of nonlinear dispersive equations.

\subsection{The equation}

We consider the cubic fourth order nonlinear Schrödinger equation (4NLS) on the circle
$\mathbb{T} = \bbR /(2\pi \bbZ)$:
\begin{equation}
\label{4NLS0}
\begin{cases}
i \dt u = \dx^4 u \pm |u|^2u\\
u|_{t=0}= u_0,
\end{cases}
\qquad (x,t)\in \mathbb{T}\times \mathbb{R},
\end{equation}
where $u$ is a complex-valued function
on $\bbT\times \bbR$.
The equation \eqref{4NLS0}
is also called
the biharmonic NLS
and it was studied in \cite{IK, Turitsyn} in the context of stability of solitons in magnetic materials.
See also \cite{Karpman, KS, BKS, FIP}
for a more general
class of fourth order NLS:
\begin{align}
i \dt u = \ld \dx^2 u + \mu \dx^4 u \pm |u|^{2}u.
\label{4NLS}
\end{align}

The equation \eqref{4NLS0} is a Hamiltonian PDE with the conserved Hamiltonian:
\[H(u)=\frac{1}{2}\int_{\bbT}|\dx^2 u|^2 dx
\pm \frac{1}{4}\int_\bbT |u|^4 dx.\]
In addition to the Hamiltonian, the flow of the equation \eqref{4NLS0}
preserves the $L^2$-norm, or the so-called ``mass'':
\begin{equation*}
M(u) = \int_{\mathbb{T}}|u|^2dx.
\end{equation*}
This mass conservation law was used in \cite{OTz1} to prove the following sharp global well-posedness result.

\begin{proposition}
The cubic 4NLS \eqref{4NLS0} is globally well-posed in $H^\s(\bbT)$ for $\s\ge 0$.
\end{proposition}

This global well-posedness result in $L^2(\bbT)$
is sharp in the sense that the cubic 4NLS~\eqref{4NLS0} is ill-posed in negative Sobolev spaces
in the sense of non-existence of solutions.
See~\cite{GO, OTz1, OW}.

The defocusing/focusing nature of the equation \eqref{4NLS0} does not play any role
in the following.
Hence, we assume that it is defocusing, \ie with the $+$ sign in \eqref{4NLS0}.

\subsection{Quasi-invariance of $\mu_s$}
Given $s > \sfrac 12$,
we consider the mean-zero Gaussian measures $\mu_s$ on $L^2(\bbT)$ with covariance operator
$2(\Id-\dx^2)^{-s}$, formally written as
\begin{align}
d \mu_s
= Z_s^{-1} e^{-\frac 12 \| u\|_{H^s}^2} du
= \prod_{n \in \bbZ} Z_{s, n}^{-1}e^{-\frac 12 \jb{n}^{2s} |\ft u_n|^2} d\ft u_n .
\label{gauss0}
\end{align}
As we see below,
the Gaussian measure $\mu_s$ is not supported on $H^s(\bbT)$, \ie $\mu_s(H^s(\bbT)) = 0$,
and we need to work in a larger space.
See \eqref{reg}.
This is due to the infinite dimensionality of the problem.

The covariance operator is diagonalized by the Fourier basis on $\mathbb{T}$
and the Gaussian measure $\mu_s$ defined above is in fact the induced probability measure
under the map\footnote{Henceforth, we drop the harmless factor of $2\pi$,
if it does not play an important role.}
\begin{align}
\o \in \O \longmapsto u^\o(x) = u(x; \o) = \sum_{n \in \bbZ} \frac{g_n(\o)}{\jb{n}^s}\,e^{inx},
\label{gauss1}
\end{align}
where $\jb{\,\cdot\,} = (1+|\cdot|^2)^{\sfrac{1}{2}}$
and
$\{ g_n \}_{n \in \bbZ}$ is a sequence of independent standard complex-valued
Gaussian random variables on a probability space $(\O, \mathcal{F}, P)$, \ie $\text{Var}(g_n) = 2$.
From this random Fourier series representation,
it is easy to see that
$u^\o$ in \eqref{gauss1} lies in $H^{\s}(\bbT)$
almost surely
if and only if
\begin{align}
\s < s -\sfrac 12.
\label{reg}
\end{align}
Lastly, note that, for the same range of $\s$,
the triplet $(H^s, H^\s, \mu_s)$ forms an abstract Wiener space.
See \cite{GROSS, Kuo2}.

In the following, we continue to study the transport
property of the Gaussian measure $\mu_s$ under the dynamics of the cubic 4NLS \eqref{4NLS0}.
Before proceeding further, recall the following definition of quasi-invariant measures;
given a measure space $(X,\mu)$,
we say that the measure is {\it quasi-invariant} under a measurable transformation $T:X\to X$ if $\mu$ and the pushforward of $\mu$ under $T$, defined by $T_*\mu = \mu\circ T^{-1}$, are
equivalent, \ie mutually absolutely continuous with respect to each other.

Our first result improves the quasi-invariance result in \cite{OTz1} to the optimal range of Sobolev exponents.

\begin{theorem}\label{THM:1}
Let $s > \sfrac 12$. Then, the Gaussian measure $\mu_s$ is quasi-invariant under the flow of
the cubic 4NLS
\eqref{4NLS0}.
\end{theorem}

Theorem \ref{THM:1} improves the main result
in \cite{OTz1},
where the first and the third authors proved quasi-invariance of $\mu_s$
under \eqref{4NLS0}
for $s > \sfrac 34$.
Moreover, the regularity $s > \sfrac 12$ is optimal
since when $ s= \sfrac 12$, the Gaussian measure $\mu_s$ is
supported in $H^{-\eps}(\bbT)\setminus\nobreak L^2(\bbT)$, while
the cubic 4NLS \eqref{4NLS0} is ill-posed in negative Sobolev spaces
in the sense of non-existence of solutions.

As shown in \cite{TzBBM}, to prove quasi-invariance of $\mu_s$, it is essential to exhibit
a smoothing of the nonlinear part of the equation.
This can be understood at an intuitive level by an analogy to the Cameron-Martin theorem:
the Gaussian measures $\mu_s$ are quasi-invariant under translations by fixed vectors in their respective Cameron Martin spaces $H^s(\bbT)$.
Since a typical element under $\mu_s$ lies in $H^\sigma(\bbT)$, $\sigma<s-\sfrac 12$, one needs to show that the nonlinear part represents a perturbation which is smoother in the Sobolev regularity.
The Cameron-Martin theorem applies only to translation by fixed vectors, but
Ramer's quasi-invariance result \cite{RA} applies to a more general nonlinear transformation
on an abstract Wiener space, although it requires the translations to be more regular.
This was applied in \cite{TzBBM} and~\cite{OTz1},
where it was noted that a direct application of Ramer's result yields a suboptimal range on $s$.
In \cite{OTz1},
we applied the normal form reduction
to the equation and exhibited $(1+\eps)$-smoothing on the nonlinearity
when $ s> 1$.
We then
proved quasi-invariance of $\mu_s$
by invoking Ramer's result.
When $\sfrac 34< s \leq 1$,
we followed the general approach
introduced by the third author
in the context of the (generalized) BBM equation \cite{TzBBM}.
This strategy combines an energy estimate with the analysis of the evolution of truncated measures.
As in \cite{OTz1}, showing a smoothing of the nonlinear part for \eqref{4NLS0} requires normal form reductions. The main improvements over \cite{OTz1} here comes from a more refined implementation of the normal form reductions, inspired by \cite{GKO}.

In the following, we first describe a rough idea behind this method introduced in~\cite{TzBBM}.
Let $\Phi(t)$ denote the solution map of \eqref{4NLS0}
sending initial data $u_0$ to the solution $u(t)$ at time $t \in \bbR$.
Suppose that we have a measurable set $A \subset L^2(\bbT)$ with $\mu_s(A) = 0$.
Fix non-zero $t \in \bbR$.
In order to prove quasi-invariance of $\mu_s$, we would like to prove $\mu_s(\Phi(t)(A)) = 0$.\footnote{By time reversibility,
this would also yield $\Phi(t)_*\mu_s(A) = \mu_s(\Phi(-t) (A)) = 0$.}
The main idea is to establish the following two properties:
\begin{enumeratei}
\item\label{enum:i}
Energy estimate (with smoothing):
\begin{align}
\frac{d}{dt} \|\Phi(t) (u)\|_{H^s}^2
\, \text{``}\leq\text{''} \, C(\| u \|_{L^2})
\| \Phi(t) u \|_{H^{s - \sfrac 12 - \eps}}^{2-\theta}
\label{P0}
\end{align}
for some $\theta > 0$,\footnote{In \cite{OTz2},
the first and third authors recently proved quasi-invariance of $\mu_s \otimes \mu_{s-1}$
on $(u, \dt u)$
under the dynamics of the two-dimensional cubic nonlinear wave equation (NLW),
where
they showed that
even when $\theta = 0$,
we can still apply Yudovich's argument in the limiting case
and establish a desired estimate of the form \eqref{Yudo}.
This was crucial in proving quasi-invariance of $\mu_s \otimes \mu_{s-1}$
under the cubic NLW on $\bbT^2$.}

\item\label{enum:ii}
A change-of-variable formula:
\begin{align*}
\mu_s(\Phi(t)(A))
& = Z_{s}^{-1}\int_{\Phi(t)(A)}
e^{-\frac 12 \|u\|_{H^s}^2} du \
\text{``}=\text{''} \
Z_s^{-1}\int_{A}
e^{-\frac 12 \|\Phi(t)(u)\|_{H^s}^2} du.
\end{align*}
\end{enumeratei}

Step \eqref{enum:i} is an example of local analysis, studying a trajectory of a single solution,
while Step \eqref{enum:ii} is an example of global analysis on the phase space.
Combining \eqref{enum:i} and \eqref{enum:ii}, we
can study the evolution of $\mu_s(\Phi(t) A)$
by estimating
$\frac d{dt} \mu_s(\Phi(t) (A))$.
In particular, by applying Yudovich's argument \cite{Y}, we obtain\footnote{Compare \eqref{Yudo}
with a much stronger estimate in Lemma \ref{LEM:meas4}.}
\begin{align}
\mu_s (\Phi(t)(A)) \leq C(t, \dl) \big(\mu_s ( A)\big)^{1-\delta}
\label{Yudo}
\end{align}
for any $\dl > 0$.
In particular, if $\mu_s(A) = 0$, then
we would have $\mu_s(\Phi(t)(A)) = 0$.
See \cite{TzBBM, OTz1, OTzXEDP}
for details.

As the quotation marks indicate,
both \eqref{enum:i} and \eqref{enum:ii} are not quite true as they are stated above.
In \cite{OTz1},
we first performed two transformations to \eqref{4NLS0}
and transformed the equation into the following renormalized equation:
\begin{align}
\dt \ft v_n
& = -i \sum_{\substack{n = n_1 - n_2 + n_3\\n \ne n_1, n_3}}
e^{-i \phi(\bar n) t} \ft v_{n_1}\cj{\ft v_{n_2}}\ft v_{n_3}
+ i |\ft v_n|^2 \ft v_n,
\label{4NLS1}
\end{align}
where the phase function $\phi(\bar n) $ is given by
\begin{align}
\phi(\bar n) = \phi(n_1, n_2, n_3, n) = n_1^4 - n_2^4 + n_3^4 - n^4.
\label{phiX}
\end{align}
Note that this reduction of \eqref{4NLS0} to \eqref{4NLS1}
via two transformations on the phase space is another instance of global analysis.
See Section \ref{SUBSEC:3.1}.
This reformulation exhibits resonant and non-resonant structure of the nonlinearity
in an explicit manner
and moreover it removes certain resonant interactions,
which was crucial in establishing an effective energy estimate in Step \eqref{enum:i}.
By applying a normal form reduction,
we introduced a {\it modified} energy $E_t = \|u(t)\|_{H^s}^2 + R_t$
for some appropriate correction term $R_t$.
See \eqref{P1}--\eqref{P3} below.
We then established
an energy estimate on the modified energy $E_t$, provided $s > \sfrac 34$.
In Step (ii),
in order to justify such a change-of-variable formula,
we considered a truncated dynamics.
Moreover, we needed to
introduce and consider
a change-of-variable formula for a {\it modified} measure associated with the modified energy
$E_t$ introduced in Step \eqref{enum:i}.

The regularity restriction $s > \sfrac 34$ in the previous paper \cite{OTz1}
comes from the energy estimate in Step \eqref{enum:i}, where we applied the normal form reduction
(namely integration by parts in time) once to the equation:
$\dt \| u \|_{H^s}^2 = \cdots$
satisfied by the $H^s$-energy functional $\| u \|_{H^s}^2$.
In the following, we prove Theorem \ref{THM:1}
by performing normal form reductions {\it infinitely many times}.
%
%
Our normal form approach is
analogous to the approach employed in
\cite{BIT, KO, GKO}.
In particular, in \cite{GKO}, the first author (with Guo and Kwon) implemented an infinite iteration
scheme of normal form reductions to
prove unconditional well-posedness
of the cubic NLS on $\bbT$ in low regularity.
In \cite{GKO}, we performed integration by parts in a successive manner,
introducing nonlinear terms of higher and higher degrees.
While the nonlinear terms thus introduced are of higher degrees,
they satisfy better estimates.
In order to keep track of all possible ways to perform integration by parts,
we introduced the notion of ordered trees.
See also~\cite{CGKO} for another example of an infinite iteration
of normal form reductions to prove unconditional well-posedness.

In establishing an improved energy estimate (Proposition \ref{PROP:energy}),
we perform an infinite iteration of normal form reductions.
It is worthwhile to note that,
unlike \cite{GKO}, we do not work at the level of the equation \eqref{4NLS0}.
Instead, we work
at the level of the evolution equation
$\dt \| u \|_{H^s}^2 = \cdots$ satisfied by the $H^s$-energy functional.
Let us first go over the computation performed in \cite{OTz1}
to show a flavor of this method.
Using \eqref{4NLS1}, we have
\begin{multline}
\frac{d}{dt} \Bigl(\frac 12\| u (t) \|_{H^s}^2 \Bigr)
= \frac{d}{dt} \Bigl(\frac 12\|v(t) \|_{H^s}^2 \Bigr)\\
= - \Re i
\sum_{n \in \bbZ}\;\;
\sum_{\substack{n = n_1 - n_2 + n_3\\n \ne n_1, n_3}}
e^{ - i \phi(\bar n) t} \jb{n}^{2s} \ft v_{n_1} \cj{\ft v_{n_2}} \ft v_{n_3} \cj{\ft v_n},
\label{P1}
\end{multline}
where $v$ is the renormalized variable as in \eqref{4NLS1}.
Then, differentiating by parts,
\ie integrating by parts without an integral symbol,\footnote{This is indeed
a Poincaré-Dulac normal form reduction
applied to the evolution equation \eqref{P1} for $\frac 12 \|v(t)\|_{H^s}^2$.
See \cite[\S1]{GKO}
for a discussion on the relation between differentiation by parts and normal form reductions.}
we obtain
\begin{multline}\label{P2}
\frac{d}{dt} \Bigl(\frac 12\|v(t) \|_{H^s}^2 \Bigr)
= \Re \frac{d}{dt} \biggl[
\sum_{n \in \bbZ}\sum_{\substack{n = n_1 - n_2 + n_3\\n \ne n_1, n_3}}
\frac{e^{ - i \phi(\bar n) t}}{ \phi(\bar n)} \jb{n}^{2s} \ft v_{n_1} \cj{\ft v_{n_2}} \ft v_{n_3} \cj{\ft v_n}\biggr] \\
- \Re
\sum_{n \in \bbZ}\sum_{\substack{n = n_1 - n_2 + n_3\\n \ne n_1, n_3}}
\frac{e^{ - i \phi(\bar n) t}}{ \phi(\bar n)} \jb{n}^{2s}
\dt \big(\ft v_{n_1} \cj{\ft v_{n_2}} \ft v_{n_3} \cj{\ft v_n}\big).
\end{multline}
This motivates us to define the first modified energy $E_t^{(1)}(v)$
with the correction term $R_t^{(1)}(v)$ by
\begin{equation}\label{P3}
\begin{split}
E^{(1)}_t(v)
= & \, \frac 12 \|v\|_{H^s}^2 + R_t^{(1)}(v)\\
: = & \, \frac 12\|v\|_{H^s}^2
- \Re
\sum_{n \in \bbZ} \sum_{\substack{n = n_1 - n_2 + n_3\\n \ne n_1, n_3}}
\frac{e^{ - i \phi(\bar n) t}}{ \phi(\bar n)} \jb{n}^{2s} \ft v_{n_1} \cj{\ft v_{n_2}} \ft v_{n_3} \cj{\ft v_n} .
\end{split}
\end{equation}
This is the modified energy used in the previous work \cite{OTz1}
(up to a constant factor).
Note that the time derivative of $E_t^{(1)}(v)$ is given by
the second term
on the right-hand side of~\eqref{P2}.

In the second step, we divide the
the second term
on the right-hand side of \eqref{P2}
into nearly resonant and non-resonant parts
and apply differentiation by parts only to the non-resonant part.
When we apply differentiation by parts as in \eqref{P2} in an iterative manner,
the time derivative may fall on any of the factors $\ft v_{n_j}$ and $\ft v_n$,
generating higher order nonlinear terms.
In general, the structure of such terms can be very complicated, depending on where the time derivative falls.
In \cite{GKO}, ordered (ternary) trees played an important role for indexing such terms.
In our case, we work on the evolution equation satisfied
by the $H^s$-energy functional
and we need to consider tree-like structures that grow in two directions.
In Section \ref{SEC:NF}, we introduce the notion of bi-trees and ordered bi-trees
for this purpose.

After $J$ steps of the normal form reductions, we arrive at
\begin{multline}\label{P4a}
\frac {d}{dt} \Bigl(\frac 12 \| v (t) \|_{H^s}^2\Bigr)
= \frac {d}{dt}\Bigl( \sum_{j = 2}^{J+1} \NN^{(j)}_0(v) (t)\Bigr)
+\sum_{j = 2}^{J+1} \NN^{(j)}_1(v)(t)\\
+ \sum_{j = 2}^{J+1} \RR^{(j)}(v)(t)
+ \NN^{(J+1)}_2(v)(t).
\end{multline}
Here, $ \NN^{(j)}_0(v)$ consists of certain $2j$-linear terms,
while
$ \NN^{(j)}_1(v)$ and $\RR^{(j)}(v)$ consist of $(2j+2)$-linear terms.
In practice, we obtain \eqref{P4a} for smooth functions
with a truncation parameter $N \in \bbN$.
Here, we can only show that the remainder term $\NN^{(J+1)}_2$ satisfies the bound of the form:
\[ \big|\NN^{(J+1)}_2\big|\leq F(N, J),\]
with the upper bound $F(N, J)$ satisfying
\[\lim_{N \to \infty} F(N, J) = \infty\]
for each fixed $J \in \bbN$.
This, however, does not cause an issue since we also show that
\[\lim_{J \to \infty} F(N, J) = 0\]
for each {\it fixed} $N \in \bbN$.
Therefore, by first taking the limit $J \to \infty$ and then $N \to \infty$,
we conclude that the error term $\NN^{(J+1)}_2$ vanishes in the limit.
See Section \ref{SUBSEC:error}.
While it is simple, this observation is crucial in an infinite iteration of the normal form reductions.

At the end of an infinite iteration of the normal form reductions,
we can rewrite \eqref{P1} as
\begin{equation}
\frac {d}{dt} \Bigl(\frac 12 \| v (t) \|_{H^s}^2 \Bigr)
= \frac {d}{dt}\bigg( \sum_{j = 2}^\infty \NN^{(j)}_0(v)(t)\bigg)
+ \sum_{j = 2}^\infty \NN^{(j)}_1(v)(t) + \sum_{j = 2}^\infty \RR^{(j)}(v)(t) ,
\label{P4}
\end{equation}
involving infinite series.
The main point of this normal form approach
is that, while
the degrees of the nonlinear terms appearing in \eqref{P4} can be arbitrarily large,
we can show that they are all bounded in $L^2(\bbT)$ (in a summable manner over $j$).
In particular, by
defining the modified energy $\EE_t(v)$
by
\begin{align}
\EE_t(v) := \frac 12 \| v(t) \|_{H^s}^2- \sum_{j = 2}^\infty \NN^{(j)}_0(v)(t),
\label{P5}
\end{align}
we see that its time derivative is bounded:
\begin{align*}
\Bigl|\frac{d}{dt}\EE_t(v) \Bigr| \leq C_s(\|u\|_{L^2}),
\end{align*}
satisfying the energy estimate \eqref{P0} in Step \eqref{enum:i} with $\theta = 2$.
See Proposition \ref{PROP:energy} below.
This is the main new ingredient for proving Theorem \ref{THM:1}.
See also the recent work \cite{OW} by the first author (with Y.\,Wang)
on an infinite iteration of normal form reductions
for establishing a crucial energy estimate
on the difference of two solutions in proving enhanced uniqueness
for the renormalized cubic 4NLS (see \eqref{4NLS0a} below) in negative Sobolev spaces.

\skpt
\begin{remark}
\begin{enumeratei}
\item
Heuristically speaking,
this infinite iteration of normal form reductions
allows us to exchange analytical difficulty
with algebraic/combinatorial difficulty.

\item
The ``correction term'' $R_t^{(1)}$ in \eqref{P3}
is nothing but the correction term in the spirit of the $I$-method \cite{CKSTT1, CKSTT2}.
In fact, at each step of normal form reductions, we obtain a correction term $\NN^{(j)}_0(v)$.
Hence, our improved energy estimate (Proposition~\ref{PROP:energy})
via an infinite iteration of normal form reductions can be basically viewed
as an implementation of the $I$-method with
an infinite sequence $\big\{\NN^{(j)}_0(v)\big\}_{j = 2}^\infty$
of correction terms.
Namely, the modified energy $\EE_t(v)$ defined in \eqref{P5}
is a modified energy of an infinite order in the $I$-method terminology.\footnote{The highest order of modified energies
used in the literature is three in the application of the $I$-method to the KdV equation \cite{CKSTT2}, corresponding
to two iterations of normal form reductions.}

\item
We point out that a finite iteration of normal form reductions is not sufficient
to go below $s > \sfrac 34$.
See (6.14) in \cite{OTz1}, showing the restriction $s - \sfrac 12 > \sfrac 14$.
\end{enumeratei}
\end{remark}

\begin{remark}
Let us briefly discuss the situation for the more general cubic fourth order NLS \eqref{4NLS}.
For this equation,
the following phase function
\begin{align*}
\phi_{\ld, \mu}(\bar n)
= -\ld(n_1^2 - n_2^2 + n_3^2 - n^2)
+ \mu(n_1^4 - n_2^4 + n_3^4 - n^4)
\end{align*}
plays an important role in the analysis.
In view of Lemma \ref{LEM:phase} below,
we have
\begin{equation}
\phi_{\ld, \mu}(\bar n) = (n_1 - n_2)(n_1-n)
\big\{- 2\ld + \mu \big(n_1^2 +n_2^2 +n_3^2 +n^2 + 2(n_1 +n_3)^2\big)\big\}.
\label{phase1}
\end{equation}

If the last factor in \eqref{phase1}
does not vanish for any $n_1, n_2, n_3, n \in \bbZ$,
then we can establish quasi-invariance of $\mu_s$
under \eqref{4NLS} for $s > \sfrac 12$
with the same proof as in \cite{OTz1} and this paper.
It suffices to note that,
while we make use of the divisor counting argument in the proof,
we only apply it to
$\mu(\bar n) = (n_1 - n_2)(n_1-n) $
and thus the integer/non-integer character of the last factor in \eqref{phase1} is irrelevant.

For example,
when $\ld \mu < 0$, the last factor in \eqref{phase1} does not vanish
and thus Theorem \ref{THM:1} applies to this case.
When $\ld \mu > 0$,
the non-resonant condition
$ 2\ld \not\in \mu \bbN$
also guarantees the non-vanishing of the last factor in \eqref{phase1}.
It seems of interest to investigate the transport property of the Gaussian measure $\mu_s$
in the resonant case $ 2\ld \in \mu \bbN$.
In this case, there are more resonant terms
and thus further analysis is required.
\end{remark}

\begin{remark}
On the one hand, the cubic 4NLS \eqref{4NLS0} is ill-posed in negative Sobolev spaces
and hence the quasi-invariance result stated in Theorem \ref{THM:1} is sharp.
On the other hand, the first author and Y.\,Wang \cite{OW} considered the following renormalized
cubic 4NLS on $\bbT$:
\begin{align}
\textstyle i \dt u = \dx^4 u + \big( |u|^2 -2 \fint |u|^2 dx\big) u,
\label{4NLS0a}
\end{align}
where $\fint_\bbT f(x) dx := \frac{1}{2\pi} \int_\bbT f(x)dx$.
In particular, they proved global well-posedness
of \eqref{4NLS0a} in $H^s(\bbT)$ for $s > -\sfrac{1}{3}$.
In a very recent work \cite{OTW}, the first and third authors with Y.\,Wang
went further and constructed global-in-time dynamics
for \eqref{4NLS0a}
almost surely with respect to the white noise, \ie the Gaussian measure $\mu_s$ with $s = 0$
supported on $H^\s(\bbT)$, $\s < -\sfrac 12$.
As a result, they proved invariance of the white noise $\mu_0$
under the renormalized cubic 4NLS~\eqref{4NLS0a}.
Invariance is of course a stronger property than quasi-invariance
and hence the white noise is in particular quasi-invariant
under~\eqref{4NLS0a}.
The question of quasi-invariance of $\mu_s$ for $s \in (0, \sfrac 12]$ under the dynamics
of the renormalized cubic 4NLS \eqref{4NLS0a} is therefore
a natural sequel of the analysis of this paper.
\end{remark}

\subsection{Non quasi-invariance under the dispersionless model}

To motivate our second result, note that, by invariance of the complex-valued Gaussian random variable~$g_n$ in~\eqref{gauss1} under rotations, it is clear that
the Gaussian measure $\mu_s$ is invariant under the linear dynamics:
\begin{equation}
\begin{cases}
i \partial_t u = \partial_x^4 u\\
u|_{t=0}= u_0,
\end{cases} \qquad (x,t)\in \mathbb{T}\times \mathbb{R}.
\label{linear}
\end{equation}
See Lemma \ref{LEM:gauss}\,\eqref{LEM:gaussi} below.
In particular, $\mu_s$ is quasi-invariant under the linear dynamics~\eqref{linear}.

In the proof of the quasi-invariance of $\mu_s$ under the cubic 4NLS \eqref{4NLS0} (Theorem~\ref{THM:1} above),
the dispersion plays an essential role.
The strong dispersion allows us to show that the nonlinear part in \eqref{4NLS0}
is a perturbation to the linear equation \eqref{linear}.
Our next result shows that the dispersion is indeed {\it essential} for Theorem \ref{THM:1} to hold.

Consider the following dispersionless model:
\begin{equation}\label{ND1}
\begin{cases}
i \partial_t u = |u|^2u\\
u|_{t=0}= u_0,
\end{cases}
\qquad (x,t)\in \mathbb{T}\times \mathbb{R}.
\end{equation}
Recall that there is an explicit solution formula for \eqref{ND1} given by:
\begin{equation}\label{formula}
u(x,t)=e^{- it|u_0(x)|^2}u_0(x)
\end{equation}
at least for continuous initial data
such that the pointwise product makes sense.

Let $s > \sfrac 12$.
Then, it is easy to see that the random function $u^\o$ in \eqref{gauss1} is continuous almost surely.
Indeed, by the equivalence of Gaussian moments
and the mean value theorem, we have
\begin{align*}
\bbE\big[|u^\o(x) - u^\o(y)|^p\big]
& \leq C_p
\Big(\bbE\big[|u^\o(x) - u^\o(y)|^2\big]\Big)^{\sfrac{p}{2}}
\sim \bigg(\sum_{n \in \bbZ}\frac{1}{\jb{n}^{2s - 2\eps}}\bigg)^{\sfrac{p}{2}}
|x - y|^{\eps p}\\
& \les |x - y|^{\eps p },
\end{align*}
provided that $\eps > 0$ is sufficiently small such that $2s - 2\eps > 1$.
Now, by choosing $p \gg1 $ such that $\eps p > 1$,
we can apply Kolmogorov's continuity criterion
and conclude that~$u^\o$ in~\eqref{gauss1} is almost surely continuous when $s > \sfrac 12$.
This in particular implies that
the solution formula \eqref{formula} is well defined for initial data distributed according to $\mu_s$, $s > \sfrac 12$,
and the corresponding solutions exist globally in time.
We denote by~$\wt \Phi(t)$
the solution map for the dispersionless model \eqref{ND1}.

We now state our second result.

\begin{theorem}\label{THM:sing}
Let $s > \sfrac 12$.
Then,
given $t \ne 0$,
the pushforward measure $\wt \Phi(t)_* \mu_s$
under the dynamics of the dispersionless model \eqref{ND1}
is not absolutely continuous with respect to the Gaussian measure $\mu_s$.
Namely, the Gaussian measure $\mu_s$ is not quasi-invariant
under the dispersionless dynamics \eqref{ND1}.
\end{theorem}

This is a sharp contrast with the quasi-invariance result
for the cubic 4NLS in Theorem~\ref{THM:1}
and for the cubic NLS
for $s \in \bbN$ (see \cite[Rem.\,1.4]{OTz1}).
In particular, Theorem~\ref{THM:sing} shows that dispersion is essential
for establishing quasi-invariance of $\mu_s$.

We prove this negative result in Theorem \ref{THM:sing}
by establishing
that typical elements under $\mu_s$, $s>\sfrac 12 $, possess an almost surely constant modulus of continuity at each point. This is the analogue of the classical law of the iterated logarithm for the Brownian motion.
We show that this modulus of continuity is
destroyed with a positive probability by the nonlinear transformation \eqref{formula}
for any non-zero time $t \in \bbR \setminus \{0\}$.

Our proof is based on three basic tools:
the Fourier series representation of the (fractional)
Brownian loops, the law of the iterated logarithm,
and the solution formula \eqref{formula} to the dispersionless model \eqref{ND1}.
We will use three different versions of the law of the iterated logarithm,
depending on
(i) $s = 1$ corresponding to the Brownian/Ornstein-Uhlenbeck loop,
(ii) $\sfrac 12 < s < \sfrac 32$, corresponding to the fractional Brownian loop,
and (iii) the critical case $s = \sfrac 32$.
In Cases (i) and (ii), we make use of the mutual absolute continuity
of the function $u$ given in the random Fourier series \eqref{gauss1}
and the (fractional) Brownian motion on $[0, 2\pi)$
to deduce the law of the iterated logarithm for the random function $u$.
In Case~(iii), we directly establish the relevant
law of the iterated logarithm for $u$ in \eqref{gauss1}.\footnote{We could apply
this argument to directly establish the relevant law of the iterated logarithm
in Cases~(i) and (ii) as well.} See Proposition~\ref{PROP:LL}.

On the one hand,
the law of the iterated logarithm
yields
almost sure constancy of the modulus of continuity at time $t = 0$.
On the other hand,
we combine this almost sure constancy of the modulus of continuity at time $t = 0$
and the solution formula \eqref{formula}
to show that the modulus of continuity at non-zero time $t\ne 0$
does not satisfy the conclusion of the law of the iterated logarithm
with a positive probability.
Lastly, for $s > \sfrac 32$, we reduce the proof to one of Cases (i), (ii), or (iii)
by differentiating the random function.

\begin{remark}

The existence of a quasi-invariant measure
shows a delicate persistence
property of the dynamics.
In particular, this persistence property due to the quasi-invariance
is stronger than the persistence of regularity\footnote{
In the scaling sub-critical case,
by persistence of regularity, we mean the following;
if one proves local well-posedness in $H^{s_0}$ for some $s_0 \in \bbR$
and if $u_0$ lies in a smoother space $H^s$ for some $s > s_0$,
then the corresponding solution remains smoother
and lies in $C([-T, T] ;H^s)$,
where
the local existence time $T>0$ depends only on the $H^{s_0}$-norm of the initial condition $u_0$.
}
obtained by the usual well-posedness theory.
While the dispersionless model \eqref{ND1}
enjoys the persistence of regularity in $H^\s(\bbT)$, $\s > \sfrac 12$,
Theorem \ref{THM:sing} shows that the Gaussian measure $\mu_s$
is not quasi-invariant
under the dynamics of \eqref{ND1}.
\end{remark}

\skpt
\begin{remark}\label{REM:eps}
\begin{enumeratei}
\item
For $\eps \in \bbR$,
consider the following 4NLS:
\begin{equation}
i \dt u = \eps \dx^4 u + |u|^2u.
\label{eNLS}
\end{equation}
For smooth initial data,
it is easy to show that,
on the unit time interval $[0, 1]$,
the corresponding solutions to the small dispersion 4NLS,
\ie \eqref{eNLS} with small $\eps \ne0$,
converge to those to the dispersionless model~\eqref{ND1}
as $\eps \to 0$.
See \cite[Lem.\,4.1]{OW0}.
In this sense,
the small dispersion 4NLS~\eqref{eNLS} with $|\eps|\ll 1$
is ``close'' to the dispersionless model~\eqref{ND1}.

On the other hand, there is a dichotomy
in the statistical behavior of solutions
to the small dispersion 4NLS \eqref{eNLS}
and to the dispersionless model \eqref{ND1}.
When $\eps \ne 0$, one can easily adapt the proof of Theorem \ref{THM:1}
and prove quasi-invariance of the Gaussian measure $\mu_s$.
When $\eps = 0$, however, Theorem \ref{THM:sing} shows that $\mu_s$ is not quasi-invariant under
\eqref{ND1}.
This shows
a dichotomy between quasi-invariance for $\eps \ne 0$
and non quasi-invariance for $\eps = 0$,
while there is a good approximation property
for the deterministic dynamics of the dispersionless model \eqref{ND1}
by that of the small dispersion 4NLS \eqref{eNLS} with $|\eps|\ll1$.

\item
We mention recent work
\cite{OTsTz, FT}
on establishing quasi-invariance of the Gaussian measure $\mu_s$
for Schrödinger-type equations with less dispersion.
In particular, Forlano-Trenberth \cite{FT} studied
the following fractional NLS:
\begin{equation}
i \dt u = (-\dx^2)^{\sfrac{\al}{2}} u + |u|^2u
\label{fNLS}
\end{equation}
and proved quasi-invariance of $\mu_s$ (for some non-optimal range of $s >s_\al$),
provided that $\al > 1$.
When $\al = 1$,
the equation \eqref{fNLS} corresponds to the half-wave equation,
which does not possess any dispersion.
See \cite{GG, Poc, GLPR}.
It would be of interest to study the transport properties
of the Gaussian measure $\mu_s$ under \eqref{fNLS}
for $0 < \al < 1$.
We also point out that
\eqref{fNLS}
for $0 < \al < 1$ also appears as a model in the study of one-dimensional wave turbulence
\cite{MMT}
and hence is a natural model for statistical study of its solutions.
\end{enumeratei}
\end{remark}

\subsection{Organization of the paper}

In Section~\ref{SEC:notations}, we introduce some notations.
In Section~\ref{SEC:3}, we prove Theorem \ref{THM:1},
assuming the improved energy estimate (Proposition \ref{PROP:energy}).
We then present the proof of Proposition \ref{PROP:energy}
in Section \ref{SEC:NF}
by implementing an infinite iteration of normal form reductions.
Lastly, by studying the random Fourier series \eqref{gauss1} and the relevant law
of the iterated logarithm,
we prove
non quasi-invariance of
the Gaussian measure $\mu_s$
under the dispersionless model (Theorem \ref{THM:sing}) in Section \ref{SEC:sing}.

\subsubsection*{Acknowledgements}
The authors are also grateful to the anonymous referees for their helpful comments that have improved the presentation of this paper.

\section{Notations}
\label{SEC:notations}

Given $N \in \bbN$,
we use $\bfP_{\leq N}$ to denote the Dirichlet projection
onto the frequencies $\{|n|\leq N\}$
and set $\bfP_{> N} := \Id - \bfP_{\leq N}$.
When $N = \infty$, it is understood that $\bfP_{\le N} = \Id$.
Define $E_N$ and $E_N^\perp$ by
\begin{align*}
E_N & = \bfP_{\leq N} L^2(\bbT) = \text{span}\{e^{inx}: |n|\leq N\},\\
E_N^\perp & = \bfP_{>N} L^2(\bbT) =
\text{span}\{e^{inx}: |n|> N\}.
\end{align*}

Given $ s> \sfrac{1}{2}$, let $\mu_s$ be the Gaussian measure on $L^2(\bbT)$
defined in \eqref{gauss0}.
Then, we can write $\mu_s$ as
\[
\mu_s = \mu_{s, N}\otimes \mu_{s, N}^\perp,
\]
where $ \mu_{s, N}$ and $\mu_{s, N}^\perp$
are the
marginal distributions of $\mu_s$ restricted onto $E_N$ and~$E_N^\perp$, respectively.
In other words,
$ \mu_{s, N}$ and $\mu_{s, N}^\perp$
are
the induced probability measures
under the following maps:
\begin{align*}
& u_N :\o \in \O \longmapsto u_N (x; \o) = \sum_{|n|\leq N} \frac{g_n(\o)}{\jb{n}^s}\,e^{inx}, \\
& u_N^\perp: \o \in \O \longmapsto u_N^\perp(x; \o) = \sum_{|n|>N} \frac{g_n(\o)}{\jb{n}^s}\,e^{inx},
\end{align*}
respectively.
Formally, we can write $ \mu_{s, N}$ and $\mu_{s, N}^\perp$
as
\begin{align}
d \mu_{s, N} = Z_{s, N}^{-1} e^{-\frac 12 \| \bfP_{\leq N} u_N\|_{H^s}^2} d u_N
\quad \text{and} \quad
d \mu_{s, N}^\perp = \ft Z_{s,N }^{-1} e^{-\frac 12 \| \bfP_{>N} u_N^\perp \|_{H^s}^2} d u_N^\perp.
\label{G4}
\end{align}
Given $r > 0$, we also define a probability measure $\mu_{s, r}$
with an $L^2$-cutoff
by
\begin{align*}
d \mu_{s, r} = Z_{s, r}^{-1}\ind_{\{ \| v\|_{L^2 } \leq r\}} d\mu_s.
\end{align*}

Given a function $v \in L^2(\bbT)$,
we simply use $v_n$ to denote the Fourier coefficient~$\ft v_n$ of~$v$,
when there is no confusion.
This shorthand notation is especially useful in Section~\ref{SEC:NF}.

We use $a+$ (and $a-$) to denote $a + \eps$ (and $a - \eps$, respectively) for arbitrarily small $\eps \ll 1$,
where an implicit constant is allowed to depend on $\eps > 0$
(and it usually diverges as $\eps \to 0$).
Given $x \in \bbR$,
we use
$\lfloor x \rfloor $ to denote the integer part
of $x$.

In view of the time reversibility of the equations \eqref{4NLS0} and \eqref{ND1},
we only consider positive times in the following.

\section{Proof of Theorem \ref{THM:1}:
Quasi-invariance of \texorpdfstring{$\mu_s$}{mus} under the cubic 4NLS}
\label{SEC:3}

In this section, we present the proof of Theorem \ref{THM:1}.
The main new ingredient is the improved energy estimate (Proposition \ref{PROP:energy})
whose proof is postponed to Section~\ref{SEC:NF}.
The remaining part of the proof follows closely
the presentation in \cite{OTz1}
and thus we keep our discussion concise.

\subsection{Basic reduction of the problem}
\label{SUBSEC:3.1}

We first go over the basic reduction of the problem from \cite{OTz1}.
Given $t \in \bbR$, we define a gauge transformation $\GG_t$ on $L^2(\bbT)$
by setting
\[
\mathcal{G}_t [f ]: = e^{ 2 i t \fint |f|^2} f.
\]
Given a function $u \in C(\bbR; L^2(\bbT))$,
we define $\GG$ by setting
\[\GG[u](t) : = \GG_t[u(t)].\]
Note that $\GG$ is invertible
and its inverse is given by $\GG^{-1}[u](t) = \GG_{-t}[u(t)]$.

Given a solution $u \in C(\bbR; L^2(\bbT))$ to \eqref{4NLS0},
let $\wt u = \GG[u]$.
Then, it follows from the mass conservation
that $\wt u$ is a solution to the following renormalized fourth order NLS:
\begin{equation}
\textstyle i \dt \wt u = \dx^4 \wt u +\big( |\wt u |^{2} - 2 \fint_\bbT |\wt u |^2dx \big) \wt u.
\label{NLS4}
\end{equation}
This is precisely the renormalized cubic 4NLS in \eqref{4NLS0a}.

Let $S(t) = e^{-it \dx^4}$ be the solution operator for the linear fourth order Schrödinger equation~\eqref{linear}.
Denoting by $v= S(-t) \wt u$ the interaction representation of $\wt u$,
we see that $v$ satisfies
the following equation for $\{v_n\}_{n \in \bbZ}$:
\begin{equation}
\label{NLS5}
\begin{split}
\dt v_n
& = -i \sum_{\Gamma(n)} e^{-i \phi(\bar n) t} v_{n_1}\cj{v_{n_2}}v_{n_3}
+ i |v_n|^2 v_n\\
& =: \NN(v)_n + \RR(v)_n,
\end{split}
\end{equation}
where the phase function $\phi(\bar n)$ is as in \eqref{phiX}
and the plane $\Gamma(n)$ is given by
\begin{align}
\Gamma(n)
= \{(n_1, n_2, n_3) \in \bbZ^3:\,
n = n_1 - n_2 + n_3 \text{ and } n_1, n_3 \ne n\}.
\label{Gam1}
\end{align}

Recall that the phase function $\phi(\bar n)$ admits
the following factorization.
See \cite{OTz1} for the proof.

\begin{lemma}\label{LEM:phase}
Let $n = n_1 - n_2 + n_3$.
Then, we have
\begin{align*}
\phi(\bar n) = (n_1 - n_2)(n_1-n)
\big( n_1^2 +n_2^2 +n_3^2 +n^2 + 2(n_1 +n_3)^2\big).
\end{align*}
\end{lemma}

It follows from Lemma \ref{LEM:phase} that
$\phi(\bar n) \ne 0$ on $\Gamma(\bar n)$.
Namely, $\NN(v)$ and $\RR(v)$
on the right-hand side of \eqref{NLS5}
correspond to the non-resonant and resonant parts, respectively.
It follows from Lemma \ref{LEM:phase} that
there is a strong smoothing property on the non-resonant term $\NN(v)$
due to the fast oscillation caused by $\phi(\bar n)$.

Given $t, \tau \in \bbR$, let
$\Phi(t):L^2(\bbT)\to L^2(\bbT)$ be the solution map for \eqref{4NLS0}
and
$ \Psi(t, \tau):L^2(\bbT)\to L^2(\bbT)$ be the solution map for \eqref{NLS5},\footnote{Note that \eqref{NLS5}
is non-autonomous.
We point out that this non-autonomy does not play an essential
role in the remaining part of the paper, since all the estimates hold uniformly in $t \in \bbR$.
}
sending initial data at time~$\tau$
to solutions at time $t$.
When $\tau =0$, we may denote $\Psi(t, 0)$ by $\Psi(t)$ for simplicity.
Then, from $v = S(-t)\circ \GG[u]$, we have
\begin{equation}
\Phi(t) = \GG^{-1} \circ S(t) \circ \Psi(t).
\label{nonlin2}
\end{equation}
Recall the following lemma from \cite{OTz1}.

\skpt
\begin{lemma}\label{LEM:gauss}
\begin{enumeratei}
\item\label{LEM:gaussi}
Let $s>\sfrac 12$
and $t \in \bbR$. Then, the Gaussian measure $\mu_s$ defined in \eqref{gauss0} is invariant under
the linear map $S(t)$
and the gauge transformation $\GG_t$.

\item\label{LEM:gaussii}
Let $(X, \mu)$ be a measure space.
Suppose that $T_1$ and $T_2$
are measurable maps on $X$ into itself
such that $\mu$ is quasi-invariant under $T_j$ for each $j = 1, 2$.
Then, $\mu$ is quasi-invariant under $T = T_1 \circ T_2$.
\end{enumeratei}
\end{lemma}

When $s = 1$, Lemma \ref{LEM:gauss}\,\eqref{LEM:gaussi} basically follows
from Theorem 3.1 in \cite{NRBSS} which exploits the properties of the Brownian loop
under conformal mappings.
For general $s > \sfrac 12$,
such approach does not seem to be appropriate.
See \cite[\S4]{OTz1} for the proof of Lemma~\ref{LEM:gauss}.

In view of this lemma,
Theorem \ref{THM:1} follows once we prove quasi-invariance of $\mu_s$ under $\Psi(t)$.
Therefore, we focus our attention to \eqref{NLS5}
in the following.

\subsection{Truncated dynamics}

Let us first introduce the following truncated approximation to \eqref{NLS5}:
\begin{equation}
\label{NLS6}
\dt v_n =
\ind_{|n|\leq N}\biggl\{-i
\sum_{\Gamma_N(n)} e^{-i \phi(\bar n) t} v_{n_1}\cj{v_{n_2}}v_{n_3}
+ i |v_n|^2 v_n\biggr\},
\end{equation}
where $\Gamma_N(n)$ is defined by
\begin{align*}
\Gamma_N(n) & = \Gamma(n) \cap \{ (n_1, n_2, n_3) \in \bbZ^3: |n_j|\leq N\}\\
& = \{(n_1, n_2, n_3) \in \bbZ^3:\,
n = n_1 - n_2 + n_3, \, n_1, n_3, \ne n,
\text{ and } |n_j| \leq N\}.
\end{align*}
Note that \eqref{NLS6} is an infinite dimensional system of ODEs
for the Fourier coefficients $\{ v_n \}_{n \in \bbZ}$,
where the flow is constant on the high frequencies $\{|n|> N\}$.
We also consider the following finite dimensional system of ODEs:
\begin{equation}
\dt v_n =
-i
\sum_{\Gamma_N(n)} e^{-i \phi(\bar n) t} v_{n_1}\cj{v_{n_2}}v_{n_3}
+ i |v_n|^2 v_n, \qquad |n| \leq N,
\label{NLS7}
\end{equation}
with $v|_{t = 0} = \bfP_{\leq N}v|_{t = 0}$,
\ie $v_n|_{t=0} = 0$ for $|n| > N$.

Given $t, \tau \in \bbR$,
denote by $\Psi_N(t, \tau)$ and $\wt \Psi_N(t, \tau)$
the solution maps of \eqref{NLS6}
and \eqref{NLS7},
sending initial data at time $\tau$
to solutions at time $t$, respectively.
For simplicity, we set
\begin{align}\Psi_N(t) = \Psi_N(t, 0)
\qquad \text{and}
\qquad
\wt \Psi_N(t) = \wt \Psi_N(t, 0)
\label{Psit}
\end{align}
when $\tau = 0$.
Then, we have the following relations:
\begin{align*}
\Psi_N(t, \tau) = \wt \Psi_N (t, \tau) \bfP_{\leq N} + \bfP_{>N}
\qquad \text{and}
\qquad
\bfP_{\leq N} \Psi_N(t, \tau) = \wt \Psi_N (t, \tau) \bfP_{\leq N}.
\end{align*}

We now recall the following approximation property
of the truncated dynamics \eqref{NLS6}.

\begin{lemma}[Proposition 6.21/B.3 in \cite{OTz1}]\label{LEM:approx}

Given $R>0$, let $A \subset B_R$ be a compact set in $L^2(\bbT)$.
Fix $t \in \bbR$.
Then, for any $\eps > 0$, there exists $N_0 = N_0(t, R, \eps) \in \bbN$ such that we have
\begin{align*}
\Psi(t) (A)\subset \Psi_N(t) (A + B_\eps)
\end{align*}
for all $N \geq N_0$.
Here, $B_r$ denotes
the ball in $ L^2(\bbT)$ of radius $r$ centered at the origin.
\end{lemma}

\subsection{Energy estimate}\label{SUBSEC:energy}

In this subsection, we state a crucial energy estimate.
The main goal is to establish an energy estimate of the form \eqref{P0}
by introducing a suitable modified energy functional.
We achieve this goal by performing normal form reductions
infinitely many times and
thus constructing an infinite sequence of correction terms.

Let $N \in \bbN \cup\{\infty\}$.
In the following,
we simply say that $v$ is a solution
to \eqref{NLS7}
if $v$ is a solution to \eqref{NLS7} when $N \in \bbN$
and to \eqref{NLS5} when $N = \infty$.

\begin{proposition}\label{PROP:energy}
Let $\sfrac 12< s < 1$.
Then, given $N \in \bbN \cup\{\infty\}$,
there exist multilinear forms
$\big\{\NN_{0, N}^{(j)}\big\}_{j = 2}^\infty$,
$\big\{\NN_{1, N}^{(j)}\big\}_{j = 2}^\infty$,
and
$\big\{\RR_N^{(j)}\big\}_{j = 2}^\infty$
such that
\[
\frac {d}{dt} \Bigl(\frac 12 \| v (t) \|_{H^s}^2 \Bigr)
= \frac {d}{dt}\bigg( \sum_{j = 2}^\infty \NN^{(j)}_{0, N}(v)(t)\bigg)
+ \sum_{j = 2}^\infty \NN^{(j)}_{1, N}(v)(t) + \sum_{j = 2}^\infty \RR_N^{(j)}(v)(t),
\]
for any solution $v \in C(\bbR; H^s(\bbT))$ to \eqref{NLS7}.
Here,
$\NN_{0, N}^{(j)}$ are $2j$-linear forms,
while
$\NN_{1, N}^{(j)}$ and $\RR_N^{(j)}$ are $(2j+2)$-linear forms, satisfying the following bounds on $L^2(\bbT)$;
there exist positive constants $C_0(j)$, $C_1(j)$, and $C_2(j)$
decaying faster than any exponential rate\footnote{In fact,
by slightly modifying the proof, we can make $C_0(j)$, $C_1(j)$, and $C_2(j)$
decay as fast as we want as $j \to \infty$.}
as $j \to \infty$
such that
\begin{align*}
| \NN^{(j)}_{0, N}(v)(t)|
& \les
C_0(j) \|v\|_{L^2}^{2j},
\\
| \NN^{(j)}_{1, N}(v)(t)|
& \les
C_1(j) \|v\|_{L^2}^{2j+2},
\\
| \RR^{(j)}_N(v)(t)|
& \les
C_2(j) \|v\|_{L^2}^{2j+2},
\end{align*}
for $j = 2, 3, \dots$.
Note that these constants
$C_0(j)$, $C_1(j)$, and $C_2(j)$
are independent of the cutoff size $N \in \bbN \cup \{\infty\}$
and $t \in \bbR$.

Define the modified energy $\EE_{N, t}(v)$
by
\begin{align}
\EE_{N, t}(v) := \frac 12 \| v(t) \|_{H^s}^2- \sum_{j = 2}^\infty \NN^{(j)}_{0, N}(v)(t).
\label{energy6}
\end{align}
Then,
the following energy estimate holds:
\begin{align}
\Bigl|\frac{d}{dt}\EE_{N, t}(v) \Bigr| \leq C_s(\|v\|_{L^2})
\label{energy7}
\end{align}
for any solution $v \in C(\bbR; H^s(\bbT))$ to \eqref{NLS7},
uniformly in $N \in \bbN \cup \{\infty\}$
and $t \in \bbR$.
\end{proposition}

In the remaining part of this section, we continue with the proof of Theorem \ref{THM:1},
assuming Proposition~\ref{PROP:energy}.
We present the proof of Proposition~\ref{PROP:energy}
in Section~\ref{SEC:NF}.
See Lemmas~\ref{LEM:N^J+1_0}
and~\ref{LEM:N^J+1_1}.
In the following, we simply denote $\EE_{\infty, t}$
by $\EE_{ t}$
and drop the subscript $N = \infty$ from the multilinear forms,
when $N = \infty$.
For example, we write
$ \NN^{(j)}_{0}$ for $\NN^{(j)}_{0, \infty}$.

\subsection{Weighted Gaussian measures}

Let $s > \sfrac 12$.
As in \cite{OTz1}, we would like to define the weighted Gaussian measures
associated with the modified energies $\EE_{N, t}(v)$ and $\EE_t(v)$:\footnote{Noting that we have $\bfP_{\leq N} v = v$ for all solutions to \eqref{NLS7},
we have $\EE_{N, t}(\bfP_{\leq N} v) = \EE_{N, t}( v)$.
In the following, we explicitly insert $\bfP_{\leq N}$ for clarity.
A similar comment applies to $\NN^{(j)}_{0, N}$.}
\begin{equation}
\label{Gibbs1}
\begin{split}
d \rho_{s, N, r, t }
& = \text{``}Z_{s, N, r}^{-1} \ind_{\{ \| v\|_{L^2 } \leq r\}} e^{- \EE_{N, t}(\bfP_{\leq N} v)} dv\text{''}\\
& =
Z_{s, N, r}^{-1} F_{N, r, t} d\mu_s
\end{split}
\end{equation}
and
\begin{equation}
\label{Gibbs1a}
\begin{split}
d \rho_{s, r, t}
& = \text{``}Z_{s, r}^{-1} \ind_{\{ \| v\|_{L^2 } \leq r\}} e^{- \EE_t( v)} dv\text{''}\\
& =
Z_{s, r}^{-1} F_{r, t} d\mu_s,
\end{split}
\end{equation}
where $\EE_{N, t}$ is the modified energy defined in \eqref{energy6}
and
$F_{N, r, t }$ and $F_{r, t }$ are given by
\begin{align}
F_{N, r, t }(v) & := \ind_{\{ \| v\|_{L^2 } \leq r\}} \exp\bigg( \sum_{j = 2}^\infty \NN^{(j)}_{0, N}(\bfP_{\leq N} v)(t)\bigg),
\label{Gibbs2}\\
F_{ r, t }(v) & =
F_{ r,\infty, t }(v): =
\ind_{\{ \| v\|_{L^2 } \leq r\}} \exp\bigg( \sum_{j = 2}^\infty \NN^{(j)}_0( v)(t)\bigg).\notag
\end{align}

It follows from Proposition \ref{PROP:energy}
that
\[
\ind_{\{ \| v\|_{L^2 } \leq r\}}
\exp\bigg( \sum_{j = 2}^\infty \big|\NN^{(j)}_{0, N}( \bfP_{\leq N}v)\big|\bigg) \leq \exp\bigg(\sum_{j = 2}^\infty
C_0(j) r^{2j}\bigg)
\leq C(s, r),
\]
uniformly in $N \in \mathbb{N} \cup\{\infty\}$ and $t \in \bbR$.
Hence, we have
\begin{align*}
Z_{s, N, r} = \int_{H^{s-\frac{1}{2}-}} F_{N, r, t}(v) d\mu_s \leq C(s, r),
\end{align*}
uniformly in $N \in \mathbb{N}\cup\{\infty\}$ and $t \in \bbR$.
See also Remark \ref{REM:t1} below.
This shows that
$\rho_{s, N, r, t}$ and
$\rho_{s, r, t}$ in \eqref{Gibbs1} and \eqref{Gibbs1a} are well defined probability measures
on $H^{s-\sfrac 12 -\eps}(\bbT)$, $\eps > 0$.
Moreover, the following lemma immediately follows from the computation above
as in \cite[Prop.\,6.2\,\&\,Cor.\,6.3]{OTz1}.
See Section \ref{SUBSEC:finite}.

\begin{lemma}\label{LEM:Gibbs}
Let $s > \sfrac 12$ and $r > 0$.
\begin{enumeratei}
\item\label{LEM:Gibbsi}
Given any finite $p \geq 1$,
$F_{N, r, t}(v)$ converges to $F_{r, t}(v)$ in $L^p (\mu_s)$,
uniformly in $t\in \bbR$, as $N \to \infty$.

\item\label{LEM:Gibbsii}
For any $\g > 0$, there exists $N_0 \in \bbN$ such that
\[ | \rho_{s, N, r, t}(A) - \rho_{s, r, t}(A)| < \g \]
for any $N \geq N_0$
and any measurable set
$A \subset L^2(\bbT)$,
uniformly in $t\in \bbR$.
\end{enumeratei}
\end{lemma}

\begin{remark}\label{REM:t1}
The normalizing constants
$Z_{s, N, r}$ and $Z_{s, r}$
a priori depend on $t \in \bbR$.
It is, however, easy to see that
they are indeed independent of $t \in \bbR$
by (i) noticing that
the correction terms $\big\{\NN^{(j)}_{0, N}\big\}_{j = 2}^\infty$ defined in
Proposition \ref{PROP:energy}
is in fact autonomous in terms of $\wt u (t)= S(t) v(t)$
and
(ii) the invariance of $\mu_s$ under $S(t)$ (Lemma \ref{LEM:gauss}).
See also Remark \ref{REM:tdepend} below.
The same comment applies to the normalizing constant
$\ft Z_{s, N, r}$ defined in \eqref{norm1}.
\end{remark}

\subsection{A change-of-variable formula}

Next, we go over an important global aspect of the proof of Theorem \ref{THM:1}.
Given $N \in \bbN$, let $d L_N = \prod_{|n|\leq N} d\ft u_n $ denote the Lebesgue
measure on $E_N \cong \bbC^{2N+1}$.
Then,
from \eqref{Gibbs1}
and \eqref{Gibbs2} with \eqref{G4}, we have
\begin{align*}
d \rho_{s, N, r, t}
& = Z_{s, N, r}^{-1}
\ind_{\{ \| v\|_{L^2 } \leq r\}} e^{ \sum_{j = 2}^\infty \NN^{(j)}_{0, N}(\bfP_{\leq N} v)}
d\mu_s\notag\\
& =\ft Z_{s, N, r}^{-1}
\ind_{\{ \| v\|_{L^2 } \leq r\}} e^{- \EE_{N, t}(\bfP_{\leq N} v)}
dL_N \otimes
d\mu_{s, N}^\perp,
\end{align*}
where $\ft Z_{s, N, r}$ is a normalizing constant defined by
\begin{equation}
\label{norm1}
\ft Z_{s, N, r} = \int_{L^2} \ind_{\{ \| v\|_{L^2 } \leq r\}}
e^{- \EE_{N, t}( \bfP_{\leq N} v)} d L_N \otimes d\mu_{s, N}^\perp.
\end{equation}
Then, proceeding as in \cite{OTz1}
and exploiting invariance of $L_N$ under the map $\wt \Psi_N(t, \tau)$ for~\eqref{NLS7},
we have the following change-of-variable formula.

\begin{lemma}\label{LEM:meas1}
Let $s > \sfrac 12$, $N \in \bbN$, and $r > 0$.
Then, we have
\begin{align*}
\rho_{s, N, r, t}(\Psi_N(t, \tau)(A))
& = Z_{s, N, r}^{-1}\int_{\Psi_N(t, \tau)(A)} \ind_{\{ \| v\|_{L^2 } \leq r\}} e^{\sum_{j = 2}^\infty \NN^{(j)}_{0, N}( \bfP_{\leq N} v)} d\mu_s(v)
\notag \\
& = \ft Z_{s, N, r}^{-1}\int_{A} \ind_{\{ \| v\|_{L^2 } \leq r\}}
e^{- \EE_{N, t}( \bfP_{\leq N} \Psi_N(t, \tau) (v))} d L_N \otimes d\mu_{s, N}^\perp
\end{align*}
for any $t, \tau \in \bbR$ and any measurable set
$A \subset L^2(\bbT)$.
Here, $\Psi_N(t, \tau)$ is the solution map to~\eqref{NLS6}
defined in \eqref{Psit}.
\end{lemma}

\subsection{On the measure evolution property
and the proof of Theorem \ref{THM:1}}\label{SUBSEC:evo}

In this subsection, we
use the energy estimate (Proposition \ref{PROP:energy})
and the change-of-variable formula (Lemma \ref{LEM:meas1})
to establish a growth estimate
on the truncated weighted Gaussian measure $\rho_{s, N, r, t}$
under $\Psi_N(t) = \Psi_N(t, 0)$ for \eqref{NLS6}.
Thanks to the improved energy estimate,
the following estimates are simpler than those presented in \cite{OTz1}.

\begin{lemma}\label{LEM:meas2}
Let $ \sfrac 12 < s < 1$.
Then, given $r > 0$,
there exists $C = C(r)>0$ such that
\begin{align}
\frac{d}{dt} \rho_{s, N, r, t}(\Psi_N(t) (A))
\leq C \rho_{s, N, r, t} (\Psi_N(t)(A))
\label{meas2}
\end{align}
for any $N \in \bbN$, any $t \in \bbR$,
and any measurable set
$A \subset L^2(\bbT)$.
As a consequence, we have the following estimate;
given $t \in \bbR$ and $r > 0$,
there exists $C = C(t, r) > 0$
such that
\begin{align}
\rho_{s, N, r, t} (\Psi_N(t) (A)) \leq C\rho_{s, N, r, t} ( A)
\label{meas3}
\end{align}
for
any $N \in \bbN$
and any measurable set
$A \subset L^2(\bbT)$.
\end{lemma}

\begin{proof}
As in \cite{TzV1, TzV2, TzBBM, OTz1},
the main idea of the proof of Lemma \ref{LEM:meas2}
is to reduce the analysis to that at $t = 0$ in the spirit of the classical Liouville theorem
on Hamiltonian dynamics.
Let $t_0 \in \bbR$.
By the definition of
$\Psi(t, \tau)$, Lemma \ref{LEM:meas1}, and Proposition~\ref{PROP:energy}, we have
\begin{align*}
\frac{d}{dt}\rho_{s, N, r, t}& (\Psi_N(t)(A))\Big|_{t = t_0}\notag\\
&\hspace*{-3mm} = \frac{d}{dt}\rho_{s, N, r, t_0+t} \big(\Psi_N(t_0+ t, t_0)(\Psi_N(t_0)(A))\big)\Big|_{t = 0}\notag \\
&\hspace*{-3mm} = \ft Z_{s, N, r}^{-1}
\frac{d}{dt} \int_{\Psi_N(t_0) (A)} \ind_{\{ \| v\|_{L^2 } \leq r\}}
e^{- \EE_{N, t_0+t}( \bfP_{\leq N} \Psi_N(t_0+ t, t_0) (v))} d L_N \otimes d\mu_{s, N}^\perp \Big|_{t = 0} \notag\\
&\hspace*{-3mm} = -
\int_{\Psi_N(t_0)( A)}
\frac{d}{dt} \EE_{N, t_0+t}\big( \bfP_{\leq N} \Psi_N(t_0 + t, t_0) (v)\big)\Big|_{t = 0}
d \rho_{s, N, r, t_0}\\
&\hspace*{-3mm} \leq C_r
\rho_{s, N, r, t_0}(\Psi_N(t_0) (A)).
\end{align*}
This proves \eqref{meas2}.
The second estimate \eqref{meas3} follows from a direct integration of \eqref{meas2}.
\end{proof}

As in \cite{OTz1},
we can upgrade
Lemma \ref{LEM:meas2}
to the untruncated measure $\rho_{s, r, t}$.

\begin{lemma}\label{LEM:meas4}
Let $ \sfrac 12< s < 1$.
Then, given $t \in \bbR$ and $r > 0$,
there exists $C = C(t, r) > 0$
such that
\begin{align*}
\rho_{s, r, t} (\Psi(t) (A)) \leq C\rho_{s, r, t} ( A)
\end{align*}
for any measurable set
$A \subset L^2(\bbT)$.
\end{lemma}

This lemma follows from
the approximation properties of $\Psi_N(t)$ to $\Psi(t)$ (Lem\-ma~\ref{LEM:approx})
and $\rho_{s, N, r, t}$ to $\rho_{s, r, t}$ (Lemma \ref{LEM:Gibbs}\,\eqref{LEM:Gibbsii})
along with some limiting argument.
See \cite[Lem.\,6.10]{OTz1}
for the details of the proof.

Once we have Lemma \ref{LEM:meas4},
the proof of Theorem \ref{THM:1} follows
just as in \cite{OTz1}.
We present its proof for the convenience of readers.
Recall that in view of
\eqref{nonlin2} and
Lemmas \ref{LEM:gauss},
that it suffices to prove that $\mu_s$
is quasi-invariant under $\Psi(t)$, \ie under the dynamics of \eqref{NLS5}.

Fix $t \in \bbR$. Let $A \subset L^2(\bbT)$ be a measurable set
such that $\mu_s(A) = 0$.
Then, for any $r > 0$, we have
\[\mu_{s, r}(A) = 0.\]
By the mutual absolute continuity
of $\mu_{s, r}$ and $\rho_{s, r, t}$,
we obtain
\[\rho_{s, r, t}(A) = 0\]
for any $r > 0$.
Then, by Lemma \ref{LEM:meas4}, we have
\[\rho_{s, r, t}(\Psi(t) (A)) = 0.\]
By invoking the mutual absolute continuity
of $\mu_{s, r}$ and $\rho_{s, r, t}$ once again,
we have
\[\mu_{s, r}(\Psi(t) (A)) = 0.\]
Then, the dominated convergence theorem yields
\[\mu_{s}\big(\Psi(t) (A)\big)
= \lim_{r \to \infty}
\mu_{s, r}\big(\Psi(t) (A)\big) = 0.\]
This proves Theorem \ref{THM:1},
assuming Proposition \ref{PROP:energy}.
In the next section, we implement an infinite iteration
of normal form reductions and prove the improved energy estimate
(Proposition \ref{PROP:energy}).

\section{Proof of Proposition \ref{PROP:energy}: Normal form reductions}
\label{SEC:NF}

In this section, we present the proof of Proposition \ref{PROP:energy}
by implementing an infinite iteration scheme
of normal form reductions.
This procedure allows us to construct an infinite sequences of correction terms
and thus build the desired modified energies $\EE_{N, t}(v)$
and $\EE_{t}(v)$ in \eqref{energy6}.

Our main goal is to obtain an effective estimate on
the growth of the $H^s$-norm of a solution $v$ to the truncated equation \eqref{NLS7},
independent of $N \in \bbN$.
For simplicity of presentation, however, we work on the equation \eqref{NLS5}
without the frequency cutoff $\ind_{|n|\leq N}$
in the following.
We point out that the same normal form reductions
and estimates hold
for the truncated equation \eqref{NLS7}, uniformly in $N \in \bbN$,
with straightforward modifications:
(i) set
$\ft v_n = 0$ for all $|n| >N$
and
(ii) the multilinear forms for \eqref{NLS7}
are obtained by inserting
the frequency cutoff $\ind_{|n|\leq N}$ in appropriate places.\footnote{Using the bi-trees introduced
in Section \ref{SUBSEC:tree} below, it follows
from \eqref{NLS7} that
we simply need to insert the frequency cutoff $\ind_{|n^{(j)}|\leq N}$
on the parental frequency $n^{(j)}$ assigned to each non-terminal node $a \in \TT^0$.}
In the following, we introduce multilinear forms
such as
$\NN_0^{(j)}$, $\NN_1^{(j)}$, and $\RR^{(j)}$
for the untruncated equation \eqref{NLS5}.
With a small modification,
these multilinear forms give rise to
$\NN_{0, N}^{(j)}$, $\NN_{1, N}^{(j)}$, and $\RR_N^{(j)}$,
$N \in \bbN$,
for the truncated equation \eqref{NLS7},
appearing in Proposition \ref{PROP:energy}.
See Section \ref{SUBSEC:finite}.

\subsection{First few steps of normal form reductions}
\label{SUBSEC:41}

In the following, we describe the first few steps
of normal form reductions.
We keep the following discussion only at a formal level
since its purpose is to show the complexity of the problem
and the necessity of effective book-keeping notations
that we introduce in Section \ref{SUBSEC:tree}.
We will present the full procedure in Sections \ref{SUBSEC:NF} and \ref{SUBSEC:error}.

Let $v \in C(\bbR; H^\infty(\bbT))$
be a global solution to \eqref{NLS5}.\footnote{While we work
with \eqref{NLS5} without a frequency cutoff in the following,
it follows from the uniform boundedness of the frequency truncation operator $\bfP_{\leq N}$
that our argument and estimates also hold for \eqref{NLS7}, uniformly in $N \in \bbN$.
Noting that any solution to \eqref{NLS7} (for some $N \in \bbN$) is smooth,
the following computation can be easily justified for solutions to \eqref{NLS7}.
}
With $\phi(\bar n)$ and $\Gamma(n)$ as in~\eqref{phiX} and~\eqref{Gam1},
we have\footnote{Recall our convention of using $v_n$ to denote the Fourier coefficient $\ft v_n$.}
\begin{equation}
\begin{split}
\frac{d}{dt}\Bigl(\frac{1}{2} \|v(t) \|_{H^s}^2 \Bigr)
& = - \Re i
\sum_{n \in \bbZ} \sum_{\Gamma(n)}
\jb{n}^{2s}
e^{ - i \phi(\bar n) t} v_{n_1} \cj{ v_{n_2}} v_{n_3} \cj{ v_n}\\
& = : \NN^{(1)}(v)(t).
\label{XN^0}
\end{split}
\end{equation}
In view of Lemma \ref{LEM:phase} with \eqref{Gam1},
we have $|\phi(\bar n)| \geq 1 $
in the summation above.
Then,
by performing a normal form reduction,
namely,
differentiating by parts as in \eqref{P2},
we obtain
\begin{equation}\label{XN^1}
\begin{split}
\NN^{(1)}(v)(t)
& = \Re \frac{d}{dt} \bigg[
\sum_{n \in \bbZ}\sum_{\Gamma(n)}
\frac{e^{ - i \phi(\bar n) t}}{ \phi(\bar n)} \jb{n}^{2s} v_{n_1} \cj{ v_{n_2}} v_{n_3} \cj{ v_n}\bigg] \\
& \hphantom{X}
- \Re
\sum_{n \in \bbZ}\sum_{\Gamma(n)}
\frac{e^{ - i \phi(\bar n) t}}{ \phi(\bar n)} \jb{n}^{2s}
\dt \big( v_{n_1} \cj{ v_{n_2}} v_{n_3} \cj{ v_n}\big)\\
& = \Re \frac{d}{dt} \bigg[
\sum_{n \in \bbZ}\sum_{\Gamma(n)}
\frac{e^{ - i \phi(\bar n) t}}{ \phi(\bar n)} \jb{n}^{2s} v_{n_1} \cj{ v_{n_2}} v_{n_3} \cj{ v_n}\bigg] \\
& \hphantom{X}
- \Re
\sum_{n \in \bbZ}\sum_{\Gamma(n)}
\frac{e^{ - i \phi(\bar n) t}}{ \phi(\bar n)} \jb{n}^{2s}
\Big\{ \RR(v)_{n_1}
\cj{ v_{n_2}} v_{n_3} \cj{ v_n}\\
& \hphantom{XXX}
+ v_{n_1} \cj{\RR(v)_{n_2}}
v_{n_3} \cj{ v_n}
+ v_{n_1}
\cj{ v_{n_2}} \RR(v)_{n_3} \cj{ v_n}
+ v_{n_1}\cj{ v_{n_2}} v_{n_3}
\cj{\RR(v)_{n}}\Big\}
\\
& \hphantom{X}
- \Re
\sum_{n \in \bbZ}\sum_{\Gamma(n)}
\frac{e^{ - i \phi(\bar n) t}}{ \phi(\bar n)} \jb{n}^{2s}
\Big\{ \NN(v)_{n_1}
\cj{ v_{n_2}} v_{n_3} \cj{ v_n}\\
& \hphantom{XXX}
+ v_{n_1} \cj{\NN(v)_{n_2}}
v_{n_3} \cj{ v_n}
+ v_{n_1}
\cj{ v_{n_2}} \NN(v)_{n_3} \cj{ v_n}
+ v_{n_1}\cj{ v_{n_2}} v_{n_3}
\cj{\NN(v)_{n}}\Big\}
\\
& =: \dt \NN_0^{(2)}(v)(t) + \RR^{(2)}(v)(t) + \NN^{(2)}(v)(t).
\end{split}
\end{equation}
In the second equality, we
applied the product rule
and used the equation \eqref{NLS5}
to replace $\dt v_{n_j}$ (and $\dt v_n$, respectively) by
the resonant part $\RR(v)_{n_j}$
(and $\RR(v)_{n}$, respec\-tively) and the non-resonant part $\NN(v)_{n_j}$
(and $\NN(v)_{n}$, respectively).

As we see below,
we can estimate the boundary term $\NN_0^{(2)}$
and the contribution $\RR^{(2)}$ from the resonant part
in a straightforward manner.

\begin{lemma}
\label{LEM:N^2_0}
Let $\NN^{(2)}_0$ and $\RR^{(2)}$
be as in \eqref{XN^1}.
Then, we have
\[
| \NN^{(2)}_0(v)|\les \|v\|_{L^2}^4,\qquad
| \RR^{(2)}(v)|\les \|v\|_{L^2}^6.
\]
\end{lemma}

See Lemma \ref{LEM:N^J+1_0}
(with $J = 1$)
for the proof.

It remains to treat
the last term $\NN^{(2)}$ in \eqref{XN^1}.
For an expository purpose,
we only consider the first term among the four terms in $\NN^{(2)}$
in the following.
A full consideration is given in Section \ref{SUBSEC:NF}
once we introduce proper notations in Section \ref{SUBSEC:tree}.
With~\eqref{NLS5}, we have
\begin{equation}
\label{XN^1_2}
\NN^{(2)}(v)(t)
 = \Re i
\sum_{n \in \bbZ}\;\sum_{\Gamma(n)}\;
\sum_{\substack{n_1 = m_1 - m_2 + m_3\\n_1 \ne m_1, m_3}}\hspace*{-4.5mm}
\frac{e^{ - i (\phi_1 + \phi_2) t}}{ \phi_1} \jb{n}^{2s}
( v_{m_1}\cj{ v_{m_2}} v_{m_3} )
\cj{ v_{n_2}} v_{n_3} \cj{ v_n},
\end{equation}
where
$\phi_1 = \phi(\bar n)$ and
$\phi_2 = \phi(m_1, m_2, m_3, n_1) = m_1^4 - m_2^4 + m_3^4 - n_1^4$
denote the phase functions
from the first and second ``generations''.\footnote{In Section \ref{SUBSEC:tree},
we make a precise definition of what we mean by ``generation''.}
It turns out that we can not establish a direct 6-linear estimate
on $\NN^{(2)}$ in \eqref{XN^1_2}.

We divide the frequency space in \eqref{XN^1_2} into
\begin{equation} \label{C1}
C_1 = \big\{ |\phi_1 + \phi_2| \leq 6^3\big\}
\end{equation}
and its complement $C_1^c$.\footnote{Clearly, the number $6^3$ in \eqref{C1}
does not make any difference at this point.
However, we insert it to match with \eqref{Cj}.
See also \eqref{C2}.}
We then write $\NN^{(2)}$ as
\begin{equation}
\NN^{(2)} = \NN^{(2)}_1 + \NN_2^{(2)},
\label{N2}
\end{equation}
where $\NN^{(2)}_1$ is the restriction of $\NN^{(2)}$
onto $C_1$
and
$\NN_2^{(2)} := \NN^{(2)} - \NN^{(2)}_1$.
On the one hand,
we can estimate the contribution $\NN^{(2)}_1$ from $C_1$
in an effective manner (Lemma~\ref{LEM:N^J+1_1} with $J = 1$)
thanks to the frequency restriction on $C_1$.
On the other hand,
$\NN_2^{(2)}$ can not be handled as it is and
thus we apply the second step of normal form reductions
to $\NN^{(2)}_2$.
After differentiation by parts with \eqref{NLS5} as in \eqref{XN^1},
we arrive~at
\begin{align*}
\NN^{(2)}_2 (v)
& = \dt \NN^{(3)}_0(v) + \RR^{(3)}(v) + \NN^{(3)}(v),
\end{align*}
where
$ \NN^{(3)}_0$ is a 6-linear form
and
$\RR^{(3)}$ and $\NN^{(3)}$
are 8-linear forms, corresponding to
the contributions
from the resonant part $\RR(v)$ and the non-resonant part
$\NN(v)$ upon the substitution of \eqref{NLS5}.
See \eqref{N^2} below for the precise computation.

As in the previous step,
we can estimate $\NN^{(3)}_0$ and $\RR^{(3)}$ in a straightforward manner
(Lemma~\ref{LEM:N^J+1_0} with $J = 2$).
On the other hand,
we can not estimate $\NN^{(3)}$ as it is
and hence we need to split it as
\begin{equation}
\NN^{(3)} = \NN^{(3)}_1 + \NN^{(3)}_2,
\label{N3}
\end{equation}
where $\NN^{(3)}_1$ is the restriction of $\NN^{(3)}$
onto
\begin{equation} \label{C2}
C_2 = \big\{ |\phi_1 + \phi_2 + \phi_3| \le 8^3\big\}
\end{equation}
and
$\NN_2^{(3)} := \NN^{(3)} - \NN^{(3)}_1$.
Here, $\phi_j$, $j = 1, 2, 3$, denotes
the phase function from the $j$th ``generation''.
As we see below,
$\NN^{(3)}_1$ satisfies a good 8-linear estimate (Lemma~\ref{LEM:N^J+1_1} with $J = 2$)
thanks to the frequency restriction on $C_2$.
On the other hand,
$\NN_2^{(3)}$ can not be handled as it is and
thus we apply the third step of normal form reductions
to~$\NN^{(3)}_2$.
In this way, we iterate normal form reductions in an indefinite manner.

As we iteratively apply normal form reductions,
the degrees of the multilinear terms increase linearly.
After $J$ steps,
we obtain
the multilinear terms
$\NN_0^{(J+1)}$
of degree $2J+2$ and $\RR^{(J+1)}$ and $\NN^{(J+1)}$
of degree $2J+4$.
See \eqref{N^J+1}.
As in the first and second steps described above,
we also divide $\NN^{(J+1)}$ into ``good'' and ``bad'' parts
and apply another normal form reduction
to the bad part of degree $2J+4$, where time differentiation
can fall on any of the $2J+4$ factors.
An easy computation shows that the number of terms
grows factorially (see~\eqref{cj1})
and hence we need to introduce an effective way to handle this combinatorial complexity.
In Section \ref{SUBSEC:tree},
we introduce indexing notation by bi-trees,
which allows us to denote a factorially growing number of multilinear terms
in a concise manner.

\subsection{Notations: index by ordered bi-trees}
\label{SUBSEC:tree}

In \cite{GKO}, the first author with Guo and Kwon implemented
an infinite iteration of normal form reductions
to study the cubic NLS on $\bbT$,
where differentiation by parts was applied to the evolution equation satisfied by
the interaction representation.
In \cite{GKO}, (ternary) trees and ordered trees played
an important role
for indexing various multilinear terms and frequencies arising in the general steps of the Poincaré-Dulac normal form reductions.

Our main goal here is to implement an infinite iteration scheme
of normal form reduction applied to the $H^s$-energy functional\footnote{More precisely,
to the evolution equation satisfied by the $H^s$-energy functional.}
$ \| v(t)\|_{H^s}^2$, as we saw above.
In particular, we need tree-like structures that grow in two directions.
For this purpose, we introduce the notion of bi-trees and ordered bi-trees
in the following.
Once we replace trees and ordered trees
by bi-trees and ordered bi-trees,
other related notions can be defined
in a similar manner as in \cite{GKO}
with certain differences to be noted.

\begin{definition} \label{DEF:tree1}
Given a partially ordered set $\TT$ with partial order $\leq$,
we say that $b \in \TT$
with $b \leq a$ and $b \ne a$
is a child of $a \in \TT$,
if $b\leq c \leq a$ implies
either $c = a$ or $c = b$.
If the latter condition holds, we also say that $a$ is the parent of $b$.
\end{definition}

As in \cite{GKO},
our trees in this paper
refer to a particular subclass of usual trees with the following properties.

\skpt
\begin{definition} \label{DEF:tree2}
\begin{enumeratei}
\item
A tree $\TT $ is a finite partially ordered set satisfying
the following properties:
\begin{enumerate}

\item[(a)] Let $a_1, a_2, a_3, a_4 \in \TT$.
If $a_4 \leq a_2 \leq a_1$ and
$a_4 \leq a_3 \leq a_1$, then we have $a_2\leq a_3$ or $a_3 \leq a_2$,

\item[(b)]
A node $a\in \TT$ is called terminal, if it has no child.
A non-terminal node $a\in \TT$ is a node
with exactly three ordered\footnote{For example,
we simply label the three children as $a_1, a_2$, and $a_3$
by moving from left to right in the planar graphical representation of the tree $\TT$.
As we see below, we assign the Fourier coefficients of the interaction representation $v$ at $a_{1}$ and $a_{3}$,
while we assign the complex conjugate of
the Fourier coefficients of $v$ at the second child $a_{2}$.} children denoted by $a_1, a_2$, and $a_3$,

\item[(c)] There exists a maximal element $r \in \TT$ (called the root node) such that $a \leq r$ for all $a \in \TT$,

\item[(d)] $\TT$ consists of the disjoint union of $\TT^0$ and $\TT^\infty$,
where $\TT^0$ and $\TT^\infty$
denote the collections of non-terminal nodes and terminal nodes, respectively.
\end{enumerate}

\item
A {\it bi-tree} $\TT = \TT_1 \cup \TT_2$ is
a disjoint union of two trees $\TT_1$ and $\TT_2$,
where the root nodes $r_j$ of $\TT_j$, $j = 1, 2$, are joined by an edge.
A bi-tree
$\TT$ consists of the disjoint union of $\TT^0$ and $\TT^\infty$,
where $\TT^0$ and $\TT^\infty$
denote the collections of non-terminal nodes and terminal nodes, respectively.
By convention, we assume that the root node $r_1$ of the tree $\TT_1$ is non-terminal,
while the root node $r_2$ of the tree $\TT_2$ may be terminal.

\item
Given a bi-tree $\TT = \TT_1 \cup \TT_2$,
we define a projection $\Pi_j$, $j = 1, 2$, onto a tree
by setting
\begin{align}
\Pi_j(\TT) = \TT_j.
\label{proj1}
\end{align}
In Figure \ref{FIG:1},
$\Pi_1(\TT)$ corresponds to the tree on the left under the root node $r_1$,
while
$\Pi_2(\TT)$ corresponds to the tree on the right under the root node $r_2$.
\end{enumeratei}
\end{definition}

Note that the number $|\TT|$ of nodes in a bi-tree $\TT$ is $3j+2$ for some $j \in \mathbb{N}$,
where $|\TT^0| = j$ and $|\TT^\infty| = 2j + 2$.
Let us denote the collection of trees in the $j$th generation
(namely, with $j$ parental nodes) by $BT(j)$, \ie
\begin{equation*}
BT(j) := \{ \TT : \TT \text{ is a bi-tree with } |\TT| = 3j+2 \}.
\end{equation*}
\begin{figure}[htb]
\vspace*{-2\baselineskip}
\[ \<T1>
\qquad\qquad
\<T2>
\qquad \qquad
\<T3>\]
\caption{Examples of bi-trees of the $j$th generation, $j = 1, 2, 3$.}
\label{FIG:1}
\end{figure}

Next, we introduce the notion of ordered bi-trees,
for which we keep track of how a bi-tree ``grew''
into a given shape.

\skpt
\begin{definition} \label{DEF:tree3}
\begin{enumeratei}
\item\label{DEF:tree3i}
We say that a sequence $\{ \TT_j\}_{j = 1}^J$ is a chronicle of $J$ generations,
if
\begin{enumerate}
\item[(a)] $\TT_j \in BT(j)$ for each $j = 1, \dots, J$,
\item[(b)] $\TT_{j+1}$ is obtained by changing one of the terminal
nodes in $\TT_j$ into a non-terminal node (with three children), $j = 1, \dots, J - 1$.
\end{enumerate}
Given a chronicle $\{ \TT_j\}_{j = 1}^J$ of $J$ generations,
we refer to $\TT_J$ as an {\it ordered bi-tree} of the $J$th generation.
We denote the collection of the ordered trees of the $J$th generation
by $\mathfrak{BT}(J)$.
Note that the cardinality of $\mathfrak{BT}(J)$ is given by
$ |\mathfrak{BT}(1)| = 1$ and
\begin{equation}
\label{cj1}
|\mathfrak{BT}(J)| = 4\cdot 6 \cdot 8 \cdot \cdots \cdot 2J
= 2^{J-1} \cdot J!=: c_J,
\quad J \geq 2.
\end{equation}

\item\label{DEF:tree3ii}
Given an ordered bi-tree $\TT_J \in \mathfrak{BT}(J)$ as above,
we define projections $\pi_j$, $j = 1, \dots, J-1$,
onto the previous generations
by setting
\begin{align*}
\pi_j(\TT_J) = \TT_j \in \mathfrak{BT}(j).
\end{align*}
\end{enumeratei}
\end{definition}

We stress that the notion of ordered bi-trees comes with associated chronicles.
For example,
given two ordered bi-trees $\TT_J$ and $\wt{\TT}_J$
of the $J$th generation,
it may happen that $\TT_J = \wt{\TT}_J$ as bi-trees (namely as planar graphs)
according to Definition \ref{DEF:tree2},
while $\TT_J \ne \wt{\TT}_J$ as ordered bi-trees according to Definition \ref{DEF:tree3}.
In the following, when we refer to an ordered bi-tree $\TT_J$ of the $J$th generation,
it is understood that there is an underlying chronicle $\{ \TT_j\}_{j = 1}^J$.

Given a bi-tree $\TT$,
we associate each terminal node $a \in \TT^\infty$ with the Fourier coefficient (or its complex conjugate) of the interaction representation
$v$ and sum over all possible frequency assignments.
In order to do this, we introduce index functions,
assigning integers to {\it all} the nodes in $\TT$ in a consistent manner.

\skpt
\begin{definition} \label{DEF:tree4}
\begin{enumeratei}
\item
Given a bi-tree $\TT = \TT_1\cup \TT_2$,
we define an index function $\bmn: \TT \to \mathbb{Z}$ such that
\begin{itemize}

\item[(a)] $n_{r_1} = n_{r_2}$, where $r_j$ is the root node of the tree $\TT_j$, $j = 1, 2$,

\item[(b)] $n_a = n_{a_1} - n_{a_2} + n_{a_3}$ for $a \in \TT^0$,
where $a_1, a_2$, and $a_3$ denote the children of $a$,

\item[(c)] $\{n_a, n_{a_2}\} \cap \{n_{a_1}, n_{a_3}\} = \emptyset$ for $a \in \TT^0$,
\end{itemize}
where we identified $\bmn: \TT \to \mathbb{Z}$
with $\{n_a \}_{a\in \TT} \in \mathbb{Z}^\TT$.
We use
$\mathfrak{N}(\TT) \subset \mathbb{Z}^\TT$ to denote the collection of such index functions $\bmn$
on $\TT$.

Given $N \in \mathbb{N}$,
we define a subcollection
$\mathfrak{N}_N(\TT) \subset \mathfrak{N}(\TT)$
by imposing $|n_a| \leq N$ for any $a \in \TT$.
We also define
$\mathfrak{N}_N^0(\TT) \subset \mathfrak{N}(\TT)$
by imposing $|n_a| \leq N$ for any non-terminal nodes $a \in \TT^0$.

\item
Given a tree $\TT$, we also define
an index function $\bmn: \TT \to \mathbb{Z}$
by omitting the condition (a)
and denote by
$\mathfrak{N}(\TT) \subset \mathbb{Z}^\TT$ the collection of index functions $\bmn$
on $\TT$.
\end{enumeratei}
\end{definition}

\skpt
\begin{remark} \label{REM:terminal}
\begin{enumeratei}
\item
In view of the consistency condition (a),
we can refer to $n_{r_1} = n_{r_2}$
as the frequency at the root node without ambiguity.
We shall simply denote it by $n_r$ in the following.

\item
Just like index functions for (ordered) trees considered in \cite{GKO},
an index function $\bmn = \{n_a\}_{a\in\TT}$ for a bi-tree $\TT$ is completely determined
once we specify the values $n_a \in \bbZ$ for all the terminal nodes $a \in \TT^\infty$.
An index function $\bn$
for a bi-tree $\TT = \TT_1 \cup \TT_2$
is basically a pair $(\bn_1, \bn_2)$
of index functions $\bn_j$ for the trees $\TT_j$, $j = 1, 2$, (omitting
the non-resonance condition in \cite[Def.\,3.5 (iii)]{GKO}),
satisfying the consistency condition (a): $n_{r_1} = n_{r_2}$.

\item
Given a bi-tree $\TT \in \mathfrak{BT}(J)$ and $n \in \bbZ$, consider
the summation of all possible frequency assignments
$\{ \bn \in \mathfrak{N}(\TT): n_r = n\}$.
While $|\TT^\infty| = 2J + 2$,
there are $2J$ free variables in this summation.
Namely, the condition $n_r = n$ reduces two summation variables.
It is easy to see this by separately considering
the cases $\Pi_2(\TT) = \{r_2\}$
and $\Pi_2(\TT) \ne \{r_2\}$.
\end{enumeratei}
\end{remark}

Given an ordered bi-tree
$\TT_J$ of the $J$th generation with a chronicle $\{ \TT_j\}_{j = 1}^J$
and associated index functions $\bmn \in \mathfrak{N}(\TT_J)$,
we would like to keep track of the ``generations'' of frequencies.
In the following, we use superscripts to denote such generations of frequencies.

Fix $\bmn \in \mathfrak{N}(\TT_J)$.
Consider $\TT_1$ of the first generation.
Its nodes consist of the two root nodes $r_1$, $r_2$,
and the children $r_{11}, r_{12}, $ and $r_{13}$ of the first root node $r_1$.
See Figure~\ref{FIG:1}.
We define the first generation of frequencies by
\[\big(n^{(1)}, n^{(1)}_1, n^{(1)}_2, n^{(1)}_3\big) :=(n_{r_1}, n_{r_{11}}, n_{r_{12}}, n_{r_{13}}).\]
From Definition \ref{DEF:tree4}, we have
\begin{equation*}
n^{(1)} = n_{r_2}, \quad n^{(1)} = n^{(1)}_1 - n^{(1)}_2 + n^{(1)}_3, \quad n^{(1)}_2\ne n^{(1)}_1, n^{(1)}_3.
\end{equation*}

Next, we construct an ordered bi-tree $\TT_2$ of the second generation
from $\TT_1$ by
changing one of its terminal nodes $a \in \TT^\infty_1 = \{ r_2, r_{11}, r_{12}, r_{13}\}$
into a non-terminal node.
Then, we define
the second generation of frequencies by setting
\[\big(n^{(2)}, n^{(2)}_1, n^{(2)}_2, n^{(2)}_3\big) :=(n_a, n_{a_1}, n_{a_2}, n_{a_3}).\]
Note that we have $n^{(2)} = n^{(1)}$ or $n_k^{(1)}$ for some $k \in \{1, 2, 3\}$,
\begin{equation*}
n^{(2)} = n^{(2)}_1 - n^{(2)}_2 + n^{(2)}_3, \quad n^{(2)}_2\ne n^{(2)}_1, n^{(2)}_3,
\end{equation*}
where the last identities follow from Definition \ref{DEF:tree4}.
This extension of $\TT_1 \in \mathfrak{BT}(1)$ to $\TT_2\in \mathfrak{BT}(2)$
corresponds to introducing a new set of frequencies
after the first differentiation by parts,
where the time derivative may fall on any of $v_n$ and $v_{n_j}$, $j = 1, 2, 3$.\footnote{The complex conjugate signs on $v_n$ and $v_{n_j}$ do not play any significant role.
Hereafter, we drop the complex conjugate sign,
when it does not play any important role.}

In general, we construct an ordered bi-tree $\TT_j$
of the $j$th generation from $\TT_{j-1}$ by
changing one of its terminal nodes $a \in \TT^\infty_{j-1}$
into a non-terminal node.
Then, we define
the $j$th generation of frequencies by
\[\big(n^{(j)}, n^{(j)}_1, n^{(j)}_2, n^{(j)}_3\big) :=(n_a, n_{a_1}, n_{a_2}, n_{a_3}).\]
As before, it follows from Definition \ref{DEF:tree4} that
\begin{equation*}
n^{(j)} = n^{(j)}_1 - n^{(j)}_2 + n^{(j)}_3, \quad n^{(j)}_2\ne n^{(j)}_1, n^{(j)}_3.
\end{equation*}
Given an ordered bi-tree $\TT$, we denote by $B_j = B_j(\TT)$
the set of all possible frequencies in the $j$th generation.

We denote by $\phi_j$ the
phase function
for the frequencies introduced at the $j$th generation:
\begin{align*}
\phi_j & = \phi_j \big( n^{(j)}, n^{(j)}_1, n^{(j)}_2, n^{(j)}_3\big)
:= \big(n_1^{(j)}\big)^4 - \big(n_2^{(j)}\big)^4 + \big(n_3^{(j)}\big)^4
- \big(n^{(j)}\big)^4.
\end{align*}
Note that we have $|\phi_1| \geq 1$ in view of Definition \ref{DEF:tree4} and Lemma \ref{LEM:phase}.
We also denote by~$\mu_j$
the phase function corresponding to the usual cubic NLS (at the $j$th generation):
\begin{align*}
\mu_j & = \mu_j \big( n^{(j)}, n^{(j)}_1, n^{(j)}_2, n^{(j)}_3\big)
:= \big(n_1^{(j)}\big)^2 - \big(n_2^{(j)}\big)^2 + \big(n_3^{(j)}\big)^2
- \big(n^{(j)}\big)^2\notag \\
& =-2 \big(n^{(j)} - n_1^{(j)}\big) \big(n^{(j)} - n_3^{(j)}\big).
\end{align*}
Then, by Lemma \ref{LEM:phase}, we have
\begin{align}
|\phi_j| \sim (n^{(j)}_\text{max})^2
\cdot| \big(n^{(j)} - n_1^{(j)}\big) \big(n^{(j)} - n_3^{(j)}\big)|
\sim (n^{(j)}_\text{max})^2 \cdot |\mu_j|,
\label{MU2}
\end{align}
where $n^{(j)}_\text{max}: = \max\big(|n^{(j)}|,
|n_1^{(j)}|, |n_2^{(j)}|, |n_3^{(j)}| \big)$.

Given an ordered bi-tree $\TT \in \mathfrak{BT}(J)$ for some $J \in \bbN$,
define $C_j \subset \mathfrak{N}(\TT)$ by
\begin{equation} \label{Cj}
C_j = \big\{ |\wt{\phi}_{j+1}| \leq (2j+4)^3\big\} ,
\end{equation}
where $\wt{\phi}_j$ is defined by
\begin{align}
\wt{\phi}_j = \sum_{k = 1}^j \phi_k.
\label{Cj2}
\end{align}
In Section \ref{SUBSEC:NF},
we perform normal form reductions in an iterative manner.
At each step,
we divide multilinear forms into
``nearly resonant'' part (corresponding to the frequencies belonging to $C_j$) and highly non-resonant part
(corresponding to the frequencies belonging to $C_j^c$)
and apply a normal form reduction only to the highly non-resonant part.

\subsection{Arithmetic lemma}
\label{SUBSEC:sum}

As we see in Section \ref{SUBSEC:NF},
normal form reductions generate multilinear forms of higher and higher degrees,
where we need to sum over all possible
ordered bi-trees in $\mathfrak{BT}(J)$.
The main issue is then to control the rapidly growing
cardinality $c_J = |\mathfrak{BT}(J)| $ defined in~\eqref{cj1}.
On the one hand, we utilize the divisor counting estimate (see \eqref{divisor} below)
as in \cite{GKO}.
On the other hand,
we split the argument into two parts.
The following lemma shows the heart of the matter
in the multilinear estimates presented in Section \ref{SUBSEC:NF}.
This allows us to show that there is a sufficiently fast decay
at each step of normal form reductions.

\begin{lemma}\label{LEM:sum1}
Let $ s< 1$ and $J \in \bbN$.
Then, the following estimates hold:
\begin{enumeratei}
\item\mbox{}\vspace*{-10pt}
\begin{equation}
\sup_{\TT_J \in \mathfrak{BT}(J)}
\sup_{n \in \bbZ} \sum_{\substack{\bmn \in \mathfrak{N}(\TT_J)\\{ n}_r = n}}
\ind_{{\scriptscriptstyle\bigcap}_{j = 1}^{J-1} C_j^c}
\frac{\jb{n}^{4s}}{|\phi_1|^{2}}
\prod_{j = 2}^J \frac{1}{|\wt{\phi}_j|^2}
\lesssim \frac{1}{\prod_{j = 2}^J(2j+2)^{3-}},
\label{sum1}
\end{equation}
\item\mbox{}\vspace*{-10pt}
\begin{equation}
\sup_{\TT_{J+1} \in \mathfrak{BT}(J+1)}
\sup_{n \in \bbZ} \sum_{\substack{\bmn \in \mathfrak{N}(\TT_{J+1})\\{ n}_r = n}}\hspace*{-2mm}
\ind_{\bigl({\scriptscriptstyle\bigcap}_{j = 1}^{J-1} C_j^c\bigr) \cap C_J}
\frac{\jb{n}^{4s}}{|\phi_1|^{2}}
\prod_{j = 2}^J \frac{1}{|\wt{\phi}_j|^2}
\lesssim \frac{J^{3+}}{\prod_{j = 2}^J(2j+2)^{3-}}.
\label{sum2}
\end{equation}
\end{enumeratei}
\end{lemma}

Before proceeding further, let us
recall the following arithmetic fact \cite{HW}.
Given $n \in \bbN$, the number $d(n)$ of the divisors of $n$
satisfies
\begin{align}
d(n) \leq C_\dl n^\dl
\label{divisor}
\end{align}
for any $\dl > 0$.
This divisor counting estimate will be used iteratively in the following proof.

\skpt
\begin{proof}[Proof of Lemma \ref{LEM:sum1}]
\begin{enumeratei}
\item\label{pf:sum1}
We first consider the case $J = 1$.
In this case,
from Lemma \ref{LEM:phase} with $s \leq 1$, we have
\begin{align}
\text{LHS of }\eqref{sum1}
\leq
\sup_{n \in \bbZ} \sum_{\substack{n_1, n_3 \in \bbZ\\n_1, n_3 \ne n\\|\phi_1|\geq 1}}
\frac{\jb{n}^{4s}}{|\phi_1|^{2}}
\les \sup_{n \in \bbZ} \sum_{\substack{n_1, n_3 \in \bbZ\\n_1, n_3 \ne n}}
\frac{1}{|(n-n_1)(n-n_3)|^{2}}
\les 1.
\label{sum1a}
\end{align}

Next, we consider the case $J \geq 2$.
Fix $\TT_J \in \mathfrak{BT}(J)$.
For simplicity of notations,
we drop the supremum
over $\TT_J \in \mathfrak{BT}(J)$ in the following
with the understanding that the implicit constants are independent
of $\TT_J \in \mathfrak{BT}(J)$.
A similar comment applies to the proof of the estimate \eqref{sum2} presented in \eqref{pf:sum2} below.

The main idea is to apply the divisor counting argument in an iterative manner.
It~follows from the divisor counting estimate \eqref{divisor}
with the factorization of $\phi_j$ (Lemma \ref{LEM:phase}) that
for fixed $n^{(j)}$ and $\phi_j$,
there are at most $O(|\phi_j|^{0+})$ many choices for $n^{(j)}_1$, $n^{(j)}_2$, and $n^{(j)}_3$
on $B_j$.
Also, note that $\phi_j$ is determined by $\wt{\phi}_1, \dots, \wt{\phi}_j$
and
\begin{equation} \label{mujj}
|\phi_j| \leq \max(|\wt{\phi}_{j-1}|, |\wt{\phi}_j|)
\end{equation}
since $\phi_j = \wt{\phi}_{j}-\wt{\phi}_{j-1}$.
In the following, we apply the divisor counting argument
to sum over the frequencies in $B_J$, $B_{J-1}$, \dots, $B_2$.
From Definition \ref{DEF:tree3}\,\eqref{DEF:tree3ii} and \eqref{Cj}, we have
\begin{align*}
\text{LHS of }\eqref{sum1}
& =
\sup_{n\in \bbZ}
\sum_{\substack{\bmn \in \mathfrak{N}(\pi_1(\TT_J))\\{ n}_r = n\\|\phi_1| \geq 1}}
\frac{\jb{n}^{4s}}{|\phi_1|^{2}}
\sum_{\substack{\psi_2 \in \bbZ\\ |\psi_2| > 6^3 }} \sum_{\substack{B_2\\ \wt \phi_2 = \psi_2}}
\frac{1}{|\psi_2|^2}
\, \cdots
\sum_{\substack{\psi_J \in \bbZ\\ |\psi_J| > (2J+2)^3 }} \sum_{\substack{B_J\\ \wt \phi_J = \psi_J}}
\frac{1}{|\psi_J|^2}.
\notag \\
\intertext{By applying the divisor counting argument in $B_J$ with \eqref{mujj},
we have}
& \les
\sup_{n \in \bbZ}
\sum_{\substack{\bmn \in \mathfrak{N}(\pi_1(\TT_J))\\{ n}_r = n\\|\phi_1| \geq 1}}
\frac{\jb{n}^{4s}}{|\phi_1|^{2}}
\sum_{\substack{\psi_2 \in \bbZ\\ |\psi_2| > 6^3 }} \sum_{\substack{B_2\\ \wt \phi_2 = \psi_2}}
\frac{1}{|\psi_2|^2} \, \cdots \notag\\
& \hphantom{XX}
\sum_{\substack{\psi_{J-1} \in \bbZ\\ |\psi_{J-1}| > (2J)^3 }} \sum_{\substack{B_{J-1}\\ \wt \phi_{J-1} = \psi_{J-1}}}
\frac{1}{|\psi_{J-1}|^2}
\sum_{\substack{\psi_J \in \bbZ\\ |\psi_J| > (2J+2)^3 }}
\frac{1}{|\psi_J|^{2}} |\psi_{J-1}|^{0+} |\psi_J|^{0+}.
\intertext{By iteratively applying the divisor counting argument in $B_{J-1}$, \dots, $B_2$,
we have}
& \les
\sup_{n \in \bbZ}
\sum_{\substack{\bmn \in \mathfrak{N}(\pi_1(\TT_J))\\{ n}_r = n\\|\phi_1| \geq 1}}
\frac{\jb{n}^{4s}}{|\phi_1|^{2-}}
\sum_{\substack{\psi_2, \dots, \psi_J \in \bbZ\\ |\psi_j| > (2j+2)^3\\j = 2, \dots, J}}
\prod_{j = 2}^J\frac{1}{|\psi_j|^{2-}}\notag\\
& \lesssim \frac{1}{\prod_{j = 2}^J(2j+2)^{3-}},
\end{align*}
where the last inequality follows from
\eqref{sum1a}.

\item\label{pf:sum2}
Fix $\TT_{J+1} \in \mathfrak{BT}(J+1)$.
We proceed with the divisor counting argument as in~\eqref{pf:sum1}.
From~\eqref{Cj}, we have
$|\phi_{J+1}| \les |\wt{\phi}_J| + J^3$ on $C_J$ and thus
for fixed $n^{(J+1)}$ and $\phi_{J+1}$,
there are at most
$O(J^{0+} |\wt \phi_{J}|^{0+})$
many choices for $n^{(J+1)}_1$, $n^{(J+1)}_2$, and $n^{(J+1)}_3$
on $B_{J+1}$.
Also, on $C_J$,
there are at most $O\big(J^3\big)$ many choices
for $\wt{\phi}_{J+1}$.
Hence,
for fixed~$\wt{\phi}_J$,
there are also at most $O\big(J^3\big)$ many choices
for $\phi_{J+1} = \wt{\phi}_{J+1} - \wt{\phi}_J$
on $C_J$.
Then, the contribution to \eqref{sum2} in this case is estimate by
\begin{align*}
\text{LHS of } \eqref{sum2}
& =
\sup_{n \in \bbZ}
\sum_{\substack{\bmn \in \mathfrak{N}(\pi_1(\TT_{J+1}))\\{ n}_r = n\\|\phi_1| \geq 1}}
\frac{\jb{n}^{4s}}{|\phi_1|^{2}}
\sum_{\substack{\psi_2 \in \bbZ\\ |\psi_2| > 6^3 }} \sum_{\substack{B_2\\ \wt \phi_2 = \psi_2}}
\frac{1}{|\psi_2|^2}
\, \cdots \\
& \hspace*{3cm}
\sum_{\substack{\psi_J \in \bbZ\\ |\psi_J| > (2J+2)^3 }} \sum_{\substack{B_J\\ \wt \phi_J = \psi_J}}
\frac{1}{|\psi_J|^2}
\sum_{\substack{B_{J+1}\\ |\phi_{J+1}+\psi_J| \leq (2J+4)^3}} 1.
\\
\intertext{By applying the divisor counting argument in $B_{J+1}$,
we have}
&\les J^{3+} \sup_{n \in \bbZ}
\sum_{\substack{\bmn \in \mathfrak{N}(\pi_1(\TT_{J+1}))\\{ n}_r = n\\|\phi_1| \geq 1}}
\frac{\jb{n}^{4s}}{|\phi_1|^{2}}
\sum_{\substack{\psi_2 \in \bbZ\\ |\psi_2| > 6^3 }} \sum_{\substack{B_2\\ \wt \phi_2 = \psi_2}}
\frac{1}{|\psi_2|^2}
\, \cdots
\\
& \hspace*{5cm}
\sum_{\substack{\psi_J \in \bbZ\\ |\psi_J| > (2J+2)^3 }} \sum_{\substack{B_J\\ \wt \phi_J = \psi_J}}
\frac{1}{|\psi_J|^2}
|\psi_J|^{0+}.
\intertext{By iteratively applying the divisor counting argument in $B_{J}$, \dots, $B_2$
and then applying~\eqref{sum1a},
we have}
& \les
J^{3+}
\sup_{n \in \bbZ}
\sum_{\substack{\bmn \in \mathfrak{N}(\pi_1(\TT_{J+1}))\\{ n}_r = n\\|\phi_1|\geq 1} }
\frac{\jb{n}^{4s}}{|\phi_1|^{2-}}
\sum_{\substack{\psi_J \in \bbZ\\ |\psi_J|>(2j+2)^3\\ j = 2, \dots, J} }
\prod_{j = 2}^J \frac{1}{|\psi_j|^{2-}}
\\
& \lesssim \frac{J^{3+}
}{\prod_{j = 2}^J(2j+2)^{3-}}.
\end{align*}
This proves \eqref{sum2}.\qedhere
\end{enumeratei}
\end{proof}

\begin{remark}
In \cite{GKO}, the authors applied the divisor counting argument even to the frequencies of the first generation.
On the other hand, we did not apply
the divisor counting argument to the frequencies of the first generation
in the proof of Lemma \ref{LEM:sum1} above.
Instead, we simply used \eqref{sum1a} to control the first generation.
By using only the factor $\mu_1 = -2 (n^{(1)} - n_1^{(1)})(n^{(1)} - n_3^{(1)})$
(and not the entire $\phi_1$)
for the summation,
\eqref{sum1a}
allows us to exhibit the required smoothing
in Proposition \ref{PROP:energy}.
\end{remark}

\subsection{Normal form reductions}
\label{SUBSEC:NF}

With the notations introduced in Section \ref{SUBSEC:tree},
let us revisit the discussion in Section \ref{SUBSEC:41}
and then discuss the general $J$th step.
We first implement
a formal infinite iteration scheme of normal form reductions
without justifying switching of limits and summations.
We justify formal computations at the end of this subsection.
Let $v \in C(\bbR; H^\infty(\bbT))$
be a global solution to \eqref{NLS5}.
Using the notations introduced in Section \ref{SUBSEC:tree},
we write \eqref{XN^0} as
\[
\frac{d}{dt}\Bigl(\frac{1}{2} \|v(t) \|_{H^s}^2 \Bigr)
 =
- \Re i
\sum_{\TT_1 \in \mathfrak{BT}(1)}
\sum_{\bmn \in \mathfrak{N}(\TT_1)}
\jb{n_r}^{2s}
e^{ - i \phi_1 t } \prod_{a \in \TT^\infty_1} v_{n_{a}}
= : \NN^{(1)}(v)(t).
\]
By performing a normal form reduction,
we then obtain
\begin{equation}
\label{N^1}
\begin{split}
\NN^{(1)}(v)(t)
& =
\Re \dt \bigg[
\sum_{\TT_1 \in \mathfrak{BT}(1)}
\sum_{\bmn \in \mathfrak{N}(\TT_1)}
\frac{ \jb{n_r}^{2s} e^{- i \phi_1 t } }{\phi_1}
\prod_{a \in \TT_1^\infty} v_{n_{a}}
\bigg] \\
& \hphantom{X}
- \Re
\sum_{\TT_1 \in \mathfrak{BT}(1)}
\sum_{\bmn \in \mathfrak{N}(\TT_1)}
\frac{ \jb{n_r}^{2s} e^{- i \phi_1 t } }{\phi_1}
\dt\bigg(\prod_{a \in \TT^\infty_1} v_{n_{a}}\bigg)
\\
& = \Re \dt \bigg[
\sum_{\TT_1 \in \mathfrak{BT}(1)}
\sum_{\bmn \in \mathfrak{N}(\TT_1)}
\frac{ \jb{n_r}^{2s} e^{- i \phi_1 t } }{\phi_1}
\prod_{a \in \TT^\infty_1} v_{n_{a}}
\bigg] \\
& \hphantom{X}
- \Re
\sum_{\TT_1 \in \mathfrak{BT}(1)}
\sum_{b\in \TT_1^\infty}
\sum_{\bmn \in \mathfrak{N}(\TT_1)}
\frac{ \jb{n_r}^{2s} e^{- i \phi_1 t } }{\phi_1}
\RR(v)_{n_b}
\prod_{a \in \TT^\infty_1\setminus\{b\}} v_{n_{a}} \\
& \hphantom{X}
+ \Re i
\sum_{\TT_2 \in \mathfrak{BT}(2)}
\sum_{\bmn \in \mathfrak{N}(\TT_2)}
\frac{ \jb{n_r}^{2s} e^{- i ( \phi_1+ \phi_2) t } }{\phi_1}
\prod_{a \in \TT^\infty_2} v_{n_{a}}
\\
& =: \dt \NN_0^{(2)}(v)(t) + \RR^{(2)}(v)(t) + \NN^{(2)}(v)(t).
\end{split}
\end{equation}
Compare \eqref{N^1} with \eqref{XN^1}.
In the second equality, we
applied the product rule
and used the equation \eqref{NLS5}
to replace $\dt v_{n_b}$ by
the resonant part $\RR(v)_{n_b}$ and the non-resonant part $\NN(v)_{n_b}$.
In substituting the non-resonant part $\NN(v)_{n_b}$,
we turned the terminal node $b \in \TT^\infty_1$
into a non-terminal node with three children $b_1, b_2,$ and $b_3$,
which corresponds to extending the tree $\TT_1 \in \mathfrak{BT}(1)$
(and $\bmn\in \mathfrak{N}(\TT_1)$)
to $\TT_2 \in \mathfrak{BT}(2)$
(and to $\bmn\in \mathfrak{N}(\TT_2)$, respectively).

\skpt
\begin{remark} \label{REM:tdepend}
\begin{enumeratei}
\item
Strictly speaking, the phase factor appearing in $\NN^{(2)}(v)$
may be $\phi_1 - \phi_2$
when the time derivative falls on the terms with the complex conjugate.
In the following, however, we simply write it as $\phi_1 + \phi_2$ since
it does not make any difference in our analysis.
Also, we often replace $\pm 1$ and $\pm i$ by $1$
for simplicity when they do not play an important role.
Lastly, for notational simplicity,
we drop the real part symbol
on multilinear forms
with the understanding that all the multilinear forms
appear with
the real part symbol.

\item
Due to the presence of
$e^{-i \phi_1 t}$
in their definitions,
the multilinear forms such as $\NN_0^{(2)}(v)$
are non-autonomous in $t$.
Therefore, strictly speaking,
they should be denoted as $\NN_0^{(2)}(t)(v(t))$.
In the following, however, we establish nonlinear estimates
on these multilinear forms, uniformly in $t \in \bbR$,
by simply using $|e^{-i \phi_1 t}| = 1$.
Hence,
we simply suppress such $t$-dependence
when there is no confusion.
The same comment applies to other multilinear forms.
Note that this convention was already used in Proposition \ref{PROP:energy}.

It is worthwhile to note that the multilinear forms introduced in this section
are non-autonomous when they are expressed in terms
of the interaction representation~$v$, solving \eqref{NLS5}.
When they are expressed in terms of the original solution $u$ to \eqref{4NLS0}
(or $\wt u$ to \eqref{NLS4}), however,
it is easy to see that these multilinear terms are indeed autonomous.
\end{enumeratei}
\end{remark}

Thanks to Lemma~\ref{LEM:sum1},
the terms $\NN_0^{(2)}$ and $\RR^{(2)}$
can be estimated
in a straightforward manner;
see Lemma~\ref{LEM:N^J+1_0} below.
We split
$\NN^{(2)}$ as in \eqref{N2}.
As mentioned above,
the good part $\NN^{(2)}_1$
is handled
in an effective manner (Lemma \ref{LEM:N^J+1_1})
thanks to the frequency restriction on $C_1$.
We then apply the second step of normal form reductions
to $\NN^{(2)}_2$
and obtain
\begin{equation}
\begin{split}
\NN^{(2)}_2 (v)
& = \dt \bigg[\sum_{\TT_2 \in \mathfrak{BT}(2)}
\sum_{\bmn \in \mathfrak{N}(\TT_2)}
\ind_{C_1^c} \frac{\jb{n_r}^{2s}e^{- i( \phi_1 + \phi_2 )t } }{\phi_1(\phi_1+\phi_2)}
\prod_{a \in \TT^\infty_2} v_{n_{a}} \bigg] \\
& \hphantom{X} -
\sum_{\TT_2 \in \mathfrak{BT}(2)}
\sum_{b \in\TT^\infty_2}
\sum_{\bmn \in \mathfrak{N}(\TT_2)}
\ind_{C_1^c}\frac{\jb{n_r}^{2s}e^{- i( \phi_1 + \phi_2)t } }{\phi_1(\phi_1+\phi_2)}
\, \RR(v)_{n_b}
\hspace*{-2mm}\prod_{a \in \TT^\infty_2 \setminus \{b\}}\hspace*{-2mm} v_{n_{a}} \\
%
& \hphantom{X}
- \sum_{\TT_3 \in \mathfrak{BT}(3)}\sum_{\bmn \in \mathfrak{N}(\TT_3)}
\ind_{C_1^c}\frac{\jb{n_r}^{2s}e^{- i( \phi_1 + \phi_2 +\phi_3)t } }{\phi_1(\phi_1+\phi_2)}
\,\prod_{a \in \TT^\infty_3} v_{n_{a}}\\
& =: \dt \NN^{(3)}_0(v) + \RR^{(3)}(v) + \NN^{(3)}(v).
\label{N^2}
\end{split}
\end{equation}
As in the previous step,
we can estimate $\NN^{(3)}_0$ and $\RR^{(3)}$ in a straightforward manner
(Lemma~\ref{LEM:N^J+1_0}),
while we split $\NN^{(3)}$
into the good part $\NN^{(3)}_1$ and the bad part $\NN^{(3)}_2$ as in \eqref{N3},
where $\NN^{(3)}_1$ is the restriction of $\NN^{(3)}$
onto $C_2$ defined in \eqref{Cj}.
We then apply the third step of normal form reductions
to the bad part $\NN^{(3)}_2$.
In this way, we iterate normal form reductions in an indefinite manner.

After the $J$th step, we have
\begin{equation}
\begin{split} \label{N^J+1}
\NN^{(J)}_2 (v)
& = \dt \bigg[
\sum_{\TT_J \in \mathfrak{BT}(J)}
\sum_{\bmn \in \mathfrak{N}(\TT_J)}
\ind_{\bigcap_{j = 1}^{J-1} C_j^c}
\frac{\jb{n_r}^{2s}e^{- i \wt{\phi}_Jt } }{\prod_{j = 1}^J \wt{\phi}_j}
\, \prod_{a \in \TT^\infty_J} v_{n_{a}}
\bigg] \\
& \hphantom{X}
- \hspace*{-2mm}\sum_{\TT_{J} \in \mathfrak{BT}(J)}
\sum_{b \in\TT^\infty_J}
\sum_{\bmn \in \mathfrak{N}(\TT_J)}
\ind_{\bigcap_{j = 1}^{J-1} C_j^c}
\frac{\jb{n_r}^{2s}e^{- i \wt{\phi}_Jt } }{\prod_{j = 1}^J \wt{\phi}_j}
\, \RR(v)_{n_b}
\hspace*{-3mm}\prod_{a \in \TT^\infty_J \setminus \{b\}}\hspace*{-3mm} v_{n_{a}} \\
& \hphantom{X}
- \hspace*{-2mm}\sum_{\TT_{J+1} \in \mathfrak{BT}(J+1)}
\sum_{\bmn \in \mathfrak{N}(\TT_{J+1})}
\ind_{\bigcap_{j = 1}^{J-1} C_j^c}
\frac{\jb{n_r}^{2s}e^{- i \wt{\phi}_{J+1}t } }{\prod_{j = 1}^J \wt{\phi}_j}
\prod_{a \in \TT^\infty_{J+1}}\hspace*{-2mm} v_{n_{a}}\\
& =: \dt \NN^{(J+1)}_0 (v)+ \RR^{(J+1)}(v) + \NN^{(J+1)}(v),
\end{split}
\end{equation}
where
$\wt{\phi}_J$ is as in \eqref{Cj2}.
In the following, we first estimate $\NN^{(J+1)}_0$ and $\RR^{(J+1)}$
by applying Cauchy-Schwarz inequality
and then applying the divisor counting argument (Lemma \ref{LEM:sum1}).

\begin{lemma}\label{LEM:N^J+1_0}
Let $\NN^{(J+1)}_0$ and $\RR^{(J+1)}$ be as in \eqref{N^J+1}.
Then, for any $s < 1$,
we have
\begin{align}
| \NN^{(J+1)}_0(v)| \lesssim
\frac{c_J}{\prod_{j = 2}^J(2j+2)^{\frac{3}{2}-}}
\|v\|_{L^2}^{2J+2},
\label{N^J+1_0-1}\\
| \RR^{(J+1)}(v)|
\lesssim
\frac{ (2J+2)\cdot c_J}{\prod_{j = 2}^J(2j+2)^{\frac{3}{2}-}}
\|v\|_{L^2}^{2J+4}.
\label{N^J+1_r}
\end{align}
\end{lemma}

\begin{proof}
We split the proof into the following two cases:
\[ \text{(i) } \Pi_2(\TT_J) = \{r_2\}
\qquad \text{and} \qquad \text{(ii) }
\Pi_2(\TT_J) \ne \{r_2\},\]
where $\Pi_2$ denotes the projection defined in \eqref{proj1}.

\subsubsection*{Case \textup{(i)}}
We first consider the case $\Pi_2(\TT_J) = \{r_2\}$.
Recall that
for general $J \in \bbN$,
we need to control the rapidly growing
cardinality
$c_J = |\mathfrak{BT}(J)| $ defined in \eqref{cj1}.
By Cauchy-Schwarz inequality and
Lemma \ref{LEM:sum1}, we have
\begin{align*}
| \NN^{(J+1)}_0(v)|
& \lesssim \|v\|_{L^2}
\sum_{\substack{\TT_J \in \mathfrak{BT}(J)\\\Pi_2(\TT_J) = \{r_2\}}}
\Bigg\{ \sum_{n \in \bbZ}
\bigg(
\sum_{\substack{\bmn \in \mathfrak{N}(\TT_J)\\{ n}_r = n}}
\ind_{\bigcap_{j = 1}^{J-1} C_j^c}
\frac{\jb{n}^{4s}}{|\phi_1|^{2}}
\prod_{j = 2}^J \frac{1}{|\wt{\phi}_j|^2}
\bigg) \\[-5pt]
& \hspace*{5cm} \times
\bigg(\sum_{\substack{\bmn \in \mathfrak{N}(\TT_J)\\{ n}_r = n}}
\prod_{a \in \TT^\infty_J\setminus\{r_2\}} |v_{n_a}|^2
\bigg)\Bigg\}^{\sfrac{1}{2}} \\
& \lesssim \frac{c_J}{\prod_{j = 2}^J(2j+2)^{\frac{3}{2}-}}\,
\|v\|_{L^2}^{2J+2}.
\end{align*}

\subsubsection*{Case \textup{(ii)}}
Next, we consider the case $\Pi_2(\TT_J) \ne \{r_2\}$.
In this case, we need to modify the argument above since
the frequency $n_r = n$ does not correspond to a terminal node.
Note that
$\TT_J^\infty = \Pi_1(\TT_J)^\infty \cup \Pi_2(\TT_J)^\infty$
and
\begin{align}
\sum_{\substack{\bmn \in \mathfrak{N}(\TT_J)\\{ n}_r = n}}
\prod_{a \in \TT^\infty_J} |v_{n_a}|^2
= \prod_{j = 1}^2 \bigg(\sum_{\substack{\bmn \in \mathfrak{N}(\Pi_j(\TT_J))\\{ n}_{r_j} = n}}
\prod_{a_j \in \Pi_j(\TT_J)^\infty} |v_{n_{a_j}}|^2
\bigg).
\label{TT1}
\end{align}
Then,
by Cauchy-Schwarz inequality and Lemma \ref{LEM:sum1}
with \eqref{TT1},
we have
\begin{align*}
| \NN^{(J+1)}_0(v)|
& \lesssim
\sum_{\substack{\TT_J \in \mathfrak{BT}(J)\\\Pi_2(\TT_J) = \{r_2\}}}
\sum_{n \in \bbZ}
\Bigg\{
\bigg(
\sum_{\substack{\bmn \in \mathfrak{N}(\TT_J)\\{ n}_r = n}}
\ind_{\bigcap_{j = 1}^{J-1} C_j^c}
\frac{\jb{n}^{4s}}{|\phi_1|^{2}}
\prod_{j = 2}^J \frac{1}{|\wt{\phi}_j|^2}
\bigg) \\[-8pt]
& \hspace*{5cm} \times
\bigg(\sum_{\substack{\bmn \in \mathfrak{N}(\TT_J)\\{n}_r = n}}
\prod_{a \in \TT^\infty_J} |v_{n_a}|^2
\bigg)\Bigg\}^{\sfrac{1}{2}} \\
&\hspace*{-12mm} \les
\frac{c_J}{\prod_{j = 2}^J(2j+2)^{\frac{3}{2}-}}
\sup_{\substack{\TT_J \in \mathfrak{BT}(2)\\ \Pi_2(\TT_J) \ne \{r_2\}}}
\sum_{n \in \bbZ} \Bigg\{ \prod_{j = 1}^2
\bigg( \sum_{\substack{\bmn \in \mathfrak{N}(\Pi_j(\TT_J))\\{ n}_{r_j} = n}}
\prod_{a_j \in \Pi_j(\TT_J)^\infty} |v_{n_{a_j}}|^2\bigg)\Bigg\}^{\sfrac{1}{2}}.
\intertext{By Cauchy-Schwarz inequality in $n$, }
& \hspace*{-12mm}\lesssim \frac{c_J}{\prod_{j = 2}^J(2j+2)^{\frac{3}{2}-}}
\sup_{\substack{\TT_J \in \mathfrak{BT}(2)\\ \Pi_2(\TT_J) \ne \{r_2\}}}
\prod_{j = 1}^2 \Bigg\{
\bigg( \sum_n \sum_{\substack{\bmn \in \mathfrak{N}(\Pi_j(\TT_J))\\{ n}_{r_j} = n}}
\prod_{a_j \in \Pi_j(\TT_J)^\infty} |v_{n_{a_j}}|^2\bigg)\Bigg\}^{\sfrac{1}{2}} \\
& \hspace*{-12mm}\lesssim \frac{c_J}{\prod_{j = 2}^J(2j+2)^{\frac{3}{2}-}}
\|v\|_{L^2}^{2J+2}.
\end{align*}
This proves the first estimate \eqref{N^J+1_0-1}.

The second estimate \eqref{N^J+1_r}
follows from \eqref{N^J+1_0-1} and $\l^2_n\subset\l^6_n$,
noting that, given $\TT_J \in \mathfrak{BT}(J)$, we have $\#\{b: b \in \TT^\infty_J\} = 2J+2$.
\end{proof}

Next, we treat $\NN^{(J+1)}$ in \eqref{N^J+1}.
As before, we write
\begin{equation} \label{N^J+1_ZZ}
\NN^{(J+1)} = \NN^{(J+1)}_1 + \NN^{(J+1)}_2,
\end{equation}
where $\NN^{(J+1)}_1$ is the restriction of $\NN^{(J+1)}$
onto $C_J$ defined in \eqref{Cj}
and
$\NN^{(J+1)}_2 := \NN^{(J+1)} - \NN^{(J+1)}_1$.
In the following lemma, we estimate the first term
in \eqref{N^J+1_1}:
\begin{equation}
\NN^{(J+1)}_1 (v)
= - \hspace*{-4mm}\sum_{\TT_{J+1} \in \mathfrak{BT}(J+1)}
\sum_{\bmn \in \mathfrak{N}(\TT_{J+1})}
\ind_{(\bigcap_{j = 1}^{J-1} C_j^c) \cap C_J}
\frac{\jb{n_r}^{2s}e^{- i \wt{\phi}_{J+1}t } }{\prod_{j = 1}^J \wt{\phi}_j}
\hspace*{-2mm}\prod_{a \in \TT^\infty_{J+1}}\hspace*{-2mm} v_{n_{a}}.
\label{N^J+1_X}
\end{equation}
Then, we apply a normal form reduction
once again to the second term $\NN^{(J+1)}_2$
as in \eqref{N^J+1}.
In Section \ref{SUBSEC:error},
we show that the error term
$\NN^{(J+1)}_2$ tends to~$0$ as~$J \to \infty$.

\begin{lemma}\label{LEM:N^J+1_1}
Let $\NN^{(J+1)}_1$ be as in \eqref{N^J+1_X}.
Then, for any $s< 1$,
we have
\begin{align}
| \NN^{(J+1)}_1(v)|
\lesssim
\frac{ J^{\frac{3}{2}+}\cdot c_{J+1}}{\prod_{j = 2}^J(2j+2)^{\frac{3}{2}-}}
\|v\|_{L^2}^{2J+4}.
\label{N^J+1_1}
\end{align}
\end{lemma}

\begin{proof}
We only discuss the case $\Pi_2(\TT_{J+1}) = \{r_2\}$
since the modification for the case $\Pi_2(\TT_{J+1}) \ne \{r_2\}$ is straightforward as in
the proof of Lemma \ref{LEM:N^J+1_0}.
By Cauchy-Schwarz inequality and
Lemma \ref{sum1}, we have
\begin{align*}
| \NN^{(J+1)}_1(v)|
& \lesssim \|v\|_{L^2}
\sum_{\substack{\TT_{J+1} \in \mathfrak{BT}(J+1)\\\Pi_2(\TT_{J+1}) = \{r_2\}}}
\Bigg\{ \sum_{n \in \bbZ}
\bigg(
\sum_{\substack{\bmn \in \mathfrak{N}(\TT_{J+1})\\{ n}_r = n}}
\ind_{(\bigcap_{j = 1}^{J-1} C_j^c) \cap C_J}
\frac{\jb{n}^{4s}}{|\phi_1|^{2}}
\prod_{j = 2}^J \frac{1}{|\wt{\phi}_j|^2}
\bigg) \notag \\
& \hspace*{5cm} \times
\bigg(\sum_{\substack{\bmn \in \mathfrak{N}(\TT_{J+1})\\{ n}_r = n}}
\prod_{a \in \TT^\infty_{J+1}\setminus\{r_2\}} |v_{n_a}|^2
\bigg)\Bigg\}^{\sfrac{1}{2}} \notag \\
& \les \frac{ J^{\frac{3}{2}+}\cdot c_{J+1}}{\prod_{j = 2}^J(2j+2)^{\frac{3}{2}-}}
\|v\|_{L^2}^{2J+4}.
\end{align*}
This proves \eqref{N^J+1_1}.
\end{proof}

\begin{remark}

A notable difference from \cite{GKO} appears in our definition of $C_j$
in \eqref{Cj};
on the one hand, the cutoff size on $\wt \phi_{j+1}$ in \cite{GKO}
depended on $\wt \phi_j$ and $\phi_1$.
On the other hand, our choice of the cutoff size on $\wt \phi_{j+1}$ in \eqref{Cj}
is independent of~$\wt \phi_j$ or~$\phi_1$,
thus providing simplification of the argument.

Another difference appears in the first step of the normal form reductions.
On the one hand,
we simply applied the first normal form reduction in \eqref{N^1}
without introducing a cutoff on the phase function $\phi_1$.
On the other hand,
in \cite{GKO}, a cutoff of the form\footnote{In \cite{GKO},
this parameter was denoted by $N$. Here, we use $K$ to avoid
the confusion with the frequency truncation parameter $N \in \bbN$.} $|\phi_1| >K$ was introduced
to separate the first multilinear term
into the nearly resonant and non-resonant parts.
The use of this extra parameter $K = K(\|u_0\|_{L^2})$
allowed the authors to show that the local existence time
can be given by $T \sim \|u_0\|_{L^2}^{-\al}$ for some $\al > 0$.
See \cite{GKO} for details.
Since our argument only requires the summability
(in $J$) of the multilinear forms,
we do not need to introduce this extra parameter.
\end{remark}

We conclude this subsection by briefly discussing how to justify all the formal steps
performed in the normal form reductions.
In particular, we need to justify
\begin{enumeratei}

\item
the application of the product rule and

\item
switching time derivatives and summations
\end{enumeratei}
Suppose that a solution $v$ to \eqref{NLS5} lies
in $C(\bbR; H^{\sfrac{1}{6}}(\bbT))$.
Then,
from \eqref{NLS5}, we have
\begin{equation*}
\|\dt v_n\|_{C_T \ell^\infty_n}
\leq \big\| \FF^{-1}(|\ft v_n|)\big\|_{C_T L^3_x}^3
\les \big\| \FF^{-1}(|\ft v_n|)\big\|_{C_T H^{\sfrac{1}{6}}_x}^3
= \| v\|_{C_T H^{\sfrac{1}{6}}_x}^3
\end{equation*}
for each $T > 0$, where $C_TB_x = C([-T, T]; B_x)$.
Hence, $\dt v_n \in C([-T, T]; \ell^\infty_n)$,
justifying (i) the application of the product rule.
Note that given $N \in \bbN$, any solution~$v$ to \eqref{NLS7} belongs
to $C(\bbR; H^\infty(\bbT))$
and hence (i) is justified.
Moreover, the summations over spatial frequencies in the normal form reductions
applied to solutions to~\eqref{NLS7}
are all finite
and therefore, (ii) the switching time derivatives and summations
over spatial frequencies trivially hold true
for \eqref{NLS7}.
In general,
the proof of Lemma \ref{LEM:N^J+1_0}
shows that the summation defining $\NN_0^{(j)}$
converges (absolutely and uniformly in time).
Then, Lemma 5.1 in~\cite{GKO} allows us to switch
the time derivative with the summations
as temporal distributions, thus
justifying differentiation by parts.

\subsection{On the error term $\NN_2^{(J+1)}$}
\label{SUBSEC:error}

In this subsection, we prove that $\NN_2^{(J+1)}$
in \eqref{N^J+1_ZZ}
tends to 0 as $J \to \infty$ under some regularity assumption on $v$.
From \eqref{N^J+1}, we have
\begin{align} \label{X1}
\NN_2^{(J+1)}(v)
& =- \sum_{\TT_{J+1} \in \mathfrak{BT}(J+1)}
\sum_{\bmn \in \mathfrak{N}(\TT_{J+1})}
\ind_{\bigcap_{j = 1}^{J} C_j^c}
\frac{\jb{n_r}^{2s}e^{- i \wt{\phi}_{J+1}t } }{\prod_{j = 1}^J \wt{\phi}_j}
\,\prod_{a \in \TT^\infty_{J+1}} v_{n_{a}}.
\end{align}

\begin{lemma}\label{LEM:N^J+1_2}
Let $\NN^{(J+1)}_2$ be as in \eqref{X1}.
Then, given $v\!\in\!H^s(\bbT)$, $s\!>\!\sfrac 12$,
we have
\begin{align}
| \NN^{(J+1)}_2(v)| \longrightarrow 0,
\label{X1a}
\end{align}
as $J \to \infty$.
\end{lemma}

We point out that one can actually prove Lemma \ref{LEM:N^J+1_2}
under a weaker regularity assumption $s \geq \sfrac 16$.
See \cite{OW}.
For our purpose, however, we only need to prove the vanishing of the error term
$\NN^{(J+1)}_2$ for sufficiently regular functions;
our main objective is to obtain an energy estimate
(on the modified energy $\EE_{N, t}$ defined in \eqref{energy6})
for solutions to the truncated equation \eqref{NLS7}.
Given $N \in \bbN$, any solution $v$ to \eqref{NLS7} belongs
to $C(\bbR; H^\infty(\bbT))$.
Therefore, while
the convergence speed in \eqref{X1a} depends on $N \in \bbN$,
the final energy estimate
\eqref{energy7} holds with an implicit constant {\it independent} of $N \in \bbN$.

\begin{proof}

Given $\bmn \in \mathfrak{N}(\TT_{J+1})$,
it follows from Definition \ref{DEF:tree4} and the triangle inequality that
there exists $C_0 > 0$ such that
\begin{align}
|n_r| \leq C_0^J |n_{b_k}|
\label{X2}
\end{align}
for (at least)
two terminal nodes $b_1, b_2 \in \TT_{J+1}^\infty$
Then, by Young's inequality (placing $v_{n_{b_k}}$ in $\l^2_n$, $k = 1, 2$,
and the rest in $\l^1_n$) with \eqref{cj1}, \eqref{Cj}, and \eqref{X2}, we have
\begin{align*}
|\NN_2^{(J+1)}(v)|
& \les \frac{C_0^{2sJ}\cdot c_J}{\prod_{j = 2}^J(2j+2)^3}\\[-5pt]
&\hspace*{5mm}\times\sup_{\TT_{J+1} \in \mathfrak{BT}(J+1)}
\sum_{\bmn \in \mathfrak{N}(\TT_{J+1})}
\bigg(\prod_{k = 1}^2 \jb{n_{b_k}}^{s}|v_{n_{b_k}}|\bigg)
\bigg(\prod_{a \in \TT^\infty_{J+1}\setminus{\{b_1, b_2\}}} |v_{n_{a}}|\bigg)\\
& \les \frac{C_0^{2sJ}\cdot c_J}{\prod_{j = 2}^J(2j+2)^3}
\|v\|_{H^s}^{2J+4} \longrightarrow 0,
\end{align*}
as $J \to \infty$.
\end{proof}

\subsection{Improved energy bound}
\label{SUBSEC:improved}

We are now ready to establish
the improved energy estimate \eqref{energy7}.
Let $v$ be a smooth global solution\footnote{In fact, it suffices to assume that $v \in C(\bbR; H^{\sfrac{1}{6}}(\bbT))$.
See \cite{GKO, OW}.} to \eqref{NLS5}.
Then, by applying the normal form reduction $J$ times, we obtain\footnote{Once again, we are replacing $\pm 1$
and $\pm i$
by 1 for simplicity since they play no role in our analysis.}
\[
\dt \Bigl(\frac 12 \| v \|_{H^s}^2 \Bigr)
= \dt \sum_{j = 2}^{J+1} \NN^{(j)}_0(v)
+ \sum_{j = 2}^{J+1} \NN^{(j)}_1(v) + \sum_{j = 2}^{J+1} \RR^{(j)}(v)
+ \NN^{(J+1)}_2.
\]
Thanks to Lemma \ref{LEM:N^J+1_2},
by taking the limit as $J \to \infty$, we obtain
\[
\dt \Bigl(\frac 12 \| v \|_{H^s}^2 \Bigr)
= \dt \sum_{j = 2}^\infty \NN^{(j)}_0(v)
+ \sum_{j = 2}^\infty \NN^{(j)}_1(v) + \sum_{j = 2}^\infty \RR^{(j)}(v).
\]
In other words, defining the modified energy $\EE_t(v)$
by
\[
\EE_t(v) := \frac 12 \| v (t) \|_{H^s}^2- \sum_{j = 2}^\infty \NN^{(j)}_0(v)(t),
\]
we have
\[
\dt \EE_t(v)
= \sum_{j = 2}^\infty \NN^{(j)}_1(v)(t) + \sum_{j = 2}^\infty \RR^{(j)}(v)(t).
\]
Suppose that $\|v\|_{C(\bbR; L^2)} \leq r$.
Then, applying Lemmas \ref{LEM:N^J+1_0} and \ref{LEM:N^J+1_1} with \eqref{cj1},
we obtain
\begin{align*}
|\dt \EE_t(v) |
& \les
\sum_{j =2}^\infty
\frac{c_{j-1}}{\prod_{k = 2}^{j-1}(2k+2)^{\frac{3}{2}-}}
r^{2j}
+ \sum_{j =2}^\infty
\frac{ j^{\frac{3}{2}+}\cdot c_{j}}{\prod_{k = 2}^{j-1}(2k+2)^{\frac{3}{2}-}}
r^{2j+2}\\
& \hspace*{4.1cm}+ \sum_{j =2}^\infty
\frac{ j\cdot c_{j-1}}{\prod_{k = 2}^{j-1}(2k+2)^{\frac{3}{2}-}}
r^{2j+2}\\
& \leq C(r).
\end{align*}
In view of the boundedness of the
frequency projections
and noting that any solution to~\eqref{NLS7} is in $H^\infty(\bbT)$,
the same energy estimate holds for solutions to the truncated equation~\eqref{NLS7},
uniformly in $N \in \bbN$.

\subsection{On the proof of Lemma \ref{LEM:Gibbs}}
\label{SUBSEC:finite}

We conclude this section with a brief discussion on the proof of Lemma \ref{LEM:Gibbs}.
First, note that Lemma~\ref{LEM:Gibbs}\,(ii) is an immediate corollary
of Lemma~\ref{LEM:Gibbs}\,(i).
Moreover, Lemma~\ref{LEM:Gibbs}\,(i) follows from Egoroff's theorem
once we prove that
\[\mathfrak{S}_N( v) = \sum_{j = 2}^\infty \NN^{(j)}_{0, N}( v)\]
converges almost surely to
\[\mathfrak{S}_\infty(v) = \sum_{j = 2}^\infty \NN^{(j)}_0( v).\]
See \cite[Prop.\,6.2]{OTz1}.
In fact, one can show that
$\mathfrak{S}_N( v)$ converges to $\mathfrak{S}_\infty( v)$
for any $v \in L^2(\bbT)$.

Recall from \eqref{N^J+1} that
$\NN^{(j)}_0( v)$ consists of a sum of the multilinear forms
associated with ordered bi-trees $\TT_{j-1} \in \mathfrak{BT}(j-1)$.
Given $N \in \bbN$, the multilinear form
$\NN^{(j)}_{0, N}( v)$
is obtained
in a similar manner
with the following modifications:

\begin{enumeratei}
\item
We set
$ v_n = 0$ for all $|n| >N$.
This corresponds to setting
$v_{n_a} = 0$ for all $|n| >N$ and all terminal nodes $a \in \TT_{j-1}^\infty$.

\item
In view of \eqref{NLS7},
we also set
$v_{n_a} = 0$ for all $|n| >N$ and all parental nodes in $\TT_{j-1}$.
This amounts to setting
$v_{n_a} = 0$ for all $|n| >N$ and all non-terminal nodes $a \in \TT_{j-1}^0$.
\end{enumeratei}
In particular, we have
\begin{align}
\NN^{(j)}_{0, N} ( v)
= \sum_{\TT_{j-1} \in \mathfrak{BT}(j-1)}
\sum_{\bmn \in \mathfrak{N}_N(\TT_{j-1})}
\ind_{\bigcap_{k = 1}^{j-2} C_k^c}
\frac{\jb{n_r}^{2s}e^{- i \wt{\phi}_{j-1}t } }{\prod_{k = 1}^{j-1} \wt{\phi}_k}
\, \prod_{a \in \TT^\infty_{j-1}} v_{n_{a}},
\label{NFX}
\end{align}
where $ \mathfrak{N}_N(\TT_{j-1})$ is as in Definition \ref{DEF:tree4}.
Namely,
$\NN^{(j)}_{0, N} (v)$ is obtained from $\NN^{(j)}_{0} (v)$
by simply truncating
all the frequencies (including the ``hidden''\footnote{Namely, the parental frequencies
at the non-terminal nodes do not appear explicitly in the sum in~\eqref{NFX}
but they implicitly appear through the relation in Definition \ref{DEF:tree4}.} parental frequencies)
by $N \in \bbN$.

Write
\begin{equation}
\label{NFX1}
\begin{split}
\NN^{(j)}_0( v) - \NN^{(j)}_{0, N}( v)
& = \Big\{\NN^{(j)}_0( v) - \wt \NN^{(j)}_{0, N}( v)\Big\}
+ \Big\{\wt \NN^{(j)}_{0, N}( v) - \NN^{(j)}_{0, N}( v)\Big\} \\
& =: \1_j + \II_j,
\end{split}
\end{equation}
where $\wt \NN^{(j)}_{0, N} ( v)$ is
obtained from
$\NN^{(j)}_{0, N} ( v)$
by replacing
$ \bmn \in \mathfrak{N}_N(\TT_{j-1})$
with
$ \bmn \in \mathfrak{N}_N^0(\TT_{j-1})$.
\ie we are truncating only the parental frequencies
at non-terminal nodes $a \in \TT^0_{j-1}$.
Then, by writing
\[ \II_j = \wt \NN^{(j)}_{0, N}( v) - \wt \NN^{(j)}_{0, N}( \bfP_{\leq N} v), \]
it follows from the multilinearity and the boundedness in $L^2(\bbT)$ (Lemma \ref{LEM:N^J+1_0})
that the second term $\II_j$ tends to 0 as $N \to \infty$,
by simply writing the difference in a telescoping sum.
More precisely, we write $\II$ as a telescoping sum, replacing
$2j$ factors of $v_{n_a}$, $a \in \TT^\infty_{j-1}$, into
$2j$ factors of $(\bfP_{\leq N} v)_{n_a}$.
This
introduces
$2j$ differences, each containing exactly one
factor of $v - \bfP_{\leq N} v$
(tending to 0 as $N \to \infty$).
We then simply apply Lemma~\ref{LEM:N^J+1_0} on each difference.

Similarly, we can show that $\1_j$ in \eqref{NFX1} tends to 0 as $N \to \infty$
by writing the difference in a telescoping sum.
Namely, noting only the difference
between $\NN^{(j)}_0( v)$ and $ \wt \NN^{(j)}_{0, N}( v)$
is the frequency cutoffs $\ind_{|n_a| \leq N}$ at each non-terminal node $a \in \TT^0_{j-1}$,
we introduce $j-1$ differences by adding the frequency cutoff $\ind_{|n_a| \leq N}$ at each non-terminal node
in a sequential manner.
By construction, each of the $j-1$ differences
has one non-terminal node $a_* \in \TT^0_{j-1}$ with the restriction $|n_{a_*}| > N$.
Then, from Definition \ref{DEF:tree4},
we see that there exists at least one terminal node $b \in \TT^\infty_{j-1}$
which is a descendants of $a_*$ such that
\begin{align*}
|n_b| \geq C_0^{-J} |n_{a_*}| > C_0^{-J}N
\end{align*}
for some $C_0 > 0$ (compare this with \eqref{X2}).
This forces each of the $j-1$ differences in the telescoping sum
to tend to 0 as $N \to \infty$,
and hence $\1_j$ in \eqref{NFX1} tends to 0 as $N \to 0$.

Therefore,
$\NN^{(j)}_{0, N}( v)$ converges to
$ \NN^{(j)}_0( v)$ as $n \to \infty$ for any $v \in L^2(\bbT)$.
Finally, in view of the fast decay in $j$ in Lemma \ref{LEM:N^J+1_0},
the convergence of $\mathfrak{S}_N( v)$ to $\mathfrak{S}_\infty( v)$
follows from
the dominated convergence theorem.

\section{Proof of Theorem \ref{THM:sing}:
Non quasi-invariance under \texorpdfstring{the~dispersionless~model}{dm}}
\label{SEC:sing}

In this section, we present the proof of
Theorem \ref{THM:sing}. The basic ingredients are
the Fourier series representation of the (fractional)
Brownian loops, the law of the iterated logarithm,
and the solution formula \eqref{formula} to the dispersionless model \eqref{ND1}.
More precisely,
we show that, while the Gaussian random initial data distributed according to $\mu_s$
satisfies the law of the iterated logarithm,
the solution given by \eqref{formula} does not satisfy
the law of the iterated logarithm for any non-zero time.
We divide the argument into three cases:
(i) $s = 1$ corresponding to
the Brownian/Ornstein-Uhlenbeck loop,
(ii) $\sfrac 12 < s < \sfrac 32$,
corresponding to the fractional Brownian loop
(and $s > \sfrac 32$ with $s \notin \sfrac 12 + \bbN$),
and (iii) $s \in \sfrac 12 + \bbN$: the critical case.
For simplicity, we set $t = 1$ in the following. The proof for non-zero $t \ne1$ follows in a similar manner.

\subsection{Brownian/Ornstein-Uhlenbeck loop}\label{SUBSEC:5.1}
We first consider the $s= 1$ case.
Under the law of the random Fourier series
\begin{equation}\label{BL}
u(x) =u(x; \o) = \sum_{n\in \mathbb{Z}}\frac{g_n(\o)}{\jb{n}}\,e^{inx}
\end{equation}
corresponding to the Gaussian measure $\mu_1$,
$\Re u$ and $\Im u$ are independent stationary Ornstein-Uhlenbeck (OU) processes (in $x$)
on $ [0,2\pi)$. Recall that the law of this process can be written as
\begin{equation}\label{rep1}
u(x) \stackrel{\mathrm{d}}{=} \bfP_{\ne 0}w(x) + g_0
= w(x) -\fint_0^{2\pi}w(y) dy + g_0,
\end{equation}
where $w$ is a complex OU bridge with $w(0)=w(2\pi)=0$
and $g_0$ is a standard complex-valued Gaussian random variable
(independent from $w$).

We now recall the law of the iterated logarithm for the Brownian motion (see \cite[I.16.1]{RW}):
\begin{proposition}\label{LEM:LIL1}
Let $B(t)$ be a standard Brownian motion on $\bbR_+$. Then, for each $t \geq 0$,
\begin{equation}\label{LIL1}
\limsup_{h\downarrow 0} \frac{B(t+h)-B(t)}{\sqrt{2h\log\log \sfrac{1}{h}}}=1,
\end{equation}
almost surely.
\end{proposition}

It follows from
the representation \eqref{rep1},
the absolute continuity\footnote{The absolute continuity property
claimed here can be easily seen by the Fourier series representations
of the Brownian motion/bridge (with \eqref{BL} and \eqref{rep1})
and Kakutani's theorem (Lemma \ref{LEM:Kakutani} below).
For example, the Brownian motion $B(t)$ on $[0, 2\pi)$ has
the following Fourier-Wiener series
\[B(t) = g_0 t + \sum_{n \in \bbZ \setminus \{0\}} \frac{g_n}{n}\, e^{int}.\] }
of the Brownian bridge with respect to the Brownian motion
on any interval $[0,\gamma)$, $\gamma<2\pi$,
and the absolute continuity of the OU bridge with respect to the Brownian bridge also on any interval $[0,\gamma)$,
that the limit \eqref{LIL1} also holds for $\Re u$ and $\Im u$ on $[0, 2 \pi)$.

Define $\psi$ by
\[\psi(h) = \sqrt{2h\log\log \sfrac{1}{h}}, \quad 0<h<1.\]
Let $0 \leq x < 2\pi$.
As a corollary to Proposition \ref{LEM:LIL1}, we have
\begin{align}
\label{LIL2}
& \limsup_{h\downarrow 0} \frac{\Re u(x+h)-\Re u(x)}{\psi(h)}=1
\end{align}
almost surely.

In the following,
by a direct calculation,
we show that
$\Re[e^{-i |u|^2}u]$ does not satisfy \eqref{LIL2} with a positive probability.
This will show that
the pushforward measure $\wt \Phi(t)_* \mu_s$
under the dynamics of the dispersionless model \eqref{ND1}
is not absolutely continuous with respect to the Gaussian measure $\mu_s$.

On the one hand, we have
\begin{multline*}
\Re [e^{-i|u(y) |^2} u(y)] -\Re[e^{-i|u(x)|^2}u(x)]\notag \\
 =(\Re u(y)-\Re u(x))\cos |u(y)|^2+(\cos|u(y)|^2-\cos|u(x)|^2)\Re u(x)\\
+(\Im u(y)-\Im u(x))\sin |u(y)|^2+(\sin |u(y)|^2-\sin|u(x)|^2)\Im u(x).
\end{multline*}
On the other hand,
by the Taylor expansion
with $\eta(x, y) = |u(y)|^2-|u(x)|^2$, we have
\begin{align*}
\cos |u(y)|^2 &=\cos |u(x)|^2 -\sin |u(x)|^2 \cdot \eta(x, y) + O\big(\eta^2(x, y)\big), \\
\sin |u(y)|^2 &= \sin |u(x)|^2 +\cos |u(x)|^2 \cdot \eta(x, y)+O\big(\eta^2(x, y)\big).
\end{align*}
Putting together, we obtain
\begin{align}
\Re [e^{- i|u(y)|^2}& u(y)] -\Re[e^{- i|u(x)|^2}u(x)] \notag\\
& = (\Re u(y)-\Re u(x))\cos |u(y)|^2
- \sin |u(x)|^2 \cdot \eta(x, y) \Re u(x) \notag \\
& \hphantom{X}
+ (\Im u(y)-\Im u(x))\sin |u(y)|^2
+\cos|u(x)|^2\cdot \eta(x, y) \Im u(x)\notag \\
& \hphantom{X}
+ O\big(\eta^2(x, y)\big)\cdot(|\Re u(x)| +|\Im u(x)|)\notag \\
& = (\Re u(y)-\Re u(x))\cos |u(y)|^2\notag \\
& \hphantom{X}
- \sin |u(x)|^2 \cdot\big\{(\Re u(y) - \Re u(x))(\Re u(y)+\Re u(x)) \big\} \Re u(x)\label{LIL4} \\
& \hphantom{X}
-\sin |u(x)|^2 \cdot\big\{(\Im u(y) - \Im u(x))(\Im u(y)+\Im u(x)) \big\} \Re u(x) \notag \\
& \hphantom{X}
+(\Im u(y)-\Im u(x))\sin |u(y)|^2\notag \\
& \hphantom{X}
+\cos|u(x)|^2\cdot\big\{(\Re u(y)-\Re u(x))(\Re u(y)+\Re u(x))\big\} \Im u(x)\notag \\
& \hphantom{X}
+\cos|u(x)|^2\cdot\big\{(\Im u(y)-\Im u(x))(\Im u(y)+\Im u(x))\big\} \Im u(x)\notag \\
& \hphantom{X}
+ O\big(\eta^2(x, y)\big)\cdot(|\Re u(x)| +|\Im u(x)|).
\notag
\end{align}

Fix $0 \leq x < 2\pi$.
Let $\{h_n = h_n(\o)\}_{n \in \bbN}$ be a (random) sequence achieving the limit supremum in \eqref{LIL2}
almost surely.
Then, for this sequence $\{h_n\}_{n \in \bbN}$, we have
\begin{align}
& \limsup_{n \to \infty} \frac{|\Im u(x+h_n)-\Im u(x)|}{\psi(h_n)}\leq 1
\label{LIL3}
\end{align}
almost surely.
Divide the expression in \eqref{LIL4} by $\psi(h_n)$,
after replacing $y$ by $x+h_n$.
Then, by taking the limit as $n \to \infty$
and applying \eqref{LIL2} and \eqref{LIL3}, we have
\begin{multline}\label{LIL5}
\limsup_{n\rightarrow \infty}
\frac{\Re [e^{- i|u(x+h_n)|^2}u(x+h_n)]-\Re[e^{-i|u(x)|^2}u(x)]}{\psi(h_n)}\\
\hspace*{-4cm}\ge - 2\sin |u(x)|^2 \cdot (\Re u(x))^2\\
\shoveright{- \big|\cos|u(x)|^2\big| -\big|\sin |u(x)|^2\big|
- 2\cdot \big|\sin |u(x)|^2 \cdot \Im u(x)\Re u(x)\big|}\\
- 2\cdot \big|\cos |u(x)|^2\cdot \Re u(x) \Im u(x)\big|
- 2\cdot\big| \cos |u(x)|^2\big| (\Im u(x))^2,
\end{multline}
almost surely.

Fix $M \gg1 $ by
\begin{align}
M^2= - \frac{\pi}{2}+2k\pi,
\label{LIL6}
\end{align}
for some large $k \in \bbN$ (to be chosen later).
Given $\eps>0$, define the set
\[A =\big\{\o \in \O: |\Re u(x;\o )-M|\le \eps , \ |\Im u(x;\o)|\le \eps \big\}.\]
Noting that under the law of the OU loop, $\Re u(x)$ and $\Im u(x)$ are independent Gaussian
random variables, we have
\begin{equation*}
P(A) \ge \delta(M,\eps)>0
\end{equation*}
for any $\eps>0$.
By choosing $\eps> 0$ sufficiently small such that $\eps M \ll 1$,
we have
\begin{align}
\big||u(x)|^2-M^2\big| \leq 2 \eps (M+\eps) = o(1).
\label{LIL7}
\end{align}
Then, from \eqref{LIL6} and \eqref{LIL7}, we have
\begin{align*}
\text{RHS of }\eqref{LIL5}
& \geq 2\big|\sin|u(x)|^2 \big| \cdot M (M-3\eps)- 2 (1 + \eps (M + 2\eps))\\
& \geq M^2
\end{align*}
on $A$.
By choosing $M \gg1$, we see that the set
\[A_1 = \left\{\limsup_{h\downarrow 0} \frac{\Re[e^{- i|u(x+h)|^2}u(x+h)]-\Re[e^{-i|u(x)|^2]}u(x)]}{\psi(h)}=1\right\}\]
does not have probability 1 under the law of $u$.
Therefore, $\mu_1$ is not quasi-invariant
under the flow of the dispersionless model \eqref{ND1}.

\subsection{Fractional Brownian motion}\label{SUBSEC:5.2}

In this subsection, we
extend the previous result to the distribution of the random Fourier series
\begin{align}
u_s(x) & = \sum_{n\in \mathbb{Z}} \frac{g_n}{\jb{n}^s}\,e^{inx}, \label{fBM}
\end{align}
corresponding to $\mu_s$.
For $\sfrac{1}{2}<s<\sfrac{3}{2}$, the series \eqref{fBM} is related to
a {\it fractional Brownian motion}.
Recall that a fractional Brownian motion with Hurst parameter~$H$, $0<H<1$,
is the Gaussian process $B^H(t)$, $t\ge 0 $ with stationary increments and covariance
\begin{equation}\label{COV}
\mathbb{E}[B^H(t_1)B^H(t_2)] = \frac{\rho(H)}{2}\,(t_1^{2H}+t_2^{2H}-|t_2-t_1|^{2H}),
\end{equation}
where
\[\rho(H) = \mathbb{E}\big[\big(B^H(1)\big)^2\big] = -2\s \frac{\cos (\pi H)}{\pi}\,\Gamma(-2H) \ \text{ when } H \ne \sfrac12
\quad \text{and}\quad \rho(\sfrac 12)=1.\]
When $H=\sfrac 12$, a fractional Brownian motion becomes the standard Brownian motion.
In the following, we only consider the case $H \ne \sfrac 12$.

It is known that there is a subtle issue on building a series representation for
a fractional Brownian motion $B^H$.
Instead, we consider the following series\footnote{As mentioned in Section \ref{SEC:1},
we drop the factor of $2\pi$.}
\[\ft {B}^H (t) = \wt g_0 t+\sqrt{2}\sum_{n\ge 1} \Bigl(\wt g_n\frac{\cos(nt)-1}{n^{H+\sfrac12}}+\wt g_n'
\frac{\sin(nt)}{n^{H+\sfrac 12}}\Bigr),\]
for $t\in [0,2\pi]$, where $\wt{g}_n$ and $\wt{g}_n'$ are now independent {\it real-valued} standard Gaussians.
Then, Picard \cite{picard} showed the following result
on the relation between $B^H$ and $\ft B^H$.

\begin{lemma}[{\cite[Th.\,24\,\&\,27, \S6]{picard}}]\label{picardthm}
The processes $B^H(t)$ and $\ft {B}^H(t)$ can be coupled in such a way that\vspace*{-3pt}
\[B^H(t)-\widehat{B}^H(t)\]
is a $C^\infty$-function on $(0,2\pi]$.
Moreover, if $H \ne \sfrac 12$, then the laws of $B^H$ and $\ft B^H$
are equivalent on $[0, T]$ for $T < 2\pi$ (and mutually singular if $T = 2\pi$).
\end{lemma}

Recall Kakutani's criterion \cite{Kakutani} in the Gaussian case.

\begin{lemma}\label{LEM:Kakutani}
Let $\{g_n\}_{n\in \bbN}$ and $\{\wt{g}_n\}_{n\in \bbN}$
be two sequences of centered Gaussian random variables with variances
$\mathbb{E}[g_n^2]=\sigma_n^2>0$ and $\mathbb{E}[\wt{g}_n^2] = \wt{\sigma}_n^2>0$.
Then, the laws of the sequences $\{g_n\}_{n\in \bbN}$ and $\{\tilde{g}_n\}_{n\in \bbN}$ are equivalent if and only if
\[\sum_{n\in \bbN} \Bigl( \frac{\wt{\s}^2_n}{\s^2_n}-1 \Bigr)^2<\infty.\vspace*{-3pt}\]
\end{lemma}

From \eqref{fBM}, we have\vspace*{-3pt}
\begin{multline}
\Re u_s(x)\\[-7pt]
= \Re g_0 + \sum_{n \geq 1}
\Bigl( \frac{\Re g_n+\Re g_{-n}}{\jb{n}^s}\cos (n x)
+ \frac{-\Im g_n +\Im g_{-n}}{\jb{n}^s}\sin(nx) \Bigr).
\label{fBM2}
\end{multline}
Then,
applying Lemma \ref{LEM:Kakutani} to the sequences
\[\Bigl\{\wt{g}_0 t, \ \sqrt{2}\frac{\wt g_n}{n^{H+\sfrac12}},\ \sqrt{2}\frac{\wt g_n'}{n^{H+\sfrac12}} \Bigr\}\quad
\text{and}
\quad \Bigl\{ \Re g_0, \ \frac{\Re g_n+\Re g_{-n}}{\jb{n}^s}, \ \frac{-\Im g_n +\Im g_{-n}}{\jb{n}^s} \Bigr\},\]
we see that if $s=H+\sfrac12$, then the series \eqref{fBM2}
for $\Re u_s$
and\vspace*{-3pt}
\begin{equation}
\wt B^H := \ft {B}^H- \fint_0^{2\pi}(\ft {B}^H(\alpha)-\wt g_0\alpha)d \al
\label{fBM2a}
\end{equation}
have laws that are mutually absolutely continuous.
Therefore, in view of the computation above
and Lemma \ref{picardthm} with \cite[Th.\,35]{picard},
we see that
the laws of $B^H$ and $\Re u_s$
are equivalent on $[0, T]$ for $T < 2\pi$.
The same holds for $\Im u_s$.

We use the following version of the law of the iterated logarithm for Gaussian processes with stationary increments \cite[Th.\,7.2.15]{marcusrosen}.
First, recall the following definition.
We say that
a function $f$ is called a normalized regularly varying function at zero with index $\al >0$ if it can be written in
the form\vspace*{-3pt}
\[ f(x) = Cx^\al \exp \bigg(\int_1^x \frac{\eps(u)}{u} du \bigg)\]
for some constant $C \ne 0$
and $\lim_{u \to 0} \eps(u) = 0$.

\begin{proposition}\label{thm:LIL}
Let $G=\big\{G(x): x\in [0,2\pi]\big\}$ be a Gaussian process with stationary increments and let
\[\s^2(h) = \mathbb{E}\big[|G(h)-G(0)|^2\big].\]
If $\s^2(h)$ is a normalized regularly varying function at zero with index $0<\alpha<2$, then
\begin{equation*}
\lim_{\dl\rightarrow 0} \sup_{|h|\le \dl} \frac{|G(h)-G(0)|}{\sqrt{2\s^2(|h|)\log\log \sfrac1{|h|}}}=1
\end{equation*}
almost surely.
\end{proposition}

Using the covariance \eqref{COV}, we have
\begin{align}
\s^2(h) = \bbE\big[|B^H(h)-B^H(0)|^2\big] = Ch^{2H}.
\label{sigma}
\end{align}
Hence, Proposition \ref{thm:LIL}
holds for $G(x)=B^H(x)$, $H<1$.
Then, by the absolute continuity, the conclusion of Proposition \ref{thm:LIL}
with 0 replaced by any $x \in (0, 2\pi)$
also holds for $\Re u_s$ and $\Im u_s$; for any $\sfrac 12<s<\sfrac32$, we have
\begin{align}
& \lim_{\dl\rightarrow 0} \sup_{|h|\le \dl} \frac{|\Re u_s(x+h)-\Re u_s(x)|}{\sqrt{2\s^2(|h|)\log\log \sfrac1{|h|}}}=1
\label{LIL11a}
\end{align}
almost surely.
Applying the law of the iterated logarithm conditionally on the set where \eqref{LIL11a} holds,
we also have
\begin{align*}
& \lim_{\dl\rightarrow 0} \sup_{|h|\le \dl} \frac{|\Im u_s(x+h)-\Im u_s(x)|}{\sqrt{2\s^2(|h|)\log\log \sfrac1{|h|}}}\leq1
\end{align*}
almost surely.
We can now reproduce exactly the proof in Section \ref{SUBSEC:5.1} for $\frac 12<s<\frac 32$.
This proves Theorem \ref{THM:sing} for $\sfrac 12<s<\sfrac 32$.

Next, we consider the case
$s\geq \sfrac 32$ such that $s\notin \sfrac 12+\bbN$.
We consider the critical case
$s\in \sfrac 12+\bbN$ in Section \ref{SUBSEC:5.3}.
The main point is to note that
$u_s$ in \eqref{fBM} has a $C^r$-version for each integer
$r<\lfloor s-\sfrac 12\rfloor $. Indeed, we have the following:

\begin{lemma}\label{LEM:cov}
Let $X(t)$, $t\in \bbR$, be a stationary Gaussian process with the covariance function
\[\rho(t) =\int e^{i\al t} \nu(d\al). \]
If $\int |\al|^{2+\eps}\nu(d\al)<\infty$ for some $\eps>0$,
then there is a version of the process $X(t)$ such that
$\dt X(t)$ exists and is continuous.
Moreover, $\dt X(t)$ is a stationary Gaussian process with covariance
\begin{equation*}
\int e^{i\al t} \al^2 \nu(d \al).
\end{equation*}
\end{lemma}

Since we work on $\bbT$, the spectral measure $\nu(d\al)$ is
the counting measure on $\bbZ$ and
\[\rho(x) =2 \sum_{n\in \bbZ} \frac{e^{inx}}{\jb{n}^{2s}}.\]
Note that when $s>\sfrac 32$, we have
\[2\sum_{n\in \bbZ} \frac{|n|^{2+\eps}}{\jb{n}^{2s}}<\infty\]
for sufficiently small $\eps > 0$
and thus we can apply Lemma \ref{LEM:cov}.
Differentiating $u_s$ in \eqref{fBM} $\lfloor s-\sfrac 12 \rfloor$ times,
we obtain a process
\begin{align*}
\dx^r u_s(x) \stackrel{\textrm{d}}{=}
\sum_{n\in \bbZ \setminus\{0\}} \frac{ g_n}{|n|^{-r} \jb{n}^{s}}\,e^{inx}
\end{align*}
with $r = \lfloor s-\sfrac 12 \rfloor$.
Given $ s \in (\sfrac 12 + j, \sfrac 32 + j)$ for some $j \in \bbN$,
we have $s - r = s-j \in (\sfrac 12, \sfrac 32)$.
Noting that $|n|^{-r} \jb{n}^{s}\sim \jb{n}^{s-r}$, we can
apply Lemma \ref{LEM:Kakutani} to show the laws of
$\wt B^H$ defined in \eqref{fBM2a}, $\Re \dx^r u_s$,
and $\Im \dx^r u_s$ are equivalent.
Hence, proceeding as before, we can apply Proposition \ref{thm:LIL} to
$\Re \dx^r u_s$ and $\Im \dx^r u_s$.
Namely, we obtain
\begin{align}
& \lim_{\dl\rightarrow 0} \sup_{|h|\le \dl} \frac{|\Re \dx^r u_s(x+h)-\Re \dx^r u_s(x)|}{\sqrt{2\s^2(|h|)\log\log \sfrac1{|h|}}}=1
\label{LIL12}
\end{align}
almost surely.
Applying the law of the iterated logarithm conditionally on the set where \eqref{LIL12} holds,
we also have
\begin{align}
& \lim_{\dl\rightarrow 0} \sup_{|h|\le \dl} \frac{|\Im \dx^r u_s(x+h)-\Im \dx^r u_s(x)|}{\sqrt{2\s^2(|h|)\log\log \sfrac1{|h|}}}\leq 1
\label{LIL13}
\end{align}
almost surely.

With \eqref{LIL12} and \eqref{LIL13} at hand,
we can basically repeat the proof
in Section~\ref{SUBSEC:5.1}
by differentiating \eqref{LIL4}
and applying
\eqref{LIL12} and \eqref{LIL13}.
A straightforward application of the product rule to compute $\dx^r (e^{-i|u_s(x)|^2}u_s(x))$
would be computationally cumbersome.
Thus, we perform some simplification before taking derivatives.
First, note that from \eqref{LIL12} and \eqref{LIL13} with \eqref{sigma}, we have
\begin{align}
& \lim_{\dl\rightarrow 0} \sup_{|h|\le \dl}
\frac{|\Re \dx^j u_s(x+h)-\Re \dx^j u_s(x)|}{\sqrt{2\s^2(|h|)\log\log \sfrac1{|h|}}}=0,
\label{LIL14}\\
& \lim_{\dl\rightarrow 0} \sup_{|h|\le \dl} \frac{|\Im \dx^j u_s(x+h)-\Im \dx^j u_s(x)|}{\sqrt{2\s^2(|h|)\log\log \sfrac1{|h|}}}=0
\label{LIL15}
\end{align}
almost surely, for $j = 0, 1, \dots, r-1$.

In the following, we will take $r$ derivatives (in $x$) of both sides of \eqref{LIL4}
by setting \hbox{$y = x+h$}.
In view of \eqref{LIL14} and \eqref{LIL15},
we see that,
after taking $r$ derivatives,
dividing by $\sqrt{2\s^2(|h|)\log\log \sfrac1{|h|}}$,
and taking $\lim_{\dl\rightarrow 0} \sup_{|h|\le \dl}$,
the only terms in \eqref{LIL4} that survive are those terms
where all the $r$ derivatives
falls only on $\Re u_s(x+h)-\Re u_s(x)$
(or $\Im u_s(x+h)-\Im u_s(x)$)
to which we can apply \eqref{LIL12} and \eqref{LIL13}.
Therefore, we obtain
\begin{align*}
\lim_{\dl\rightarrow 0} \sup_{|h|\le \dl}
& \frac{\Re [\dx^r(e^{- i|u_s(x+h)|^2} u_s(x+h))] -\Re[\dx^r (e^{- i|u_s(x)|^2}u_s(x))]}
{\sqrt{2\s^2(|h|)\log\log \sfrac1{|h|}}}
\\
&\ge - 2\sin |u_s(x)|^2 (\Re u_s(x))^2\notag\\
&\phantom{X} - \big|\cos|u_s(x)|^2\big| -\big|\sin |u_s(x)|^2\big|
- 2\cdot \big|\sin |u_s(x)|^2 \Im u_s(x)\Re u_s(x)\big|\\
&\phantom{X}
- 2\cdot \big|\cos |u_s(x)|^2 \Im u_s(x)\Re u_s(x)\big|
- 2\cdot\big| \cos |u_s(x)|^2\big| (\Im u_s(x))^2,
\end{align*}
which is exactly the right-hand side of \eqref{LIL5}.
The rest follows as in Section \ref{SUBSEC:5.1}.

\subsection{Critical case:\,$s\in \sfrac 12+\mathbb{N}$}\label{SUBSEC:5.3}

In this case, we cannot simply apply Proposition \ref{thm:LIL} directly,
because, taking $s=\sfrac 32$ for example, we have\footnote{This computation follows from
the computations in the proof of Proposition
\ref{PROP:LL} below.}
\begin{equation}
\label{COV2}
\begin{split}
\s_{\sfrac 32}(x) = \mathbb{E}\big[|u_{\sfrac32}(x)-u_{\sfrac 32}(0)|^2\big]
&= 2\sum_{n\in \mathbb{Z}}\frac{|1-e^{inx}|^2}{(1+n^2)^{\sfrac 32}}\\
& = 2 \sum_{n=1}^\infty \frac{(1-\cos(nx))^2+\sin^2(nx)}{(1+n^2)^{\sfrac 32}} \\
& \sim x^2 \cdot \log \sfrac{1}{|x|},
\end{split}
\end{equation}
as $x \to 0$.
In particular, $\s_{\sfrac 32} (x)$ is not
a normalized regularly varying function at zero with index
$0< \al < 2$.
Hence, Proposition \ref{thm:LIL} is not applicable.

In \cite{benassi}, the authors considered the Gaussian process on $\mathbb{R}^n$ with covariance function given by the kernel of the inverse of a quite general elliptic pseudodifferential operator and studied the precise regularity of the process. In particular, they obtained a result generalizing Proposition \ref{thm:LIL} by very different methods from those in \cite{marcusrosen}.

For us, the relevant operator is $2^{-1}(\Id-\dx^2)^{s}$ on $\mathbb{T}$.
In this case, which the authors of \cite{benassi} call ``critical'' owing to the behavior \eqref{COV2} of the increments, the relevant result from \cite[Th.\,1.3\,(ii)]{benassi} reads as follows.

\begin{proposition}\label{prop: benassi}
Let $X_{\sfrac 32}$ be the stationary Gaussian process on $\mathbb{R}$
with the covariance operator $2(\textup{Id}-\dx^2 )^{-\sfrac 32}$.
Then, $X_{\sfrac 32}(x)$ has continuous sample paths.
Moreover, there exists a constant $c_{\sfrac32}>0$ such that for each $y\in \mathbb{R}$, we have
\[\limsup_{x\to y} \frac{|X_{\sfrac32}(x)-X_{\sfrac32}(y)|}{|x-y|\sqrt{\log \frac{1}{|x-y|} \log \log \log\frac{1}{|x-y|}}}=
c_{\sfrac{3}{2}}\]
almost surely.
\end{proposition}

The $\log\log$ from the classical law of the iterated logarithm and Proposition \ref{thm:LIL}
is now replaced by a factor involving the triply iterated logarithm $\log\log\log$.
In the following, we state and prove an analogue of Proposition \ref{prop: benassi}
on $\bbT$
in a direct manner. See Proposition~\ref{PROP:LL} below.
Using this almost sure constancy of the modulus of
continuity (Proposition~\ref{PROP:LL}),
we can once again repeat the argument presented
in Section~\ref{SUBSEC:5.1}.

The results in \cite{benassi} are much more general than Proposition \ref{prop: benassi}. In particular, they apply to operators with variable coefficients. In that case, the local modulus of continuity of the process can change from point to point (although it is constant across different realizations of the sample path).
In our specific case, it is possible to give a more elementary proof,
using the classical Khintchine's law of the iterated logarithm
for independent sums, that
the process $u_{\sfrac32}$ has an exact modulus of continuity almost surely.
In terms of the setting in \cite{benassi},
this simplified proof comes as no surprise since
our process $u_{\sfrac32}$ has a particularly simple representation as a sum of independent terms with respect to which the covariance operator is diagonal.

\begin{proposition}\label{PROP:LL}
Let $u_{\sfrac32}$ be given by the random Fourier series in \eqref{fBM}
with $s = \sfrac 32$.
Then, for each $y\in \mathbb{T}$, we have
\begin{align}
\limsup_{x\to y} \frac{|u_{\sfrac32}(x)-u_{\sfrac32}(y)|}
{2^{\sfrac32}|x-y|\sqrt{\log \frac{1}{|x-y|} \log\log\log \frac{1}{|x-y|}}}=1
\label{LIL16a}
\end{align}
almost surely.
\end{proposition}

Once we prove Proposition \ref{PROP:LL}, we can proceed as in Section \ref{SUBSEC:5.1}
when $s = \sfrac 32$.
For $s \in \sfrac 32 + \bbN$, the modification is straightforward following the second half of Section
\ref{SUBSEC:5.2} and thus we omit details.

\begin{proof}
Without loss of generality, set $y = 0$.
By writing
\begin{equation}
\label{LIL16}
\begin{split}
u_{\sfrac32}(x)-u_{\sfrac32}(0)
&= \sum_{n\in \bbZ\setminus\{0\}}^\infty \frac{e^{inx}-1}{\jb{n}^{\sfrac{3}{2}}}\,g_n \\
&= \sum_{n=1}^\infty \frac{\cos(nx)-1}{\jb{n}^{\sfrac{3}{2}}}\, (g_n+g_{-n})
+i \sum_{n=1}^\infty \frac{ \sin(nx)}{\jb{n}^{\sfrac{3}{2}}}\,(g_n-g_{-n}),
\end{split}
\end{equation}
we first show that the first term on the right-hand side of \eqref{LIL16} does not contribute
to the limit in \eqref{LIL16a}.
Then, we break up the second sum into $\log \sfrac{1}{|x|}$ pieces, each with variance of order 1, plus a small remainder,
and then apply the classical law of the iterated logarithm for a sum of i.i.d.~random variables.
As we see below,
the leading order contribution comes from the sum
\begin{equation}\label{sineterm}
\sum_{n=1}^{\lfloor \sfrac{1}{|x|}\rfloor }
\frac{\sin (nx)}{\jb{n}^{\sfrac{3}{2}}}\,(g_n-g_{-n}).
\end{equation}

We split the first sum on the right-hand side of \eqref{LIL16} into
$\big\{n > \lfloor \sfrac{1}{|x|}\rfloor \big\}$
and $\big\{1 \leq n \leq \lfloor \sfrac{1}{|x|}\rfloor \big\}$.
The contribution from $\big\{n > \lfloor\sfrac{1}{|x|}\rfloor\big\}$ is a mean-zero Gaussian random variable with variance
\begin{align}
\s_L^2 := 4\sum_{n=L}^\infty \frac{(\cos(nx)-1)^2}{\jb{n}^3} \les L^{-2}.
\label{V1}
\end{align}
In particular, when $L = \lfloor\sfrac{1}{|x|}\rfloor + 1$,
we have $\s_L^2 = O(x^2)$.
Then,
for $\lambda>0$, we have
\[P\biggl(\bigg|\sum_{n=L}^\infty \frac{\cos (nx)-1}{\jb{n}^{\sfrac{3}{2}}}(g_n+g_{-n})\bigg|
\ge \s_L \ld \biggr)\les e^{-c\ld^2}. \]
Taking $\ld= c \sqrt{\log L}$ for sufficiently large $c>1$,
the right-hand side is summable in~$L$.
Hence, by the Borel-Cantelli lemma and the variance bound \eqref{V1},
there exists $C>0$ such that
\[\sup_{L\ge L_0} \frac{\left|\sum^\infty_{n=L}
\frac{\cos (nx)-1}{\jb{n}^{\sfrac{3}{2}}}(g_n+g_{-n})\right|}{CL^{-1}\sqrt{\log L}}\le 1 \]
for some $L_0 = L_0(\o) < \infty$
with probability 1.
As a consequence, we obtain
\[\limsup_{x\to 0} \frac{\left|\sum_{n>\lfloor \sfrac{1}{|x|}\rfloor }
\frac{\cos(nx)-1}{\jb{n}^{\sfrac{3}{2}}}(g_n+g_{-n})\right|}{|x|\sqrt{\log \sfrac{1}{|x|}\,\log\log\log \sfrac{1}{|x|}}}=0\]
almost surely.

The contribution from $\{ 1\leq n \leq \lfloor \sfrac{1}{|x|}\rfloor \}$ to the first sum
on the right-hand side of \eqref{LIL16}
can be estimated in a similar manner by noticing
that it is a mean-zero Gaussian random variable
with variance
\[ \sum_{n=1}^{\lfloor \sfrac{1}{|x|}\rfloor} \frac{(\cos(nx)-1)^2}{\jb{n}^3}
\les x^4 \sum_{n=1}^{\lfloor \sfrac{1}{|x|}\rfloor} \frac{n^4}{\jb{n}^3}\les x^2. \]
This shows that the contribution from the first sum on the right-hand side of \eqref{LIL16}
to the limit \eqref{LIL16a} is 0.

Next, we consider the second sum on the right-hand side of \eqref{LIL16}.
The contribution from $\{ n > \lfloor \sfrac{1}{|x|}\rfloor \}$
can be estimated as above.
We split the main term in \eqref{sineterm} as follows.
Write
\begin{equation}\label{split}
\sum_{n=1}^{\lfloor \sfrac{1}{|x|}\rfloor }\frac{\sin (nx)}{\jb{n}^{\sfrac{3}{2}}}\,
(g_n-g_{-n})
= x\!\sum_{n=1}^{\lfloor \sfrac{1}{|x|}\rfloor }\frac{n}{\jb{n}^{\sfrac{3}{2}}}\,(g_n-g_{-n})
{}+\!\sum_{n=1}^{\lfloor \sfrac{1}{|x|}\rfloor}\frac{h(nx)}{\jb{n}^{\sfrac{3}{2}}}\,(g_n-g_{-n}),
\end{equation}
where
\[h(z) = \sin z - z \sim z^3(1+o(1)).\]
The second term in \eqref{split} can be treated as a remainder
by noticing that that it is a mean-zero Gaussian random variable
with variance
\begin{align*}
4\sum_{n=1}^{\lfloor \sfrac{1}{|x|}\rfloor } \frac{h^2(nx )}{\jb{n}^3}
\les x^6\sum_{n=1}^{\lfloor \sfrac{1}{|x|}\rfloor }\frac{n^6}{\jb{n}^3}
\les x^2.
\end{align*}
It remains to consider the first term in \eqref{split}.
First, define a sequence \hbox{$\{N(k)\}_{k = 0}^\infty\!\subset\!\bbN$}
by setting $N(0) = 0$ and
\[N(k) = \min\bigg\{N> N(k-1): \sum_{n=N(k-1)+1}^{N}\frac{n^2}{\jb{n}^3} \geq 1\bigg\}\]
for $k \in \bbN$.
Noting that
\begin{align}
\sum_{n=M}^N \frac{n^2}{\jb{n}^3}
= \sum_{n=M}^N \frac{1}{n} + O\bigg(\sum_{n = M}^N \frac{1}{n^3}\bigg)
= \log N - \log M +O\left(\sfrac{1}{M}\right)
\label{LIL18}
\end{align}
for $N > M \geq 1$,
we first see that $N(k) \ge C_1 e^k$.
Using \eqref{LIL18} once again,
\begin{align*}
\log N(k) + O(1)=
\sum_{n=1}^{N(k)} \frac{n^2}{\jb{n}^3}
\leq k + \sum_{n=1}^k\frac{1}{N(n)+1}
\leq k + O(1),
\end{align*}
giving $N(k) \leq C_2 e^k$.
Putting together, we have
\begin{align}
C_1 e^k \leq N(k) \leq C_2 e^k.
\label{LIL19}
\end{align}

Now, we define a sequence $\{X_{k}\}_{k\in \bbN}$ of independent Gaussian random variables by setting
\[X_k = x\sum_{n=N(k-1)+1}^{N(k)}\frac{n}{\jb{n}^{\sfrac{3}{2}}}(g_n-g_{-n}). \]
Then, we have
\[\mathbb{E}\big[|\Re X_k|^2\big] =
\mathbb{E}\big[|\Im X_k|^2\big] = x^2\big(1+ O(N(k)^{-1})\big).\]

Finally,
define $L(|x|)$ by
\[L(|x|)=\inf \left\{ k: N(k) \ge\lfloor \sfrac{1}{|x|}\rfloor \right\}.\]
Then, from \eqref{LIL19}, we have
\[L(|x|)=\log \sfrac{1}{|x|}\cdot (1+o(1)).\]
Applying Khintchine's law of the iterated logarithm to the sum
\[S(x) = \sum_{k=1}^{L(|x|)} X_k,\]
we find
\[\limsup_{x\to 0} \frac{S(x)}{2^{\sfrac32}|x|\sqrt{\log\sfrac{1}{|x|}\,\log\log\log{\sfrac{1}{|x|}}}}=1\]
almost surely.
This completes the proof of Proposition \ref{PROP:LL}.
\end{proof}

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