Assume now that $\displaystyle R_{1}\sim R_{2}$ and $\displaystyle R_{2}\sim
R_{3}$. We may write $\displaystyle R_{2}$ as above and $\displaystyle
R_{3}=\\{w_{1},\dots,w_{k}\\}$, where $\displaystyle
w_{i^{\prime}}=-r^{\prime}v_{i^{\prime}}$ and $\displaystyle
w_{j}=r^{\prime}(v_{j}-v_{i^{\prime}})$ for all $\displaystyle j\neq
i^{\prime}$ for some $\displaystyle 1\leq i^{\prime}\leq k$.

If $\displaystyle i=i^{\prime}$, then $\displaystyle
w_{i}=-r^{\prime}v_{i}=-r^{\prime}(-ru_{i})=rr^{\prime}u_{i}$, and
$\displaystyle
w_{j}=r^{\prime}(v_{j}-v_{i})=r^{\prime}r(u_{j}-u_{i})-r^{\prime}(-ru_{i})=rr^{\prime}u_{j}$
for all $\displaystyle j\neq i$. So $\displaystyle R_{3}=rr^{\prime}R_{1}$.
This implies that $\displaystyle R_{1}\sim R_{3}$.

If $\displaystyle i\neq i^{\prime}$, then $\displaystyle
w_{i}=r^{\prime}(v_{i}-v_{i^{\prime}})=r^{\prime}(-ru_{i})-r^{\prime}r(u_{i^{\prime}}-u_{i})=-rr^{\prime}u_{i^{\prime}}$,
$\displaystyle
w_{i^{\prime}}=-r^{\prime}v_{i^{\prime}}=r^{\prime}r(u_{i}-u_{i^{\prime}})$,
and $\displaystyle
w_{j}=r^{\prime}(v_{j}-v_{i^{\prime}})=r^{\prime}r(u_{j}-u_{i})-r^{\prime}r(u_{i^{\prime}}-u_{i})=rr^{\prime}(u_{j}-u_{i^{\prime}})$
for all $\displaystyle j\neq i,i^{\prime}$. This implies that $\displaystyle
R_{1}\sim R_{3}$ and the result follows. ∎

We write $\displaystyle R_{1}\lesssim R_{2}$ for some $\displaystyle
R_{1},R_{2}\in V$ if there exists $\displaystyle R_{3}\in V$ such that
$\displaystyle R_{2}\sim R_{3}$ and $\displaystyle R_{1}\subseteq R_{3}$.

Recall that for
$\displaystyle\textbf{b}=(b_{1},\dots,b_{L})\in(\mathbb{Q}^{d})^{L},$
$\displaystyle b_{i}\in\mathbb{Q}^{d}$, we denote $\displaystyle
G(\textbf{b})=\text{span}_{\mathbb{Q}}\\{b_{1},\dots,b_{L}\\}\cap\mathbb{Z}^{d}.$
The first ingredient we need to prove Theorem 5.1 is an upper bound for the
multiple averages in terms of Host-Kra seminorms. The following proposition
shows that we can somehow control the coefficients we get in the end of the
PET-induction by the initial ones.

######  Proposition 5.5 (Bounding multiple averages by averaged Host-Kra
seminorms).

Let $\displaystyle d,k,L\in\mathbb{N}^{\ast},$ $\displaystyle p_{1},\dots$,
$\displaystyle p_{k}\colon\mathbb{Z}^{L}\to\mathbb{Z}^{d}$ a non-degenerate
family of polynomials of degrees at most $\displaystyle K,$ with
$\displaystyle p_{i}(n)=\sum_{v\in\mathbb{N}^{L},|v|\leq K}b_{i,v}n^{v}$ for
some $\displaystyle b_{i,v}\in\mathbb{Q}^{d},$ and $\displaystyle
R_{v}\coloneqq\\{b_{i,v}\colon 1\leq i\leq k\\}\cup\\{{\bf 0}\\}$. Then there
exist $\displaystyle s,t_{1},\dots,t_{k}\in\mathbb{N}^{\ast}$ and polynomials
$\displaystyle\mathbb{c}_{i,m}\colon(\mathbb{Z}^{L})^{s}\to(\mathbb{Z}^{d})^{L},1\leq
i\leq k,1\leq m\leq t_{i}$ with $\displaystyle\mathbb{c}_{i,m}\not\equiv{\bf
0},$ such that the following hold

  * (i)

(Control of the coefficients) Each $\displaystyle\mathbb{c}_{i,m}$ is of the
form

$\displaystyle\mathbb{c}_{i,m}(h_{1},\dots,h_{s})=\sum_{a_{1},\dots,a_{s}\in\mathbb{N}^{L}}h^{a_{1}}_{1}\dots
h^{a_{s}}_{s}\cdot\mathbb{u}_{i,m}(a_{1},\dots,a_{s})$

for some

$\displaystyle\mathbb{u}_{i,m}(a_{1},\dots,a_{s})=(u_{i,m,1}(a_{1},\dots,a_{s}),\dots,u_{i,m,L}(a_{1},\dots,a_{s}))\in(\mathbb{Q}^{d})^{L}$

with all but finitely many terms being zero for each $\displaystyle(i,m).$ In
addition, for all $\displaystyle a_{1},\dots,a_{s}\in\mathbb{N}^{L}$ not all
equal to 0 and every $\displaystyle 1\leq i\leq k,1\leq r\leq L$, denoting

$\displaystyle
U_{i,r}(a_{1},\dots,a_{s})\coloneqq\\{u_{i,m,r}(a_{1},\dots,a_{s})\in\mathbb{Z}^{d}\colon
1\leq m\leq t_{k}\\}\cup\\{{\bf 0}\\},$

we have that there exists $\displaystyle v\in\mathbb{N}^{L},|v|>0$ such that
$\displaystyle U_{i,r}(a_{1},\dots,a_{s})\lesssim R_{v}$.

  * (ii)

(Control of the average) For every $\displaystyle\mathbb{Z}^{d}$-system
$\displaystyle\mathbb{X}=(X,\mathcal{B},\mu,(T_{g})_{g\in\mathbb{Z}^{d}})$ and
every $\displaystyle f_{1},\dots,$ $\displaystyle f_{k}\in L^{\infty}(\mu)$
bounded by $\displaystyle 1,$ we have that

(21) $\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\ \text{ F\o lner
seq.}\end{subarray}}\varlimsup_{N\to\infty}\Bigl{\|}\mathbb{E}_{n\in
I_{N}}\prod_{i=1}^{k}T_{p_{i}(n)}f_{i}\Bigr{\|}_{L^{2}(\mu)}\leq
C\cdot\min_{1\leq i\leq
k}\overline{\mathbb{E}}^{\square}_{h_{1},\dots,h_{s}\in\mathbb{Z}^{L}}\|f_{i}\|_{(G(\mathbb{c}_{i,m}(h_{1},\dots,h_{s})))_{1\leq
m\leq t_{i}}},$

where $\displaystyle C>0$ is a constant depending only on $\displaystyle
p_{1},\dots,p_{k}$.282828 One can in fact show that $\displaystyle C$ depends
only on $\displaystyle d,k,L$ and the highest degree of $\displaystyle
p_{1},\dots,p_{k}$. We will not go into more detail about this fact, since it
is unimportant for our purposes.

The second ingredient we need in order to show Theorem 5.1 (which is the main
novelty of this paper) is to estimate the right hand side of (21) using the
concatenation theorem.

######  Proposition 5.6 (Bounding averaged Host-Kra seminorms by a single
one).

Let $\displaystyle p_{1},\dots,p_{k}\colon\mathbb{Z}^{L}\to\mathbb{Z}^{d}$ be
a family of polynomials. Suppose that there exist $\displaystyle
s,t_{1},\dots,t_{k}\in\mathbb{N}^{\ast}$ and polynomials
$\displaystyle\mathbb{c}_{i,m}\colon(\mathbb{Z}^{L})^{s}\to(\mathbb{Z}^{d})^{L},1\leq
i\leq k,1\leq m\leq t_{i},$ with $\displaystyle\mathbb{c}_{i,m}\not\equiv{\bf
0}$ given by

(22)
$\begin{split}\mathbb{c}_{i,m}(h_{1},\dots,h_{s})=\sum_{a_{1},\dots,a_{s}\in\mathbb{N}^{L}}h^{a_{1}}_{1}\dots
h^{a_{s}}_{s}\cdot\mathbb{u}_{i,m}(a_{1},\dots,a_{s})\end{split}$

for some
$\displaystyle\mathbb{u}_{i,m}(a_{1},\dots,a_{s})\in(\mathbb{Q}^{d})^{L}$ with
all but finitely many terms equal to $\displaystyle{\bf 0}$ for each
$\displaystyle(i,m)$ such that the following holds: if for every
$\displaystyle\mathbb{Z}^{d}$-system
$\displaystyle(X,\mathcal{B},\mu,(T_{g})_{g\in\mathbb{Z}^{d}})$ and every
$\displaystyle f_{1},\dots,f_{k}\in L^{\infty}(\mu)$ bounded by $\displaystyle
1,$ we have that

(23) $\begin{split}\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\ \text{
F\o lner seq.}\end{subarray}}\varlimsup_{N\to\infty}\Bigl{\|}\mathbb{E}_{n\in
I_{N}}\prod_{i=1}^{k}T_{p_{i}(n)}f_{i}\Bigr{\|}_{L^{2}(\mu)}\leq
C\cdot\min_{1\leq i\leq
k}\overline{\mathbb{E}}^{\square}_{h_{1},\dots,h_{s}\in\mathbb{Z}^{L}}\|f_{i}\|_{(G(\mathbb{c}_{i,m}(h_{1},\dots,h_{s})))_{1\leq
m\leq t_{i}}},\end{split}$

where $\displaystyle C>0$ is a constant depending only on $\displaystyle
p_{1},\dots,p_{k}$, then letting

$\begin{split}H_{i,m}=\text{span}_{\mathbb{Q}}\\{G(\mathbb{u}_{i,m}(a_{1},\dots,a_{s}))\colon
a_{1},\dots,a_{s}\in\mathbb{N}^{L}\\}\cap\mathbb{Z}^{d},\end{split}$

we have that

(24) $\begin{split}\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\ \text{
F\o lner seq.}\end{subarray}}\varlimsup_{N\to\infty}\Bigl{\|}\mathbb{E}_{n\in
I_{N}}T_{p_{1}(n)}f_{1}\cdot\ldots\cdot
T_{p_{k}(n)}f_{k}\Bigr{\|}_{L^{2}(\mu)}=0\;\text{ if
$\displaystyle\;\min_{1\leq i\leq
k}\|f_{i}\|_{H^{\times\infty}_{i,1},\dots,H^{\times\infty}_{i,t_{i}}}=0$}.\end{split}$

We now use Propositions 5.5 and 5.6 to show Theorem 5.1, and leave the proofs
of Propositions 5.5 and 5.6 to Sections 6 and 7 respectively.

###### Proof of Theorem 5.1 assuming Propositions 5.5 and 5.6.

Let the set $\displaystyle R$ be defined as in Theorem 5.1. We can assume
without loss of generality that
$\displaystyle\mathbb{E}(f_{1}|Z_{\\{G(r)^{\times\infty}\\}_{r\in R}})=0$.
Suppose that $\displaystyle p_{i}(n)=\sum_{v\in\mathbb{N}^{L},|v|\leq
K}b_{i,v}n^{v}$ for some $\displaystyle b_{i,v}\in\mathbb{Q}^{d}$ and denote
$\displaystyle R_{v}=\\{b_{i,v}\colon 1\leq i\leq k\\}\cup\\{{\bf 0}\\}$ as in
Proposition 5.5. By Proposition 5.5, there exist $\displaystyle
s,t_{1},\dots,t_{k}\in\mathbb{N}^{\ast}$ and polynomials
$\displaystyle\mathbb{c}_{i,m}\colon(\mathbb{Z}^{L})^{s}\to(\mathbb{Z}^{d})^{L},1\leq
i\leq k,1\leq m\leq t_{i},$ with $\displaystyle\mathbb{c}_{i,m}\not\equiv{\bf
0}$ given by

$\displaystyle\mathbb{c}_{i,m}(h_{1},\dots,h_{s})=\sum_{a_{1},\dots,a_{s}\in\mathbb{N}^{L}}h^{a_{1}}_{1}\dots
h^{a_{s}}_{s}\cdot\mathbb{u}_{i,m}(a_{1},\dots,a_{s})$

for some
$\displaystyle\mathbb{u}_{i,m}(a_{1},\dots,a_{s})\in(\mathbb{Q}^{d})^{L}$ with
all but finitely many terms equal to $\displaystyle{\bf 0}$ for each
$\displaystyle(i,m)$ (and satisfying the additional assumptions given by
Proposition 5.5), such that (21) holds. Let

$\begin{split}H_{i,m}=\text{span}_{\mathbb{Q}}\\{G(\mathbb{u}_{i,m}(a_{1},\dots,a_{s}))\colon
a_{1},\dots,a_{s}\in\mathbb{N}^{L}\\}\cap\mathbb{Z}^{d}.\end{split}$

By Proposition 5.6,

(25) $\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\ \text{ F\o lner
seq.}\end{subarray}}\varlimsup_{N\to\infty}\Bigl{\|}\mathbb{E}_{n\in
I_{N}}T_{p_{1}(n)}f_{1}\cdot\ldots\cdot
T_{p_{k}(n)}f_{k}\Bigr{\|}_{L^{2}(\mu)}=0\;\text{ if
$\displaystyle\;\min_{1\leq i\leq
k}\|f_{i}\|_{H^{\times\infty}_{i,1},\dots,H^{\times\infty}_{i,t_{i}}}=0$.}$

On the other hand, by the description of $\displaystyle\mathbb{c}_{i,m}$,
writing

$\displaystyle\mathbb{u}_{i,m}(a_{1},\dots,a_{s})=(u_{i,m,1}(a_{1},\dots,a_{s}),\dots,u_{i,m,L}(a_{1},\dots,a_{s})),$

each $\displaystyle u_{i,m,j}(a_{1},\dots,a_{s})$ belongs to the set
$\displaystyle U_{i,r}$, which is contained in a set equivalent to one of
$\displaystyle R_{v},v\in\mathbb{N}^{L},0<|v|\leq k$. By the definition of
$\displaystyle R$, $\displaystyle u_{i,m,j}(a_{1},\dots,a_{s})=qr$ for some
$\displaystyle q\in\mathbb{Q}$ and $\displaystyle r\in R$. Since
$\displaystyle\mathbb{c}_{i,m}\not\equiv{\bf 0},$ there exists $\displaystyle
q_{m}r_{m}\in H_{1,m}\backslash\\{\mathbb{0}\\}$ for some $\displaystyle
q_{m}\in\mathbb{Q}$ and $\displaystyle r_{m}\in R$ for all $\displaystyle
1\leq m\leq t_{1}$. So $\displaystyle G(r_{m})$ is a subgroup of
$\displaystyle H_{1,m}$. By Lemma 2.4, we have that

$\displaystyle
Z_{H^{\times\infty}_{1,1},\dots,H^{\times\infty}_{1,t_{1}}}\subseteq
Z_{G(r_{1})^{\times\infty},\dots,G(r_{t_{1}})^{\times\infty}}\subseteq
Z_{\\{G(r)^{\times\infty}\\}_{r\in R}}.$

Since $\displaystyle\mathbb{E}(f_{1}|Z_{\\{G(r)^{\times\infty}\\}_{r\in
R}})=0$, we have that
$\displaystyle\mathbb{E}(f_{1}|Z_{H^{\times\infty}_{1,1},\dots,H^{\times\infty}_{1,t_{1}}})=0$,
meaning that the right hand side of (25) is 0, which implies that (11) equals
to 0.

If in addition, $\displaystyle(T_{g})_{g\in G(r)}$ is assumed to be ergodic
for all $\displaystyle r\in R$, then by Corollary 2.5, we have that
$\displaystyle Z_{\\{G(r)^{\times\infty}\\}_{r\in
R}}=Z_{(\mathbb{Z}^{d})^{\times\infty}}$ and the proof is complete. ∎

##  6\. Proof of Proposition 5.5

Our strategy to show (21) in Proposition 5.5 is the following: We first fix
the functions $\displaystyle f_{i}$ on the right hand side of (21). By a
“dimension-increment” argument (see Proposition 6.3 below), for a fixed
$\displaystyle i,$ we may assume that $\displaystyle p_{i}$ has the highest
degree among $\displaystyle p_{1},\ldots,p_{k}$, making the PET-tuple to be
standard for $\displaystyle f_{i}$. Then, Theorem 4.2 allows us to control the
left hand side of (21) by a PET-tuple of degree 1 which is also standard for
$\displaystyle f_{i}$. Finally, a Host-Kra-type inequality for linear
polynomials (see Proposition 6.1) implies that (21) holds for some polynomials
$\displaystyle\mathbb{c}_{i,m}$. Up to this point, the method we use is
similar to the one used in [17] and [21] (the main difference is that we have
a more explicit upper bound for
$\displaystyle\varlimsup_{N\to\infty}\Bigl{\|}\mathbb{E}_{n\in
I_{N}}T_{p_{1}(n)}f_{1}\cdot\ldots\cdot
T_{p_{k}(n)}f_{k}\Bigr{\|}_{L^{2}(\mu)}$ in Proposition 5.5). Our innovation
is that in order for the equation (21) to be useful for our purposes, we need
a better description of the functions $\displaystyle\mathbb{c}_{i,m}$, which
is the content of part (i) of Proposition 5.5.

We start with the linear case of Proposition 5.5 (the special case
$\displaystyle L=1$ was first proved in [15, Proposition 1]):

######  Proposition 6.1 (Host-Kra inequality for linear
$\displaystyle\mathbb{Z}^{L}$-averages).

Let $\displaystyle d,k,L\in\mathbb{N}^{\ast}$,
$\displaystyle(X,\mathcal{B},\mu,(T_{g})_{g\in\mathbb{Z}^{d}})$ a
$\displaystyle\mathbb{Z}^{d}$-system and $\displaystyle
p_{1},\dots,p_{k}\colon\mathbb{Z}^{L}\to\mathbb{Z}^{d}$ essentially distinct
and essentially non-constant polynomials of degree $\displaystyle 1.$ Suppose
that $\displaystyle p_{i}(n)=\mathbb{u}_{i}\cdot n+v_{i}$ for some
$\displaystyle\mathbb{u}_{i}\in(\mathbb{Z}^{d})^{L},v_{i}\in\mathbb{Z}^{d}$
for all $\displaystyle 1\leq i\leq k$.303030 Here for
$\displaystyle\mathbb{u}=(u_{1},\dots,u_{L})\in({\mathbb{Z}}^{d})^{L}$ and
$\displaystyle n=(n_{1},\dots,n_{L})\in{\mathbb{N}}^{L}$,
$\displaystyle\mathbb{u}\cdot n$ denotes $\displaystyle
n_{1}u_{1}+\dots+n_{L}u_{L}\in\mathbb{Z}^{d}$. Then for every $\displaystyle
f_{1},\ldots,f_{k}\in L^{\infty}(\mu)$ bounded by $\displaystyle 1,$ we have
that

$\displaystyle\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\ \text{ F\o
lner seq.}\end{subarray}}\varlimsup_{N\to\infty}\Bigl{\|}\mathbb{E}_{n\in
I_{N}}T_{p_{1}(n)}f_{1}\cdot\ldots\cdot
T_{p_{k}(n)}f_{k}\Bigr{\|}_{L^{2}(\mu)}\leq C\cdot\min_{1\leq i\leq
k}\|f_{i}\|_{G(-\mathbb{u}_{i}),(G(\mathbb{u}_{j}-\mathbb{u}_{i}))_{1\leq
j\leq k,j\neq i}},$

where $\displaystyle C$ is a constant only depending on $\displaystyle k$.
Moreover, writing
$\displaystyle\mathbb{u}_{i}=(u_{i,1},\dots,u_{i,L}),u_{i,j}\in\mathbb{Z}^{d}$
and $\displaystyle R_{e_{r}}=\\{u_{i,r}\colon 1\leq i\leq
k\\}\cup\\{{\bf{0}}\\}$, the set

$\displaystyle U_{i,r}(\emptyset)=\\{-u_{i,r},u_{j,r}-u_{i,r}\colon 1\leq
j\leq k\\}$

is equivalent to $\displaystyle R_{e_{r}}$.313131 It is not hard to verify
that the sets $\displaystyle R_{e_{r}}$ and $\displaystyle U_{i,r}(\emptyset)$
coincide with the sets $\displaystyle R_{v}$ and $\displaystyle
U_{i,r}(a_{1},\dots,a_{s})$ defined in Proposition 5.5 (in this case for
$\displaystyle s=0$) respectively. We leave the verification to the interested
reader.

###### Proof.

We prove the proposition by induction on $\displaystyle k$. When
$\displaystyle k=1$, let $\displaystyle
p_{1}(n_{1},\dots,n_{L})=u_{1,1}n_{1}+\dots+u_{1,L}n_{L}+v_{1}$ for some
$\displaystyle u_{1,1},\dots,u_{1,L},v_{1}\in\mathbb{Z}^{d}$. By repeatedly
using Theorem 2.3, and the fact that the limit exists, we have that

$\displaystyle\displaystyle\mathbb{E}_{n\in\mathbb{Z}^{L}}T_{p_{1}(n)}f_{1}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\mathbb{E}_{n_{L}\in\mathbb{Z}}\ldots\mathbb{E}_{n_{1}\in\mathbb{Z}}T_{u_{1,1}n_{1}+\dots+u_{1,L}n_{L}+v_{1}}f_{1}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\mathbb{E}(T_{v_{1}}f_{1}|I(u_{1,1})\cap\dots\cap
I(u_{1,L}))=\mathbb{E}(T_{v_{1}}f_{1}|I(G(\mathbb{u}_{1}))),$

hence,
$\displaystyle\|\mathbb{E}_{n\in\mathbb{Z}^{L}}T_{p_{1}(n)}f_{1}\|_{L^{2}(\mu)}=\|\mathbb{E}(T_{v_{1}}f_{1}|I(G(\mathbb{u}_{1})))\|_{L^{2}(\mu)}=\|\mathbb{E}(f_{1}|I(G(\mathbb{u}_{1})))\|_{L^{2}(\mu)}=\|f_{1}\|_{G(\mathbb{u}_{1})}$.

Now suppose that the conclusion holds for $\displaystyle k-1$ for some
$\displaystyle k\geq 2$. Then by Lemma 2.2, the Cauchy-Schwarz inequality, and
using the fact that $\displaystyle p_{1},\dots,p_{k}$ are of degree
$\displaystyle 1,$ we have that

$\begin{split}&\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\ \text{ F\o
lner seq.}\end{subarray}}\varlimsup_{N\to\infty}\Bigl{\|}\mathbb{E}_{n\in
I_{N}}T_{p_{1}(n)}f_{1}\cdot\ldots\cdot
T_{p_{k}(n)}f_{k}\Bigr{\|}_{L^{2}(\mu)}^{2^{k}}\\\ &\leq
4^{2^{k-1}}\overline{\mathbb{E}}^{\square}_{h\in\mathbb{Z}^{L}}\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\
\text{ F\o lner
seq.}\end{subarray}}\varlimsup_{N\to\infty}\Bigl{|}\mathbb{E}_{n\in
I_{N}}\int_{X}\prod_{i=1}^{k}T_{p_{i}(n)}f_{i}\cdot\prod_{i=1}^{k}T_{p_{i}(n+h)}f_{i}\,d\mu\Bigr{|}^{2^{k-1}}\\\
&=4^{2^{k-1}}\overline{\mathbb{E}}^{\square}_{h\in\mathbb{Z}^{L}}\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\
\text{ F\o lner
seq.}\end{subarray}}\varlimsup_{N\to\infty}\Bigl{|}\int_{X}T_{p_{k}(h)}f_{k}\cdot\mathbb{E}_{n\in
I_{N}}\prod_{i=1}^{k-1}T_{p_{i}(n)-p_{k}(n)}(f_{i}\cdot
T_{p_{i}(h)}f_{i})\,d\mu\Bigr{|}^{2^{k-1}}\\\ &\leq
4^{2^{k-1}}\overline{\mathbb{E}}^{\square}_{h\in\mathbb{Z}^{L}}\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\
\text{ F\o lner
seq.}\end{subarray}}\varlimsup_{N\to\infty}\Bigl{\|}\mathbb{E}_{n\in
I_{N}}\prod_{i=1}^{k-1}T_{p_{i}(n)-p_{k}(n)}(f_{i}\cdot
T_{p_{i}(h)}f_{i})\Bigr{\|}_{L^{2}(\mu)}^{2^{k-1}}.\end{split}$

Since $\displaystyle p_{i}(n)-p_{k}(n)=\mathbb{u}^{\prime}_{i}\cdot
n+(v_{i}-v_{k})$, where
$\displaystyle\mathbb{u}^{\prime}_{i}\coloneqq\mathbb{u}_{i}-\mathbb{u}_{k}$,
by induction hypothesis, there is a constant $\displaystyle C^{\prime}$
depending on $\displaystyle k-1$ such that

$\begin{split}&4^{2^{k-1}}\overline{\mathbb{E}}^{\square}_{h\in\mathbb{Z}^{L}}\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\
\text{ F\o lner
seq.}\end{subarray}}\varlimsup_{N\to\infty}\Bigl{\|}\mathbb{E}_{n\in
I_{N}}\prod_{i=1}^{k-1}T_{p_{i}(n)-p_{k}(n)}(f_{i}\cdot
T_{p_{i}(h)}f_{i})\Bigr{\|}_{L^{2}(\mu)}^{2^{k-1}}\\\ &\leq
4^{2^{k-1}}C^{\prime}\cdot\overline{\mathbb{E}}^{\square}_{h\in\mathbb{Z}^{L}}\Bigl{\|}f_{1}\cdot
T_{p_{1}(h)}f_{1}\Bigr{\|}_{G(-\mathbb{u}^{\prime}_{1}),\\{G(\mathbb{u}^{\prime}_{1}-\mathbb{u}^{\prime}_{j})\\}_{2\leq
j\leq k-1}}^{2^{k-1}}\\\
&=C\cdot\overline{\mathbb{E}}^{\square}_{h\in\mathbb{Z}^{L}}\Bigl{\|}f_{1}\cdot
T_{p_{1}(h)}f_{1}\Bigr{\|}_{G(\mathbb{u}_{k}-\mathbb{u}_{1}),\\{G(\mathbb{u}_{1}-\mathbb{u}_{j})\\}_{2\leq
j\leq k-1}}^{2^{k-1}}\\\
&=C\cdot\overline{\mathbb{E}}^{\square}_{h\in\mathbb{Z}^{L}}\Bigl{\|}f_{1}\cdot
T_{p_{1}(h)}f_{1}\Bigr{\|}_{\\{G(\mathbb{u}_{1}-\mathbb{u}_{j})\\}_{2\leq
j\leq
k}}^{2^{k-1}}=C\cdot\|f_{1}\|_{G(-\mathbb{u}_{1}),\\{G(\mathbb{u}_{j}-\mathbb{u}_{1})\\}_{2\leq
j\leq k}}^{2^{k}},\end{split}$

where $\displaystyle C=4^{2^{k-1}}C^{\prime}$ and we used Lemma 2.4 (iii) in
the last equality. It is clear that the constant $\displaystyle C$ depends
only on $\displaystyle k$. By symmetry,

$\displaystyle\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\ \text{ F\o
lner seq.}\end{subarray}}\varlimsup_{N\to\infty}\Bigl{\|}\mathbb{E}_{n\in
I_{N}}T_{p_{1}(n)}f_{1}\cdot\ldots\cdot
T_{p_{k}(n)}f_{k}\Bigr{\|}_{L^{2}(\mu)}\leq C\cdot\min_{1\leq i\leq
k}\|f_{i}\|_{G(-\mathbb{u}_{i}),\\{G(\mathbb{u}_{j}-\mathbb{u}_{i})\\}_{1\leq
j\leq k,j\neq i}}$

and the claim follows. ∎

Before proving the general case of Proposition 5.5, we continue with some
additional computations for our Example 1.

Second part of computations for Example 1: Recall that we are dealing with the
case $\displaystyle(T^{n^{2}+n}_{1},T^{n^{2}}_{2})$, with the PET-tuple

$\displaystyle A=(1,0,2,(f_{1},f_{2}),(p_{1},p_{2})),$

where $\displaystyle
p_{1}(n)=(n^{2}+n,0)=(n^{2}+n)e_{1},p_{2}(n)=(0,n^{2})=n^{2}e_{2},$ and
$\displaystyle e_{1}=(1,0),e_{2}=(0,1)$ and $\displaystyle e=e_{1}-e_{2}$. In
this case, $\displaystyle L=1$ and $\displaystyle d=2$, $\displaystyle
R_{1}=\\{e_{1},{\bf 0}\\}$, $\displaystyle R_{2}=\\{e_{1},e_{2},{\bf 0}\\}$
and $\displaystyle R_{v}=\\{{\bf 0}\\}$ for all $\displaystyle v>2$. Take
$\displaystyle s=3$.

By the first part of computations of Example 1, isolating $\displaystyle
f_{1},$ we have that
$\displaystyle\partial_{2}\partial_{3}\partial_{2}A=(3,7,(f_{1},\dots,f_{1}),$
$\displaystyle\mathbb{p}_{3}),$ where the tuple
$\displaystyle\mathbb{p}_{3}=(q_{1},\dots,q_{7})$ essentially equals to

$\displaystyle(-2h_{1}ne_{1},2h_{2}ne-2h_{1}e_{1},2h_{2}ne,2h_{3}ne-2h_{1}ne_{1},2h_{3}ne,2(h_{2}+h_{3})ne-2h_{1}ne_{1},2(h_{2}+h_{3})ne).$

By Propositions 6.1, 4.1 and Lemma 2.4 (iv) and the fact that Host-Kra
seminorms are $\displaystyle T_{g}$-invariant, we have that

(26) $\begin{split}\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\ \text{
F\o lner seq.}\end{subarray}}\varlimsup_{N\to\infty}\Bigl{\|}\mathbb{E}_{n\in
I_{N}}T^{n^{2}+n}_{1}f_{1}\cdot
T^{n^{2}}_{2}f_{2}\Bigr{\|}^{8}_{L^{2}(\mu)}&=S(A,2^{3})\leq C\cdot
S(\partial_{2}\partial_{3}\partial_{2}A,1)\\\ &\leq
C\cdot\overline{\mathbb{E}}^{\square}_{h_{1},h_{2},h_{3}\in\mathbb{Z}}\|f_{1}\|_{G(\mathbb{c}_{1,1}(h_{1},h_{2},h_{3})),\dots,G(\mathbb{c}_{1,7}(h_{1},h_{2},h_{3}))},\end{split}$

where $\displaystyle\mathbb{c}_{1,1}(h_{1},h_{2},h_{3})=-2h_{1}e_{1},$
$\displaystyle\mathbb{c}_{1,2}(h_{1},h_{2},h_{3})=2h_{2}e,$
$\displaystyle\mathbb{c}_{1,3}(h_{1},h_{2},h_{3})=-2h_{1}e_{1}+2h_{2}e$,
$\displaystyle\mathbb{c}_{1,4}(h_{1},h_{2},h_{3})$ $\displaystyle=2h_{3}e$,
$\displaystyle\mathbb{c}_{1,5}(h_{1},$ $\displaystyle
h_{2},h_{3})=-2h_{1}e_{1}+2h_{3}e$,
$\displaystyle\mathbb{c}_{1,6}(h_{1},h_{2},h_{3})=2(h_{2}+h_{3})e$,
$\displaystyle\mathbb{c}_{1,7}(h_{1},h_{2},$ $\displaystyle
h_{3})=-2h_{1}e_{1}+2(h_{2}+h_{3})e$. This verifies part (ii) of Proposition
5.5 for $\displaystyle i=1$. Moreover, using the notation in Proposition 5.5,
we have that

$\displaystyle\displaystyle U_{1,1}(1,0,0)$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\\{-2e_{1},{\bf 0},-2e_{1},{\bf 0},-2e_{1},{\bf
0},-2e_{1}\\}=\\{-2e_{1},{\bf 0}\\}\sim R_{1},$ $\displaystyle\displaystyle
U_{1,1}(0,1,0)$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\\{{\bf 0},2e,2e,{\bf 0},{\bf
0},2e,2e\\}=\\{2e,{\bf 0}\\}\subseteq\\{2e,-2e_{2},{\bf 0}\\}\sim R_{2},$
$\displaystyle\displaystyle U_{1,1}(0,0,1)$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\\{{\bf 0},{\bf 0},{\bf
0},2e,2e,2e,2e\\}=\\{2e,{\bf 0}\\}\subseteq\\{2e,-2e_{2},{\bf 0}\\}\sim
R_{2}.$

This verifies part (i) of Proposition 5.5 for $\displaystyle i=1$.

Similarly, by isolating $\displaystyle f_{2},$ we have that
$\displaystyle\partial_{1}\partial_{2}\partial_{1}A=(3,7,(f_{2},\dots,f_{2}),\mathbb{p}_{3}),$
where the tuple $\displaystyle\mathbb{p}_{3}$ essentially equals to

$\displaystyle(2h_{1}ne_{2},-2h_{2}ne,-2h_{2}ne+2h_{1}ne_{2},-2h_{3}ne,-2h_{3}ne+2h_{1}ne_{2},-2(h_{2}+h_{3})ne,-2(h_{2}+h_{3})ne+2h_{1}ne_{2}).$
Analogously to (26), we have

(27) $\begin{split}\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\ \text{
F\o lner seq.}\end{subarray}}\varlimsup_{N\to\infty}\Bigl{\|}\mathbb{E}_{n\in
I_{N}}T^{n^{2}+n}_{1}f_{1}\cdot
T^{n^{2}}_{2}f_{2}\Bigr{\|}^{8}_{L^{2}(\mu)}&=S(A,2^{3})\leq C\cdot
S(\partial_{1}\partial_{2}\partial_{1}A,1)\\\ &\leq
C\cdot\overline{\mathbb{E}}^{\square}_{h_{1},h_{2},h_{3}\in\mathbb{Z}}\|f_{2}\|_{G(\mathbb{c}_{2,1}(h_{1},h_{2},h_{3})),\dots,G(\mathbb{c}_{2,7}(h_{1},h_{2},h_{3}))},\end{split}$

where $\displaystyle\mathbb{c}_{2,1}(h_{1},h_{2},h_{3})=2h_{1}e_{2},$
$\displaystyle\mathbb{c}_{2,2}(h_{1},h_{2},h_{3})=-2h_{2}e+2h_{1}e_{2},$
$\displaystyle\mathbb{c}_{2,3}(h_{1},h_{2},h_{3})=-2h_{2}e$,
$\displaystyle\mathbb{c}_{2,4}(h_{1},h_{2},h_{3})=-2h_{3}e+2h_{1}e_{2}$,
$\displaystyle\mathbb{c}_{2,5}(h_{1},h_{2},h_{3})=-2h_{3}e$,
$\displaystyle\mathbb{c}_{2,6}(h_{1},h_{2},h_{3})=-2(h_{2}+h_{3})e+2h_{1}e_{2}$,
$\displaystyle\mathbb{c}_{2,7}(h_{1},h_{2},h_{3})=-2(h_{2}+h_{3})e.$ This
verifies part (ii) of Proposition 5.5 for $\displaystyle i=2$. Using the
notation in Proposition 5.5, we have that

$\displaystyle\displaystyle U_{2,1}(1,0,0)$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\\{2e_{2},2e_{2},{\bf 0},2e_{2},{\bf 0},2e_{2},{\bf
0}\\}=\\{2e_{2},{\bf 0}\\}\subseteq\\{-2e,2e_{2},{\bf 0}\\}\sim R_{2},$
$\displaystyle\displaystyle U_{2,1}(0,1,0)$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\\{{\bf 0},-2e,-2e,{\bf 0},{\bf
0},-2e,-2e\\}=\\{-2e,{\bf 0}\\}\subseteq\\{-2e,2e_{2},{\bf 0}\\}\sim R_{2},$
$\displaystyle\displaystyle U_{2,1}(0,0,1)$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\\{{\bf 0},{\bf 0},{\bf
0},-2e,-2e,-2e,-2e\\}=\\{-2e,{\bf 0}\\}\subseteq\\{-2e,2e_{2},{\bf 0}\\}\sim
R_{2}.$

This verifies part (i) of Proposition 5.5 for $\displaystyle i=2$.

We now introduce some additional notation that we will use in the general
case. Let $\displaystyle d,\ell,L\in\mathbb{N}^{\ast}$, $\displaystyle
s\in\mathbb{N}$ and $\displaystyle
q_{1},\dots,q_{\ell}\colon(\mathbb{Z}^{L})^{s+1}\to\mathbb{Z}^{d}$ be
polynomials. Denote $\displaystyle\mathbb{q}=(q_{1},\dots,q_{\ell})$, where

$\displaystyle
q_{i}(n;h_{1},\dots,h_{s})=\sum_{b,a_{1},\dots,a_{s}\in\mathbb{N}^{L}}h^{a_{1}}_{1}\dots
h^{a_{s}}_{s}n^{b}\cdot u_{i}(b;a_{1},\dots,a_{s})$

for some $\displaystyle u_{i}(b;a_{1},\dots,a_{s})\in\mathbb{Q}^{d}$ with all
but finitely many being $\displaystyle{\bf 0}$ for each $\displaystyle 1\leq
i\leq\ell$. For all $\displaystyle b,a_{1},\dots,a_{s}\in\mathbb{N}^{L}$,
denote

$\displaystyle
R_{\mathbb{q}}(b;a_{1},\dots,a_{s})\coloneqq\\{u_{i}(b;a_{1},\dots,a_{s})\colon
1\leq i\leq\ell\\}\cup\\{{\bf 0}\\}\subseteq\mathbb{Q}^{d}.$

Roughly speaking, $\displaystyle R_{\mathbb{q}}(b;a_{1},\dots,a_{s})$ records
the coefficients of $\displaystyle\mathbb{q}$ at
“level”-$\displaystyle(b;a_{1},\dots,a_{s})$ (together with the zero vector
0).

The following proposition shows that, during the PET-induction process, after
applying the vdC-operation to our expression, we can still keep track of the
coefficients of the polynomials.

######  Proposition 6.2 (vdC-operations treat the sets $\displaystyle
R_{\mathbb{q}}(b;a_{1},\dots,a_{s})$ nicely).

Let $\displaystyle d,\ell,L\in\mathbb{N}^{\ast}$, $\displaystyle
s\in\mathbb{N},$
$\displaystyle(X,\mathcal{B},\mu,(T_{g})_{g\in\mathbb{Z}^{d}})$ a
$\displaystyle\mathbb{Z}^{d}$-system, $\displaystyle
q_{1},\dots,q_{\ell}\colon(\mathbb{Z}^{L})^{s+1}\to\mathbb{Z}^{d}$ polynomials
and $\displaystyle\mathbb{q}=(q_{1},\dots,q_{\ell})$. If $\displaystyle
A=(L,s,\ell,\mathbb{q})$ with
$\displaystyle\partial_{w}A=(L,s+1,\ell^{\ast},\mathbb{q}^{\ast})$ for some
$\displaystyle\ell^{\ast}\in\mathbb{N}^{\ast},1\leq w\leq\ell$, where
$\displaystyle\mathbb{q}^{\ast}=(q^{\ast}_{1},\dots,q^{\ast}_{\ell^{\ast}})$
for some polynomials $\displaystyle
q^{\ast}_{1},\dots,q^{\ast}_{\ell^{\ast}}\colon(\mathbb{Z}^{L})^{s+2}\to\mathbb{Z}^{d},$
then for all $\displaystyle b,a_{1},\dots,a_{s+1}\in\mathbb{N}^{L}$ not all
equal to $\displaystyle{\bf 0},$ we have that

(28) $R_{\mathbb{q}^{\ast}}(b;a_{1},\dots,a_{s+1})\lesssim
R_{\mathbb{q}}(b+a_{s+1};a_{1},\dots,a_{s}).$

###### Proof.

For convenience we write
$\displaystyle\mathbb{q}^{\ast}\approx(p_{1},\dots,p_{\ell^{\prime}})$ for
some polynomials $\displaystyle p_{1},\dots,p_{\ell^{\prime}}$ if
$\displaystyle\mathbb{q}^{\ast}$ can be obtained by removing all the essential
constant polynomials from $\displaystyle p_{1},\dots,p_{\ell^{\prime}}$,
ordering the rest into groups such that two polynomials are essentially
distinct if and only if they are in different groups, and then picking one
polynomial from each group. It is not hard to see that if
$\displaystyle\mathbb{q}\approx\mathbb{q}^{\prime}$, then $\displaystyle
R_{\mathbb{q}}(b;a_{1},\dots,a_{s+1})=R_{\mathbb{q}^{\prime}}(b;a_{1},\dots,a_{s+1})$
for all $\displaystyle b,a_{1},\dots,a_{s+1}\in\mathbb{N}^{L}$ not all equal
to 0.

Denote $\displaystyle
q^{\prime}_{i}\colon(\mathbb{Z}^{L})^{s+2}\to\mathbb{Z}^{d}$, $\displaystyle
q^{\prime}_{i}(n;h_{1},\dots,h_{s+1})=q_{i}(n+h_{s+1};h_{1},\dots,h_{s})$ for
all $\displaystyle 1\leq i\leq\ell$. It suffices to show the statement for
$\displaystyle\mathbb{q}^{\ast}\approx(q^{\prime}_{1}-q_{1},q_{i}-q_{1},q^{\prime}_{i}-q_{1}\colon
i\neq 1)$ as the general case follows similarly.

Suppose that

$\displaystyle
q_{i}(n;h_{1},\dots,h_{s})=\sum_{b,a_{1},\dots,a_{s}\in\mathbb{N}^{L}}h^{a_{1}}_{1}\dots
h^{a_{s}}_{s}n^{b}\cdot u_{i}(b;a_{1},\dots,a_{s})$

for all $\displaystyle 1\leq i\leq\ell$. Then, one can immediately check that

$\displaystyle
q^{\prime}_{i}(n;h_{1},\dots,h_{s+1})=\sum_{b,a_{1},\dots,a_{s+1}\in\mathbb{N}^{L}}h^{a_{1}}_{1}\dots
h^{a_{s+1}}_{s+1}n^{b}\cdot\binom{b+a_{s+1}}{b}u_{i}(b+a_{s+1};a_{1},\dots,a_{s}).$

If $\displaystyle a_{s+1}=0$, then the coefficient of $\displaystyle
h^{a_{1}}_{1}\dots h^{a_{s}}_{s}n^{b}$ for $\displaystyle
q^{\prime}_{1}-q_{1}$ is $\displaystyle{\bf 0},$ and for both $\displaystyle
q_{i}-q_{1}$ and $\displaystyle q^{\prime}_{i}-q_{1}$ are $\displaystyle
u_{i}(b;a_{1},\dots,a_{s})-u_{1}(b;a_{1},\dots,a_{s})$. This implies that
$\displaystyle
R_{\mathbb{q}^{\ast}}(b;a_{1},\dots,a_{s},0)=R_{\mathbb{q}^{\prime}}(b;a_{1},\dots,a_{s},0)\lesssim
R_{\mathbb{q}}(b;a_{1},\dots,a_{s}),$333333 Note that $\displaystyle
R_{\mathbb{q}^{\prime}}(b;a_{1},\dots,a_{s},0)\sim
R_{\mathbb{q}}(b;a_{1},\dots,a_{s})$ if and only if one of $\displaystyle
u_{i}(b;a_{1},\dots,a_{s})$ is $\displaystyle{\bf 0}$. which proves (28).

If $\displaystyle a_{s+1}>0$, then the coefficient of $\displaystyle
h^{a_{1}}_{1}\dots h^{a_{s+1}}_{s+1}n^{b}$ for $\displaystyle
q^{\prime}_{1}-q_{1},q_{i}-q_{1}$ and $\displaystyle q^{\prime}_{i}-q_{1}$ are
$\displaystyle\binom{b+a_{s+1}}{b}u_{1}(b+a_{s+1};a_{1},\dots,a_{s})$,
$\displaystyle{\bf 0}$ and
$\displaystyle\binom{b+a_{s+1}}{b}u_{i}(b+a_{s+1};a_{1},\dots,a_{s})$
respectively. In this case $\displaystyle R_{\mathbb{q}^{\ast}}(b;a_{1},$
$\displaystyle\dots,a_{s+1})$
$\displaystyle=R_{\mathbb{q}^{\prime}}(b;a_{1},\dots,a_{s+1})\sim
R_{\mathbb{q}}(b+a_{s+1};a_{1},\dots,a_{s})$, which finishes the proof. ∎

Let $\displaystyle A$ be a PET-tuple and $\displaystyle f\in L^{\infty}(\mu)$.
If $\displaystyle A$ is semi-standard but not standard for $\displaystyle f$,
then the PET-induction does not work well enough to provide an upper bound for
$\displaystyle S(A,\kappa)$ in terms of the Host-Kra seminorms of
$\displaystyle f$. To overcome this difficulty, we use a “dimension-increment”
argument to change $\displaystyle A$ into a new PET-tuple which is standard
for $\displaystyle f$, but at the cost of increasing the dimension from
$\displaystyle L$ to $\displaystyle 2L$.343434 In the papers [17, 21], where
similar methods were used, the dimension was increased from $\displaystyle L$
to $\displaystyle 3L$ instead. In fact, this is the main reason that justifies
the multi-variable nature of the results in this article.

This “dimension-increment” argument is carried out in the following
proposition. The idea essentially comes from [17, 21], but again some
additional work needs to be done in order to keep track of the set
$\displaystyle R_{\mathbb{q}}(b;a_{1},\dots,a_{s})$:

######  Proposition 6.3 (Dimension-increasing property).

Let $\displaystyle L,d,\ell\in\mathbb{N}^{\ast},$ $\displaystyle
s\in\mathbb{N},$
$\displaystyle(X,\mathcal{B},\mu,(T_{g})_{g\in\mathbb{Z}^{d}})$ a
$\displaystyle\mathbb{Z}^{d}$-system, $\displaystyle f\in L^{\infty}(\mu)$,
$\displaystyle
q_{1},\dots,q_{\ell}\colon(\mathbb{Z}^{L})^{s+1}\to\mathbb{Z}^{d}$
polynomials, $\displaystyle g_{1},\dots,g_{\ell}\colon
X\times(\mathbb{Z}^{L})^{s}\to\mathbb{R}$ functions such that each
$\displaystyle g_{i}(\cdot;h_{1},\dots,h_{s})$ is an $\displaystyle
L^{\infty}(\mu)$ function bounded by $\displaystyle 1$ for all $\displaystyle
h_{1},\dots,h_{s}\in\mathbb{Z}^{L},1\leq i\leq\ell$, and let
$\displaystyle\mathbb{q}=(q_{1},\dots,q_{\ell})$ and
$\displaystyle\mathbb{g}=(g_{1},\dots,g_{\ell})$.

If the PET-tuple $\displaystyle A=(L,s,\ell,\mathbb{g},\mathbb{q})$ is non-
degenerate and semi-standard but not standard for $\displaystyle f$, then
there exist polynomials $\displaystyle
q^{\prime}_{1},\dots,q^{\prime}_{2\ell-1}\colon(\mathbb{Z}^{2L})^{s+1}\to\mathbb{Z}^{d}$,
functions $\displaystyle g^{\prime}_{1},\dots,g^{\prime}_{2\ell-1}\colon
X\times(\mathbb{Z}^{2L})^{s}\to\mathbb{R}$ such that each $\displaystyle
g^{\prime}_{i}(\cdot;h_{1},\dots,h_{s})$ is an $\displaystyle L^{\infty}(\mu)$
function bounded by $\displaystyle 1$ for all $\displaystyle
h_{1},\dots,h_{s}\in\mathbb{Z}^{L},1\leq i\leq 2\ell-1,$
$\displaystyle\mathbb{q}^{\prime}=(q^{\prime}_{1},\dots,q^{\prime}_{2\ell-1})$
and
$\displaystyle\mathbb{g}^{\prime}=(g^{\prime}_{1},\dots,g^{\prime}_{2\ell-1})$,
such that the PET-tuple $\displaystyle
A^{\prime}=(2L,s,2\ell-1,\mathbb{g}^{\prime},\mathbb{q}^{\prime})$ is non-
degenerate and standard for $\displaystyle f$ and $\displaystyle
S(A,2\kappa)\leq S(A^{\prime},\kappa)$ for all $\displaystyle\kappa>0$.
Moreover, for all $\displaystyle b,b^{\prime},a_{1},\dots,a_{s},$
$\displaystyle a^{\prime}_{1},\dots,a^{\prime}_{s}\in\mathbb{N}^{L}$ not all
equal to $\displaystyle{\bf 0}$, there exist $\displaystyle
b^{\prime\prime},a^{\prime\prime}_{1},\dots,a^{\prime\prime}_{s}\in\mathbb{N}^{L}$
not all equal to $\displaystyle{\bf 0},$ such that

(29)
$R_{\mathbb{q}^{\prime}}(b,b^{\prime};a_{1},\dots,a_{s},a^{\prime}_{1},\dots,a^{\prime}_{s})\sim
R_{\mathbb{q}}(b^{\prime\prime};a^{\prime\prime}_{1},\dots,a^{\prime\prime}_{s}).$

###### Proof.

Since $\displaystyle A$ is semi-standard but not standard for $\displaystyle
f$, we may assume without loss of generality that $\displaystyle
g_{1}(x;h_{1},\dots,h_{s})=f(x)$, $\displaystyle\deg(q_{1})<\deg(A)$, and
$\displaystyle\deg(q_{\ell})=\deg(A)$. For convenience denote
$\displaystyle{\bf h}=(h_{1},\dots,h_{s})$ and $\displaystyle{\bf
h}^{\prime}=(h^{\prime}_{1},\dots,h^{\prime}_{s})$. For $\displaystyle 1\leq
m\leq\ell$ we set

$\displaystyle q^{\prime}_{m}((n,n^{\prime});({\bf h},{\bf
h}^{\prime}))\coloneqq q_{m}(n;{\bf h})-q_{\ell}(n^{\prime};{\bf
h}),\quad\text{and}\quad g^{\prime}_{m}(x;({\bf h},{\bf h}^{\prime}))\coloneqq
g_{m}(x;{\bf h}),$

while for $\displaystyle 1\leq m\leq\ell-1$ we set

$\displaystyle q^{\prime}_{m+\ell}((n,n^{\prime});({\bf h},{\bf
h}^{\prime}))\coloneqq q_{m}(n^{\prime};{\bf h})-q_{\ell}(n^{\prime};{\bf
h}),\quad\text{and}\quad g^{\prime}_{m+\ell}(x;({\bf h},{\bf
h}^{\prime}))\coloneqq g_{m}(x;{\bf h}).$

Also, let
$\displaystyle\mathbb{q}^{\prime}=(q^{\prime}_{1},\dots,q^{\prime}_{2\ell-1}),$
$\displaystyle\mathbb{g}^{\prime}=(g^{\prime}_{1},\dots,g^{\prime}_{2\ell-1})$
and $\displaystyle
A^{\prime}=(2L,s,2\ell-1,\mathbb{q}^{\prime},\mathbb{g}^{\prime})$.

Since $\displaystyle\deg(q_{\ell})=\deg(A)$ and
$\displaystyle\deg(q_{1})<\deg(A)$, we have that
$\displaystyle\deg(q^{\prime}_{1})=\deg(A^{\prime})$ and moreover
$\displaystyle g^{\prime}_{1}=f$. So $\displaystyle A^{\prime}$ is standard
for $\displaystyle f$. On the other hand, since $\displaystyle A$ is non-
degenerate, one can easily see that $\displaystyle
q^{\prime}_{1},\dots,q^{\prime}_{2\ell-1}$ are essentially distinct (note that
$\displaystyle q_{\ell}(n;{\bf h})-q_{\ell}(n^{\prime};{\bf h})$ is
essentially non-constant). So $\displaystyle A^{\prime}$ is non-degenerate.

Recall that $\displaystyle\overline{\mathbb{E}}^{\square}_{{\bf
h}\in(\mathbb{Z}^{L})^{s}}=\overline{\mathbb{E}}^{\square}_{h_{1}\in\mathbb{Z}^{L}}\dots\overline{\mathbb{E}}^{\square}_{h_{s}\in\mathbb{Z}^{L}}.$
By the fact that the action is measure preserving and the Cauchy-Schwarz
inequality, we have that

$\begin{split}&S(A,2\kappa)=\overline{\mathbb{E}}^{\square}_{{\bf
h}\in(\mathbb{Z}^{L})^{s}}\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\
\text{ F\o lner
seq.}\end{subarray}}\varlimsup_{N\to\infty}\Bigl{\|}\mathbb{E}_{n\in
I_{N}}\prod_{m=1}^{\ell}T_{q_{m}(n;{\bf h})}g_{m}(x;{\bf
h})\Bigr{\|}^{2\kappa}_{L^{2}(\mu)}\\\ =&\overline{\mathbb{E}}^{\square}_{{\bf
h}\in(\mathbb{Z}^{L})^{s}}\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\
\text{ F\o lner
seq.}\end{subarray}}\varlimsup_{N\to\infty}\Bigl{|}\mathbb{E}_{n,n^{\prime}\in
I_{N}}\int_{X}\prod_{m=1}^{\ell}T_{q_{m}(n;{\bf h})}g_{m}(x;{\bf h})\cdot
T_{q_{m}(n^{\prime};{\bf h})}g_{m}(x;{\bf h})\,d\mu\Bigr{|}^{\kappa}\\\
\leq&\overline{\mathbb{E}}^{\square}_{{\bf
h}\in(\mathbb{Z}^{L})^{s}}\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\
\text{ F\o lner
seq.}\end{subarray}}\varlimsup_{N\to\infty}\Bigl{\|}\mathbb{E}_{n,n^{\prime}\in
I_{N}}\prod_{m=1}^{\ell}T_{q_{m}(n;{\bf h})-q_{\ell}(n^{\prime};{\bf
h})}g_{m}(x;{\bf h})\cdot\\\ &\hskip
270.30118pt\prod_{m=1}^{\ell-1}T_{q_{m}(n^{\prime};{\bf
h})-q_{\ell}(n^{\prime};{\bf h})}g_{m}(x;{\bf
h})\Bigr{\|}_{L^{2}(\mu)}^{\kappa}\\\ =&\overline{\mathbb{E}}^{\square}_{({\bf
h},{\bf
h}^{\prime})\in(\mathbb{Z}^{2L})^{s}}\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\
\text{ F\o lner
seq.}\end{subarray}}\varlimsup_{N\to\infty}\Bigl{\|}\mathbb{E}_{n,n^{\prime}\in
I_{N}}\prod_{m=1}^{2\ell-1}T_{q^{\prime}_{m}((n,n^{\prime});({\bf h},{\bf
h}^{\prime}))}g^{\prime}_{m}(x;({\bf h},{\bf
h}^{\prime}))\Bigr{\|}_{L^{2}(\mu)}^{\kappa}\\\
\leq&S(A^{\prime},\kappa),\end{split}$

where the last inequality holds because $\displaystyle(I_{N}\times
I_{N})_{N\in{\mathbb{N}}}$ is a Følner sequence of
$\displaystyle{\mathbb{Z}}^{L}\times{\mathbb{Z}}^{L}$. On the other hand, if

$\displaystyle q_{i}(n;{\bf
h})=\sum_{b,a_{1},\dots,a_{s}\in\mathbb{N}^{L}}h^{a_{1}}_{1}\dots
h^{a_{s}}_{s}n^{b}\cdot u_{i}(b;a_{1},\dots,a_{s})$

for some $\displaystyle u_{i}(b;a_{1},\dots,a_{s})\in\mathbb{Q}^{d}$, then for
$\displaystyle 1\leq i\leq\ell-1$, we have

$\displaystyle q^{\prime}_{i+\ell}(n,n^{\prime};{\bf h},{\bf
h}^{\prime})=\sum_{b,a_{1},\dots,a_{s}\in\mathbb{N}^{L}}h^{a_{1}}_{1}\dots
h^{a_{s}}_{s}{n^{\prime}}^{b}\cdot(u_{i}(b;a_{1},\dots,a_{s})-u_{\ell}(b;a_{1},\dots,a_{s})),$

and for $\displaystyle 1\leq i\leq\ell$,

$\displaystyle q^{\prime}_{i}(n,n^{\prime};{\bf h},{\bf
h}^{\prime})=\sum_{b,a_{1},\dots,a_{s}\in\mathbb{N}^{L}}h^{a_{1}}_{1}\dots
h^{a_{s}}_{s}(n^{b}\cdot u_{i}(b;a_{1},\dots,a_{s})-{n^{\prime}}^{b}\cdot
u_{\ell}(b;a_{1},\dots,a_{s})).$

So for all $\displaystyle
b,b^{\prime},a_{1},\dots,a_{s},a^{\prime}_{1},\dots,a^{\prime}_{s}\in\mathbb{N}^{L}$,
similar to the argument in the proof of Proposition 6.2, we have

$\displaystyle
R_{\mathbb{q}}(b;a_{1},\dots,a_{s})=R_{\mathbb{q}^{\prime}}((b,{\bf
0});(a_{1},{\bf 0}),\dots,(a_{s},{\bf 0}))\sim R_{\mathbb{q}^{\prime}}(({\bf
0},b);(a_{1},{\bf 0}),\dots,(a_{s},{\bf 0}))$

and $\displaystyle
R_{q^{\prime}}((b,b^{\prime});(a_{1},a^{\prime}_{1}),\dots,(a_{s},a^{\prime}_{s}))=\\{{\bf
0}\\}$. This implies (29) and finishes the proof. ∎

We are now ready to prove Proposition 5.5 and close this section.

###### Proof of Proposition 5.5.

Let $\displaystyle A$ denote the PET-tuple
$\displaystyle(L,0,k,(p_{1},\dots,p_{k}),(f_{1},\dots,f_{k}))$. Then for all
$\displaystyle\kappa>0$,

$\displaystyle
S(A,\kappa)=\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\ \text{ F\o
lner seq.}\end{subarray}}\varlimsup_{N\to\infty}\Bigl{\|}\mathbb{E}_{n\in
I_{N}}\prod_{m=1}^{k}T_{p_{m}(n)}f_{m}\Bigr{\|}^{\kappa}_{L^{2}(\mu)}.$

By the assumption, $\displaystyle A$ is non-degenerate. We only prove (21) for
$\displaystyle f_{1}$ as the other cases are identical.

We first assume that $\displaystyle A$ is standard for $\displaystyle f_{1}$.
By Theorem 4.2, there exist finitely many vdC-operations
$\displaystyle\partial_{\rho_{1}},\dots,\partial_{\rho_{t}}$ such that
$\displaystyle A^{\prime}=\partial_{\rho_{t}}\dots\partial_{\rho_{1}}A$ is a
non-degenerate PET-tuple which is standard for $\displaystyle f_{1}$, and
$\displaystyle\deg(A^{\prime})=1$. By Proposition 4.1, $\displaystyle
S(A,2^{t})\leq C\cdot S(A^{\prime},1)$ for some $\displaystyle C>0$ depending
only on the polynomials $\displaystyle p_{1},\dots,p_{k}$. We may assume that

$\begin{split}&S(A^{\prime},1)=\overline{\mathbb{E}}^{\square}_{h_{1},\dots,h_{s}\in\mathbb{Z}^{L}}\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\
\text{ F\o lner
seq.}\end{subarray}}\varlimsup_{N\to\infty}\Bigl{\|}\mathbb{E}_{n\in
I_{N}}\prod_{m=1}^{\ell}T_{\textbf{d}_{m}(h_{1},\dots,h_{s})\cdot
n+r_{m}(h_{1},\dots,h_{s})}g_{m}(x;h_{1},\dots,h_{s})\Bigr{\|}_{L^{2}(\mu)}\end{split}$

for some $\displaystyle s,\ell\in\mathbb{N}^{\ast}$, functions $\displaystyle
g_{1},\dots,g_{\ell}\colon X\times(\mathbb{Z}^{L})^{s}\to\mathbb{R},$ where
$\displaystyle g_{1}(\cdot;h_{1},\dots,h_{s})=f_{1},$ such that each
$\displaystyle g_{m}(\cdot;h_{1},\dots,h_{s})$ is an $\displaystyle
L^{\infty}(\mu)$ function bounded by $\displaystyle 1,$ and polynomials
$\displaystyle\textbf{d}_{m}\colon(\mathbb{Z}^{L})^{s}\to(\mathbb{Z}^{d})^{L}$
and $\displaystyle r_{m}\colon(\mathbb{Z}^{L})^{s}\to\mathbb{Z}^{d},$
$\displaystyle 1\leq m\leq\ell$, where $\displaystyle\textbf{d}_{m},r_{m}$
take value in vectors with integer coordinates because the vdC-operations send
integer-valued polynomials to integer-valued polynomials. Let
$\displaystyle\mathbb{c}_{1,1}=-\textbf{d}_{1}$ and
$\displaystyle\mathbb{c}_{1,m}=\textbf{d}_{m}-\textbf{d}_{1}$ for
$\displaystyle m\neq 1$. Since $\displaystyle A^{\prime}$ is non-degenerate,
we have that
$\displaystyle\mathbb{c}_{1,1},\dots,\mathbb{c}_{1,s}\not\equiv{\bf 0}$. By
Proposition 6.1, we have that

$\displaystyle\displaystyle S(A^{\prime},1)$ $\displaystyle\displaystyle\leq$
$\displaystyle\displaystyle
C^{\prime}\cdot\overline{\mathbb{E}}^{\square}_{h_{1},\dots,h_{s}\in\mathbb{Z}^{L}}\|T_{r_{1}(h_{1},\dots,h_{s})}f_{1}\|_{\\{G(\mathbb{c}_{1,i}(h_{1},\dots,h_{s}))\\}_{1\leq
i\leq\ell}}$ $\displaystyle\displaystyle=$ $\displaystyle\displaystyle
C^{\prime}\cdot\overline{\mathbb{E}}^{\square}_{h_{1},\dots,h_{s}\in\mathbb{Z}^{L}}\|f_{1}\|_{\\{G(\mathbb{c}_{1,i}(h_{1},\dots,h_{s}))\\}_{1\leq
i\leq\ell}}$

for some $\displaystyle C^{\prime}>0$ depending only on the polynomials
$\displaystyle p_{1},\dots,p_{k}$. Combining this with the fact that
$\displaystyle S(A,2^{t})\leq C\cdot S(A^{\prime},1)$, we get (21).

Suppose that

$\displaystyle\mathbb{c}_{1,m}(h_{1},\dots,h_{s})=\sum_{a_{1},\dots,a_{s}\in\mathbb{N}^{L}}h^{a_{1}}_{1}\dots
h^{a_{s}}_{s}\cdot\mathbb{u}_{1,m}(a_{1},\dots,a_{s}),\;\;\text{ and}$
$\displaystyle\textbf{d}_{1,m}(h_{1},\dots,h_{s})=\sum_{a_{1},\dots,a_{s}\in\mathbb{N}^{L}}h^{a_{1}}_{1}\dots
h^{a_{s}}_{s}\cdot\mathbb{v}_{1,m}(a_{1},\dots,a_{s})$

for some
$\displaystyle\mathbb{u}_{1,m}(a_{1},\dots,a_{s}),\mathbb{v}_{1,m}(a_{1},\dots,a_{s})\in(\mathbb{Q}^{d})^{L}$
with all but finitely many terms being zero for each $\displaystyle m$. Write
$\displaystyle\mathbb{u}_{i,m}(a_{1},\dots,a_{s})=(u_{i,m,1}(a_{1},\dots,a_{s}),\dots,u_{i,m,L}(a_{1},\dots,a_{s})),$
$\displaystyle\mathbb{v}_{i,m}(a_{1},\dots,a_{s})=(v_{i,m,1}(a_{1},\dots,a_{s}),$
$\displaystyle\dots,v_{i,m,L}(a_{1},\dots,a_{s})),$ and, for all
$\displaystyle 1\leq r\leq\ell,$ set

$\displaystyle
U_{1,r}(a_{1},\dots,a_{s})\coloneqq\\{u_{1,m,r}(a_{1},\dots,a_{s})\in\mathbb{Q}^{d}\colon
1\leq m\leq\ell\\}\cup\\{{\bf 0}\\};\quad\text{and}\quad$ $\displaystyle
V_{1,r}(a_{1},\dots,a_{s})\coloneqq\\{v_{1,m,r}(a_{1},\dots,a_{s})\in\mathbb{Q}^{d}\colon
1\leq m\leq\ell\\}\cup\\{{\bf 0}\\}.$

Since $\displaystyle A^{\prime}=\partial_{w_{t}}\dots\partial_{w_{1}}A$, by
repeatedly using Proposition 6.2, for all $\displaystyle
a_{1},\dots,a_{s}\in\mathbb{N}^{L}$ not all equal to $\displaystyle{\bf 0}$
and every $\displaystyle 1\leq r\leq L$, there exists $\displaystyle
v\in\mathbb{N}^{L},v\neq{\bf 0}$ such that $\displaystyle
V_{1,r}(a_{1},\dots,a_{s})\lesssim R_{v}$. By the relation between
$\displaystyle\mathbb{u}_{1,m}$ and $\displaystyle\mathbb{v}_{1,m}$, we get
$\displaystyle U_{1,r}(a_{1},\dots,a_{s})\sim V_{1,r}(a_{1},\dots,a_{s})$ and
so $\displaystyle U_{1,r}(a_{1},\dots,a_{s})\lesssim R_{v}$.

We now assume that $\displaystyle
A=(L,0,k,(p_{1},\dots,p_{k}),(f_{1},\dots,f_{k}))$ is not standard for
$\displaystyle f_{1}$. Since $\displaystyle A$ is semi-standard for
$\displaystyle f_{1}$, by Proposition 6.3, there exists a PET-tuple
$\displaystyle A^{\prime}=(2L,0,\ell,\mathbb{q},\mathbb{g})$ which is non-
degenerate and standard for $\displaystyle f_{1}$ such that $\displaystyle
S(A,2\kappa)\leq S(A^{\prime},\kappa)$ for all $\displaystyle\kappa>0$ and
(29) holds. Working with the PET-tuple $\displaystyle A^{\prime}$ instead of
$\displaystyle A$ as before (and using (29)), we get the result. ∎

##  7\. Proof of Proposition 5.6

This last section is dedicated to the proof of Proposition 5.6. We remark that
it is in this proposition where the concatenation results (Theorem 2.10 and
Corollary 2.11) are used.

Following the notation of Proposition 5.6, for every $\displaystyle{\bf
h}=(h_{1},\dots,h_{s})\in(\mathbb{Z}^{L})^{s}$ and $\displaystyle 1\leq i\leq
k$, we set

$\displaystyle W_{i,{\bf h}}\coloneqq Z_{G(\mathbb{c}_{i,1}({\bf
h})),\dots,G(\mathbb{c}_{i,t_{i}}({\bf h}))}(\mathbb{X}),$

and for every subset $\displaystyle J\subseteq(\mathbb{Z}^{L})^{s},$

$\displaystyle W_{i,J}\coloneqq\bigvee_{{\bf h}\in J}W_{i,{\bf h}}.$

We have:

######  Lemma 7.1.

Let the notations be as in Proposition 5.6. If (23) holds for every
$\displaystyle\mathbb{Z}^{d}$-system $\displaystyle(X,\mathcal{B},\mu,$
$\displaystyle(T_{g})_{g\in\mathbb{Z}^{d}})$ and every $\displaystyle
f_{1},\dots,f_{k}\in L^{\infty}(\mu)$, then for every $\displaystyle
J_{1},\dots,J_{k}\subseteq(\mathbb{Z}^{L})^{s}$ of density $\displaystyle 1,$
we have that

(30)
$\begin{split}\mathbb{E}_{n\in\mathbb{Z}^{L}}T_{p_{1}(n)}f_{1}\cdot\ldots\cdot
T_{p_{k}(n)}f_{k}=0,\;\;\text{ if
$\displaystyle\;\;\mathbb{E}(f_{i}|W_{i,J_{i}})=0$ for some $\displaystyle
1\leq i\leq k$.}\end{split}$

###### Proof.

Suppose that $\displaystyle\mathbb{E}(f_{i}|W_{i,J_{i}})=0$ for some
$\displaystyle 1\leq i\leq k$. By definition,
$\displaystyle\|f_{i}\|_{G(c_{i,1}({\bf h})),\dots,G(c_{i,t_{i}}({\bf h}))}=0$
for all $\displaystyle{\bf h}\in J_{i}$. Since $\displaystyle J_{i}$ is of
density $\displaystyle 1,$ the conclusion follows from (23). ∎

Before proving Proposition 5.6, we continue with our main example (Example 1).

Third part of computations for Example 1: We are dealing with the
$\displaystyle(T^{n^{2}+n}_{1},T^{n^{2}}_{2})$ case. Applying (26) to Lemma
7.1, we have that

(31) $\begin{split}\mathbb{E}_{n\in{\mathbb{Z}}}T^{n^{2}+n}_{1}f_{1}\cdot
T^{n^{2}}_{2}f_{2}=0,\;\;\text{ if
$\displaystyle\;\;\mathbb{E}(f_{i}|W_{i,J_{i}})=0$ for $\displaystyle i=1$ or
2,}\end{split}$

for all $\displaystyle J_{1},J_{2}\in\mathbb{Z}^{3}$ of density $\displaystyle
1,$ where

$\displaystyle W_{i,J}=\bigvee_{(h_{1},h_{2},h_{3})\in
J}W_{i,(h_{1},h_{2},h_{3})}=\bigvee_{(h_{1},h_{2},h_{3})\in
J}Z_{G(\mathbb{c}_{i,1}(h_{1},h_{2},h_{3})),\dots,G(\mathbb{c}_{i,7}(h_{1},h_{2},h_{3}))},\quad
i=1,2,$

where $\displaystyle\mathbb{c}_{i,j}\colon{\mathbb{Z}}^{3}\to{\mathbb{Z}}^{2}$
are the ones in the second part of computations for Example 1.

Recall that $\displaystyle e_{1}=(1,0),$ $\displaystyle e_{2}=(0,1),$
$\displaystyle e=e_{1}-e_{2}$. In this case, we have that $\displaystyle
H_{1,1}=\mathbb{Z}e_{1}$, $\displaystyle H_{1,3}=H_{1,5}=H_{1,7}=\mathbb{Z}e$
and $\displaystyle H_{1,2}=H_{1,4}=H_{1,6}=\mathbb{Z}^{2}$. Moreover,
$\displaystyle H_{2,1}=\mathbb{Z}e_{2}$, $\displaystyle
H_{2,2}=H_{2,4}=H_{2,6}=\mathbb{Z}e$ and $\displaystyle
H_{2,3}=H_{2,5}=H_{2,7}=\mathbb{Z}^{2}$.

Applying (26) to Lemma 7.1, we get

(32) $\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\ \text{ F\o lner
seq.}\end{subarray}}\varlimsup_{N\to\infty}\Bigl{\|}\mathbb{E}_{n\in
I_{N}}\Bigl{(}T_{1}^{n^{2}+n}f_{1}\cdot
T_{2}^{n^{2}}f_{2}-T_{1}^{n^{2}+n}\mathbb{E}(f_{1}|W_{1,\mathbb{Z}^{3}})\cdot
T_{2}^{n^{2}}f_{2}\Bigr{)}\Bigr{\|}_{L^{2}(\mu)}=0.$

Fix $\displaystyle\varepsilon>0$. Since $\displaystyle
W_{1,{\mathbb{Z}}^{3}}=\bigvee_{N=1}^{\infty}W_{1,[-N,N]^{3}}$, by
approximation, there exists a finite subset $\displaystyle I$ of
$\displaystyle\mathbb{Z}^{3}$ such that
$\displaystyle\|\mathbb{E}(f_{1}|W_{1,\mathbb{Z}^{3}})-\mathbb{E}(f_{1}|W_{1,I})\|_{L^{1}(\mu)}<\varepsilon^{2}/2$.
Since
$\displaystyle\|f_{1}\|_{L^{\infty}(\mu)},\|f_{2}\|_{L^{\infty}(\mu)}\leq 1$,
for all $\displaystyle n\in{\mathbb{Z}}$,

$\begin{split}&\quad\Bigr{\|}\Bigl{(}T_{1}^{n^{2}+n}\mathbb{E}(f_{1}|W_{1,\mathbb{Z}^{3}})\cdot
T_{2}^{n^{2}}f_{2}-T_{1}^{n^{2}+n}\mathbb{E}(f_{1}|W_{1,I})\cdot
T_{2}^{n^{2}}f_{2}\Bigr{)}\Bigr{\|}^{2}_{L^{2}(\mu)}\\\
&=\int_{X}\Bigl{(}T_{1}^{n^{2}+n}\mathbb{E}(f_{1}|W_{1,\mathbb{Z}^{3}})\cdot
T_{2}^{n^{2}}f_{2}-T_{1}^{n^{2}+n}\mathbb{E}(f_{1}|W_{1,I})\cdot
T_{2}^{n^{2}}f_{2}\Bigr{)}^{2}\,d\mu\\\
&\leq\int_{X}2\Bigl{|}T_{1}^{n^{2}+n}\mathbb{E}(f_{1}|W_{1,\mathbb{Z}^{3}})-T_{1}^{n^{2}+n}\mathbb{E}(f_{1}|W_{1,I})\Bigr{|}\,d\mu=\int_{X}2\Bigl{|}\mathbb{E}(f_{1}|W_{1,\mathbb{Z}^{3}})-\mathbb{E}(f_{1}|W_{1,I})\Bigr{|}\,d\mu<\varepsilon^{2}.\end{split}$

So,

(33) $\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\ \text{ F\o lner
seq.}\end{subarray}}\varlimsup_{N\to\infty}\Bigl{\|}\mathbb{E}_{n\in
I_{N}}\Bigl{(}T_{1}^{n^{2}+n}\mathbb{E}(f_{1}|W_{1,\mathbb{Z}^{3}})\cdot
T_{2}^{n^{2}}f_{2}-T_{1}^{n^{2}+n}\mathbb{E}(f_{1}|W_{1,I})\cdot
T_{2}^{n^{2}}f_{2}\Bigr{)}\Bigr{\|}_{L^{2}(\mu)}<\varepsilon.$

Note that $\displaystyle W_{1,I}$ is contained in the
$\displaystyle(7|I|)$-step factor

$\displaystyle W^{\prime}_{1}\coloneqq
Z_{(G(\mathbb{c}_{1,1}(h_{1},h_{2},h_{3})),\dots,G(\mathbb{c}_{1,7}(h_{1},h_{2},h_{3})))_{(h_{1},h_{2},h_{3})\in
I}}.$

We say that
$\displaystyle(h^{\prime}_{1},h^{\prime}_{2},h^{\prime}_{3})\in\mathbb{Z}^{3}$
is _good_ if for any $\displaystyle(h_{1},h_{2},h_{3})\in I$, any

$\displaystyle
g\in\\{-2h_{1}e_{1},2h_{2}e-2h_{1}e_{1},2h_{2}e,2h_{3}e-2h_{1}e_{1},2h_{3}e,2(h_{2}+h_{3})e-2h_{1}e_{1},2(h_{2}+h_{3})e\\}$

(_i.e._ , $\displaystyle g$ is the generator of one of $\displaystyle
G(\mathbb{c}_{1,1}(h_{1},h_{2},h_{3})),\dots,G(\mathbb{c}_{1,7}(h_{1},h_{2},h_{3}))$)
and any action

$\displaystyle
g^{\prime}\in\\{-2h^{\prime}_{1}e_{1},2h^{\prime}_{2}e-2h^{\prime}_{1}e_{1},2h^{\prime}_{2}e,2h^{\prime}_{3}e-2h^{\prime}_{1}e_{1},2h^{\prime}_{3}e,2(h^{\prime}_{2}+h^{\prime}_{3})e-2h^{\prime}_{1}e_{1},2(h^{\prime}_{2}+h^{\prime}_{3})e\\},$

(_i.e._ , $\displaystyle g^{\prime}$ is the generator of one of $\displaystyle
G(\mathbb{c}_{1,1}(h^{\prime}_{1},h^{\prime}_{2},h^{\prime}_{3})),\dots,G(\mathbb{c}_{1,7}(h^{\prime}_{1},h^{\prime}_{2},h^{\prime}_{3}))$)
the set

$\displaystyle
H\coloneqq\text{span}_{\mathbb{Q}}\\{g,g^{\prime}\\}\cap\mathbb{Z}^{2}$

satisfies the following:

$\displaystyle\left\\{\begin{array}[]{ll}H=\mathbb{Z}e_{1}&\;,\text{ if
}g=-2h_{1}e_{1},g^{\prime}=-2h^{\prime}_{1}e_{1}\\\ H=\mathbb{Z}e&\;,\text{ if
}g\in\\{2h_{2}e,2h_{3}e,2(h_{2}+h_{3})e\\},g^{\prime}\in\\{2h^{\prime}_{2}e,2h^{\prime}_{3}e,2(h^{\prime}_{2}+h^{\prime}_{3})e\\}\\\
H=\mathbb{Z}^{2}&\;,\text{ otherwise}\end{array}\right..$

Let $\displaystyle J$ be the set of all good tuples. Since $\displaystyle I$
is finite, it is not hard to show that $\displaystyle J$ is of density 1 (see
also the claim in the proof of Proposition 5.6). Again, applying (26) to Lemma
7.1,

(37) $\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\ \text{ F\o lner
seq.}\end{subarray}}\varlimsup_{N\to\infty}\Bigl{\|}\mathbb{E}_{n\in
I_{N}}\Bigl{(}T_{1}^{n^{2}+n}\mathbb{E}(f_{1}|W_{1,I})\cdot
T_{2}^{n^{2}}f_{2}-T_{1}^{n^{2}+n}\mathbb{E}(f_{1}|W_{1,J}\cap W_{1,I})\cdot
T_{2}^{n^{2}}f_{2}\Bigr{)}\Bigr{\|}_{L^{2}(\mu)}=0.$

So (32), (33) and (37) implies that

(38) $\begin{split}\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\ \text{
F\o lner seq.}\end{subarray}}\varlimsup_{N\to\infty}\Bigl{\|}\mathbb{E}_{n\in
I_{N}}\Bigl{(}T_{1}^{n^{2}+n}f_{1}\cdot
T_{2}^{n^{2}}f_{2}-T_{1}^{n^{2}+n}\mathbb{E}(f_{1}|W_{1,J}\cap W_{1,I})\cdot
T_{2}^{n^{2}}f_{2}\Bigr{)}\Bigr{\|}_{L^{2}(\mu)}<\varepsilon.\end{split}$

By the definition of good tuples and Corollary 2.11, we have that

$\begin{split}W_{1,J}\cap
W_{1,I}\subseteq\bigvee_{(h^{\prime}_{1},h^{\prime}_{2},h^{\prime}_{3})\in
J}W^{\prime}_{1}\cap
W_{1,(h^{\prime}_{1},h^{\prime}_{2},h^{\prime}_{3})}=\bigvee_{(h^{\prime}_{1},h^{\prime}_{2},h^{\prime}_{3})\in
J}Z_{(\mathbb{Z}e_{1})^{\times|I|},(\mathbb{Z}e)^{\times
9|I|},(\mathbb{Z}^{2})^{\times 39|I|}}\subseteq
Z_{e^{\times\infty},e_{1}^{\times\infty}}.\end{split}$

So (38) implies that

$\begin{split}\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\ \text{ F\o
lner seq.}\end{subarray}}\varlimsup_{N\to\infty}\Bigl{\|}\mathbb{E}_{n\in
I_{N}}T_{1}^{n^{2}+n}f_{1}\cdot
T_{2}^{n^{2}}f_{2}\Bigr{\|}_{L^{2}(\mu)}<\varepsilon,\;\;\text{ if
$\displaystyle\;\;\mathbb{E}(f_{1}|Z_{e_{1}^{\times\infty},e^{\times\infty}})=0$.}\end{split}$

Since $\displaystyle\varepsilon>0$ is arbitrary,

$\begin{split}\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\ \text{ F\o
lner seq.}\end{subarray}}\varlimsup_{N\to\infty}\Bigl{\|}\mathbb{E}_{n\in
I_{N}}T_{1}^{n^{2}+n}f_{1}\cdot
T_{2}^{n^{2}}f_{2}\Bigr{\|}_{L^{2}(\mu)}=0,\;\;\text{ if
$\displaystyle\;\;\mathbb{E}(f_{1}|Z_{e_{1}^{\times\infty},e^{\times\infty}})=0$.}\end{split}$

Working analogously for the $\displaystyle T_{2}^{n^{2}}f_{2}$ term, we
eventually get that

(39) $\begin{split}\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\ \text{
F\o lner seq.}\end{subarray}}\varlimsup_{N\to\infty}\Bigl{\|}\mathbb{E}_{n\in
I_{N}}T_{1}^{n^{2}+n}f_{1}\cdot
T_{2}^{n^{2}}f_{2}\Bigr{\|}_{L^{2}(\mu)}=0\;\text{ if
$\displaystyle\;\mathbb{E}(f_{i}|Z_{e_{i}^{\times\infty},e^{\times\infty}})=0\;$
for $\displaystyle i=1$ or $\displaystyle 2$.}\end{split}$

We remark that (39) is a stronger version of (14) (_i.e._ , continuation of
Example 1).

######  Remark.

As mentioned before, the characteristic factors described in Theorem 5.1 are
not the optimal ones in general, but they are good enough for the purposes of
our study.

We briefly explain the idea on proving Proposition 5.6. Under the assumptions
of Proposition 5.6, Lemma 7.1 says that one can assume that $\displaystyle
f_{1}$ is measurable with respect to the factor $\displaystyle W_{1,J_{1}}$.
However, thanks to the freedom of the choices of $\displaystyle J_{1}$, we can
use Lemma 7.1 to repeatedly choose different subsets $\displaystyle
J_{1,1},\dots$, $\displaystyle J_{1,r}$, for some $\displaystyle
r\in\mathbb{N}^{\ast}$, and assume that $\displaystyle f_{1}$ is measurable
with respect to the factor $\displaystyle W_{1,J_{1,1}}\cap
W_{1,J_{1,2}}\cap\dots\cap W_{1,J_{1,r}}$. We then employ the concatenation
theorems to estimate the intersection of $\displaystyle W_{1,J_{1,j}}$, and
find a smaller factor characterizing the multiple average we aim to study.

###### Proof of Proposition 5.6.

Let $\displaystyle(X,\mathcal{B},\mu,(T_{g})_{g\in\mathbb{Z}^{d}})$ be a
$\displaystyle\mathbb{Z}^{d}$-system, $\displaystyle f_{1},\dots,f_{k}\in
L^{\infty}(\mu)$ and $\displaystyle s,t_{1},\dots,$ $\displaystyle
t_{k},\mathbb{c}_{i,m},$ $\displaystyle 1\leq i\leq k,$ $\displaystyle 1\leq
m\leq t_{i}$ be as in the statement. By Lemma 2.13,

$\displaystyle
H_{i,m}=\text{span}_{\mathbb{Q}}\\{G(\mathbb{c}_{i,m}(h_{1},\dots,h_{s}))\colon
h_{1},\dots,h_{s}\in\mathbb{Z}^{L}\\}\cap\mathbb{Z}^{d}.$

To show (24), it suffices to show that if
$\displaystyle\mathbb{E}(f_{i}|Z_{(H_{i,1})^{\times\infty},\dots,(H_{i,t_{i}})^{\times\infty}})=0$
for some $\displaystyle 1\leq i\leq k$, then the left hand side of (24) equals
to 0. We assume without loss of generality that $\displaystyle i=1$.

For every $\displaystyle r\in\mathbb{N}$, every finite subset $\displaystyle
I\subseteq{\mathbb{Z}}^{L}$, and every tuple
$\displaystyle(J_{1},\dots,J_{r})$, where $\displaystyle
J_{i}\subseteq(\mathbb{Z}^{L})^{s}$, $\displaystyle 1\leq i\leq r$, denote

$\displaystyle A_{I}(J_{1},\dots,J_{r})\coloneqq\mathbb{E}_{n\in
I}T_{p_{1}(n)}\mathbb{E}(f_{1}|W_{1,J_{1}}\cap\dots\cap W_{1,J_{r}})\cdot
T_{p_{2}(n)}f_{2}\cdot\ldots\cdot T_{p_{k}(n)}f_{k},$

and in the degenerated case, set

$\displaystyle A_{I}(\emptyset)\coloneqq\mathbb{E}_{n\in
I}T_{p_{1}(n)}f_{1}\cdot T_{p_{2}(n)}f_{2}\cdot\ldots\cdot T_{p_{k}(n)}f_{k}.$

We say that a tuple $\displaystyle(J_{1},\dots,J_{r})$ of subsets of
$\displaystyle(\mathbb{Z}^{L})^{s}$ is _admissible_ if for every
$\displaystyle{\bf h}_{u}\in J_{u},1\leq u\leq r$ and every $\displaystyle
1\leq m\leq t_{1}$, denoting

(40) $G_{K}\coloneqq\text{span}_{\mathbb{Q}}\\{G(\mathbb{c}_{1,m}({\bf
h}_{u}))\colon u\in K\\}\cap\mathbb{Z}^{d}$

for all $\displaystyle K\subseteq\\{1,\dots,r\\}$, the following holds: for
all $\displaystyle\emptyset\neq K^{\prime}\subsetneq
K\subseteq\\{1,\dots,r\\}$ such that $\displaystyle\max\\{x\in
K^{\prime}\\}<\min\\{x\in K\backslash K^{\prime}\\}$, either $\displaystyle
G_{K^{\prime}}\subsetneq G_{K}$ or $\displaystyle
G_{K^{\prime}}=H_{1,m}$.363636 We think of this as a notion of having “full
rank”.

Fix $\displaystyle\varepsilon>0$. By (23), we have that

$\displaystyle\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\ \text{ F\o
lner
seq.}\end{subarray}}\varlimsup_{N\to\infty}\Bigl{\|}A_{I_{N}}(\emptyset)-A_{I_{N}}((\mathbb{Z}^{L})^{s})\Bigr{\|}_{L^{2}(\mu)}=0.$

By an approximation argument similar to the one obtaining (33), there exists a
finite subset $\displaystyle J_{1}^{\prime}\subseteq(\mathbb{Z}^{L})^{s}$ such
that

$\displaystyle\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\ \text{ F\o
lner
seq.}\end{subarray}}\varlimsup_{N\to\infty}\Bigl{\|}A_{I_{N}}((\mathbb{Z}^{L})^{s})-A_{I_{N}}(J^{\prime}_{1})\Bigr{\|}_{L^{2}(\mu)}<\varepsilon,$

and so

$\displaystyle\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\ \text{ F\o
lner
seq.}\end{subarray}}\varlimsup_{N\to\infty}\Bigl{\|}A_{I_{N}}(\emptyset)-A_{I_{N}}(J^{\prime}_{1})\Bigr{\|}_{L^{2}(\mu)}<\varepsilon.$

Suppose that for some $\displaystyle r\geq 1$, we have constructed finite
subsets $\displaystyle
J^{\prime}_{1},\dots,J^{\prime}_{r}\subseteq(\mathbb{Z}^{L})^{s}$ such that:

  * (i)

$\displaystyle\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\ \text{ F\o
lner
seq.}\end{subarray}}\varlimsup_{N\to\infty}\Bigl{\|}A_{I_{N}}(\emptyset)-A_{I_{N}}(J^{\prime}_{1},\dots,J^{\prime}_{r})\Bigr{\|}_{L^{2}(\mu)}<r\varepsilon$;
and

  * (ii)

$\displaystyle(J^{\prime}_{1},\dots,J^{\prime}_{r})$ is admissible.

We now construct $\displaystyle J^{\prime}_{r+1}$. We first claim that there
exists $\displaystyle J_{r+1}\subseteq(\mathbb{Z}^{L})^{s}$ of density
$\displaystyle 1$ such that
$\displaystyle(J^{\prime}_{1},\dots,J^{\prime}_{r},J_{r+1})$ is admissible.
For every $\displaystyle{\bf h}_{u}\in J^{\prime}_{u},1\leq u\leq r$,
$\displaystyle 1\leq m\leq t_{1}$ and nonempty subset $\displaystyle
K\subseteq\\{1,\dots,r\\}$, let

$\displaystyle Q_{m;{\bf h}_{1},\dots,{\bf
h}_{r};K}\coloneqq\text{span}_{\mathbb{Q}}\\{G(\mathbb{c}_{1,m}({\bf
h}_{u}))\colon u\in K\\}\cap\mathbb{Z}^{d}.$

If $\displaystyle Q_{m;{\bf h}_{1},\dots,{\bf h}_{r};K}=H_{1,m}$, we let
$\displaystyle V_{m;{\bf h}_{1},\dots,{\bf h}_{r};K}=(\mathbb{Z}^{L})^{s}$,
otherwise $\displaystyle V_{m;{\bf h}_{1},\dots,{\bf h}_{r};K}$ denotes the
set of $\displaystyle{\bf h}=(h_{1},\dots,h_{s})\in(\mathbb{Z}^{L})^{s}$ such
that $\displaystyle G(\mathbb{c}_{1,m}({\bf h}))$ is not contained in
$\displaystyle Q_{m;{\bf h}_{1},\dots,{\bf h}_{r};K}$. Let

$\displaystyle J_{r+1}\coloneqq\bigcap_{\begin{subarray}{c}{\bf h}_{u}\in
I_{u},\;1\leq u\leq r,\;1\leq m\leq
t_{1},\;K\subseteq\\{1,\dots,r\\}\end{subarray}}V_{m;{\bf h}_{1},\dots,{\bf
h}_{r};K}.$

To show that $\displaystyle(J^{\prime}_{1},\dots,J^{\prime}_{r},J_{r+1})$ is
admissible, fix $\displaystyle{\bf h}_{i}\in J^{\prime}_{i},1\leq i\leq r$,
$\displaystyle{\bf h}_{r+1}\in J_{r+1},$ $\displaystyle 1\leq m\leq t_{1}$,
and let $\displaystyle G_{K}$ be defined as in (40) for all $\displaystyle
K\subseteq\\{1,\dots,r+1\\}$. Let $\displaystyle\emptyset\neq
K^{\prime}\subsetneq K\subseteq\\{1,\dots,r+1\\}$ such that
$\displaystyle\max\\{x\in K^{\prime}\\}<\min\\{x\in K\backslash
K^{\prime}\\}$. We have the following three possible cases for $\displaystyle
r+1$:

Case (i): $\displaystyle r+1\notin K$. Then $\displaystyle r+1\notin
K^{\prime}$ and so $\displaystyle\emptyset\neq K^{\prime}\subsetneq
K\subseteq\\{1,\dots,r\\}$. Since $\displaystyle(I_{1},\dots,I_{r})$ is
admissible, either $\displaystyle G_{K^{\prime}}\subsetneq G_{K}$ or
$\displaystyle G_{K^{\prime}}=H_{1,m}$.

Case (ii): $\displaystyle r+1\in K^{\prime}$. This contradicts the assumption
that $\displaystyle\max\\{x\in K^{\prime}\\}<\min\\{x\in K\backslash
K^{\prime}\\}.$ So this case is not possible.

Case (iii): $\displaystyle r+1\in K$ but $\displaystyle r+1\notin K^{\prime}$.
Then $\displaystyle K^{\prime}\subseteq\\{1,\dots,r\\}$ and so $\displaystyle
J_{r+1}\subseteq V_{m;{\bf h}_{1},\dots,{\bf h}_{r};K^{\prime}}$. If
$\displaystyle G_{K^{\prime}}\neq H_{1,m}$, then since $\displaystyle{\bf
h}_{r+1}\in J_{r+1}\subseteq V_{m;{\bf h}_{1},\dots,{\bf h}_{r};K^{\prime}}$,
the subgroup $\displaystyle G(\mathbb{c}_{1,m}({\bf h}_{r+1}))$ (which is
contained in $\displaystyle G_{K}$ since $\displaystyle r+1\in K$) is not
contained in $\displaystyle Q_{m;{\bf h}_{1},\dots,{\bf
h}_{r};K^{\prime}}=G_{K^{\prime}}$. This implies that $\displaystyle G_{K}\neq
G_{K^{\prime}}$.

In conclusion, we have that
$\displaystyle(J^{\prime}_{1},\dots,J^{\prime}_{r},J_{r+1})$ is admissible.
The second part of the claim is that $\displaystyle J_{r+1}$ is of density
$\displaystyle 1.$ Since $\displaystyle J^{\prime}_{1},\dots,J^{\prime}_{r}$
are finite sets, it suffices to show that every $\displaystyle V_{m;{\bf
h}_{1},\dots,{\bf h}_{r};K}$ is of density $\displaystyle 1.$ If
$\displaystyle Q_{m;{\bf h}_{1},\dots,{\bf h}_{r};K}=H_{1,m}$, $\displaystyle
V_{m;{\bf h}_{1},\dots,{\bf h}_{r};K}=(\mathbb{Z}^{L})^{s}$ and we are done.
Now assume that $\displaystyle Q_{m;{\bf h}_{1},\dots,{\bf h}_{r};K}\neq
H_{1,m}$. By Lemma 2.12, the set

$\displaystyle V_{m;{\bf h}_{1},\dots,{\bf h}_{r};K}=\\{{\bf
h}\in(\mathbb{Z}^{L})^{s}\colon G(\mathbb{c}_{1,m}({\bf h}))\nsubseteq
Q_{m;{\bf h}_{1},\dots,{\bf h}_{r};K}\\}$

is either of density $\displaystyle 1,$ or is empty and

$\displaystyle Q_{m;{\bf h}_{1},\dots,{\bf
h}_{r};K}=\text{span}_{\mathbb{Q}}\\{G(\mathbb{c}_{i,m}(h_{1},\dots,h_{s}))\colon
h_{1},\dots,h_{s}\in\mathbb{Z}^{L}\\}\cap\mathbb{Z}^{d}=H_{1,m}.$

By our assumption, $\displaystyle V_{m;{\bf h}_{1},\dots,{\bf h}_{r};K}$ is of
density $\displaystyle 1.$ This finishes the proof of the claim.

By Lemma 7.1, $\displaystyle
A(J^{\prime}_{1},\dots,J^{\prime}_{r})=A(J^{\prime}_{1},\dots,J^{\prime}_{r},J_{r+1})$.
By an approximation argument, there exists a finite subset $\displaystyle
J^{\prime}_{r+1}\subseteq J_{r+1}$ such that

$\displaystyle\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\ \text{ F\o
lner
seq.}\end{subarray}}\varlimsup_{N\to\infty}\Bigl{\|}A_{I_{N}}(J^{\prime}_{1},\dots,J^{\prime}_{r},J_{r+1})-A_{I_{N}}(J^{\prime}_{1},\dots,J^{\prime}_{r},J^{\prime}_{r+1})\Bigr{\|}_{L^{2}(\mu)}<\varepsilon.$

By induction hypothesis,

$\displaystyle\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\ \text{ F\o
lner
seq.}\end{subarray}}\varlimsup_{N\to\infty}\Bigl{\|}A_{I_{N}}(\emptyset)-A_{I_{N}}(J^{\prime}_{1},\dots,J^{\prime}_{r},J^{\prime}_{r+1})\Bigr{\|}_{L^{2}(\mu)}<(r+1)\varepsilon.$

So (i) holds for $\displaystyle r+1$. Since
$\displaystyle(J^{\prime}_{1},\dots,J^{\prime}_{r},J_{r+1})$ is admissible, so
is $\displaystyle(J^{\prime}_{1},\dots,J^{\prime}_{r},J^{\prime}_{r+1})$,
hence (ii) holds for $\displaystyle r+1$. In conclusion, there exist a tuple
$\displaystyle(J^{\prime}_{1},\dots,J^{\prime}_{dt_{1}})$ of finite subsets of
$\displaystyle(\mathbb{Z}^{L})^{s}$ such that

$\displaystyle\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\ \text{ F\o
lner
seq.}\end{subarray}}\varlimsup_{N\to\infty}\Bigl{\|}A_{I_{N}}(\emptyset)-A_{I_{N}}(J^{\prime}_{1},\dots,J^{\prime}_{dt_{1}})\Bigr{\|}_{L^{2}(\mu)}<dt_{1}\varepsilon$

and $\displaystyle(J^{\prime}_{1},\dots,J^{\prime}_{dt_{1}})$ is admissible.
Note that

$\begin{split}W_{1,J^{\prime}_{1}}\cap\dots\cap
W_{1,J^{\prime}_{dt_{1}}}&=\bigcap_{u=1}^{dt_{1}}\bigvee_{\begin{subarray}{c}{\bf
h}_{u}\in J^{\prime}_{u}\end{subarray}}W_{1,{\bf h}_{u}}\\\
&=\bigcap_{u=1}^{dt_{1}}\bigvee_{\begin{subarray}{c}{\bf h}_{u}\in
J^{\prime}_{u}\end{subarray}}Z_{G(\mathbb{c}_{1,1}({\bf
h}_{u})),\dots,G(\mathbb{c}_{1,t_{1}}({\bf h}_{u}))}\\\
&\subseteq\bigcap_{u=1}^{dt_{1}}Z_{\\{G(\mathbb{c}_{1,m}({\bf
h}_{u}))\\}_{1\leq m\leq t_{1},{\bf h}_{u}\in J^{\prime}_{u}}},\end{split}$

where we used Lemma 2.4 (vii) in the last inclusion. For each $\displaystyle
1\leq u\leq dt_{1}$, pick some $\displaystyle 1\leq m_{u}\leq t_{1}$ and
$\displaystyle{\bf h}_{u}\in J^{\prime}_{u}$. Consider the set

$\displaystyle
P\coloneqq\text{span}_{\mathbb{Q}}\\{G(\mathbb{c}_{1,m_{u}}({\bf
h}_{u}))\colon 1\leq u\leq dt_{1}\\}\cap\mathbb{Z}^{d}.$

By the pigeon-hole principle, there exist $\displaystyle 1\leq m\leq t_{1}$
and $\displaystyle 1\leq u_{1}<\dots<u_{d}\leq dt_{1}$ such that
$\displaystyle m_{u_{1}}=\dots=m_{u_{d}}=m$. For all $\displaystyle 1\leq
i\leq d$, let $\displaystyle
K_{i}=\\{u_{1},\dots,u_{i}\\}\subseteq\\{1,\dots,dt_{1}\\}$ and

$\displaystyle
P_{i}\coloneqq\text{span}_{\mathbb{Q}}\\{G(\mathbb{c}_{1,m_{u}}({\bf
h}_{u}))\colon u\in K_{i}\\}\cap\mathbb{Z}^{d}.$

Since $\displaystyle(J^{\prime}_{1},\dots,J^{\prime}_{dt_{1}})$ is admissible,
for all $\displaystyle 1\leq i\leq d-1$, either $\displaystyle P_{i}=H_{1,m}$
or the dimension of $\displaystyle P_{i+1}$ is higher than that of
$\displaystyle P_{i}$. Since the dimension of $\displaystyle P_{i}$ can not
exceed $\displaystyle d$, we must have that $\displaystyle P_{i}$ contains
$\displaystyle H_{1,m}$ for some $\displaystyle 1\leq i\leq d$. As
$\displaystyle P_{i}\subseteq P$, we have that $\displaystyle P$ also contains
$\displaystyle H_{1,m}$. By Corollary 2.11,

$\begin{split}W_{1,J^{\prime}_{1}}\cap\dots\cap
W_{1,J^{\prime}_{dt_{1}}}\subseteq\bigcap_{u=1}^{dt_{1}}Z_{\\{G(\mathbb{c}_{1,m}(\textbf{h}_{u}))\\}_{1\leq
m\leq t_{1},\textbf{h}_{u}\in J^{\prime}_{u}}}\subseteq
Z_{H_{1,1}^{\times\infty},\dots,H_{1,t_{1}}^{\times\infty}}.\end{split}$

Since
$\displaystyle\mathbb{E}(f_{1}|Z_{(H_{1,1})^{\times\infty},\dots,(H_{1,t_{1}})^{\times\infty}})=0$,
$\displaystyle A(J^{\prime}_{1},\dots,J^{\prime}_{dt_{1}})=0$ and so

$\displaystyle\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\ \text{ F\o
lner
seq.}\end{subarray}}\varlimsup_{N\to\infty}\Bigl{\|}A_{I_{N}}(\emptyset)\Bigr{\|}_{L^{2}(\mu)}<dt_{1}\varepsilon.$

Since $\displaystyle\varepsilon$ is chosen arbitrary, the left hand side of
(24) is equal to
$\displaystyle\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\ \text{ F\o
lner
seq.}\end{subarray}}\varlimsup_{N\to\infty}\Bigl{\|}A_{I_{N}}(\emptyset)\Bigr{\|}_{L^{2}(\mu)}=0,$
which finishes the proof. ∎

## References

  * [1] T. Austin. On the norm convergence of nonconventional ergodic averages. _Ergodic Theory Dynam. Systems_ 30 (2010), no. 2, 321–338.
  * [2] T. Austin. Pleasant extensions retaining algebraic structure, I. _J. Anal. Math._ 125 (2015), 1–36.
  * [3] T. Austin. Pleasant extensions retaining algebraic structure, II. _J. Anal. Math._ 126 (2015), 1–111.
  * [4] D. Berend and V. Bergelson. Jointly ergodic measure-preserving transformations. _Israel J. Math._ 49 (1984), no. 4, 307–314.
  * [5] V. Bergelson. Ergodic Ramsey theory–an update. Ergodic theory of $\displaystyle\textbf{Z}^{d}$ actions (Warwick, 1993–1994), 1–61, London Math. Soc. Lecture Note Ser., 228, Cambridge Univ. Press, Cambridge, 1996.
  * [6] V. Bergelson. Weakly mixing PET. _Ergodic Theory Dynam. Systems_ 7 (1987), no. 3, 337–349.
  * [7] V. Bergelson and A. Leibman. Cubic averages and large intersections. _Recent trends in ergodic theory and dynamical systems_ , 5–19, Contemp. Math., 631, Amer. Math. Soc., Providence, RI, 2015.
  * [8] V. Bergelson, A. Leibman and Y. Son. Joint ergodicity along generalized linear functions. _Ergodic Theory Dynam. Systems_ 36 (2016), no. 7, 2044–2075.
  * [9] Q. Chu, N. Frantzikinakis and B. Host. Ergodic averages of commuting transformations with distinct degree polynomial iterates. _Proc. of the London Math. Society_ , (3), 102 (2011), 801–842.
  * [10] M. Einsiedler and T. Ward. _Ergodic theory with a view towards number theory_. Springer-Verlag, London, 259 (2011).
  * [11] H. Furstenberg. Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions. _J. Analyse Math._ 31 (1977), 204–256.
  * [12] H. Furstenberg. _Recurrence in Ergodic Theory and Combinatorial Number Theory_. Princeton University Press, Princeton, 1981.
  * [13] N. Frantzikinakis. A multidimensional Szemerédi theorem for Hardy sequences of different growth. Tran. of the A. M. S. 367, no. 8, (2015), 5653–5692.
  * [14] N. Frantzikinakis and B. Kra. Polynomial averages converge to the product of integrals. _Israel J. Math._ 148 (2005), 267–276.
  * [15] B. Host. Ergodic seminorms for commuting transformations and applications. _Studia Math._ , 195 (2009), no.1, 31–49.
  * [16] B. Host and B. Kra. Nonconventional ergodic averages and nilmanifolds. _Ann. of Math._ (2) 161 (2005), no. 1, 397–488.
  * [17] M. Johnson. Convergence of polynomial ergodic averages of several variables for some commuting transformations. _Illinois J. Math._ 53 (2009), no. 3, 865–882.
  * [18] D. Karageorgos and A. Koutsogiannis. Integer part independent polynomial averages and applications along primes, to appear in Studia Mathematica.
  * [19] A. Koutsogiannis. Integer part polynomial correlation sequences. Ergodic Theory Dynam. Systems 38 (2018), no. 4, 1525–1542.
  * [20] A. Leibman. Pointwise convergence of ergodic averages for polynomial actions of $\displaystyle{\mathbb{Z}}^{d}$ by translations on a nilmanifold. _Ergodic Theory Dynam. Systems_ 25 (2005), no. 1, 215–225.
  * [21] A. Leibman. Convergence of multiple ergodic averages along polynomials of several variables. _Israel J. Math._ 146 (2005), 303–315.
  * [22] W. Sun. Weak ergodic averages over dilated curves. Submitted for publication. arXiv: 1809.06916.
  * [23] T. Tao. Norm convergence of multiple ergodic averages for commuting transformations. _Ergodic Theory Dynam. Systems_ 28 (2008), no. 2, 657–688.
  * [24] T. Tao and T. Ziegler. Concatenation Theorems for anti-Gowers-uniform functions and Host-Kra characteristic factors. _Discrete Anal._ 2016, Paper No. 13.
  * [25] M. Walsh. Norm convergence of nilpotent ergodic averages. _Ann. of Math._ (2) 175 (2012), no. 3, 1667–1688.
  * [26] T. Ziegler. Nilfactors of $\displaystyle\mathbb{R}^{m}$-actions and configurations in sets of positive upper density in $\displaystyle\mathbb{R}^{m}$. _J. Anal. Math._ , 99 (2006), 249–266.
  * [27] T. Ziegler. Universal characteristic factors and Furstenberg averages. _J. Amer. Math. Soc._ 20 (2007), no. 1, 53–97.