# Multilinear Wiener-Wintner type ergodic averages and its application

Rongzhong Xiao  School of Mathematical Sciences, University of Science and
Technology of China, Hefei, Anhui, 230026, PR China<EMAIL_ADDRESS>
###### Abstract.

In this paper, we extend the generalized Wiener–Wintner Theorem built by Host
and Kra to the multilinear case under the hypothesis of pointwise convergence
of multilinear ergodic averages. In particular, we have the following result:

Let $(X,\mathcal{B},\mu,T)$ be a measure preserving system. Let $a$ and $b$ be
two distinct non-zero integers. Then for any $f_{1},f_{2}\in L^{\infty}(\mu)$,
there exists a full measure subset $X(f_{1},f_{2})$ of $X$ such that for any
$x\in X(f_{1},f_{2})$, and any nilsequence
$\textbf{b}=\\{b_{n}\\}_{n\in\mathbb{Z}}$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}b_{n}f_{1}(T^{an}x)f_{2}(T^{bn}x)$

exists.

###### Key words and phrases:

multilinear Wiener-Wintner type ergodic averages, nilsequence, cubic averages,
polynomial ergodic averages, Furstenberg systems.

###### 2020 Mathematics Subject Classification:

Primary: 37A30; Secondary: 37A46.

##  1\. Introduction

Throughout the paper, by a _measure preserving system_ or a _system_ , we mean
a Lebesgue space $(X,\mathcal{B},\mu)$ with an invertible measure preserving
transformation $T:X\rightarrow X$. We write a system as
$(X,\mathcal{B},\mu,T)$. A system $(X,\mathcal{B},\mu,T)$ is ergodic if the
only $T$-invariant subsets in $\mathcal{B}$ have measure $0$ or $1$.

In 1941, Wiener and Wintner strengthened the classical Birkhoff’s pointwise
ergodic theorem.

######  Theorem 1.1.

$($[21]$)$For a system $(X,\mathcal{B},\mu,T)$, and any $f\in
L^{\infty}(\mu)$, there exists a full measure subset $X(f)$ of $X$ such that
for any $x\in X(f)$, and any $t\in\mathbb{R}$, the limit

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}e^{2\pi int}f(T^{n}x)$

exists.

The core of the Wiener–Wintner Theorem is that $X(f)$ is independent of the
choice of $t$. Since $\mathbb{R}$ is uncountable, the result leaded reseachers
to focus on general phenomena in ergodic theory where samplings are good for
an uncountable number of systems. And for the convergent behavior of some
polynomial ergodic averages, the Wiener–Wintner Theorem can also help us to
understand it.

In 2009, Host and Kra built the following result, which is called generalized
Wiener–Wintner Theorem.

######  Theorem 1.2.

$($[13, Theorem 2.22]$)$For a system $(X,\mathcal{B},\mu,T)$, and any $f\in
L^{\infty}(\mu)$, there exists a full measure subset $X(f)$ of $X$ such that
for any $x\in X(f)$, and any nilsequence $($for definition, see Subsection
2.2.$)$ $\textbf{b}=\\{b_{n}\\}_{n\in\mathbb{Z}}$, the limit

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}b_{n}f(T^{n}x)$

exists.

Note that for any $t\in\mathbb{R}$, $\\{e^{2\pi int}\\}_{n\in\mathbb{Z}}$ is a
nilsequence since $\mathbb{R}/\mathbb{Z}$ is a $1$-step nilmanifold. So the
above statements generalize Wiener-Wintner theorem. In addition, Host and Kra
pointed out the following result in [14].

######  Theorem 1.3.

$($[14, Theorem 23.5]$)$Given $d\in\mathbb{N}$, let $a_{1},\cdots,a_{d}$ be
distinct non-zero integers. Let $(X,\mathcal{B},\mu,T)$ be a measure
preserving system. Then for any $f_{1},\cdots,f_{d}\in L^{\infty}(\mu)$ and
any nilsequence $\textbf{b}=\\{b_{n}\\}_{n\in\mathbb{Z}}$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}b_{n}\prod_{j=1}^{d}f_{j}(T^{a_{j}n}x)$

exists in $L^{2}(\mu)$.

The averages which are mentioned in the above theorem are called multilinear
Wiener-Wintner type ergodic averages. The above theorem guarantees the norm
convergence of multilinear Wiener-Wintner type ergodic averages. For extra
results related to the above theorem, one can see [16, Theorem B.4].

Based on Theorem 1.2 and Theorem 1.3, there is a natural question: can we
generalize Theorem 1.2 to multilinear version under the hypothesis of
pointwise convergence of multilinear ergodic averages?

For the bilinear case, one can find some results in [4, 2]. For the general
case, we obtain the following result.

######  Theorem 1.4.

Given $d\in\mathbb{N}$, let $a_{1},\cdots,a_{d}$ be distinct non-zero
integers. Let $(X,\mathcal{B},\mu,T)$ be a measure preserving system. Then for
any $h_{1},\cdots,h_{d}\in L^{\infty}(\mu)$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}\prod_{j=1}^{d}h_{j}(T^{a_{j}n}x)$

exists almost everywhere if and only if for any $f_{1},\cdots,f_{d}\in
L^{\infty}(\mu)$, there exists full measure subset $X(f_{1},\cdots,f_{d})$ of
$X$ such that for any $x\in X(f_{1},\cdots,f_{d})$, and any nilsequence
$\textbf{b}=\\{b_{n}\\}_{n\in\mathbb{Z}}$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}b_{n}\prod_{j=1}^{d}f_{j}(T^{a_{j}n}x)$

exists.

In 2015, Zorin-Kranich [22, Corollary 1.4] had pointed out the above theorem
for ergodic systems and provided an inductive proof via harmonic analsis’s
method. In this paper, the author will give a purely ergodic theoretic proof
for the above theorem based on Host-Kra factor and determine related
characteristic factor.

Based on the above theorem, [5, Theorem], [15, Theorem C] and [13, Theorem
2.24], we have the following two corollaries.

######  Corollary 1.5.

Let $(X,\mathcal{B},\mu,T)$ be a measure preserving system. Let $a$ and $b$ be
two distinct non-zero integers. Then for any $f_{1},f_{2}\in L^{\infty}(\mu)$,
there exists full measure subset $X(f_{1},f_{2})$ of $X$ such that for any
$x\in X(f_{1},f_{2})$, and any nilsequence
$\textbf{b}=\\{b_{n}\\}_{n\in\mathbb{Z}}$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}b_{n}f_{1}(T^{an}x)f_{2}(T^{bn}x)$

exists.

######  Corollary 1.6.

Given $d\in\mathbb{N}$, let $a_{1},\cdots,a_{d}$ be distinct non-zero
integers. Let $(X,\mathcal{B},\mu,T)$ be an ergodic distal measure preserving
system $($for definition, see [15, Definition 5.4]$)$. Then for any
$f_{1},\cdots,f_{d}\in L^{\infty}(\mu)$, there exists full measure subset
$X(f_{1},\cdots,f_{d})$ of $X$ such that for any $x\in X(f_{1},\cdots,f_{d})$,
we have

  * (1)

for any nilsequence $\textbf{b}=\\{b_{n}\\}_{n\in\mathbb{Z}}$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}b_{n}\prod_{j=1}^{d}f_{j}(T^{a_{j}n}x)$

exists;

  * (2)

for any system $(Y,\mathcal{D},\nu,S)$, any $k\geq 1$, and any
$g_{1},\cdots,g_{k}\in L^{\infty}(\nu)$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}\prod_{j=1}^{d}f_{j}(T^{a_{j}n}x)\prod_{i=1}^{k}S^{in}g_{i}(y)$

exists in $L^{2}(\nu)$.

We remark for the $(2)$ of Corollary 1.6, Assani and Moore [3, Theorem 1.5]
built the corresponding result for $d=2$ without requiring ergodic distal
condition.

Next, we apply Theorem 1.4 to study some polynomial ergodic averages for
multiple measure preserving transformations under lacking complete
commutativity. Due to lacking complete commutativity, we have to do some
restrictions for transformations such as zero entropy which can help us to use
disjointness between zero entropy and completely positive entropy to attack
some polynomial ergodic averages. The reason for which Theorem 1.4 can work
for some polynomial ergodic averages is the characteristic behavior of
$\infty$-step factor $($ for definition, see Remark 2.3.$)$.

Recently, Frantzikinakis and Host established a result on norm convergence of
some polynomial ergodic averages.

######  Theorem 1.7.

$($[9, Theorem 1.1]$)$Let $T,S$ be invertible measure preserving
transformations acting on a Lebesgue space $(X,\mathcal{B},\mu)$ such that
$(X,\mathcal{B},\mu,T)$ has zero entropy and let $p(n)$ be a non-constant non-
linear integer coefficents polynomial. Then for any $f,g\in L^{\infty}(\mu)$,
the limit

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^{N}f(T^{n}x)g(S^{p(n)}x)$

exists in $L^{2}(\mu)$.

The zero entropy assumption on $T$ in the above result is necessary. If there
is no zero entropy restriction for $T$, one can find a counterexample from [9,
Proof of Proposition 1.4]. Motivated by the above theorem, we establish the
following result.

######  Theorem 1.8.

Given $d\in\mathbb{N}$, let $T,S_{1},\cdots,S_{d}$ be invertible measure
preserving transformations acting on a Lebesgue space $(X,\mathcal{B},\mu)$
such that $(X,\mathcal{B},\mu,T)$ has zero entropy and $S_{1},\cdots,S_{d}$
are commuting and let $p_{1}(n),\cdots,p_{d}(n)$ be non-constant non-linear
integer coefficents polynomials with distinct degrees. Let $a$ and $b$ be two
distinct non-zero integers. Then for any $f_{1},f_{2},g_{1},\cdots,g_{d}\in
L^{\infty}(\mu)$, the limit

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^{N}f_{1}(T^{an}x)f_{2}(T^{bn}x)\prod_{j=1}^{d}g_{j}(S_{j}^{p_{j}(n)}x)$

exists in $L^{2}(\mu)$.

### Organization of the paper

In Section 2, we recall some notions and needed results. In section 3, we
introduce two useful estimations. In sections 4 and 5, we prove Theorem 1.4
and Theorem 1.8 respectively.

##  2\. Preliminaries

In this section we recall some basic notions and results.

###  2.1. Factor and joining

A _factor_ of a system $(X,\mathcal{B},\mu,T)$ is a $T$-invariant
sub-$\sigma$-algebra of $\mathcal{B}$. A _factor map_ from
$(X,\mathcal{B},\mu,T)$ to $(Y,\mathcal{D},\nu,S)$ is a measurable map
$\pi:X_{0}\rightarrow Y_{0}$ with $\pi\circ T=S\circ\pi$ and such that $\nu$
is the image of $\mu$ under $\pi$ where $X_{0}$ is a $T$-invariant full
measurable subset of $X$ and $Y_{0}$ is a $S$-invariant full measurable subset
of $Y$. In this case, $\pi^{-1}(\mathcal{D})$ is a factor of
$(X,\mathcal{B},\mu,T)$ and every factor of $(X,\mathcal{B},\mu,T)$ can be
obtained in this way.

A _joining_ of two systems $(X,\mathcal{B},\mu,T)$ and $(Y,\mathcal{D},\nu,S)$
is a probability measure $\lambda$ on $X\times Y$, invariant under $T\times S$
and whose projections on $X$ and $Y$ are $\mu$ and $\nu$ respectively. Likely,
we can define joining on more systems.

To study non-conventional ergodic averages on $(X,\mathcal{B},\mu,T)$,
_Furstenberg’s self-joining_ was introduced.

At first, we claim some notations. Given $d\in\mathbb{N}$, let $X^{d}$ denote
the space $X\times\cdots\times X(d\ times)$ where an element of $X^{d}$ can be
written as $\textbf{x}=(x_{1},\cdots,x_{d})$ and $\mathcal{B}^{\otimes d}$
denote the product $\sigma$-algebra.

Let $\mathcal{A}=\\{a_{1}n,\cdots,a_{d}n\\}$ where $a_{1},\cdots,a_{d}$ are
distinct non-zero integers. We define a measure $\mu^{\mathcal{A}}_{d}$ on
$(X^{d},\mathcal{B}^{\otimes d})$ by

$\mu^{\mathcal{A}}_{d}(A_{1}\times\cdots\times
A_{d})=\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}\int_{X}\prod_{i=1}^{d}T^{a_{i}n}1_{A_{i}}d\mu$

where $A_{i}\in\mathcal{B},1\leq i\leq d$. Existence of the above limit can be
guaranteed by [12, Theorem 1.1]. We refer to $\mu^{\mathcal{A}}_{d}$ as the
Furstenberg’s self-joining of $(X,\mathcal{B},\mu,T)$ with respect to
polynomial family $\mathcal{A}$ in the light of its historical genesis in
Furstenberg’s work on the ergodic-theoretic proof for Szemerédi’s theorem
[11]. By an easy observation, we have the following result.

######  Proposition 2.1.

$(X^{d},\mathcal{B}^{\otimes
d},\mu^{\mathcal{A}}_{d},T^{a_{1}}\times\cdots\times T^{a_{d}})$ is a measure
preserving system.

###  2.2. Nilsystem and nilsequence

Let $k\in\mathbb{N}$. A _$k$ -step nilmanifold_ $X$ is the quotient space
$G/\Gamma$ where $G$ is a $k$-step nilpotent Lie group and $\Gamma$ is a
cocompact discrete subgroup of $G$. The group $G$ acts on $X$ by left
translations. The unique invariant Borel probability measure on $X$ under the
action is called the Harr measure of $X$ denoted by $m_{X}$. If $a\in G$ and
$T_{a}:X\rightarrow X$ is the translation $x\mapsto a\cdot x$, the system
$(X,\mathcal{B}_{X},T_{a},m_{X})$ is a _$k$ -step nilsystem_ where
$\mathcal{B}_{X}$ is the Borel $\sigma$-algebra of $X$.

A _basic $k$-step nilsequence_ is the sequence $\\{f(a^{n}\cdot
x)\\}_{n\in\mathbb{Z}}$ where $f\in C(X),a\in G,x\in X$. A _$k$ -step
nilsequence_ is a uniform limit of basic $k$-step nilsequences. Clearly, all
$k$-step nilsequences is an invariant algebra of $l^{\infty}(\mathbb{Z})$
under translation.(For more details, see [14, Chapter 11]).

###  2.3. Host-Kra seminorm and local seminorm

In 2005, Host and Kra introduced Host-Kra seminorm on $(X,\mathcal{B},\mu,T)$
when they considered the norm convergence of non-conventional ergodic averages
in [12].

At first, we introdue some notations. Given $k\in\mathbb{N}$, let
$V_{k}=\\{0,1\\}^{k}$ where an element of $V_{k}$ can be written as
$\underline{\epsilon}=(\epsilon_{1},\cdots,\epsilon_{k})$, $X^{[k]}=X^{2^{k}}$
where an element of $X^{[k]}$ can be written as
$\underline{x}=(x_{\underline{\epsilon}}:\underline{\epsilon}\in V_{k})$
,$T^{[k]}=T\times\cdots\times T(2^{k}\ times)$ and
$\mathcal{B}^{\otimes[k]}=\mathcal{B}^{\otimes 2^{k}}$. Given
$N\in\mathbb{N}$, let $[N]^{k}=\\{\underline{h}=(h_{1},\cdots,h_{k}):0\leq
h_{1},\cdots,h_{k}\leq N-1\\}$.

Now, we define $\mu^{[k]}$ by induction. Let $(\Omega,\mathcal{F},m,S)$ denote
the factor associated to $\mathcal{I}(T)$ where $\mathcal{I}(T)$ is a
sub-$\sigma$-algebra of $\mathcal{B}$ consisting of all $T$-invariant subsets.
Let

$\mu^{[1]}=\int_{\Omega}\mu_{w}\times\mu_{w}dm(w)$

where

$\mu=\int_{\Omega}\mu_{w}dm(w)$

is the ergodic decomposition of $\mu$ with respect to $T$. Then we get a
measure preserving system
$(X^{[1]},\mathcal{B}^{\otimes[1]},\mu^{[1]},T^{[1]})$. Let $k\in\mathbb{N}$.
If we have defined $\mu^{[k]}$, we can define $\mu^{[k+1]}$ as follows. Let
$(\Omega_{k},\mathcal{F}_{k},m_{k},S_{k})$ denote the factor associated to
$\mathcal{I}(T^{[k]})$ where $\mathcal{I}(T^{[k]})$ is a sub-$\sigma$-algebra
of $\mathcal{B}^{\otimes[k]}$ consisting of all $T^{[k]}$-invariant subsets.
Let

$\mu^{[k+1]}=\int_{\Omega_{k}}\mu^{[k]}_{w}\times\mu^{[k]}_{w}dm_{k}(w)$

where

$\mu^{[k]}=\int_{\Omega_{k}}\mu^{[k]}_{w}dm_{k}(w)$

is the ergodic decomposition of $\mu^{[k]}$ with respect to $T^{[k]}$.

For any $f\in L^{\infty}(\mu)$ and any $k\in\mathbb{N}$, we define the _Host-
Kra seminorm_ by

$\lvert\\!|\\!|f|\\!|\\!\rvert_{k}=\Big{(}\int_{X^{[k]}}\prod_{\underline{\epsilon}\in
V_{k}}C^{|\underline{\epsilon}|}f(x_{\underline{\epsilon}})d\mu^{[k]}(\underline{x})\Big{)}^{\frac{1}{2^{k}}}$

where $C$ is a complex conjugation operator and
$|\underline{\epsilon}|=\epsilon_{1}+\cdots+\epsilon_{k}$.

The follwoing results on Host-Kra seminorm will be used in the proof.

######  Theorem 2.2.

$($[14, Lemma 8.12, Proposition 8.16 and Theorem 9.7]$)$For a system
$(X,\mathcal{B},\mu,T)$, $k\geq 1$ and any $f\in L^{\infty}(\mu)$, we have :

  * (1)

$\lvert\\!|\\!|f|\\!|\\!\rvert_{k+1}^{2^{k+1}}=\lim_{H\rightarrow\infty}\frac{1}{H}\sum_{h=0}^{H-1}\lvert\\!|\\!|f\cdot
T^{h}\overline{f}|\\!|\\!\rvert_{k}^{2^{k}}$;

  * (2)

$\lvert\\!|\\!|f|\\!|\\!\rvert_{k}\leq\lvert\\!|\\!|f|\\!|\\!\rvert_{k+1}$;

  * (3)

there exists a factor $\mathcal{Z}_{k}(T)$, called $k$-step factor of
$(X,\mathcal{B},\mu,T)$, such that $\lvert\\!|\\!|f|\\!|\\!\rvert_{k+1}=0$ if
and only if $\mathbb{E}(f|\mathcal{Z}_{k}(T))=0$.

######  Remark 2.3.

From the above, we can get a sequence of factors
$\\{\mathcal{Z}_{k}(T)\\}_{k\geq 1}$ of $(X,\mathcal{B},\mu,T)$. And, for any
$k\in\mathbb{N}$, we have $\mathcal{Z}_{k}(T)\subset\mathcal{Z}_{k+1}(T)$. So
we can define factor $\mathcal{Z}_{\infty}(T)$ called $\infty$-step factor by
letting it be the smallest $\sigma$-algebra containing $\bigcup_{k\geq
1}\mathcal{Z}_{k}(T)$. $($For more details, see [14, Chapter 8,9,16]$)$.

######  Proposition 2.4.

$($[18, Lemma 8]$)$For a system $(X,\mathcal{B},\mu,T)$, $k\geq 1$, $f\in
L^{\infty}(\mu)$ which takes real value and $a\in\mathbb{Z}\backslash\\{0\\}$,
one has

$\lim_{H\rightarrow\infty}\frac{1}{H}\sum_{h=0}^{H-1}\lvert\\!|\\!|f\cdot
T^{ah}{f}|\\!|\\!\rvert_{k}^{2^{k}}\leq|a|\cdot\lvert\\!|\\!|f|\\!|\\!\rvert_{k+1}^{2^{k+1}}.$

######  Proposition 2.5.

$($[18, Proposition 5]$)$For a system $(X,\mathcal{B},\mu,T)$, $d\geq 1$,
$f_{1},\cdots,f_{d}\in L^{\infty}(\mu)$ which take real value and are bounded
by one and distinct $a_{1},\cdots,a_{d}\in\mathbb{Z}\backslash\\{0\\}$, there
exists a constant $A(a_{1},\cdots,a_{d})>0$ such that

$\limsup_{N\rightarrow\infty}\left\|\frac{1}{N}\sum_{n=0}^{N-1}\prod_{i=1}^{d}T^{a_{i}n}f_{i}\right\|_{2}\leq
A\min_{1\leq i\leq d}\lvert\\!|\\!|f_{i}|\\!|\\!\rvert_{d+1}.$

Next, we reproduce local seminorm of bounded sequences which was defined by
Host and Kra when they generalized Wiener–Wintner Theorem to nilsequences in
[13].

######  Definition 2.6.

$($[13, Definition 2.1]$)$Let $k\in\mathbb{N}$ and
$\textbf{a}=\\{a_{n}\\}_{n\in\mathbb{Z}}$ be a bounded sequence. We say that
_a satisfies $\mathcal{P}(k)$_ if for any
$\underline{h}=(h_{1},\cdots,h_{k})\in\mathbb{Z}^{k}$, the limit

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}\prod_{\underline{\epsilon}\in
V_{k}}C^{|\underline{\epsilon}|}a_{n+\underline{h}\cdot\underline{\epsilon}}$

exists where
$\underline{h}\cdot\underline{\epsilon}=\sum_{i=1}^{k}\epsilon_{i}h_{i}$. We
denote the limit by $c_{\underline{h}}(\textbf{a})$.

By [13, Proposition 2.2], we know the limit

$\lim_{H\rightarrow\infty}\frac{1}{H^{k}}\sum_{\underline{h}\in[H]^{k}}c_{\underline{h}}(\textbf{a})$

exists and is non-negative. Then we can define _local seminorm_
$\lvert\\!|\\!|\textbf{a}|\\!|\\!\rvert_{k}$ of a by

$\lvert\\!|\\!|\textbf{a}|\\!|\\!\rvert_{k}=\Big{(}\lim_{H\rightarrow\infty}\frac{1}{H^{k}}\sum_{\underline{h}\in[H]^{k}}c_{\underline{h}}(\textbf{a})\Big{)}^{\frac{1}{2^{k}}}.$

The follwoing results on local seminorm will be used in the proof.

######  Proposition 2.7.

$($[13, Corollary 2.14]$)$Let $k\geq 2$,
$\textbf{b}=\\{b_{n}\\}_{n\in\mathbb{Z}}$ be a $(k-1)$-step nilsequence and
$\delta>0$. There exists $c(\textbf{b},\delta)>0$ such that for any bounded
sequence $\textbf{a}=\\{a_{n}\\}_{n\in\mathbb{Z}}$ which satisfies
$\mathcal{P}(k)$, then

$\limsup_{N\rightarrow\infty}\Big{|}\frac{1}{N}\sum_{n=0}^{N-1}a_{n}b_{n}\Big{|}\leq
c\lvert\\!|\\!|\textbf{a}|\\!|\\!\rvert_{k}+\delta\left\|\textbf{a}\right\|_{\infty}$

where $\left\|\textbf{a}\right\|_{\infty}=\sup_{n\in\mathbb{Z}}|a_{n}|$.

######  Proposition 2.8.

$($[13, Corollary 5.10]$)$Let $k\geq 2$ and
$\textbf{a}=\\{a_{n}\\}_{n\in\mathbb{Z}}$ be a bounded sequence. Assume that
for any $\delta>0$, there exists a $(k-1)$-step nilsequence
$\textbf{r}=\\{r_{n}\\}_{n\in\mathbb{Z}}$ such that $\textbf{a}-\textbf{r}$
satisfies $\mathcal{P}(k)$ and
$\lvert\\!|\\!|\textbf{a}-\textbf{r}|\\!|\\!\rvert_{k}<\delta$. Then for any
$(k-1)$-step nilsequence $\textbf{b}=\\{b_{n}\\}_{n\in\mathbb{Z}}$, the limit

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}a_{n}b_{n}$

exists.

###  2.4. Furstenberg systems for sequences

Here, we recall the notion of the Furstenberg systems for bounded real
sequences$($For details, see [8]$)$.

######  Definition 2.9.

Let $\\{N_{k}\\}_{k\geq 1}$ be a strictly increasing sequence of positive
integers and $I$ be a closed bounded interval on $\mathbb{R}$. We say that a
sequence $\textbf{z}=\\{z_{n}\\}_{n\in\mathbb{Z}}$ which takes value on $I$
admits a correlation on sequence $\\{N_{k}\\}_{k\geq 1}$ if the limit

$\lim_{k\rightarrow\infty}\frac{1}{N_{k}}\sum_{n=1}^{N_{k}}\prod_{j=1}^{s}z_{n+n_{j}}$

exists for any $s\in\mathbb{N}$ and any
$(n_{1},\cdots,n_{s})\in\mathbb{Z}^{s}$.

Let $\Omega=I^{\mathbb{Z}}$. The element of $\Omega$ can be written
$w=(w(n))_{n\in\mathbb{Z}}$. So sequence z can be viewed as an element of
$\Omega$. The shift $\sigma$ on $\Omega$ is defined by $(\sigma(w))(n)=w(n+1)$
for any $n\in\mathbb{Z}$. If z admits a correlation on sequence
$\\{N_{k}\\}_{k\geq 1}$, then the limit

$\lim_{k\rightarrow\infty}\frac{1}{N_{k}}\sum_{n=1}^{N_{k}}\delta_{\sigma^{n}\textbf{z}}$

exists in the weak-$*$ topology. We denote the limit by $\nu$.

We say that system $(\Omega,\mathcal{F},\nu,\sigma)$ is the _Furstenberg
system_ associated with z on sequence $\\{N_{k}\\}_{k\geq 1}$ where
$\mathcal{F}$ is the Borel $\sigma$-algebra of $\Omega$.

Let $F_{0}$ the $0$-th coordinate projection of $\Omega$. That is,
$F_{0}:\Omega\rightarrow I,w\mapsto w(0)$. Clearly,

$\lim_{k\rightarrow\infty}\frac{1}{N_{k}}\sum_{n=1}^{N_{k}}\prod_{j=1}^{s}z_{n+n_{j}}=\int\prod_{j=1}^{s}\sigma^{n_{j}}F_{0}d\nu$

for any $s\in\mathbb{N}$ and any $(n_{1},\cdots,n_{s})\in\mathbb{Z}^{s}$.

###  2.5. Van der Corput’s lemma

######  Lemma 2.10.

$($[17, Lemma 1.3.1]$)$Let $\\{u_{n}\\}_{n\in\mathbb{Z}}$ be a bounded complex
sequence. For any $N,H\geq 1$ with $H\leq N$, we have

$H^{2}\Big{|}\sum_{n=1}^{N}u_{n}\Big{|}^{2}\leq
H(N+H-1)\sum_{n=1}^{N}|u_{n}|^{2}+2(N+H-1)\sum_{h=1}^{H-1}(H-h)\Re\sum_{n=1}^{N}u_{n}\overline{u}_{n+h}.$

##  3\. Two lemmas

Before the statements of two lemmas, we introdue some notations. By
$A\lesssim_{c_{1},\cdots,c_{d}}B$, it means that there exists an implicit
constant $C>0$ depending on $c_{1},\cdots,c_{d}$ such that $A\leq CB$. Let
$X,Y$ be two non-empty sets and
$f:X\rightarrow\mathbb{C},g:Y\rightarrow\mathbb{C}$ be two functions. By
$f\otimes g$, it means a function $f\otimes g:X\times
Y\rightarrow\mathbb{C},(x,y)\mapsto f(x)g(y)$.

###  3.1. An estimation for cubic averages

Let $k\in\mathbb{N}$ and $V_{k}^{*}=V_{k}\backslash\\{(0,\cdots,0)\\}$. Define
$\phi:V_{k}^{*}\rightarrow\\{1,2,\cdots,2^{k}-1\\},\underline{\epsilon}=(\epsilon_{1},\cdots,\epsilon_{k})\mapsto\sum_{i=1}^{k}\epsilon_{i}\cdot
2^{i-1}$. For any $1\leq i\leq k$, let $V_{k,i}^{*}=\\{\underline{\epsilon}\in
V_{k}^{*}:\epsilon_{i}=0\\}$. For any $1\leq i,j\leq k$ with $i\neq j$, let
$A^{j}_{i}=V_{k,i}^{*}\backslash V_{k,j}^{*}$.

Let $\\{a_{j,n}\\}_{1\leq j\leq 2^{k}-1,n\geq 0}$ be a collection of bounded
real sequences to be bounded by one. Let $N\in\mathbb{N}$. Define _$k$ -step
cubic averages_

$C(N;a_{1,n},\cdots,a_{2^{k}-1,n})=\frac{1}{N^{k}}\sum_{\underline{h}\in[N]^{k}}\prod_{\underline{\epsilon}\in
V_{k}^{*}}a_{\phi(\underline{\epsilon}),\underline{\epsilon}\cdot\underline{h}}.$

For example, when $k=3$,

$C(N;a_{1,n},\cdots,a_{7,n})=\frac{1}{N^{3}}\sum_{\underline{h}\in[N]^{3}}a_{1,h_{1}}a_{2,h_{2}}a_{3,h_{1}+h_{2}}a_{4,h_{3}}a_{5,h_{3}+h_{1}}a_{6,h_{3}+h_{2}}a_{7,h_{3}+h_{2}+h_{1}}.$

When $k=2$, Assani built the following estimation.

######  Proposition 3.1.

$($[1, Lemma 5]$)$Let
$\\{a_{n}\\}_{n\in\mathbb{Z}},\\{b_{n}\\}_{n\in\mathbb{Z}},\\{c_{n}\\}_{n\in\mathbb{Z}}$
be three bound real sequences to be bounded by one. For any $N\in\mathbb{N}$,
we have

$\Big{|}\frac{1}{N^{2}}\sum_{n,m=0}^{N-1}a_{n}b_{m}c_{n+m}\Big{|}^{2}\leq\frac{1}{N}\sum_{n=0}^{N-1}\Big{|}\frac{1}{N}\sum_{m=0}^{N-1}b_{m}c_{n+m}\Big{|}^{2}\leq\sup_{t}\Big{|}\frac{1}{N}\sum_{m=0}^{2(N-1)}e^{2\pi
imt}c_{m}\Big{|}^{2}.$

Actually, Assani also gave an estimation for $3$-step cubic averages in [1,
Lemma 6] and pointed out that one can build similar result for usual case.
Here, we give the specific statements of general case and provide a proof for
the completeness. The essence of the following is to let one index vanish.

######  Lemma 3.2.

Given $k,N\in\mathbb{N}$ with $k>2$, let $\\{a_{j,n}\\}_{1\leq j\leq
2^{k}-1,n\in\mathbb{Z}}$ be a collection of bounded real sequences to be
bounded by one. Then

$|C(N;a_{1,n},\cdots,a_{2^{k}-1,n})|^{2}\leq\frac{C}{N^{k}}\sum_{h_{1},\cdots,h_{k-2}=0}^{N-1}\sup_{t}\Big{|}\sum_{h_{k}=0}^{N-1}e^{2\pi
ih_{k}t}\prod_{\underline{\epsilon}\in
A^{k}_{k-1}}a_{\phi(\underline{\epsilon}),\epsilon_{k}h_{k}+\sum_{i=1}^{k-2}\epsilon_{i}h_{i}}\Big{|}^{2}$

where $C$ is an absolutely constant which can take value 2.

###### Proof.

$\displaystyle|C(N;a_{1,n},\cdots,a_{2^{k}-1,n})|^{2}$
$\displaystyle\leq\frac{1}{N^{k-1}}\sum_{h_{1},\cdots,h_{k-1}=0}^{N-1}\Big{|}\frac{1}{N}\sum_{h_{k}=0}^{N-1}\prod_{\underline{\epsilon}\in
V_{k}^{*}\backslash
V_{k,k}^{*}}a_{\phi(\underline{\epsilon}),\sum_{i=1}^{k}\epsilon_{i}h_{i}}\Big{|}^{2}$
$\displaystyle=\frac{1}{N^{k-1}}\sum_{h_{1},\cdots,h_{k-1}=0}^{N-1}\Big{|}\int_{\mathbb{T}}\Big{(}\frac{1}{N}\sum_{h_{k}=0}^{N-1}e^{-2\pi
ih_{k}t}\prod_{\underline{\epsilon}\in
A^{k}_{k-1}}a_{\phi(\underline{\epsilon}),\epsilon_{k}h_{k}+\sum_{i=1}^{k-2}\epsilon_{i}h_{i}}\Big{)}\cdot$
$\displaystyle\ \ \ \Big{(}\sum_{m=0}^{2(N-1)}e^{2\pi
imt}\prod_{\underline{\epsilon}\in V_{k}^{*}\backslash(V_{k,k-1}^{*}\cup
V_{k,k}^{*})}a_{\phi(\underline{\epsilon}),m+\sum_{i=1}^{k-2}\epsilon_{i}h_{i}}\Big{)}e^{-2\pi
ih_{k-1}t}dt\Big{|}^{2}$
$\displaystyle\leq\frac{1}{N^{k-1}}\sum_{h_{1},\cdots,h_{k-2}=0}^{N-1}\int_{\mathbb{T}}\Big{|}\Big{(}\frac{1}{N}\sum_{h_{k}=0}^{N-1}e^{-2\pi
ih_{k}t}\prod_{\underline{\epsilon}\in
A^{k}_{k-1}}a_{\phi(\underline{\epsilon}),\epsilon_{k}h_{k}+\sum_{i=1}^{k-2}\epsilon_{i}h_{i}}\Big{)}\cdot$
$\displaystyle\ \ \ \Big{(}\sum_{m=0}^{2(N-1)}e^{2\pi
imt}\prod_{\underline{\epsilon}\in V_{k}^{*}\backslash(V_{k,k-1}^{*}\cup
V_{k,k}^{*})}a_{\phi(\underline{\epsilon}),m+\sum_{i=1}^{k-2}\epsilon_{i}h_{i}}\Big{)}\Big{|}^{2}dt$
$\displaystyle\leq\frac{1}{N^{k-1}}\sum_{h_{1},\cdots,h_{k-2}=0}^{N-1}\Big{(}(2(N-1)+1)\sup_{t}\Big{|}\frac{1}{N}\sum_{h_{k}=0}^{N-1}e^{-2\pi
ih_{k}t}\prod_{\underline{\epsilon}\in
A^{k}_{k-1}}a_{\phi(\underline{\epsilon}),\epsilon_{k}h_{k}+\sum_{i=1}^{k-2}\epsilon_{i}h_{i}}\Big{|}^{2}\Big{)}$
$\displaystyle\leq\frac{2}{N^{k-2}}\sum_{h_{1},\cdots,h_{k-2}=0}^{N-1}\sup_{t}\Big{|}\frac{1}{N}\sum_{h_{k}=0}^{N-1}e^{-2\pi
ih_{k}t}\prod_{\underline{\epsilon}\in
A^{k}_{k-1}}a_{\phi(\underline{\epsilon}),\epsilon_{k}h_{k}+\sum_{i=1}^{k-2}\epsilon_{i}h_{i}}\Big{|}^{2}$

where the first inequality comes from Jesen’s inequality and the second
inequality comes from Parseval’s identity. This finishes the proof. ∎

###  3.2. An extimation for multivariable non-conventional ergodic averages

######  Lemma 3.3.

Let $(X,\mathcal{B},\mu,T)$ be a system, $d\in\mathbb{N}$,
$f_{1},\cdots,f_{d}$ be bounded real valued functions to be bounded by one,
$a_{1},\cdots,a_{d}$ be distinct non-zero integers, and
$\mu^{\mathcal{A}}_{d}$ be Furstenberg self-joining of $(X,\mathcal{B},\mu,T)$
with respect to polynomial family $\mathcal{A}=\\{a_{1}n,\cdots,a_{d}n\\}$.
Then

$\int_{X^{d}}\limsup_{N\rightarrow\infty}\sup_{t}\Big{|}\frac{1}{N}\sum_{n=1}^{N}e^{2\pi
int}\prod_{i=1}^{d}f_{i}(T^{a_{i}n}x_{i})\Big{|}d\mu^{\mathcal{A}}_{d}(\textbf{x})\lesssim_{a_{1},\cdots,a_{d}}\min_{1\leq
i\leq d}\lvert\\!|\\!|f_{i}|\\!|\\!\rvert_{d+3}.$

###### Proof.

Let $C>2$ and fix it. By Lemma 2.10, we have

$\displaystyle\int_{X^{d}}\limsup_{N\rightarrow\infty}\sup_{t}\Big{|}\frac{1}{N}\sum_{n=1}^{N}e^{2\pi
int}\prod_{i=1}^{d}f_{i}(T^{a_{i}n}x_{i})\Big{|}^{2}d\mu^{\mathcal{A}}_{d}(\textbf{x})$
$\displaystyle\leq\frac{C}{H_{1}}+\frac{C}{H_{1}}\sum_{h_{1}=1}^{H_{1}-1}\int_{X^{d}}\limsup_{N\rightarrow\infty}\Big{|}\frac{1}{N}\sum_{n=1}^{N}\prod_{i=1}^{d}T^{a_{i}n}(f_{i}\cdot
T^{a_{i}h_{1}}f_{i})(x_{i})\Big{|}d\mu^{\mathcal{A}}_{d}(\textbf{x})$
$\displaystyle\leq\frac{C}{H_{1}}+\frac{C}{H_{1}}\sum_{h_{1}=1}^{H_{1}-1}\Big{(}\int_{X^{d}}\limsup_{N\rightarrow\infty}\Big{|}\frac{1}{N}\sum_{n=1}^{N}\prod_{i=1}^{d}T^{a_{i}n}(f_{i}\cdot
T^{a_{i}h_{1}}f_{i})(x_{i})\Big{|}^{2}d\mu^{\mathcal{A}}_{d}(\textbf{x})\Big{)}^{\frac{1}{2}}$

for any $H_{1}\in\mathbb{N}$. By Proposition 2.1, Birkhoff’s pointwise ergodic
theorem and Lemma 2.10, we have

$\displaystyle\int_{X^{d}}\limsup_{N\rightarrow\infty}\sup_{t}\Big{|}\frac{1}{N}\sum_{n=1}^{N}e^{2\pi
int}\prod_{i=1}^{d}f_{i}(T^{a_{i}n}x_{i})\Big{|}^{2}d\mu^{\mathcal{A}}_{d}(\textbf{x})$
$\displaystyle\leq\frac{C}{H_{1}}+\frac{C}{H_{1}}\sum_{h_{1}=1}^{H_{1}-1}\Big{(}\limsup_{N\rightarrow\infty}\int_{X^{d}}\Big{|}\frac{1}{N}\sum_{n=1}^{N}\prod_{i=1}^{d}T^{a_{i}n}(f_{i}\cdot
T^{a_{i}h_{1}}f_{i})(x_{i})\Big{|}^{2}d\mu^{\mathcal{A}}_{d}(\textbf{x})\Big{)}^{\frac{1}{2}}$
$\displaystyle\leq\frac{C}{H_{1}}+\frac{C}{H_{1}}\sum_{h_{1}=1}^{H_{1}-1}\Big{(}\limsup_{N\rightarrow\infty}\int_{X^{d}}\frac{C}{H_{2}}+\frac{2(N+H_{2}-1)}{H_{2}N}\sum_{h_{2}=1}^{H_{2}-1}\frac{H_{2}-h_{2}}{H_{2}}\cdot$
$\displaystyle\ \ \
\frac{1}{N}\sum_{n=1}^{N}\prod_{i=1}^{d}T^{a_{i}n}(f_{i}\cdot
T^{a_{i}h_{1}}f_{i})(x_{i})T^{a_{i}(n+h_{2})}(f_{i}\cdot
T^{a_{i}h_{1}}f_{i})(x_{i})d\mu^{\mathcal{A}}_{d}(\textbf{x})\Big{)}^{\frac{1}{2}}$

for any $H_{2}\in\mathbb{N}$. By definition of $\mu^{\mathcal{A}}_{d}$, we
know

$\displaystyle\int_{X^{d}}\limsup_{N\rightarrow\infty}\sup_{t}\Big{|}\frac{1}{N}\sum_{n=1}^{N}e^{2\pi
int}\prod_{i=1}^{d}f_{i}(T^{a_{i}n}x_{i})\Big{|}^{2}d\mu^{\mathcal{A}}_{d}(\textbf{x})$
$\displaystyle\leq\frac{C}{H_{1}}+\frac{C}{H_{1}}\sum_{h_{1}=1}^{H_{1}-1}\Big{(}\frac{C}{H_{2}}+\limsup_{N\rightarrow\infty}\Big{(}\frac{2(N+H_{2}-1)}{H_{2}N}\sum_{h_{2}=1}^{H_{2}-1}\frac{H_{2}-h_{2}}{H_{2}}\cdot$
$\displaystyle\ \ \
\frac{1}{N}\sum_{n=1}^{N}\lim_{M\rightarrow\infty}\frac{1}{M}\sum_{m=1}^{M}\int_{X}\prod_{i=1}^{d}T^{a_{i}(n+m)}(f_{i}\cdot
T^{a_{i}h_{1}}f_{i}\cdot T^{a_{i}h_{2}}f_{i}\cdot
T^{a_{i}(h_{1}+h_{2})}f_{i})d\mu(x)\Big{)}\Big{)}^{\frac{1}{2}}$
$\displaystyle\leq\frac{C}{H_{1}}+\frac{C}{H_{1}}\sum_{h_{1}=1}^{H_{1}-1}\Big{(}\frac{C}{H_{2}}+\frac{C}{H_{2}}\sum_{h_{2}=1}^{H_{2}-1}\limsup_{N\rightarrow\infty}\frac{1}{N}\cdot$
$\displaystyle\ \ \
\sum_{n=1}^{N}\limsup_{M\rightarrow\infty}\left\|\frac{1}{M}\sum_{m=1}^{M}\prod_{i=1}^{d}T^{a_{i}(n+m)}(f_{i}\cdot
T^{a_{i}h_{1}}f_{i}\cdot T^{a_{i}h_{2}}f_{i}\cdot
T^{a_{i}(h_{1}+h_{2})}f_{i})\right\|_{2}\Big{)}^{\frac{1}{2}}.$

By Proposition 2.5, we have

$\displaystyle\int_{X^{d}}\limsup_{N\rightarrow\infty}\sup_{t}\Big{|}\frac{1}{N}\sum_{n=1}^{N}e^{2\pi
int}\prod_{i=1}^{d}f_{i}(T^{a_{i}n}x_{i})\Big{|}^{2}d\mu^{\mathcal{A}}_{d}(\textbf{x})$
$\displaystyle\lesssim_{a_{1},\cdots,a_{d}}\frac{1}{H_{1}}+\frac{1}{H_{1}}\sum_{h_{1}=1}^{H_{1}-1}\Big{(}\frac{1}{H_{2}}+\frac{1}{H_{2}}\sum_{h_{2}=1}^{H_{2}-1}\min_{1\leq
i\leq d}\lvert\\!|\\!|f_{i}\cdot T^{a_{i}h_{1}}f_{i}\cdot
T^{a_{i}h_{2}}(f_{i}\cdot
T^{a_{i}h_{1}}f_{i})|\\!|\\!\rvert_{d+1}\Big{)}^{\frac{1}{2}}.$

By Proposition 2.4 and taking $\limsup$ for $H_{1},H_{2}$ respectively, we
have

$\displaystyle\int_{X^{d}}\limsup_{N\rightarrow\infty}\sup_{t}\Big{|}\frac{1}{N}\sum_{n=1}^{N}e^{2\pi
int}\prod_{i=1}^{d}f_{i}(T^{a_{i}n}x_{i})\Big{|}^{2}d\mu^{\mathcal{A}}_{d}(\textbf{x})$
$\displaystyle\lesssim_{a_{1},\cdots,a_{d}}\min_{1\leq i\leq
d}\limsup_{H_{1}\rightarrow\infty}\frac{1}{H_{1}}\sum_{h_{1}=1}^{H_{1}-1}\Big{(}\limsup_{H_{2}\rightarrow\infty}\frac{1}{H_{2}}\sum_{h_{2}=1}^{H_{2}-1}\lvert\\!|\\!|(f_{i}T^{a_{i}h_{1}}f_{i})T^{a_{i}h_{2}}(f_{i}T^{a_{i}h_{1}}f_{i})|\\!|\\!\rvert_{d+1}\Big{)}^{\frac{1}{2}}$
$\displaystyle\lesssim_{a_{1},\cdots,a_{d}}\min_{1\leq i\leq
d}\limsup_{H_{1}\rightarrow\infty}\frac{1}{H_{1}}\sum_{h_{1}=1}^{H_{1}-1}\lvert\\!|\\!|f_{i}\cdot
T^{a_{i}h_{1}}f_{i}|\\!|\\!\rvert_{d+2}$
$\displaystyle\lesssim_{a_{1},\cdots,a_{d}}\min_{1\leq i\leq
k}\lvert\\!|\\!|f_{i}|\\!|\\!\rvert_{d+3}^{2}.$

Therefore,

$\int_{X^{d}}\limsup_{N\rightarrow\infty}\sup_{t}\Big{|}\frac{1}{N}\sum_{n=1}^{N}e^{2\pi
int}\prod_{i=1}^{d}f_{i}(T^{a_{i}n}x_{i})\Big{|}d\mu^{\mathcal{A}}_{d}(\textbf{x})\lesssim_{a_{1},\cdots,a_{d}}\min_{1\leq
i\leq d}\lvert\\!|\\!|f_{i}|\\!|\\!\rvert_{d+3}.$

This finishes the proof. ∎

##  4\. Proof of Theorem 1.4

###### Proof of Theorem 1.4.

Note that all constant sequences are nilsequences. So we only need to verify
the following fact: Assume that for any $h_{1},\cdots,h_{d}\in
L^{\infty}(\mu)$, we have

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}\prod_{j=1}^{d}h_{j}(T^{a_{j}n}x)$

exists almost everywhere. Then for any real valued bounded
$f_{1},\cdots,f_{d}$ to be bounded by $\frac{1}{2}$, there exists full measure
subset $X(f_{1},\cdots,f_{d})$ of $X$ such that for any $x\in
X(f_{1},\cdots,f_{d})$, and any nilsequence
$\textbf{b}=\\{b_{n}\\}_{n\in\mathbb{Z}}$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}b_{n}\prod_{j=1}^{d}f_{j}(T^{a_{j}n}x)$

exists.

Choose a positive integer $k$ greater than one arbitrarily and fix it. Next,
we show that the above fact holds for all $(k-1)$-step nilsequences. Once we
have done it, we can finish the whole proof because the intersection of
countable full measure subsets still is of full measure.

For any $1\leq j\leq d$, let

$f_{j}=\mathbb{E}(f_{j}|\mathcal{Z}_{d+k}(T))+(f_{j}-\mathbb{E}(f_{j}|\mathcal{Z}_{d+k}(T))).$

Based on this decomposition, the rest can be divided into two steps.

######  Step 1.

At this step, we verify the following: If we can find $j$ from
$\\{1,\cdots,d\\}$ such that $f_{j}$ has zero conditional expectation with
respect to $\mathcal{Z}_{d+k}(T)$, then there exists full measure subset
$X(f_{1},\cdots,f_{d})$ of $X$ such that for any $x\in X(f_{1},\cdots,f_{d})$,
and any $(k-1)$-step nilsequence $\textbf{b}=\\{b_{n}\\}_{n\in\mathbb{Z}}$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}b_{n}\prod_{j=1}^{d}f_{j}(T^{a_{j}n}x)=0.$

The assumption tells us that there exists a full measure subset $X_{1}$ of $X$
such that for any $x\in X_{1}$, any $\underline{h}\in\mathbb{Z}^{k}$, we can
define $c_{\underline{h}}(x)$ by

$c_{\underline{h}}(x)=\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}\prod_{\underline{\epsilon}\in
V_{k}}a_{n+\underline{h}\cdot\underline{\epsilon}}(x)$

where for any $n\in\mathbb{Z}$, $a_{n}(x)=\prod_{j=1}^{d}f_{j}(T^{a_{j}n}x)$.
By [13, Proposition 2.2], for any $x\in X_{1}$, we can define
$\lvert\\!|\\!|\textbf{a}(x)|\\!|\\!\rvert_{k}$ by

$\lvert\\!|\\!|\textbf{a}(x)|\\!|\\!\rvert_{k}=\Big{(}\lim_{H\rightarrow\infty}\frac{1}{H^{k}}\sum_{\underline{h}\in[H]^{k}}c_{\underline{h}}(x)\Big{)}^{\frac{1}{2^{k}}}.$

Next, we verify that for $\mu$-a.e. $x\in X_{1}$,
$\lvert\\!|\\!|\textbf{a}(x)|\\!|\\!\rvert_{k}=0$. The rest proof of this step
is divided into two parts.

Part I: $k>2$. Let $\mu^{\mathcal{A}}_{d}$ be Furstenberg self-joining of
$(X,\mathcal{B},\mu,T)$ with respect to polynomial family
$\mathcal{A}=\\{a_{1}n,\cdots,a_{d}n\\}$. By the definition of
$\mu^{\mathcal{A}}_{d}$ and [13, Proposition 2.2], we have

$\displaystyle\int_{X}\lim_{H\rightarrow\infty}\frac{1}{H^{k}}\sum_{\underline{h}\in[H]^{k}}c_{\underline{h}}(x)d\mu(x)$
$\displaystyle=\lim_{H\rightarrow\infty}\frac{1}{H^{k}}\sum_{\underline{h}\in[H]^{k}}\int_{X}\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}\prod_{\underline{\epsilon}\in
V_{k}}a_{n+\underline{h}\cdot\underline{\epsilon}}(x)d\mu(x)$
$\displaystyle=\lim_{H\rightarrow\infty}\frac{1}{H^{k}}\sum_{\underline{h}\in[H]^{k}}\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}\int_{X}\prod_{j=1}^{d}T^{a_{j}n}(\prod_{\underline{\epsilon}\in
V_{k}}T^{a_{j}(\underline{h}\cdot\underline{\epsilon})}f_{j})d\mu(x)$
$\displaystyle=\lim_{H\rightarrow\infty}\frac{1}{H^{k}}\sum_{\underline{h}\in[H]^{k}}\int_{X^{d}}(\prod_{\underline{\epsilon}\in
V_{k}}T^{a_{1}(\underline{h}\cdot\underline{\epsilon})}f_{1})\otimes\cdots\otimes(\prod_{\underline{\epsilon}\in
V_{k}}T^{a_{d}(\underline{h}\cdot\underline{\epsilon})}f_{d})d\mu^{\mathcal{A}}_{d}(\textbf{x}).$

By Fatou’s Lemma, we know

$\displaystyle\int_{X}\lim_{H\rightarrow\infty}\frac{1}{H^{k}}\sum_{\underline{h}\in[H]^{k}}c_{\underline{h}}(x)d\mu(x)$
$\displaystyle\leq\Big{(}\int_{X^{d}}\limsup_{H\rightarrow\infty}\Big{|}\frac{1}{H^{k}}\sum_{\underline{h}\in[H]^{k}}(\prod_{\underline{\epsilon}\in
V_{k}}T^{a_{1}(\underline{h}\cdot\underline{\epsilon})}f_{1})\otimes\cdots\otimes(\prod_{\underline{\epsilon}\in
V_{k}}T^{a_{d}(\underline{h}\cdot\underline{\epsilon})}f_{d})\Big{|}^{2^{k-2}}d\mu^{\mathcal{A}}_{d}(\textbf{x})\Big{)}^{\frac{1}{2^{k-2}}}.$

Note that

$\displaystyle\frac{1}{H^{k}}\sum_{\underline{h}\in[H]^{k}}(\prod_{\underline{\epsilon}\in
V_{k}}T^{a_{1}(\underline{h}\cdot\underline{\epsilon})}f_{1})\otimes\cdots\otimes(\prod_{\underline{\epsilon}\in
V_{k}}T^{a_{d}(\underline{h}\cdot\underline{\epsilon})}f_{d})$
$\displaystyle=\Big{(}f_{1}\otimes\cdots\otimes
f_{d}\Big{)}\frac{1}{H^{k}}\sum_{\underline{h}\in[H]^{k}}\prod_{\underline{\epsilon}\in
V_{k}^{*}}G_{\phi(\underline{\epsilon}),\underline{h}\cdot\underline{\epsilon}}$

where $\\{G_{i,n}\\}_{1\leq i\leq 2^{k}-1,n\in\mathbb{Z}}$ is a family of
function sequences and for any $1\leq i\leq 2^{k}-1,n\in\mathbb{Z}$,
$G_{i,n}=T^{a_{1}n}f_{1}\otimes\cdots\otimes T^{a_{d}n}f_{d}$.

Let $C>2,\tilde{C}>(1024)^{k}C^{1024k}$ and fix them. By Lemma 3.2, we know
that

$\displaystyle\Big{(}\frac{1}{H^{k}}\sum_{\underline{h}\in[H]^{k}}(\prod_{\underline{\epsilon}\in
V_{k}}T^{a_{1}(\underline{h}\cdot\underline{\epsilon})}f_{1})\otimes\cdots\otimes(\prod_{\underline{\epsilon}\in
V_{k}}T^{a_{d}(\underline{h}\cdot\underline{\epsilon})}f_{d})\Big{)}^{2}$
$\displaystyle\leq\frac{C}{H^{k-2}}\sum_{h_{1},\cdots,h_{k-2}=0}^{H-1}\sup_{t}\Big{|}\frac{1}{H}\sum_{h_{k}=0}^{H-1}e^{2\pi
ih_{k}t}\prod_{\underline{\epsilon}\in
A_{k-1}^{k}}G_{\phi(\underline{\epsilon}),h_{k}\epsilon_{k}+\sum_{i=1}^{k-2}h_{i}\epsilon_{i}}\Big{|}^{2}.$

By an easy computation, for $k\geq 4$, we have

$\displaystyle
H\Big{|}\frac{1}{H^{k}}\sum_{\underline{h}\in[H]^{k}}(\prod_{\underline{\epsilon}\in
V_{k}}T^{a_{1}(\underline{h}\cdot\underline{\epsilon})}f_{1})\otimes\cdots\otimes(\prod_{\underline{\epsilon}\in
V_{k}}T^{a_{d}(\underline{h}\cdot\underline{\epsilon})}f_{d})\Big{|}^{2^{k-2}}$
$\displaystyle\leq\sum_{h_{1}=0}^{H-1}\Big{(}\frac{1}{H}\sum_{h_{2}=0}^{H-1}\Big{(}\cdots\Big{(}\frac{1}{H}\sum_{h_{k-2}=0}^{H-1}\sup_{t}\Big{|}\frac{1}{H}\sum_{h_{k}=0}^{H-1}e^{2\pi
ih_{k}t}\cdot$ $\displaystyle\ \ \ \prod_{\underline{\epsilon}\in
A_{k-1}^{k}}G_{\phi(\underline{\epsilon}),h_{k}\epsilon_{k}+\sum_{i=1}^{k-2}h_{i}\epsilon_{i}}\Big{|}^{2}\Big{)}^{2}\cdots\Big{)}^{2}\Big{)}^{2}.$

Choose sufficiently large $H$ and fix it. By Lemma 2.10, we have for any
$1\leq K\leq H$ and any $(h_{1},\cdots,h_{k-2})\in[H]^{k-2}$,

$\displaystyle\sup_{t}\Big{|}\frac{1}{H}\sum_{h_{k}=0}^{H-1}e^{2\pi
ih_{k}t}\prod_{\underline{\epsilon}\in
A_{k-1}^{k}}G_{\phi(\underline{\epsilon}),h_{k}\epsilon_{k}+\sum_{i=1}^{k-2}h_{i}\epsilon_{i}}\Big{|}^{2}$
$\displaystyle\leq\frac{C}{K}+\frac{C}{K}\sum_{s=1}^{K}\Big{|}\frac{1}{H}\sum_{n=0}^{H-1}\Big{(}\prod_{\underline{\epsilon}\in
A_{k-1}^{k}}G_{\phi(\underline{\epsilon}),n+\sum_{i=1}^{k-2}h_{i}\epsilon_{i}}\Big{)}\cdot\Big{(}\prod_{\underline{\epsilon}\in
A_{k-1}^{k}}G_{\phi(\underline{\epsilon}),n+s+\sum_{i=1}^{k-2}h_{i}\epsilon_{i}}\Big{)}\Big{|}.$

Then we have

$\displaystyle\frac{1}{H}\sum_{h_{k-2}=0}^{H-1}\sup_{t}\Big{|}\frac{1}{H}\sum_{h_{k}=0}^{H-1}e^{2\pi
ih_{k}t}\prod_{\underline{\epsilon}\in
A_{k-1}^{k}}G_{\phi(\underline{\epsilon}),h_{k}\epsilon_{k}+\sum_{i=1}^{k-2}h_{i}\epsilon_{i}}\Big{|}^{2}$
$\displaystyle\leq\frac{C}{K}+\frac{C}{K}\sum_{s=1}^{K}\frac{1}{H}\sum_{h_{k-2}=0}^{H-1}\Big{|}\frac{1}{H}\sum_{n=0}^{H-1}\prod_{\underline{\epsilon}\in
A_{k-1}^{k}}G_{\phi(\underline{\epsilon}),n+\sum_{i=1}^{k-2}h_{i}\epsilon_{i}}\prod_{\underline{\epsilon}\in
A_{k-1}^{k}}G_{\phi(\underline{\epsilon}),n+s+\sum_{i=1}^{k-2}h_{i}\epsilon_{i}}\Big{|}$
$\displaystyle\leq\frac{C}{K}+\frac{C}{K}\sum_{s=1}^{K}\Big{(}\frac{1}{H}\sum_{h_{k-2}=0}^{H-1}\Big{|}\frac{1}{H}\sum_{n=0}^{H-1}\prod_{\underline{\epsilon}\in
A_{k-1}^{k}}G_{\phi(\underline{\epsilon}),n+\sum_{i=1}^{k-2}h_{i}\epsilon_{i}}\prod_{\underline{\epsilon}\in
A_{k-1}^{k}}G_{\phi(\underline{\epsilon}),n+s+\sum_{i=1}^{k-2}h_{i}\epsilon_{i}}\Big{|}^{2}\Big{)}^{\frac{1}{2}}$

for any $(h_{1},\cdots,h_{k-3})\in[H]^{k-3}$. For any $1\leq s\leq K$, by
Proposition 3.1, we know

$\displaystyle\frac{1}{H}\sum_{h_{k-2}=0}^{H-1}\Big{|}\frac{1}{H}\sum_{n=0}^{H-1}\prod_{\underline{\epsilon}\in
A_{k-1}^{k}}G_{\phi(\underline{\epsilon}),n+\sum_{i=1}^{k-2}h_{i}\epsilon_{i}}\prod_{\underline{\epsilon}\in
A_{k-1}^{k}}G_{\phi(\underline{\epsilon}),n+s+\sum_{i=1}^{k-2}h_{i}\epsilon_{i}}\Big{|}^{2}$
$\displaystyle\leq\sup_{t}\Big{|}\frac{1}{H}\sum_{m=0}^{2(H-1)}e^{2\pi
imt}\Big{(}\prod_{\underline{\epsilon}\in
V_{k-3}^{*}}T^{a_{1}(m+\sum_{i=1}^{k-3}h_{i}\epsilon_{i})}f_{1}\otimes\cdots\otimes\prod_{\underline{\epsilon}\in
V_{k-3}^{*}}T^{a_{d}(m+\sum_{i=1}^{k-3}h_{i}\epsilon_{i})}f_{d}\Big{)}\cdot$
$\displaystyle\ \ \ \Big{(}\prod_{\underline{\epsilon}\in
V_{k-3}^{*}}T^{a_{1}(m+s+\sum_{i=1}^{k-3}h_{i}\epsilon_{i})}f_{1}\otimes\cdots\otimes\prod_{\underline{\epsilon}\in
V_{k-3}^{*}}T^{a_{d}(m+s+\sum_{i=1}^{k-3}h_{i}\epsilon_{i})}f_{d}\Big{)}\Big{|}^{2}.$

Therefore,

$\displaystyle\frac{1}{H}\sum_{h_{k-2}=0}^{H-1}\sup_{t}\Big{|}\frac{1}{H}\sum_{h_{k}=0}^{H-1}e^{2\pi
ih_{k}t}\prod_{\underline{\epsilon}\in
A_{k-1}^{k}}G_{\phi(\underline{\epsilon}),h_{k}\epsilon_{k}+\sum_{i=1}^{k-2}h_{i}\epsilon_{i}}\Big{|}^{2}$
$\displaystyle\leq\frac{C}{K}+\frac{C}{K}\sum_{s=1}^{K}\sup_{t}\Big{|}\frac{1}{H}\sum_{m=0}^{2(H-1)}e^{2\pi
imt}\cdot$ $\displaystyle\ \ \ \Big{(}\prod_{\underline{\epsilon}\in
V_{k-3}^{*}}T^{a_{1}(m+\sum_{i=1}^{k-3}h_{i}\epsilon_{i})}f_{1}\otimes\cdots\otimes\prod_{\underline{\epsilon}\in
V_{k-3}^{*}}T^{a_{d}(m+\sum_{i=1}^{k-3}h_{i}\epsilon_{i})}f_{d}\Big{)}\cdot$
$\displaystyle\ \ \ \Big{(}\prod_{\underline{\epsilon}\in
V_{k-3}^{*}}T^{a_{1}(m+s+\sum_{i=1}^{k-3}h_{i}\epsilon_{i})}f_{1}\otimes\cdots\otimes\prod_{\underline{\epsilon}\in
V_{k-3}^{*}}T^{a_{d}(m+s+\sum_{i=1}^{k-3}h_{i}\epsilon_{i})}f_{d}\Big{)}\Big{|}.$

If $k=3$, by Proposition 2.4, Lemma 3.3 and taking $\limsup$ for $H,K$ in
turn, we have

$\int_{X}\lim_{H\rightarrow\infty}\frac{1}{H^{3}}\sum_{\underline{h}\in[H]^{3}}c_{\underline{h}}(x)d\mu(x)\lesssim_{a_{1},\cdots,a_{d}}\min_{1\leq
i\leq d}\lvert\\!|\\!|f_{i}|\\!|\\!\rvert_{d+4}.$

If $k>3$, we go on working.

For any $(h_{1},\cdots,h_{k-4})\in[H]^{k-4}$, by Lemma 2.10 and Proposition
3.1, we have

$\displaystyle\frac{1}{H}\sum_{h_{k-3}=0}^{H-1}\Big{(}\frac{1}{H}\sum_{h_{k-2}=0}^{H-1}\sup_{t}\Big{|}\frac{1}{H}\sum_{h_{k}=0}^{H-1}e^{2\pi
ih_{k}t}\prod_{\underline{\epsilon}\in
A_{k-1}^{k}}G_{\phi(\underline{\epsilon}),h_{k}\epsilon_{k}+\sum_{i=1}^{k-2}h_{i}\epsilon_{i}}\Big{|}^{2}\Big{)}^{2}$
$\displaystyle\leq\frac{C^{2}}{K^{2}}+\frac{4C^{2}}{K}+\frac{4C^{3}}{K_{1}}+\frac{4C^{2}}{K}\sum_{s=1}^{K}\frac{C}{K_{1}}\sum_{t_{1}=1}^{K_{1}}\sup_{t}\Big{|}\frac{1}{2H-1}\sum_{m=0}^{4(H-1)}e^{2\pi
imt}\cdot$ $\displaystyle\ \ \ \Big{(}\prod_{\underline{\epsilon}\in
V_{k-4}^{*}}T^{a_{1}(m+\sum_{i=1}^{k-4}h_{i}\epsilon_{i})}f_{1}\otimes\cdots\otimes\prod_{\underline{\epsilon}\in
V_{k-4}^{*}}T^{a_{d}(m+\sum_{i=1}^{k-4}h_{i}\epsilon_{i})}f_{d}\Big{)}\cdot$
$\displaystyle\ \ \ \Big{(}\prod_{\underline{\epsilon}\in
V_{k-4}^{*}}T^{a_{1}(m+t_{1}+\sum_{i=1}^{k-4}h_{i}\epsilon_{i})}f_{1}\otimes\cdots\otimes\prod_{\underline{\epsilon}\in
V_{k-4}^{*}}T^{a_{d}(m+t_{1}+\sum_{i=1}^{k-4}h_{i}\epsilon_{i})}f_{d}\Big{)}\cdot$
$\displaystyle\ \ \ \Big{(}\prod_{\underline{\epsilon}\in
V_{k-4}^{*}}T^{a_{1}(m+s+\sum_{i=1}^{k-4}h_{i}\epsilon_{i})}f_{1}\otimes\cdots\otimes\prod_{\underline{\epsilon}\in
V_{k-4}^{*}}T^{a_{d}(m+s+\sum_{i=1}^{k-4}h_{i}\epsilon_{i})}f_{d}\Big{)}\cdot$
$\displaystyle\ \ \ \Big{(}\prod_{\underline{\epsilon}\in
V_{k-4}^{*}}T^{a_{1}(m+s+t_{1}+\sum_{i=1}^{k-4}h_{i}\epsilon_{i})}f_{1}\otimes\cdots\otimes\prod_{\underline{\epsilon}\in
V_{k-4}^{*}}T^{a_{d}(m+s+t_{1}+\sum_{i=1}^{k-4}h_{i}\epsilon_{i})}f_{d}\Big{)}\Big{|}$

for any $1\leq K_{1}\leq H$. In the above process, the index $h_{k-3}$
vanishes and new index $t_{1}$ is introduced. If $k=4$, we can stop at here.
If $k>4$, we repeat the process until index $h_{1}$ vanishes. Finally, if
$k\geq 4$, for any $1\leq K,K_{1},\cdots,K_{k-3}\leq H$, we have

$\displaystyle\Big{|}\frac{1}{H^{k}}\sum_{\underline{h}\in[H]^{k}}(\prod_{\underline{\epsilon}\in
V_{k}}T^{a_{1}(\underline{h}\cdot\underline{\epsilon})}f_{1})\otimes\cdots\otimes(\prod_{\underline{\epsilon}\in
V_{k}}T^{a_{d}(\underline{h}\cdot\underline{\epsilon})}f_{d})\Big{|}^{2^{k-2}}$
$\displaystyle\leq
R(K,K_{1},\cdots,K_{k-3})+\tilde{C}\frac{1}{K}\sum_{s=1}^{K}\frac{1}{K_{1}}\sum_{t_{1}=1}^{K_{1}}\cdots\frac{1}{K_{k-3}}\sum_{t_{k-3}=1}^{K_{k-3}}\sup_{t}\Big{|}\frac{1}{2^{k-2}(H-1)+1}\cdot$
$\displaystyle\ \ \ \sum_{m=0}^{2^{k-2}(H-1)}e^{2\pi
imt}\Big{(}\prod_{\underline{\epsilon}\in
V_{k-2}}T^{a_{1}(m+\underline{\epsilon}\cdot\underline{k})}f_{1}\Big{)}\otimes\cdots\otimes\Big{(}\prod_{\underline{\epsilon}\in
V_{k-2}}T^{a_{d}(m+\underline{\epsilon}\cdot\underline{k})}f_{d}\Big{)}\Big{|}$

where $\underline{k}=(t_{k-3},\cdots,t_{1},s),$ and

$\lim_{K\rightarrow\infty}\cdots\lim_{K_{k-3}\rightarrow\infty}R(K,K_{1},\cdots,K_{k-3})=0.$

By Proposition 2.4, Lemma 3.3 and taking $\limsup$ for
$H,K_{k-3},\cdots,K_{1},K$ in turn, we have

$\int_{X}\lim_{H\rightarrow\infty}\frac{1}{H^{k}}\sum_{\underline{h}\in[H]^{k}}c_{\underline{h}}(x)d\mu(x)\lesssim_{a_{1},\cdots,a_{d}}\min_{1\leq
i\leq d}\lvert\\!|\\!|f_{i}|\\!|\\!\rvert_{d+k+1}.$

Part II: $k=2$. By Proposition 3.1 and Lemma 3.3, we have

$\displaystyle\int_{X}\lim_{H\rightarrow\infty}\frac{1}{H^{k}}\sum_{\underline{h}\in[H]^{k}}c_{\underline{h}}(x)d\mu(x)$
$\displaystyle\leq\Big{(}\int_{X^{d}}\limsup_{H\rightarrow\infty}\Big{|}\frac{1}{H^{2}}\sum_{\underline{h}\in[H]^{2}}(\prod_{\underline{\epsilon}\in
V_{2}}T^{a_{1}(\underline{h}\cdot\underline{\epsilon})}f_{1})\otimes\cdots\otimes(\prod_{\underline{\epsilon}\in
V_{2}}T^{a_{d}(\underline{h}\cdot\underline{\epsilon})}f_{d})\Big{|}^{2}d\mu^{\mathcal{A}}_{d}(\textbf{x})\Big{)}^{\frac{1}{2}}$
$\displaystyle\leq\Big{(}\int_{X^{d}}\limsup_{H\rightarrow\infty}\Big{|}\frac{1}{H^{2}}\sum_{h_{1},h_{2}=0}^{H-1}(T^{a_{1}h_{1}}f_{1}T^{a_{1}h_{2}}f_{1}T^{a_{1}(h_{1}+h_{2})}f_{1})\otimes\cdots\otimes$
$\displaystyle\ \ \
(T^{a_{d}h_{1}}f_{d}T^{a_{d}h_{2}}f_{d}T^{a_{d}(h_{1}+h_{2})}f_{d})\Big{|}^{2}d\mu^{\mathcal{A}}_{d}(\textbf{x})\Big{)}^{\frac{1}{2}}$
$\displaystyle\leq\Big{(}4\int_{X^{d}}\limsup_{H\rightarrow\infty}\sup_{t}\Big{|}\frac{1}{2H-1}\sum_{m=0}^{2(H-1)}e^{2\pi
imt}(T^{a_{1}m}f_{1})\otimes\cdots\otimes(T^{a_{d}m}f_{d})\Big{|}^{2}d\mu^{\mathcal{A}}_{d}(\textbf{x})\Big{)}^{\frac{1}{2}}$
$\displaystyle\lesssim_{a_{1},\cdots,a_{d}}\min_{1\leq i\leq
d}\lvert\\!|\\!|f_{i}|\\!|\\!\rvert_{d+3}.$

To sum up, for any $k\geq 2$, we have

$\int_{X}\lim_{H\rightarrow\infty}\frac{1}{H^{k}}\sum_{\underline{h}\in[H]^{k}}c_{\underline{h}}(x)d\mu(x)\lesssim_{a_{1},\cdots,a_{d}}\min_{1\leq
i\leq d}\lvert\\!|\\!|f_{i}|\\!|\\!\rvert_{d+k+1}.$

By Theorem 2.2, we know that for $\mu$-a.e. $x\in X_{1}$,
$\lvert\\!|\\!|\textbf{a}(x)|\\!|\\!\rvert_{k}=0$. By Proposition 2.7, we know
that there exists full measure subset $X(f_{1},\cdots,f_{d})$ of $X$ such that
for any $x\in X(f_{1},\cdots,f_{d})$, and any $(k-1)$-step nilsequence
$\textbf{b}=\\{b_{n}\\}_{n\in\mathbb{Z}}$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}b_{n}\prod_{j=1}^{d}f_{j}(T^{a_{j}n}x)=0.$

######  Step 2.

At this step, we verify the following: If all $f_{j},1\leq j\leq d$ are
measurable with respect to $\mathcal{Z}_{d+k}(T)$, then there exists full
measure subset $X(f_{1},\cdots,f_{d})$ of $X$ such that for any $x\in
X(f_{1},\cdots,f_{d})$, and any $(k-1)$-step nilsequence
$\textbf{b}=\\{b_{n}\\}_{n\in\mathbb{Z}}$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}b_{n}\prod_{j=1}^{d}f_{j}(T^{a_{j}n}x)$

exists.

By [14, Theorem 16.10], we know that for any $1\leq j\leq d$ and any
$r\in\mathbb{N}$, we can find a function sequence $\\{g_{j,m}\\}_{m\geq 1}$
such that for each $m\geq 1$, the followings hold:

  * (1)

$\left\|g_{j,m}\right\|_{\infty}\leq\left\|f_{j}\right\|_{\infty}$;

  * (2)

$\left\|g_{j,m}-f_{j}\right\|_{1}\leq\frac{1}{4^{m+1+r}Cd}$ where
$C=d^{2^{k+d+1}}$;

  * (3)

for $\mu$-a.e. $x\in X$, $\\{g_{j,m}(T^{a_{j}n}x)\\}_{n\in\mathbb{Z}}$ is a
$(d+k)$-step nilsequence.

Let $s=d+k+1$. By Birkhoff’s pointwise ergodic theorem and our assumption, we
can find a full measure subset $X_{0}$ of $X$ satisfies the followings:

  * (1)

for any $x\in X_{0}$, any $n\in\mathbb{Z}$ and any $1\leq j\leq d$,
$|f_{j}(T^{a_{j}n}x)|\leq\frac{1}{2}$;

  * (2)

for any $x\in X_{0}$, any $n\in\mathbb{Z}$, any $1\leq j\leq d$ and any $m\geq
1$, $|g_{j,m}(T^{a_{j}n}x)|\leq\frac{1}{2}$;

  * (3)

for any $x\in X_{0}$, any $1\leq j\leq d$ and any $m\geq 1$,
$\\{g_{j,m}(T^{a_{j}n}x)\\}_{n\in\mathbb{Z}}$ is a $(d+k)$-step nilsequence;

  * (4)

for any $x\in X_{0}$, any $1\leq j\leq d$ and any $m\geq 1$, we have

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}|f_{j}(T^{a_{j}n}x)-g_{j,m}(T^{a_{j}n}x)|=\mathbb{E}(|f_{j}-g_{j,m}|\big{|}\mathcal{I}_{a_{j}})(x)$

where $\mathcal{I}_{a_{j}}$ is a sub-$\sigma$-algebra generated by all
$T^{a_{j}}$-invariant subsets;

  * (5)

for any $x\in X_{0}$, any $m\geq 1$ and any $\underline{h}\in\mathbb{Z}^{s}$,
the limit

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}\prod_{\underline{\epsilon}\in
V_{s}}a^{(m)}_{n+\underline{h}\cdot\underline{\epsilon}}$

exists where

$\displaystyle a_{n}^{(m)}$
$\displaystyle=\prod_{j=1}^{d}f_{j}(T^{a_{j}n}x)-\prod_{j=1}^{d}g_{j,m}(T^{a_{j}n}x)$
$\displaystyle=\sum_{i=1}^{d}\prod_{j=1}^{i-1}f_{j}(T^{a_{j}n}x)\cdot(f_{i}(T^{a_{i}n}x)-g_{i,m}(T^{a_{i}n}x))\cdot\prod_{l=i+1}^{d}g_{l,m}(T^{a_{l}n}x)$

for any $n\in\mathbb{Z}$. we denote the limit by $c_{\underline{h}}^{(m)}(x)$;

  * (6)

for any $x\in X_{0}$, any $m\geq 1$ and any $\underline{h}\in\mathbb{Z}^{s}$,

$|c_{\underline{h}}^{(m)}(x)|\leq
C\sum_{j=1}^{d}\mathbb{E}\Big{(}|f_{j}-g_{j,m}|\big{|}\mathcal{I}_{a_{j}}\Big{)}(x).$

By [13, Proposition 2.2], for any $m\geq 1$, we have

$\displaystyle\int_{X}\lim_{H\rightarrow\infty}\frac{1}{H^{s}}\sum_{\underline{h}\in[H]^{s}}c^{(m)}_{\underline{h}}(x)d\mu(x)$
$\displaystyle=\lim_{H\rightarrow\infty}\frac{1}{H^{s}}\sum_{\underline{h}\in[H]^{s}}\int_{X}c^{(m)}_{\underline{h}}(x)d\mu(x)$
$\displaystyle\leq\limsup_{H\rightarrow\infty}\frac{1}{H^{s}}\sum_{\underline{h}\in[H]^{s}}\int_{X}C\sum_{j=1}^{d}\mathbb{E}\big{(}|f_{j}-g_{j,m}|\big{|}\mathcal{I}_{a_{j}}\Big{)}(x)d\mu(x)$
$\displaystyle\leq C\sum_{j=1}^{d}\left\|f_{j}-g_{j,m}\right\|_{1}$
$\displaystyle=\frac{1}{4^{m+r+1}}.$

Then for any $m\geq 1$,

$\mu(\\{x\in
X_{0}:\lim_{H\rightarrow\infty}\frac{1}{H^{s}}\sum_{\underline{h}\in[H]^{s}}c^{(m)}_{\underline{h}}(x)<\frac{1}{2^{m+1+r}}\\})\geq
1-\frac{1}{2^{m+1+r}}.$

Let

$X_{m}=\\{x\in
X_{0}:\lim_{H\rightarrow\infty}\frac{1}{H^{s}}\sum_{\underline{h}\in[H]^{s}}c^{(m)}_{\underline{h}}(x)\leq\frac{1}{2^{m+1+r}}\\}.$

For any $m\geq 1$, let $Y_{r}=\bigcap_{m\geq 1}X_{m}$, then $\mu(Y_{r})\geq
1-\frac{1}{2^{r}}$. Let $Y=\bigcup_{r\geq 1}Y_{r}$. Then $\mu(Y)=1$.

Note that the product of finite $(d+k)$-step nilsequences is still a
$(d+k)$-step nilsequence and every $(k-1)$-step nilsequence can be viewed as a
$(d+k)$-step nilsequence.

By the definition of $Y$ and Proposition 2.8, we know that for any $x\in Y$,
and any $(k-1)$-step nilsequence $\textbf{b}=\\{b_{n}\\}_{n\in\mathbb{Z}}$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=0}^{N-1}b_{n}\prod_{j=1}^{d}f_{j}(T^{a_{j}n}x)$

exists. This finishes the whole proof. ∎

##  5\. Proof of Theorem 1.8

Before the proof, we provide a lemma. Based on [10, Theorem 2.8] and [19,
Theorem 1.1], we can get the following lemma by repeating the arguments of the
proof of [9, Proposition 3.2].

######  Lemma 5.1.

Given $d\in\mathbb{N}$, let $S_{1},\cdots,S_{d}$ be invertible measure
preserving transformations acting on a Lebesgue space $(X,\mathcal{B},\mu)$
such that $S_{1},\cdots,S_{d}$ are commuting and let
$p_{1}(n),\cdots,p_{d}(n)$ be non-constant integer coefficents polynomials
with $\deg p_{1}>\deg p_{2}>\cdots>\deg p_{d}\geq 2$. Then for any real valued
$g_{1},\cdots,g_{d}\in L^{\infty}(\mu)$, and any strictly increasing sequence
of positive integers $\\{N_{k}\\}_{k\geq 1}$, there exists a subsequence
$\\{N^{\prime}_{k}\\}_{k\geq 1}$ of $\\{N_{k}\\}_{k\geq 1}$ such that for
$\mu$-a.e $x\in X$, the sequence
$\\{\prod_{j=1}^{d}g_{j}(S_{j}^{p_{j}(n)}x)\\}_{n\in\mathbb{Z}}$ admits a
correlation on sequence $\\{N^{\prime}_{k}\\}_{k\geq 1}$. Moreover, if for
some $j\in\\{1,\cdots,d\\}$,
$\mathbb{E}(g_{j}|\mathcal{Z}_{\infty}(S_{j}))=0$, then the $0$-th coordinate
projection $F_{0}:\Omega\rightarrow\mathbb{R}$ has zero conditional
expectation with respect to the Pinsker factor111Every system
$(Y,\mathcal{D},\nu,S)$ has a maximal zero entropy factor. The factor is
called Pinsker factor of $(Y,\mathcal{D},\nu,S)$. of the corresponding
Furstenberg system.

Now, we begin to prove the Theorem 1.8. The idea of this proof is from the
proof of [9, Proposition 4.1,4.2] partly.

###### Proof of Theorem 1.8.

Without loss of generality, we can assume that $f_{1},f_{2},g_{1},\cdots,\\\
g_{d}$ are bounded real valued functions which take value on interval
$[-1,1]$. Let $\Omega=[-1,1]^{\mathbb{Z}}$. Let $\mathcal{F}$ be the Borel
$\sigma$-algebra of $\Omega$. The rest proof is divided into two parts.

Part I: We verify the following: If we can find $j$ from $\\{1,\cdots,d\\}$
such that $g_{j}$ has zero conditional expectation with respect to
$\mathcal{Z}_{\infty}(S_{j})$, then

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^{N}f_{1}(T^{an}x)f_{2}(T^{bn}x)\prod_{j=1}^{d}g_{j}(S_{j}^{p_{j}(n)}x)=0$

in $L^{2}(\mu)$.

Suppose that the above result fails. Then there exist $\epsilon>0$ and a
strictly increasing sequence of positive integers $\\{N_{k}\\}_{k\geq 1}$ such
that for any $k\in\mathbb{N}$,

$\left\|\frac{1}{N_{k}}\sum_{n=1}^{N_{k}}f_{1}(T^{an}x)f_{2}(T^{bn}x)\prod_{j=1}^{d}g_{j}(S_{j}^{p_{j}(n)}x)\right\|_{2}\geq\epsilon.$

By Lemma 5.1, there exists a subsequence $\\{N^{\prime}_{k}\\}_{k\geq 1}$ of
$\\{N_{k}\\}_{k\geq 1}$ such that for $\mu$-a.e $x\in X$, the sequence
$\\{\prod_{j=1}^{d}g_{j}(S_{j}^{p_{j}(n)}x)\\}_{n\in\mathbb{Z}}$ admits a
correlation on sequence $\\{N^{\prime}_{k}\\}_{k\geq 1}$. Moreover, the $0$-th
coordinate projection $F_{0}:\Omega\rightarrow\mathbb{R}$ has zero conditional
expectation with respect to the Pinsker factor of the corresponding
Furstenberg system $(\Omega,\mathcal{F},\nu_{x},\sigma)$.

By [9, Proposition 2.5] and [20, Theorem 4.13.(ii)], for $\mu$-a.e $x\in X$,
the sequence $\\{f_{1}(T^{an}x)\\}_{n\in\mathbb{Z}}$ admits a correlation on
sequence $\\{N^{\prime}_{k}\\}_{k\geq 1}$ and the corresponding Furstenberg
system $(\Omega,\mathcal{F},\lambda_{x}^{a},\sigma)$ is ergodic and has zero
entropy. Let $G_{0}^{a}$ be its $0$-th coordinate projection. Likely, for
$\mu$-a.e $x\in X$, the sequence $\\{f_{2}(T^{bn}x)\\}_{n\in\mathbb{Z}}$
admits a correlation on sequence $\\{N^{\prime}_{k}\\}_{k\geq 1}$ and the
corresponding Furstenberg system $(\Omega,\mathcal{F},\lambda_{x}^{b},\sigma)$
is ergodic and has zero entropy. Let $G_{0}^{b}$ be its $0$-th coordinate
projection.

So we can find a full measure subset $X_{0}$ of $X$ such that for any $x\in
X_{0}$, the previous properties hold. By our assumption, there exists a subset
$X_{1}$ of $X_{0}$ with $\mu(X_{1})>0$ such that for any $x\in X_{1}$, there
exists a subsequence $\\{N^{\prime}_{x,k}\\}_{k\geq 1}$ of
$\\{N^{\prime}_{k}\\}_{k\geq 1}$ such that

$\lim_{k\rightarrow\infty}\frac{1}{N^{\prime}_{x,k}}\sum_{n=1}^{N^{\prime}_{x,k}}f_{1}(T^{an}x)f_{2}(T^{bn}x)\prod_{j=1}^{d}g_{j}(S_{j}^{p_{j}(n)}x)$

exists and is non-zero.

Let $x\in X_{1}$ be fixed at here. Let $w_{a},w_{b},z\in\Omega$ defined by
$w_{a}(n)=f_{1}(T^{an}x),w_{b}(n)\\\
=f_{2}(T^{bn}x),z(n)=\prod_{j=1}^{d}g_{j}(S_{j}^{p_{j}(n)}x)$. Then there
exists a subsequence $\\{N^{\prime\prime}_{x,k}\\}_{k\geq 1}$ of
$\\{N^{\prime}_{x,k}\\}_{k\geq 1}$ such that the sequence

$\Big{\\{}\frac{1}{N^{\prime\prime}_{x,k}}\sum_{n=1}^{N^{\prime\prime}_{x,k}}\delta_{(\sigma^{n}w_{a},\sigma^{n}w_{b})}\Big{\\}}_{k\geq
1}$

has weak-$*$ limit $\lambda_{x}$ and the sequence

$\Big{\\{}\frac{1}{N^{\prime\prime}_{x,k}}\sum_{n=1}^{N^{\prime\prime}_{x,k}}\delta_{(\sigma^{n}w_{a},\sigma^{n}w_{b},\sigma^{n}z)}\Big{\\}}_{k\geq
1}$

has weak-$*$ limit $\rho_{x}$. Clealy, $\rho_{x}$ is a joining of
$(\Omega,\mathcal{F},\nu_{x},\sigma)$ and
$(\Omega\times\Omega,\mathcal{F}^{\otimes 2},\lambda_{x},\sigma\times\sigma)$.
By [7, Fact 4.4.3], we know that $(\Omega\times\Omega,\mathcal{F}^{\otimes
2},\lambda_{x},\sigma\times\sigma)$ has zero entropy since $\lambda_{x}$ is a
joining of $(\Omega,\mathcal{F},\lambda_{x}^{a},\sigma)$ and
$(\Omega,\mathcal{F},\lambda_{x}^{b},\sigma)$. By [9, Proposition 2.1], we
have

$\lim_{k\rightarrow\infty}\frac{1}{N^{\prime\prime}_{x,k}}\sum_{n=1}^{N^{\prime\prime}_{x,k}}f_{1}(T^{an}x)f_{2}(T^{bn}x)\prod_{j=1}^{d}g_{j}(S_{j}^{p_{j}(n)}x)=\int
F_{0}\otimes G_{0}^{a}\otimes G_{0}^{b}d\rho_{x}=0.$

It is a contradiction. So

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^{N}f_{1}(T^{an}x)f_{2}(T^{bn}x)\prod_{j=1}^{d}g_{j}(S_{j}^{p_{j}(n)}x)=0$

in $L^{2}(\mu)$.

Part II: we verify the following: If for any $1\leq j\leq d$, $g_{j}$ is
measurable with respect to $\mathcal{Z}_{\infty}(S_{j})$, then

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^{N}f_{1}(T^{an}x)f_{2}(T^{bn}x)\prod_{j=1}^{d}g_{j}(S_{j}^{p_{j}(n)}x)$

exists almost everywhere.

By [14, Theorem 14.15,16.10], we know that for any $1\leq j\leq d$, we can
find a function sequence $\\{h_{j,m}\\}_{m\geq 1}$ such that for each $m\geq
1$, the followings hold:

  * (1)

$\left\|h_{j,m}-g_{j}\right\|_{d+3}\leq\frac{1}{4^{m+1}}$;

  * (2)

for $\mu$-a.e. $x\in X$, $\\{h_{j,m}(S_{j}^{p_{j}(n)}x)\\}_{n\in\mathbb{Z}}$
is a nilsequence.

Note that product of finite nilsequences is still a nilsequence. By Corollary
1.5, we know for any $m\geq 1$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^{N}f_{1}(T^{an}x)f_{2}(T^{bn}x)\prod_{j=1}^{d}h_{j,m}(S_{j}^{p_{j}(n)}x)$

exists almost everywhere. By repeating the arguments of the proof of [6,
Corollary 2.2], we know that

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^{N}f_{1}(T^{an}x)f_{2}(T^{bn}x)\prod_{j=1}^{d}g_{j}(S_{j}^{p_{j}(n)}x)$

exists almost everywhere. This finishes the proof. ∎

## Acknowledgement

The author is supported by NNSF of China (11971455, 12031019, 12090012). The
author’s thanks go to Professor Song Shao and Professor Xiangdong Ye for their
useful suggestions.

## References

  * [1] I. Assani.  Pointwise convergence of ergodic averages along cubes.  Journal d’Analyse Mathématique, 110(1):241–269, 2010.
  * [2] I. Assani, D. Duncan, and R. Moore.  Pointwise characteristic factors for Wiener-Wintner double recurrence theorem.  Ergodic Theory Dyn. Syst., 36(4):1037–1066, 2016.
  * [3] I. Assani and R. Moore.  A good universal weight for nonconventional ergodic averages in norm.  Ergodic Theory Dyn. Syst., 37(4):1009–1025, 2017.
  * [4] I. Assani and R. Moore.  Extension of Wiener-Wintner double recurrence theorem to polynomials.  J. Anal. Math., 134(2):597–613, 2018.
  * [5] J. Bourgain.  Double recurrence and almost sure convergence.  J. Reine Angew. Math., 404:140–161, 1990.
  * [6] J. Derrien and E. Lesigne.  A pointwise polynomial ergodic theorem for exact endomorphisms and K-systems.  Ann. Inst. Henri Poincaré, Probab. Stat., 32(6):765–778, 1996\.
  * [7] T. Downarowicz.  Entropy in dynamical systems, volume 18 of New Math. Monogr. Cambridge: Cambridge University Press, 2011.
  * [8] N. Frantzikinakis and B. Host.  The logarithmic Sarnak conjecture for ergodic weights.  Ann. Math. (2), 187(3):869–931, 2018.
  * [9] N. Frantzikinakis and B. Host.  Multiple recurrence and convergence without commutativity.  arXiv preprint arXiv:2111.01518, to appear in the Journal of the London Mathematical Society., 2023.
  * [10] N. Frantzikinakis and B. Kuca.  Joint ergodicity for commuting transformations and applications to polynomial sequences.  arXiv preprint arXiv:2207.12288, 2022.
  * [11] H. Furstenberg.  Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions.  J. Anal. Math., 31:204–256, 1977.
  * [12] B. Host and B. Kra.  Nonconventional ergodic averages and nilmanifolds.  Ann. Math. (2)., 161(1):397–488, 2005.
  * [13] B. Host and B. Kra.  Uniformity seminorms on $\ell^{\infty}$ and applications.  J. Anal. Math., 108:219–276, 2009.
  * [14] B. Host and B. Kra.  Nilpotent structures in ergodic theory, volume 236 of Math. Surv. Monogr. Providence, RI: American Mathematical Society (AMS), 2018.
  * [15] W. Huang, S. Shao, and X. Ye.  Pointwise convergence of multiple ergodic averages and strictly ergodic models.  J. Anal. Math., 139(1):265–305, 2019.
  * [16] W. Huang, S. Shao, and X. Ye.  Polynomial furstenberg joinings and its applications.  arXiv preprint arXiv:2301.07881, 2023.
  * [17] L. Kuipers and H. Niederreiter.  Uniform distribution of sequences.  Pure and Applied Mathematics. New York etc.: John Wiley & Sons, a Wiley-Interscience Publication. xiv, 390 p. £ 13.00 (1974)., 1974.
  * [18] A. Leibman.  Convergence of multiple ergodic averages along polynomials of several variables.  Isr. J. Math., 146:303–315, 2005.
  * [19] M. N. Walsh.  Norm convergence of nilpotent ergodic averages.  Ann. Math. (2)., 175(3):1667–1688, 2012.
  * [20] P. Walters.  An introduction to ergodic theory, volume 79 of Grad. Texts Math. Springer, Cham, 1982.
  * [21] N. Wiener and A. Wintner.  Harmonic analysis and ergodic theory.  Am. J. Math., 63:415–426, 1941.
  * [22] P. Zorin-Kranich.  A uniform nilsequence wiener-wintner theorem for bilinear ergodic averages.  arXiv preprint arXiv:1504.04647, 2015.