# Seminorms for multiple averages along polynomials and applications to joint
ergodicity

Sebastián Donoso, Andreas Koutsogiannis and Wenbo Sun  Universidad de
O’Higgins, Instituto de Ciencias de la Ingeniería, Av. Libertador Bernardo
O’Higgins 611, Rancagua, 2841959, Chile<EMAIL_ADDRESS>The Ohio
State University, Department of mathematics, 231 West 18th Avenue, Columbus,
OH 43210-1174, USA<EMAIL_ADDRESS>The Ohio State University,
Department of mathematics, 231 West 18th Avenue, Columbus, OH 43210-1174, USA
<EMAIL_ADDRESS>
###### Abstract.

Exploiting the recent work of Tao and Ziegler on the concatenation theorem on
factors, we find explicit characteristic factors for multiple averages along
polynomials on systems with commuting transformations, and use them to study
the criteria of joint ergodicity for sequences of the form
$\displaystyle(T^{p_{1,j}(n)}_{1}\cdot\ldots\cdot
T^{p_{d,j}(n)}_{d})_{n\in\mathbb{Z}},$ $\displaystyle 1\leq j\leq k$, where
$\displaystyle T_{1},\dots,T_{d}$ are commuting measure preserving
transformations on a probability measure space and $\displaystyle p_{i,j}$ are
integer polynomials. To be more precise, we provide a sufficient condition for
such sequences to be jointly ergodic. We also give a characterization for
sequences of the form $\displaystyle(T^{p(n)}_{i})_{n\in\mathbb{Z}},1\leq
i\leq d$ to be jointly ergodic, answering a question due to Bergelson.

###### Key words and phrases:

Multiple averages, polynomials, characteristic factors, concatenation theorem

###### 2010 Mathematics Subject Classification:

Primary: 37A05; Secondary: 37A30, 28A99

The first author is supported by Fondecyt Iniciación en Investigación Grant
11160061.

##  1\. introduction

###  1.1. Characteristic factors for multiple averages

Let $\displaystyle\mathbb{X}=(X,\mathcal{B},\mu,T)$ be a measure preserving
$\displaystyle\mathbb{Z}$-system.111 By this we mean that
$\displaystyle(X,\mathcal{B},\mu)$ is a probability space and $\displaystyle
T$ is an invertible measure preserving transformation, _i.e._ ,
$\displaystyle\mu(T^{-1}A)=\mu(A)$ for all $\displaystyle A\in\mathcal{B}$. We
also denote such a system as
$\displaystyle(X,\mathcal{B},\mu,(S_{g})_{g\in\mathbb{Z}})$ later in this
paper, where $\displaystyle S_{n}=T^{n},$ _i.e._ , the composition of
$\displaystyle T$ with itself $\displaystyle n$ times. When $\displaystyle T$
is ergodic (_i.e._ , the measure of any $\displaystyle T$-invariant set is
$\displaystyle 0$ or $\displaystyle 1$), the von Neumann ergodic theorem (see
for example [10, Theorem 2.21]) asserts that for all $\displaystyle f\in
L^{2}(\mu)$, the $\displaystyle L^{2}(\mu)$ limit of the “time average”
$\displaystyle\frac{1}{N}\sum_{n=0}^{N-1}T^{n}f$ equals to the “natural” one,
namely the “space limit” $\displaystyle\int_{X}f\,d\mu$.

In the past decades, the $\displaystyle L^{2}$-limit behavior of the “multiple
averages” became a central topic in ergodic theory. Several authors have
studied averages for a single transformation $\displaystyle T$, as

(1) $\frac{1}{N}\sum_{n=0}^{N-1}T^{p_{1}(n)}f_{1}\cdot\ldots\cdot
T^{p_{k}(n)}f_{k},$

averages for several (usually commuting) $\displaystyle T_{i}$’s, as

(2) $\frac{1}{N}\sum_{n=0}^{N-1}T_{1}^{p_{1}(n)}f_{1}\cdot\ldots\cdot
T_{k}^{p_{k}(n)}f_{k}$

and even more general averages as

(3)
$\frac{1}{N}\sum_{n=0}^{N-1}\prod_{i=1}^{m}T_{i}^{p_{i,1}(n)}f_{1}\cdot\ldots\cdot\prod_{i=1}^{m}T_{i}^{p_{i,k}(n)}f_{k}$

for some $\displaystyle m,k\in\mathbb{N}^{\ast}$, integer valued sequences
$\displaystyle(p_{i}(n))_{n\in\mathbb{N}}$ and $\displaystyle f_{i}\in
L^{\infty}(\mu),$ $\displaystyle 1\leq i\leq k$. Fruitful results has been
obtained, which include, but are not limited to [1, 4, 6, 8, 9, 14, 15, 16,
17, 18, 19, 21, 23, 25, 27]. In particular, it was proved by Walsh [25]
(following the ideas of Tao [23]) that the multiple (uniform) averages, as in
(3), converge in the $\displaystyle L^{2}$ sense for any integer valued
polynomials $\displaystyle p_{i}$ when $\displaystyle T_{1},\ldots,T_{d}$ span
a nilpotent group. However, the result in [25] does not give any description
or information about the limit. In general, very little is known about the
limit of multiple averages.

The existing results employ the idea of _characteristic factors_ , which
intends to reduce the average under study to a more tractable one. For a
single transformation $\displaystyle T$ and for linear $\displaystyle
p_{i}$’s, the main content of [16] is the introduction of some seminorms that
control the behavior of the average (1) and are characterized by _nilsystems_.
These seminorms were also used by Leibman (in [20]) to bound the limit of (1)
for polynomial $\displaystyle p_{i}$’s (always in the context of a single
transformation). For several commuting transformations, Host (in [15])
introduced similar seminorms to bound the limit of (2) for linear
$\displaystyle p_{i}$’s but in that case there was still no clear connection
to nilsystems (see also [22, 24] for slight generalizations of these
seminorms). When considering non linear polynomials $\displaystyle p_{i}$’s,
even less is known and even simple cases can be very intricate. For instance,
Austin in [2, 3] found precise characteristic factors for some specific cases
of quadratic polynomials for $\displaystyle k=2$ (and linear polynomials for
$\displaystyle k=3$).

In this paper, under a further development of a recent result by Tao and
Ziegler ([24]) on concatenation (intersection) of factors, we provide an upper
bound for the limit of (3) for any $\displaystyle m,k\in{\mathbb{N}}^{\ast}$
and polynomials $\displaystyle p_{i,j}$ taking integer values at integers by
using some seminorms on the system (generically called _Host-Kra seminorms_),
which to the best of our knowledge, has never been studied before in this
generality. We state here a simplified more aesthetic one-parameter version of
our main result, and refer the readers to Theorem 5.1 below for the result in
its full generality:

######  Theorem 1.1 (Bounding multiple averages along polynomials by
seminorms).

Let $\displaystyle d,k,K\in\mathbb{N}^{\ast}$ and $\displaystyle
p_{1},\dots,p_{k}\colon\mathbb{Z}\to\mathbb{Z}^{d}$ be a family of polynomials
of degrees at most $\displaystyle K$ such that $\displaystyle
p_{i},p_{i}-p_{j}$ are not constant for all $\displaystyle 1\leq i,j\leq k$,
$\displaystyle i\neq j$, where $\displaystyle p_{i}(n)=\sum_{0\leq v\leq
K}b_{i,v}n^{v}$ for some $\displaystyle b_{i,v}\in\mathbb{Q}^{d}$. Denote the
set of the coefficients and pairwise differences of the coefficients
(excluding $\displaystyle{\bf 0}$) of the polynomials with

$\displaystyle R=\bigcup_{0<v\leq K}\\{b_{i,v},b_{i,v}-b_{i^{\prime},v}\colon
1\leq i,i^{\prime}\leq k\\}\backslash\\{{\bf 0}\\}.$

Let $\displaystyle(X,\mathcal{B},\mu,(T_{g})_{g\in\mathbb{Z}^{d}})$ be a
$\displaystyle\mathbb{Z}^{d}$-system (see Section 1.4 for the definition). If
the Host-Kra seminorm
$\displaystyle\|f_{i}\|_{{\\{G(r)^{\times\infty}\\}_{r\in R}}}$ (see Section 2
for definitions) of $\displaystyle f_{i}$ equals to 0 for some $\displaystyle
1\leq i\leq k$, then

$\displaystyle\lim_{N-M\to\infty}\frac{1}{N-M}\sum_{n=M}^{N-1}T_{p_{1}(n)}f_{1}\cdot\ldots\cdot
T_{p_{k}(n)}f_{k}=0.$

######  Remark.

Unlike the conventional “finite-step” Host-Kra seminorms, the seminorms we use
such as $\displaystyle\|\cdot\|_{{\\{G(r)^{\times\infty}\\}_{r\in R}}}$ are
“infinite-step” ones. It is an interesting question to ask whether one can
replace the “infinite-step” seminorms in the main theorems of this paper by
“finite-step” ones.

###  1.2. The joint ergodicity property

An interesting application of Theorem 1.1 and its stronger version Theorem 5.1
is that they can be used to study joint ergodicity problems, also allowing us
to answer a question due to Bergelson. Back to the description of the limit of
(3), there are interesting cases where the limit has a “simple” description.
In [6], Bergelson showed that if $\displaystyle(X,\mathcal{B},\mu,T)$ is a
weakly mixing system (meaning that $\displaystyle T\times T$ is ergodic for
$\displaystyle\mu\times\mu$) 222 In this case we also say that $\displaystyle
T$ is a weakly mixing transformation. and $\displaystyle p_{1},\dots,p_{k}$
are polynomials such that $\displaystyle p_{i},p_{i}-p_{j}$ are non-constant
for all $\displaystyle 1\leq i,j\leq k,i\neq j$, then the $\displaystyle
L^{2}(\mu)$ limit of (1) is the “expected” one, namely the “multiple space
limit” $\displaystyle\prod_{i=1}^{k}\int_{X}f_{i}\,d\mu$.333 This result was
previously obtained by Furstenberg (in [11]) in the special case where
$\displaystyle p_{i}(n)=in$, $\displaystyle i=1,\ldots,k$. One can think of
this result as a strong independence property of the sequences
$\displaystyle(T^{p_{i}(n)})_{n\in\mathbb{Z}},1\leq i\leq k$ in the weakly
mixing case. This naturally leads to the following definition of joint
ergodicity, in which we demand the average to converge to the expected limit.

######  Definition.

Let $\displaystyle d,k,L\in\mathbb{N}^{\ast}$, $\displaystyle
p_{1},\dots,p_{k}\colon\mathbb{Z}^{L}\to\mathbb{Z}^{d}$ be functions, and
$\displaystyle(X,\mathcal{B},\mu,(T_{g})_{g\in\mathbb{Z}^{d}})$ be a
$\displaystyle\mathbb{Z}^{d}$-system. We say that the tuple
$\displaystyle(T_{p_{1}(n)},\dots,T_{p_{k}(n)})_{n\in\mathbb{Z}^{L}}$ is
_jointly ergodic_ for $\displaystyle\mu$ if for every $\displaystyle
f_{1},\dots,f_{k}\in L^{\infty}(\mu)$ and every Følner sequence
$\displaystyle(I_{N})_{N\in\mathbb{N}}$ of $\displaystyle\mathbb{Z}^{L}$,444 A
sequence of finite subsets $\displaystyle(I_{N})_{N\in\mathbb{N}}$ of
$\displaystyle\mathbb{Z}^{L}$ with the property
$\displaystyle\lim_{N\to\infty}|I_{N}|^{-1}\cdot|(g+I_{N})\triangle I_{N}|=0$
for all $\displaystyle g\in\mathbb{Z}^{L},$ is called _Følner sequence_ in
$\displaystyle\mathbb{Z}^{L}$. we have that

(4) $\lim_{N\to\infty}\frac{1}{|I_{N}|}\sum_{n\in
I_{N}}T_{p_{1}(n)}f_{1}\cdot\ldots\cdot
T_{p_{k}(n)}f_{k}=\int_{X}f_{1}\,d\mu\cdot\ldots\cdot\int_{X}f_{k}\,d\mu,$

where the limit is taken in $\displaystyle L^{2}(\mu)$. When $\displaystyle
k=1$, we also say that $\displaystyle(T_{p_{1}(n)})_{n\in\mathbb{Z}^{L}}$ is
_ergodic_ for $\displaystyle\mu$ instead.555 The main reason we change from
single-variable $\displaystyle p_{i}$’s to multi-variable ones and give the
definition in this generality is technical. More specifically, we will deal
with multi-variable integer valued polynomials, since our arguments, even for
single-variable polynomials, naturally lead to multi-variable ones (for
details, see the “dimension-increment” method, explained before Proposition
6.3).

For $\displaystyle d,L\in\mathbb{N}^{\ast}$, we say that $\displaystyle
q\colon\mathbb{Z}^{L}\to\mathbb{Z}^{d}$ is an _integer-valued polynomial_ if
$\displaystyle q=(q_{1},\ldots,q_{d}),$ where each $\displaystyle q_{i}$ is an
integer polynomial (meaning that it takes integer values at integers) of
$\displaystyle L$ variables. The polynomial $\displaystyle q$ is _non-
constant_ if some $\displaystyle q_{i}$ is non-constant. A family of
polynomials $\displaystyle
p_{1},\dots,p_{k}\colon\mathbb{Z}^{L}\to\mathbb{Z}^{d}$ is _non-degenerate_ if
they are _essentially non-constant_ (meaning that each $\displaystyle p_{i}$
is not a constant polynomial) and _essentially distinct_ (meaning that
$\displaystyle p_{i}-p_{j}$ is essentially non-constant for all $\displaystyle
1\leq i,j\leq k,i\neq j$).666 Throughout this paper, when we write “a
polynomial $\displaystyle p\colon\mathbb{Z}^{L}\to\mathbb{Z}^{d}$”, we
implicitly assume that $\displaystyle p$ is integer-valued, hence, in general,
$\displaystyle p$ has rational coefficients. Using this new language, it
follows from [6] that if $\displaystyle T$ is weakly mixing and $\displaystyle
p_{1},\dots,p_{k}\colon\mathbb{Z}\to\mathbb{Z}$ is a non-degenerate family of
polynomials, then
$\displaystyle(T^{p_{1}(n)},\dots,T^{p_{k}(n)})_{n\in\mathbb{Z}}$ is jointly
ergodic for $\displaystyle\mu$. Later, it was proved by Frantzikinakis and Kra
(in [14]) that if $\displaystyle
p_{1},\dots,p_{k}\colon\mathbb{Z}\to\mathbb{Z}$ is an independent family of
polynomials (_i.e._ , every linear combination along integers of the
$\displaystyle p_{i}$’s is non-constant) and $\displaystyle T$ is _totally
ergodic_ (_i.e._ , $\displaystyle T^{n}$ is ergodic for all $\displaystyle
n\in\mathbb{Z}\backslash\\{0\\}$), then the tuple
$\displaystyle(T^{p_{1}(n)},\dots,T^{p_{k}(n)})_{n\in\mathbb{Z}}$ is jointly
ergodic for $\displaystyle\mu$ (for integer part of real valued strongly
independent polynomials, see [18]). By combining existing results, we have the
following proposition:

######  Proposition 1.2.

Let $\displaystyle d,k,L\in\mathbb{N}^{\ast}$ and $\displaystyle
p_{1},\dots,p_{k}\colon\mathbb{Z}^{L}\to\mathbb{Z}^{d}$ be a non-degenerate
family of polynomials. Let
$\displaystyle(X,\mathcal{B},\mu,(T_{g})_{g\in\mathbb{Z}^{d}})$ be a
$\displaystyle\mathbb{Z}^{d}$-system such that:

  * (i)

$\displaystyle T_{g}$ is ergodic for $\displaystyle\mu$ for all $\displaystyle
g\in\mathbb{Z}^{d}\setminus\\{{\bf 0}\\}$; and

  * (ii)

$\displaystyle(T_{p_{1}(n)}\times\dots\times
T_{p_{k}(n)})_{n\in\mathbb{Z}^{L}}$ is ergodic for $\displaystyle\mu^{\otimes
k}.$777 $\displaystyle\mu^{\otimes k}$ is the product measure
$\displaystyle\mu\otimes\cdots\otimes\mu$ on $\displaystyle X^{k}.$

Then $\displaystyle(T^{p_{1}(n)},\dots,T^{p_{k}(n)})_{n\in\mathbb{Z}^{L}}$ is
jointly ergodic for $\displaystyle\mu$.

Proposition 1.2 is a direct corollary of Proposition 2.10 of [17], Theorem
10.1 of [16] (see also Theorem 2.6 below), and Theorem B of [20] (see also
Theorem 2.9 below). We leave the details of the proof to the interested
readers.

We remark that in all the aforementioned results, one needs to make rather
strong assumptions for the system, more specifically that either the
transformation is weakly mixing or that infinitely many transformations
$\displaystyle T_{g}$ are ergodic. It is then natural to ask if one can obtain
joint ergodicity results under weaker conditions, _e.g._ , assuming that only
finitely many transformations (or sequences of transformations with specific
iterates) are ergodic, and finally, if there are any cases in which the
sufficient condition is also necessary. In this direction, it is worth
mentioning two results related to our study.

Let $\displaystyle d\in\mathbb{N}^{\ast}$ and
$\displaystyle(X,\mathcal{B},\mu,T_{1},\dots,T_{d})$ be a measure preserving
system with commuting transformations.888 Here, as in the expression (2),
$\displaystyle(X,\mathcal{B},\mu,T_{1},\dots,T_{d})$ can be understood as an
abbreviation for the $\displaystyle\mathbb{Z}^{d}$-system
$\displaystyle(X,\mathcal{B},\mu,(S_{g})_{g\in\mathbb{Z}^{d}})$, where
$\displaystyle
T_{1}=S_{(1,0,\dots,0)},T_{2}=S_{(0,1,0,\dots,0)},\dots,T_{d}=S_{(0,\dots,0,1)}$.
It was proved by Berend and Bergelson (in [4]) that the tuple
$\displaystyle(T^{n}_{1},\dots,T^{n}_{d})_{n\in\mathbb{Z}}$ is jointly ergodic
for $\displaystyle\mu$ if and only if $\displaystyle T_{i}T^{-1}_{j}$ is
ergodic for $\displaystyle\mu$ for all $\displaystyle 1\leq i,j\leq d,i\neq j$
and $\displaystyle T_{1}\times\dots\times T_{d}$ is ergodic for
$\displaystyle\mu^{\otimes d}$. Recently, it was proved by Bergelson, Leibman
and Son (in [8]) that if $\displaystyle
p_{1},\dots,p_{d}\colon\mathbb{Z}\to\mathbb{Z}$ are generalized linear
functions (_i.e._ , functions of the form $\displaystyle
p(n)=[\alpha_{1}n+\alpha_{2}],\;[\alpha_{3}[\alpha_{1}n+\alpha_{2}]]$, _etc._
, where $\displaystyle[\cdot]$ denotes the integer part, or floor, function),
then the tuple
$\displaystyle(T^{p_{1}(n)}_{1},\dots,T^{p_{d}(n)}_{d})_{n\in\mathbb{Z}}$ is
jointly ergodic for $\displaystyle\mu$ if and only if the sequence
$\displaystyle(T^{p_{i}(n)}_{i}T^{-p_{j}(n)}_{j})_{n\in\mathbb{Z}}$ is ergodic
for $\displaystyle\mu$ for all $\displaystyle 1\leq i,j\leq d,i\neq j$, and
the sequence $\displaystyle(T^{p_{1}(n)}_{1}\times\dots\times
T^{p_{d}(n)}_{d})_{n\in\mathbb{Z}}$ is ergodic for $\displaystyle\mu^{\otimes
d}$. Note that both results, while being characterizations, hold under only
the ergodicity assumption for finitely many transformations and sequences of
transformations.

In this paper, we study joint ergodicity properties for sequences of
transformations with polynomial iterates. The following is our first
application of Theorems 1.1 and 5.1:

######  Theorem 1.3.

Let $\displaystyle d,k,K,L\in\mathbb{N}^{\ast}$ and $\displaystyle
p_{1},\dots,p_{k}\colon\mathbb{Z}^{L}\to\mathbb{Z}^{d}$ be a non-degenerate
family of polynomials of degrees at most $\displaystyle K$. Suppose that
$\displaystyle p_{i}(n)=\sum_{v\in\mathbb{N}^{L},|v|\leq K}b_{i,v}n^{v}$ for
some $\displaystyle b_{i,v}\in\mathbb{Q}^{d}$.999 For $\displaystyle
n=(n_{1},\dots,n_{L})\in\mathbb{Z}^{L}$ and $\displaystyle
v=(v_{1},\dots,v_{L})\in\mathbb{N}^{L}$, $\displaystyle n^{v}$ denotes the
quantity $\displaystyle n^{v_{1}}_{1}\cdot\ldots\cdot n^{v_{L}}_{L}$, and
$\displaystyle|v|=v_{1}+\dots+v_{L}$. Denote the set of the coefficients and
pairwise differences of the coefficients (excluding $\displaystyle{\bf 0}$) of
the polynomials with

$\displaystyle R=\bigcup_{0<|v|\leq
K}\\{b_{i,v},b_{i,v}-b_{i^{\prime},v}\colon 1\leq i,i^{\prime}\leq
k\\}\backslash\\{{\bf 0}\\}.$

Let $\displaystyle(X,\mathcal{B},\mu,(T_{g})_{g\in\mathbb{Z}^{d}})$ be a
$\displaystyle\mathbb{Z}^{d}$-system such that:

  * (i)

For all $\displaystyle r\in R$, denoting $\displaystyle
G(r)\coloneqq\text{span}_{\mathbb{Q}}\\{r\\}\cap{\mathbb{Z}}^{d}$ (see also
the relation (7) in the corresponding definition in Subsection 2.5), the
action $\displaystyle(T_{g})_{g\in G(r)}$ is ergodic for
$\displaystyle\mu$;101010 For a subgroup $\displaystyle H$ of
$\displaystyle{\mathbb{Z}}^{d}$, $\displaystyle(T_{g})_{g\in G(r)}$ is
_ergodic_ for $\displaystyle\mu$ if every $\displaystyle A\in\mathcal{B}$
which is invariant under $\displaystyle T_{g}$ for all $\displaystyle g\in
G(r)$ is of $\displaystyle\mu$-measure $\displaystyle 0$ or $\displaystyle 1$.
and

  * (ii)

$\displaystyle(T_{p_{1}(n)}\times\dots\times
T_{p_{k}(n)})_{n\in\mathbb{Z}^{L}}$ is ergodic for $\displaystyle\mu^{\otimes
k}$.

Then $\displaystyle(T_{p_{1}(n)},\dots,T_{p_{k}(n)})_{n\in\mathbb{Z}^{L}}$ is
jointly ergodic for $\displaystyle\mu$.

We remark that Theorem 1.3 is stronger than Proposition 1.2 since we only
require finitely many $\displaystyle T_{g}$’s to be ergodic, _i.e._ , those
$\displaystyle g$’s belonging to $\displaystyle R$, and the set $\displaystyle
R$ has an explicit expression.

######  Example 1.

Let $\displaystyle(X,\mathcal{B},\mu,T_{1},T_{2})$ be a system with two
commuting transformations and assume that $\displaystyle(T^{n^{2}+n}_{1}\times
T^{n^{2}}_{2})_{n\in\mathbb{Z}}$ is ergodic for $\displaystyle\mu\times\mu$.
Then Theorem 1.3 implies that if $\displaystyle T_{1},T_{2},T_{1}T^{-1}_{2}$
are ergodic for $\displaystyle\mu$, then
$\displaystyle(T^{n^{2}+n}_{1},T^{n^{2}}_{2})_{n\in\mathbb{Z}}$ is jointly
ergodic for $\displaystyle\mu$.

Conversely, the joint ergodicity of
$\displaystyle(T^{n^{2}+n}_{1},T^{n^{2}}_{2})_{n\in\mathbb{Z}}$ implies the
ergodicity of $\displaystyle(T^{n^{2}+n}_{1})_{n\in\mathbb{Z}}$ and
$\displaystyle(T^{n^{2}}_{2})_{n\in\mathbb{Z}}$ for $\displaystyle\mu$, which
in turn implies the ergodicity of $\displaystyle T_{1}$ and $\displaystyle
T_{2}$ for $\displaystyle\mu$. However, the fact that
$\displaystyle(T^{n^{2}+n}_{1},T^{n^{2}}_{2})_{n\in\mathbb{Z}}$ is jointly
ergodic for $\displaystyle\mu$ does not necessarily imply that $\displaystyle
T_{1}T^{-1}_{2}$ is ergodic (take for instance $\displaystyle T_{1}=T_{2}=T$
where $\displaystyle T$ is a weakly mixing transformation).

Throughout this paper, Example 1 will be our main example via which we
demonstrate how our method works. Note that annoyingly enough, the expression
of the limit of the average of the sequence $\displaystyle
T_{1}^{n^{2}+n}f_{1}\cdot T_{2}^{n^{2}}f_{2}$ for bounded $\displaystyle
f_{1}$ and $\displaystyle f_{2}$ cannot be immediately found from known
results, despite the fact that the polynomials $\displaystyle
p_{1}(n)=n^{2}+n$ and $\displaystyle p_{2}(n)=n^{2}$ are essentially distinct.

The second application of Theorems 1.1 and 5.1 is the following theorem, which
provides necessary and sufficient conditions for joint ergodicity of the
polynomial sequences $\displaystyle T_{i}^{p(n)},1\leq i\leq d$. This
generalizes the result from [4] and answers a question due to Bergelson:111111
Personal communication.

######  Theorem 1.4.

Let $\displaystyle d,L\in\mathbb{N}^{\ast}$, $\displaystyle
p\colon\mathbb{Z}^{L}\to\mathbb{Z}$ be a polynomial and
$\displaystyle(X,\mathcal{B},\mu,T_{1},\dots,T_{d})$ be a system with
commuting transformations. Then
$\displaystyle(T_{1}^{p(n)},\dots,T_{d}^{p(n)})_{n\in\mathbb{Z}^{L}}$ is
jointly ergodic for $\displaystyle\mu$ if and only if both of the following
conditions are satisfied:

  * (i)

$\displaystyle T_{i}T^{-1}_{j}$ is ergodic for $\displaystyle\mu$ for all
$\displaystyle 1\leq i,j\leq d,$ $\displaystyle i\neq j$; and

  * (ii)

$\displaystyle((T_{1}\times\dots\times T_{d})^{p(n)})_{n\in\mathbb{Z}^{L}}$ is
ergodic for $\displaystyle\mu^{\otimes d}$.

As an immediate example, for a system
$\displaystyle(X,\mathcal{B},\mu,T_{1},T_{2})$ with two commuting
transformations, the sequence
$\displaystyle(T^{n^{2}}_{1},T^{n^{2}}_{2})_{n\in\mathbb{Z}}$ is jointly
ergodic for $\displaystyle\mu$ if and only if $\displaystyle T_{1}T^{-1}_{2}$
is ergodic for $\displaystyle\mu$ and $\displaystyle(T^{n^{2}}_{1}\times
T^{n^{2}}_{2})_{n\in\mathbb{Z}}$ is ergodic for $\displaystyle\mu\times\mu$.

One might wonder if there are better descriptions of condition (ii) of Theorem
1.4. In Section 3, we provide several criteria and equivalent conditions of
(ii), related to the eigenvalues of the system.

Based on the work of [4, 8] and the main results of this paper, we have a
natural conjecture:

######  Conjecture 1.5.

Let $\displaystyle d,k,L\in\mathbb{N}^{\ast}$, $\displaystyle
p_{1},\dots,p_{k}\colon\mathbb{Z}^{L}\to\mathbb{Z}^{d}$ be polynomials and
$\displaystyle(X,\mathcal{B},\mu,(T_{g})_{g\in\mathbb{Z}^{d}})$ be a
$\displaystyle\mathbb{Z}^{d}$-system. Then
$\displaystyle(T_{p_{1}(n)},\dots,T_{p_{k}(n)})_{n\in\mathbb{Z}^{L}}$ is
jointly ergodic for $\displaystyle\mu$ if and only if both of the following
conditions are satisfied:

  * (i)

$\displaystyle(T_{p_{i}(n)-p_{j}(n)})_{n\in\mathbb{Z}^{L}}$ is ergodic for
$\displaystyle\mu$ for all $\displaystyle 1\leq i,j\leq k,$ $\displaystyle
i\neq j$; and

  * (ii)

$\displaystyle(T_{p_{1}(n)}\times\dots\times
T_{p_{k}(n)})_{n\in\mathbb{Z}^{L}}$ is ergodic for $\displaystyle\mu^{\otimes
k}$.

###  1.3. Method and Organization

Section 2 contains all the background material and Section 3 the conditions
equivalent to (ii) of Theorem 1.4 (see Proposition 3.2).

In order to prove the joint ergodicity results of this paper, we introduce a
characterization theorem (Theorem 5.1, the stronger version of Theorem 1.1) in
Section 5, which allows us to study joint ergodicity properties under the
assumption that all the functions $\displaystyle f_{1},\dots,f_{k}$ are
measurable with respect to certain Host-Kra characteristic factors (see
Section 2 for definitions). Once Theorem 5.1 is proven, a standard argument
using results from [16, 21] (or via Theorems 2.6 and 2.9 – see below) yields
the main results of this paper. The proofs of Theorems 1.3 and 1.4, under the
assumption of the validity of Theorem 5.1, are enclosed in Section 5 as well.
In the same section, we also introduce the two main ingredients for proving
Theorem 5.1, namely Propositions 5.5 (which we prove in Section 6) and 5.6
(which we prove in Section 7).

To obtain the characterization theorem (Theorem 5.1), we employ the by now
classical “PET induction” (first introduced in [6]), which allows us to
convert the average in (4) to a special case where every $\displaystyle
p_{i}(n)$ is a linear function by repeatedly applying the van der Corput lemma
(Lemma 2.2). Adaptations of this method have been extensively studied in the
past in [9, 17, 21] too. We explain it in detail in Section 4 tailored for our
purposes.

There are two major difficulties to carry out the PET induction in proving
Theorem 5.1 though. The first is that although PET induction variations used
in the past allow us to eventually reduce the left hand side of (4) to an
expression with linear iterates, they provide no information on the
coefficients of these iterates, which is a crucial detail in describing the
set $\displaystyle R$ defined in Theorem 1.3. To overcome this difficulty, we
introduce a new alteration of this technique in Section 6 (see the proof of
Proposition 5.5) which allows us to keep track of the coefficients of the
polynomials when we iteratively apply van der Corput (vdC) operations.

The second, and perhaps the most important problem is how to bound the left
hand side of (4) by some Host-Kra-type seminorm of each function
$\displaystyle f_{i}$. It turns out that for a general non-degenerate family
of polynomials $\displaystyle
p_{1},\dots,p_{k}\colon\mathbb{Z}^{L}\to\mathbb{Z}^{d},$ we can use the PET
induction to bound the left hand side of (4) by an averaged Host-Kra seminorm,
as the righthand sides of (21) and (23) (see Section 5). The problem-goal now
is to bound such an averaged seminorm effectively by a single one. In the
past, in analogous situations, issues like these were resolved under
additional restrictions, such as the assumption that $\displaystyle d=1$
([6]), that all $\displaystyle T_{g}$’s are ergodic ([14, 17]), or that
$\displaystyle p_{1},\dots,p_{k}$ have different (and positive) degrees ([9]).
In this paper, we address this difficulty in Section 7 (see the proof of
Proposition 5.6) in its full generality. Our method is based on the recent
work of Tao and Ziegler on the concatenation theorem ([24]).

###  Acknowledgements.

We thank Vitaly Bergelson for bringing the problem that we are addressing in
Theorem 1.4 to our attention, and also for providing useful advices. The
second author thanks the hospitality of the Center for Mathematical Modeling
(CMM) of the University of Chile, where this work started. The first author
thanks the hospitality of the Ohio State University, where this work was
finished.

###  1.4. Definitions and notations

We denote with $\displaystyle\mathbb{N}^{\ast},$ $\displaystyle\mathbb{N},$
$\displaystyle\mathbb{Z},$ $\displaystyle\mathbb{Q}$ and
$\displaystyle\mathbb{R}$ the set of positive integers, non-negative integers,
integers, rational numbers and real numbers, respectively. If $\displaystyle
X$ is a set, and $\displaystyle d\in\mathbb{N}^{\ast}$, $\displaystyle X^{d}$
denotes the Cartesian product $\displaystyle X\times\cdots\times X$ of
$\displaystyle d$ copies of $\displaystyle X$.

We say that a tuple
$\displaystyle(X,\mathcal{B},\mu,(T_{g})_{g\in{\mathbb{Z}}^{d}})$ is a
_$\displaystyle{\mathbb{Z}}^{d}$ -measure preserving system_ (or a
_$\displaystyle{\mathbb{Z}}^{d}$ -system_) if
$\displaystyle(X,\mathcal{B},\mu)$ is a probability space and $\displaystyle
T_{g}\colon X\to X$ are measurable, measure preserving transformations on
$\displaystyle X$ such that $\displaystyle T_{(0,\dots,0)}=id$ and
$\displaystyle T_{g}\circ T_{h}=T_{gh}$ for all $\displaystyle
g,h\in{\mathbb{Z}}^{d}$. The system is _ergodic_ if for any $\displaystyle
A\in\mathcal{B}$ such that $\displaystyle T_{g}A=A$ for all $\displaystyle
g\in{\mathbb{Z}}^{d}$, we have that $\displaystyle\mu(A)\in\\{0,1\\}$.

We say that $\displaystyle(Y,\mathcal{D},\nu,(S_{g})_{g\in{\mathbb{Z}}^{d}})$
is a _factor_ of
$\displaystyle(X,\mathcal{B},\mu,(T_{g})_{g\in{\mathbb{Z}}^{d}})$ if there
exists a measurable map
$\displaystyle\pi\colon(X,\mathcal{B},\mu)\to(Y,\mathcal{D},\nu)$ such that
$\displaystyle\mu(\pi^{-1}(A))=\nu(A)$ for all $\displaystyle
A\in\mathcal{D}$, and that $\displaystyle\pi\circ T_{g}=S_{g}\circ\pi$ for all
$\displaystyle g\in{\mathbb{Z}}^{d}$. A factor
$\displaystyle(Y,\mathcal{D},\nu,(S_{g})_{g\in{\mathbb{Z}}^{d}})$ of
$\displaystyle(X,\mathcal{B},\mu,(T_{g})_{g\in{\mathbb{Z}}^{d}})$ can be
identified as a sub-$\displaystyle\sigma$-algebra
$\displaystyle\mathcal{B}^{\prime}$ of $\displaystyle\mathcal{B}$ or a
subspace $\displaystyle V$ of $\displaystyle L^{2}(\mu)$ by setting
$\displaystyle\mathcal{B}^{\prime}\coloneqq\pi^{-1}(\mathcal{D})$ or
$\displaystyle V\coloneqq L^{2}(\nu)\circ\pi$. Given two
$\displaystyle\sigma$-algebras $\displaystyle\mathcal{B}_{1}$ and
$\displaystyle\mathcal{B}_{2}$, their _joining_
$\displaystyle\mathcal{B}_{1}\vee\mathcal{B}_{2}$ is the
$\displaystyle\sigma$-algebra generated by $\displaystyle B_{1}\cap B_{2}$ for
all $\displaystyle B_{1}\in\mathcal{B}_{1}$ and $\displaystyle
B_{2}\in\mathcal{B}_{2}$, _i.e._ , the smallest $\displaystyle\sigma$-algebra
containing both $\displaystyle\mathcal{B}_{1}$ and
$\displaystyle\mathcal{B}_{2}$. This definition extends to a countable
collection of $\displaystyle\sigma$-algebras $\displaystyle\mathcal{B}_{i}$,
$\displaystyle i\in{\mathbb{N}}$ and we denote it by
$\displaystyle\bigvee_{i=0}^{\infty}\mathcal{B}_{i}.$

We will denote with $\displaystyle e_{i}$ the vector which has $\displaystyle
1$ as its $\displaystyle i$th coordinate and $\displaystyle 0$’s elsewhere. We
use in general lower-case letters to symbolize both numbers and vectors but
bold letters to symbolize vectors of vectors to highlight this exact fact, in
order to make the content more reader-friendly. The only exception to this
convention is the vector $\displaystyle{\bf 0}$ (_i.e._ , the vector with
coordinates only $\displaystyle 0$’s) which we always symbolize in bold.

####  1.4.1. Notation on averaging

Throughout this article, we use the following notations about averages. Let
$\displaystyle(a(n))_{n\in{\mathbb{Z}}^{L}}$ be a sequence of real numbers, or
a sequence of measurable functions on a probability space
$\displaystyle(X,\mathcal{B},\mu)$. Denote

$\displaystyle\displaystyle\mathbb{E}_{n\in A}a(n)$
$\displaystyle\displaystyle\coloneqq\frac{1}{|A|}\sum_{n\in
A}a({n}),\;\;\text{ where A is a finite subset of}\;\;\mathbb{Z}^{L},$
$\displaystyle\displaystyle\overline{\mathbb{E}}^{\square}_{n\in{\mathbb{Z}}^{L}}a(n)$
$\displaystyle\displaystyle\coloneqq\varlimsup_{N\to\infty}\mathbb{E}_{n\in[-N,N]^{L}}a(n),\text{}$
$\displaystyle\displaystyle\overline{\mathbb{E}}_{n\in\mathbb{Z}^{L}}a(n)$
$\displaystyle\displaystyle\coloneqq\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\
\text{ F\o lner seq.}\end{subarray}}\varlimsup_{N\to\infty}\mathbb{E}_{n\in
I_{N}}a(n),$
$\displaystyle\displaystyle\mathbb{E}^{\square}_{n\in{\mathbb{Z}}^{L}}a(n)$
$\displaystyle\displaystyle\coloneqq\lim_{N\to\infty}\mathbb{E}_{n\in[-N,N]^{L}}a(n)\;\;\text{
(provided that the limit exists)},$
$\displaystyle\displaystyle\mathbb{E}_{n\in\mathbb{Z}^{L}}a(n)$
$\displaystyle\displaystyle\coloneqq\lim_{N\to\infty}\mathbb{E}_{n\in
I_{N}}a(n)\text{ (provided that the limit exists for all F\o lner sequences
$\displaystyle(I_{N})_{N\in{\mathbb{N}}}$)}.$

We also consider _iterated_ averages. Let
$\displaystyle(a(h_{1},\dots,h_{s}))_{h_{1},\dots,h_{s}\in{\mathbb{Z}}^{L}}$
be a multi-parameter sequence. We denote

$\displaystyle\overline{\mathbb{E}}_{h_{1},\dots,h_{s}\in\mathbb{Z}^{L}}a(h_{1},\dots,h_{s})\coloneqq\overline{\mathbb{E}}_{h_{1}\in\mathbb{Z}^{L}}\ldots\overline{\mathbb{E}}_{h_{s}\in\mathbb{Z}^{L}}a(h_{1},\dots,h_{s})$

and adopt similar conventions for
$\displaystyle\mathbb{E}_{h_{1},\dots,h_{s}\in\mathbb{Z}^{L}}$,
$\displaystyle\overline{\mathbb{E}}^{\square}_{h_{1},\dots,h_{s}\in\mathbb{Z}^{L}}$
and $\displaystyle\mathbb{E}^{\square}_{h_{1},\dots,h_{s}\in\mathbb{Z}^{L}}$.

######  Convention.

Throughout this paper, all the limits of measurable functions on a measure
preserving system are taken in $\displaystyle L^{2}$ (unless otherwise
stated). Even though all the expressions with polynomial iterates that we will
encounter converge (in $\displaystyle L^{2}$) by [25], we don’t a priori
postulate any existence of such limits throughout the whole article.

##  2\. Background material

###  2.1. The van der Corput lemma

The main tool in reducing the complexity of polynomial families and running
the PET induction is the van der Corput lemma (and its variations), whose
original proof can be found in [6]. We state a convenient for us version that
can be easily deduced from the one in [6].

######  Lemma 2.1 ([6]).

Let $\displaystyle\mathcal{H}$ be a Hilbert space, $\displaystyle
a\colon{\mathbb{Z}}^{L}\to\mathcal{H}$ be a sequence bounded by $\displaystyle
1$, and $\displaystyle(I_{N})_{N\in{\mathbb{N}}}$ be a Følner sequence in
$\displaystyle{\mathbb{Z}}^{L}$. Then

$\displaystyle\displaystyle\varlimsup_{N\to\infty}\left\|\mathbb{E}_{n\in
I_{N}}a(n)\right\|^{2}$ $\displaystyle\displaystyle\leq
4\overline{\mathbb{E}}^{\square}_{h\in\mathbb{Z}^{L}}\varlimsup_{N\to\infty}|\mathbb{E}_{n\in
I_{N}}\langle a(n+h),a(n)\rangle|.$

We also need the following variation of Lemma 2.1:

######  Lemma 2.2.

Let $\displaystyle\mathcal{H}$ be a Hilbert space,
$\displaystyle(a(n;h_{1},\ldots,h_{s}))_{(n;h_{1},\ldots,h_{s})\in({\mathbb{Z}}^{L})^{s+1}}$131313
We use this unorthodox notation to separate the variable $\displaystyle n$
from the $\displaystyle h_{i}$’s. The variable $\displaystyle n$ plays a
different role later. be a sequence bounded by $\displaystyle 1$ in
$\displaystyle\mathcal{H}$, and $\displaystyle(I_{N})_{N\in{\mathbb{N}}}$ be a
Følner sequence in $\displaystyle{\mathbb{Z}}^{L}$. Then for
$\displaystyle\kappa\in\mathbb{N}$,

$\displaystyle\displaystyle\overline{\mathbb{E}}^{\square}_{h_{1},\ldots,h_{s}\in{\mathbb{Z}}^{L}}\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\
\text{ F\o lner
seq.}\end{subarray}}\varlimsup_{N\to\infty}\left\|\mathbb{E}_{n\in
I_{N}}a(n;h_{1},\ldots,h_{s})\right\|^{2\kappa}$
$\displaystyle\displaystyle\leq
4^{\kappa}\overline{\mathbb{E}}^{\square}_{h_{1},\ldots,h_{s},h_{s+1}\in{\mathbb{Z}}^{L}}\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\
\text{ F\o lner
seq.}\end{subarray}}\varlimsup_{N\to\infty}\left|\mathbb{E}_{n\in
I_{N}}\left\langle
a(n+h_{s+1};h_{1},\ldots,h_{s}),a(n;h_{1},\ldots,h_{s})\right\rangle\right|^{\kappa}$

###### Proof.

For fixed $\displaystyle h_{1},\ldots,h_{s}$, we apply Lemma 2.1 for
$\displaystyle a(n)=a(n;h_{1},\ldots,h_{s})$ and $\displaystyle h=h_{s+1}.$ By
Jensen’s inequality, we have

$\begin{split}&\quad\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\
\text{ F\o lner
seq.}\end{subarray}}\varlimsup_{N\to\infty}\left\|\mathbb{E}_{n\in
I_{N}}a(n;h_{1},\ldots,h_{s})\right\|^{2\kappa}\\\ &\leq
4^{\kappa}\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\ \text{ F\o lner
seq.}\end{subarray}}\left(\overline{\mathbb{E}}^{\square}_{h_{s+1}\in{\mathbb{Z}}^{L}}\varlimsup_{N\to\infty}\left|\mathbb{E}_{n\in
I_{N}}\left\langle
a(n+h_{s+1};h_{1},\ldots,h_{s}),a(n;h_{1},\ldots,h_{s})\right\rangle\right|\right)^{\kappa}\\\
&\leq 4^{\kappa}\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\ \text{
F\o lner
seq.}\end{subarray}}\overline{\mathbb{E}}^{\square}_{h_{s+1}\in{\mathbb{Z}}^{L}}\varlimsup_{N\to\infty}\left|\mathbb{E}_{n\in
I_{N}}\left\langle
a(n+h_{s+1};h_{1},\ldots,h_{s}),a(n;h_{1},\ldots,h_{s})\right\rangle\right|^{\kappa}\\\
&\leq
4^{\kappa}\overline{\mathbb{E}}^{\square}_{h_{s+1}\in{\mathbb{Z}}^{L}}\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\
\text{ F\o lner
seq.}\end{subarray}}\varlimsup_{N\to\infty}\left|\mathbb{E}_{n\in
I_{N}}\left\langle
a(n+h_{s+1};h_{1},\ldots,h_{s}),a(n;h_{1},\ldots,h_{s})\right\rangle\right|^{\kappa}.\end{split}$

The conclusion follows by taking the limsup of the averages over
$\displaystyle h_{s},\ldots,h_{1}.$ ∎

###  2.2. Host-Kra characteristic factors

The use of Host-Kra characteristic factors is a fundamental tool in studying
problems related to multiple averages. They were first introduced in [16] for
ergodic $\displaystyle{\mathbb{Z}}$-systems (see also [27]) and later for
$\displaystyle{\mathbb{Z}}^{d}$-systems in [15]. In this paper, we need to use
a slightly more general version of Host-Kra characteristic factors, which is
similar to the one used in [22].

For a $\displaystyle{\mathbb{Z}}^{d}$-measure preserving system
$\displaystyle{\bf{X}}=(X,\mathcal{B},\mu,(T_{g})_{g\in{\mathbb{Z}}^{d}})$ and
a subgroup $\displaystyle H$ of $\displaystyle{\mathbb{Z}}^{d}$,
$\displaystyle\mathcal{I}(H)$ denotes the sub-$\displaystyle\sigma$-algebra of
$\displaystyle(T_{h})_{h\in H}$-invariant sets, _i.e._ , sets $\displaystyle
A\in\mathcal{B}$ such that $\displaystyle T_{h}A=A$ for all $\displaystyle
h\in H$. Let $\displaystyle\mathcal{A}$ be an invariant
sub-$\displaystyle\sigma$-algebra of $\displaystyle\mathcal{B}$, the measure
$\displaystyle\mu\times_{\mathcal{A}}\mu$ denotes the relative independent
product of $\displaystyle\mu$ with itself over $\displaystyle\mathcal{A}$.
That is, $\displaystyle\mu\times_{\mathcal{A}}\mu$ is the measure defined on
the product space $\displaystyle X\times X$ as

$\displaystyle\int_{X\times X}f\otimes
g~{}d(\mu\times_{\mathcal{A}}\mu)=\int_{X}\mathbb{E}(f|\mathcal{A})\mathbb{E}(g|\mathcal{A})d\mu$

for all $\displaystyle f,g\in L^{\infty}(\mu)$.

Let $\displaystyle H_{1},\dots,H_{k}$ be subgroups of
$\displaystyle{\mathbb{Z}}^{d}$. Define

$\displaystyle\mu_{H_{1}}=\mu\times_{\mathcal{I}(H_{1})}\mu$

and for $\displaystyle k>1,$

$\displaystyle\mu_{H_{1},\dots,H_{k}}=\mu_{H_{1},\dots,H_{k-1}}\times_{\mathcal{I}(H_{k}^{[k-1]})}\mu_{H_{1},\dots,H_{k-1}},$

where $\displaystyle H^{[k-1]}_{k}$ denotes the subgroup of
$\displaystyle({\mathbb{Z}}^{d})^{2^{k-1}}$ consisting of all the elements of
the form $\displaystyle h_{k}\times\dots\times h_{k}$ ($\displaystyle 2^{k-1}$
copies of $\displaystyle h_{k}$) for some $\displaystyle h_{k}\in H_{k}$.
Define the characteristic factor $\displaystyle
Z_{H_{1},\dots,H_{k}}(\mathbb{X})$ (or $\displaystyle Z_{H_{1},\dots,H_{k}}$
when there is no confusion) to be the sub-$\displaystyle\sigma$-algebra of
$\displaystyle\mathcal{B}$ such that

$\displaystyle\mathbb{E}(f|Z_{H_{1},\dots,H_{k}})=0\text{ if and only if
}\|f\|_{H_{1},\dots,H_{k}}^{2^{k}}\coloneqq\int_{X^{[k]}}f^{\otimes
2^{k}}\,d\mu_{H_{1},\dots,H_{k}}=0,$

where $\displaystyle f^{\otimes 2^{k}}=f\otimes\cdots\otimes f$ and
$\displaystyle X^{[k]}=X\times\cdots\times X$ ($\displaystyle 2^{k}$ copies of
$\displaystyle f$ and $\displaystyle X$ respectively). Similar to the proof of
Lemma 4 of [15] (or Lemma 4.3 of [16]), one can show that $\displaystyle
Z_{H_{1},\dots,H_{k}}$ is well defined. Note that when $\displaystyle k=1$,
$\displaystyle Z_{H_{1}}=\mathcal{I}(H_{1})$. When we have $\displaystyle k$
copies of $\displaystyle H,$ we write $\displaystyle Z_{H^{\times k}}\coloneqq
Z_{H,\dots,H},$ and $\displaystyle
Z_{H^{\times\infty}}\coloneqq\bigvee_{k=1}^{\infty}Z_{H^{\times k}}$.

######  Convention.

For convenience, we adopt a flexible way to write the Host-Kra characteristic
factors combining the aforementioned notation. For example, if $\displaystyle
A=\\{H_{1},H_{2}\\}$, then the notation $\displaystyle Z_{A,H_{3},H^{\times
2}_{4},(H_{i})_{i=5,6}}$ refers to $\displaystyle
Z_{H_{1},H_{2},H_{3},H_{4},H_{4},H_{5},H_{6}}$, and $\displaystyle
Z_{H_{1},H^{\times\infty}_{2},H^{\times\infty}_{3}}$ refers to
$\displaystyle\bigvee_{k=1}^{\infty}Z_{H_{1},H^{\times k}_{2},H^{\times
k}_{3}}$.141414 Or, equivalently
$\displaystyle\bigvee_{k_{1}=1}^{\infty}\bigvee_{k_{2}=1}^{\infty}Z_{H_{1},H^{\times
k_{1}}_{2},H^{\times k_{2}}_{3}}$. We adopt a similar flexibility for the
subscripts of the seminorms.

When each $\displaystyle H_{i}$ is generated by a single element
$\displaystyle g_{i}$, we write
$\displaystyle\|\cdot\|_{g_{1},\dots,g_{d}}\coloneqq\|\cdot\|_{H_{1},\dots,H_{d}}$
and $\displaystyle Z_{g_{1},\dots,g_{d}}\coloneqq Z_{H_{1},\dots,H_{d}}$ in
short.

For the rest of the section,
$\displaystyle\mathbb{X}=(X,\mathcal{B},\mu,(T_{g})_{g\in\mathbb{Z}^{d}})$
will denote, as usual, a $\displaystyle\mathbb{Z}^{d}$-system.

Let $\displaystyle H$ be a subgroup of $\displaystyle{\mathbb{Z}}^{d}$ and
$\displaystyle(a(g))_{g\in H}$ be a sequence on a Hilbert space. If for all
Følner sequences $\displaystyle(I_{N})_{N\in{\mathbb{N}}}$ of $\displaystyle
H$, the limit $\displaystyle\lim_{N\to\infty}\mathbb{E}_{g\in I_{N}}a(g)$
exists, we then use $\displaystyle\mathbb{E}_{g\in H}a(g)$ to denote this
limit.151515 The fact that the limit exists for all Følner sequences actually
implies that the limit is the same for all of them. The following theorem is
classical (see for example [10, Theorem 8.13]):

######  Theorem 2.3 (Mean ergodic theorem for
$\displaystyle\mathbb{Z}^{d}$-actions).

For every $\displaystyle f\in L^{2}(\mu)$ and every subgroup $\displaystyle H$
of $\displaystyle{\mathbb{Z}}^{d}$, the limit $\displaystyle\mathbb{E}_{g\in
H}T_{g}f$ exists in $\displaystyle L^{2}(\mu)$ and equals to
$\displaystyle\mathbb{E}(f|\mathcal{I}(H))$ (or
$\displaystyle\mathbb{E}(f|Z_{H})$).

The following are some basic properties of the Host-Kra seminorms:

######  Lemma 2.4.

Let $\displaystyle H_{1},\dots,H_{k},H^{\prime}$ be subgroups of
$\displaystyle{\mathbb{Z}}^{d}$ and $\displaystyle f\in L^{\infty}(\mu)$.

  * (i)

For every permutation
$\displaystyle\sigma\colon\\{1,\dots,k\\}\to\\{1,\dots,k\\}$, we have that

$\displaystyle
Z_{H_{1},\dots,H_{k}}(\mathbb{X})=Z_{H_{\sigma(1)},\dots,H_{\sigma(k)}}(\mathbb{X}).$

  * (ii)

If $\displaystyle\mathcal{I}(H_{j})=\mathcal{I}(H^{\prime})$, then
$\displaystyle
Z_{H_{1},\dots,H_{j},\dots,H_{k}}(\mathbb{X})=Z_{H_{1},\dots,H_{j-1},H^{\prime},H_{j+1},\dots,H_{k}}(\mathbb{X})$

  * (iii)

For $\displaystyle k\geq 2$ we have that

$\displaystyle\|f\|^{2^{k}}_{H_{1},\dots,H_{k}}=\mathbb{E}_{g\in
H_{k}}\Bigl{\|}f\cdot T_{g}f\Bigr{\|}^{2^{k-1}}_{H_{1},\dots,H_{k-1}},$

while for $\displaystyle k=1,$

$\displaystyle\|f\|^{2}_{H_{1}}=\mathbb{E}_{g\in H_{1}}\int_{X}f\cdot
T_{g}f\,d\mu.$

  * (iv)

Let $\displaystyle k\geq 2$. If $\displaystyle H^{\prime}\leq H_{j}$ is of
finite index, then

$\displaystyle
Z_{H_{1},\dots,H_{j},\dots,H_{k}}(\mathbb{X})=Z_{H_{1},\dots,H_{j-1},H^{\prime},H_{j+1},\dots,H_{k}}(\mathbb{X}).$

  * (v)

If $\displaystyle H^{\prime}\leq H_{j}$, then $\displaystyle
Z_{H_{1},\dots,H_{j},\dots,H_{k}}(\mathbb{X})\subseteq
Z_{H_{1},\dots,H_{j-1},H^{\prime},H_{j+1},\dots,H_{k}}(\mathbb{X})$.

  * (vi)

For $\displaystyle k\geq 1$,
$\displaystyle\|f\|_{H_{1},\ldots,H_{k-1}}\leq\|f\|_{H_{1},\ldots,H_{k-1},H_{k}}$
and thus $\displaystyle Z_{H_{1},\ldots,H_{k-1}}(\mathbb{X})\subseteq
Z_{H_{1},\ldots,H_{k-1},H_{k}}(\mathbb{X}).$

  * (vii)

For $\displaystyle k\geq 1$, if $\displaystyle
H_{1}^{\prime},\ldots,H_{k}^{\prime}$ are subgroups of
$\displaystyle{\mathbb{Z}}^{d}$, then $\displaystyle
Z_{H_{1},\ldots,H_{k}}(\mathbb{X})\vee
Z_{H_{1}^{\prime},\ldots,H_{k}^{\prime}}(\mathbb{X})\subseteq
Z_{H_{1}^{\prime},\ldots,H_{k}^{\prime},H_{1},\ldots,H_{k}}(\mathbb{X}).$

###### Proof.

(i) and (ii) follow from [22, Lemma 2.2] (for (i), see also [15]).

To show (iii), if $\displaystyle k\geq 2$, then

$\displaystyle\displaystyle\|f\|^{2^{k}}_{H_{1},\dots,H_{k}}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\int_{X^{[k]}}f^{\otimes
2^{k}}\,d\mu_{H_{1},\dots,H_{k}}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\int_{X^{[k-1]}}f^{\otimes
2^{k-1}}\cdot\mathbb{E}(f^{\otimes
2^{k-1}}|\mathcal{I}(H^{[k-1]}_{k}))\,d\mu_{H_{1},\dots,H_{k-1}}$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle\mathbb{E}_{g\in
H_{k}}\int_{X^{[d-1]}}f^{\otimes 2^{k-1}}\cdot(T_{g}f)^{\otimes
2^{k-1}}\,d\mu_{H_{1},\dots,H_{k-1}}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\mathbb{E}_{g\in H_{k}}\Bigl{\|}f\cdot
T_{g}f\Bigr{\|}^{2^{k-1}}_{\mathbb{X},H_{1},\dots,H_{k-1}},$

where we invoke the mean ergodic theorem (Theorem 2.3) in the third equality.
Similarly, for $\displaystyle k=1$,

$\begin{split}\|f\|^{2}_{H_{1}}=\int_{X^{2}}f\otimes
f\,d\mu_{H_{1}}=\int_{X}f\cdot\mathbb{E}(f|\mathcal{I}(H_{1}))\,d\mu=\mathbb{E}_{g\in
H_{1}}\int_{X}f\cdot T_{g}f\,d\mu.\end{split}$

We now prove (iv). By (i), we may assume without loss of generality that
$\displaystyle j=k$. Suppose that $\displaystyle
H_{k}=\sqcup_{i=1}^{l}g_{i}H^{\prime}$ for some $\displaystyle l>0$ and
$\displaystyle g_{i}\in{\mathbb{Z}}^{d},1\leq i\leq l$. We may assume that
$\displaystyle g_{1}=e_{G}$. Let $\displaystyle(I_{N})_{N\in\mathbb{N}}$ be
any Følner sequence on $\displaystyle H^{\prime}$. We claim that
$\displaystyle(I_{N}\cdot\\{g_{1},\dots,g_{l}\\})_{N\in\mathbb{N}}$ is a
Følner sequence in $\displaystyle H_{k}$. Indeed, by the elementary inclusion
$\displaystyle(A\cup B)\triangle C\subseteq(A\triangle C)\cup(B\triangle C)$
it follows that 161616 We use multiplicative notation for convenience.

$\displaystyle\displaystyle(I_{N}\cdot\\{g_{1},\dots,g_{l}\\})\triangle
g(I_{N}\cdot\\{g_{1},\dots,g_{l}\\})\subseteq\bigcup_{1\leq i,j\leq
l}I_{N}g_{i}\triangle gI_{N}g_{j}=\bigcup_{1\leq i,j\leq l}g_{i}I_{N}\triangle
g_{j}gI_{N},$

and since $\displaystyle|I_{N}|^{-1}\cdot|g_{i}I_{N}\triangle
g_{j}gI_{N}|=|I_{N}|^{-1}\cdot|I_{N}\triangle(g_{i}^{-1}g_{j}g)I_{N}|\to 0$ as
$\displaystyle N\to\infty,$ the claim follows.

By (iii), we have that

(5) $\begin{split}\|f\|^{2^{k}}_{H_{1},\dots,H_{k-1},H_{k}}&=\mathbb{E}_{g\in
H_{k}}\Bigl{\|}f\cdot T_{g}f\Bigr{\|}^{2^{k-1}}_{H_{1},\dots,H_{k-1}}\\\
&=\lim_{N\to\infty}\frac{1}{l|I_{N}|}\sum_{i=1}^{l}\sum_{g\in
I_{N}}\Bigl{\|}f\cdot T_{g_{i}g}f\Bigr{\|}^{2^{k-1}}_{H_{1},\dots,H_{k-1}}\\\
&\geq\lim_{N\to\infty}\frac{1}{l|I_{N}|}\sum_{g\in I_{N}}\Bigl{\|}f\cdot
T_{g}f\Bigr{\|}^{2^{k-1}}_{H_{1},\dots,H_{k-1}}\\\
&=\frac{1}{l}\|f\|^{2^{k}}_{H_{1},\dots,H_{k-1},H^{\prime}}.\end{split}$

On the other hand, since $\displaystyle\mathcal{I}({H}^{[k-1]}_{k})$ is a sub
$\displaystyle\sigma$-algebra of
$\displaystyle\mathcal{I}({H^{\prime}}^{[k-1]})$, by the Cauchy-Schwarz
inequality,

(6)
$\begin{split}\|f\|^{2^{k}}_{H_{1},\dots,H_{k-1},H^{\prime}}&=\int_{X^{[k]}}f^{\otimes
2^{k}}\,d\mu_{H_{1},\dots,H_{k-1},H^{\prime}}\\\ &=\int_{X^{[k-1]}}f^{\otimes
2^{k-1}}\cdot\mathbb{E}(f^{\otimes
2^{k-1}}|\mathcal{I}({H^{\prime}}^{[k-1]}))\,d\mu_{H_{1},\dots,H_{k-1}}\\\
&=\int_{X^{[k-1]}}\Bigl{|}\mathbb{E}(f^{\otimes
2^{k-1}}|\mathcal{I}({H^{\prime}}^{[k-1]}))\Bigr{|}^{2}\,d\mu_{H_{1},\dots,H_{k-1}}\\\
&\geq\int_{X^{[k-1]}}\Bigl{|}\mathbb{E}(f^{\otimes
2^{k-1}}|\mathcal{I}({H}^{[k-1]}_{k}))\Bigr{|}^{2}\,d\mu_{H_{1},\dots,H_{k-1}}\\\
&=\int_{X^{[k-1]}}f^{\otimes 2^{k-1}}\cdot\mathbb{E}(f^{\otimes
2^{k-1}}|\mathcal{I}({H}^{[k-1]}_{k}))\,d\mu_{H_{1},\dots,H_{k-1}}\\\
&=\int_{X^{[k]}}f^{\otimes
2^{k}}\,d\mu_{H_{1},\dots,H_{k-1},H_{k}}=\|f\|^{2^{k}}_{H_{1},\dots,H_{k-1},H_{k}}.\end{split}$

Therefore,
$\displaystyle\|f\|_{H_{1},\dots,H_{k-1},H_{k}}=0\Leftrightarrow\|f\|_{H_{1},\dots,H_{k-1},H^{\prime}}=0$,
and the conclusion follows.

(v) Since
$\displaystyle\|f\|^{2^{k}}_{H_{1},\dots,H_{k-1},H_{k}}\leq\|f\|^{2^{k}}_{H_{1},\dots,H_{k-1},H^{\prime}}$
by (6) whenever $\displaystyle H^{\prime}$ is a subgroup of $\displaystyle
H_{k}$, we have that $\displaystyle
Z_{H_{1},\dots,H_{k-1},H_{k}}(\mathbb{X})\subseteq
Z_{H_{1},\dots,H_{k-1},H^{\prime}}(\mathbb{X})$. So (v) follows from (i).

(vi) Similarly to (iii), and by Jensen inequality we have

$\displaystyle\displaystyle\|f\|^{2^{k}}_{H_{1},\dots,H_{k-1},H_{k}}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\int_{X^{[k]}}f^{\otimes
2^{k}}\,d\mu_{H_{1},\dots,H_{k}}$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\int_{X^{[k-1]}}\mathbb{E}(f^{\otimes
2^{k-1}}|\mathcal{I}(H^{[k-1]}_{k}))^{2}\,d\mu_{H_{1},\dots,H_{k-1}}$
$\displaystyle\displaystyle\geq$
$\displaystyle\displaystyle\Bigl{(}\int_{X^{[k-1]}}\mathbb{E}(f^{\otimes
2^{k-1}}|\mathcal{I}(H^{[k-1]}_{k}))\,d\mu_{H_{1},\dots,H_{k-1}}\Bigr{)}^{2}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\Bigl{(}\int_{X^{[k-1]}}f^{\otimes
2^{k-1}}\,d\mu_{H_{1},\dots,H_{k-1}}\Bigr{)}^{2}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\|f\|_{H_{1},\ldots,H_{k-1}}^{2^{k}},$

from where the conclusion follows.

(vii) Applying (vi) several times, we get that both $\displaystyle
Z_{H_{1},\ldots,H_{k}}(\mathbb{X})$ and$\displaystyle
Z_{H_{1}^{\prime},\ldots,H_{k}^{\prime}}(\mathbb{X})$ are
sub-$\displaystyle\sigma$-algebras of $\displaystyle
Z_{H_{1}^{\prime},\ldots,H_{k}^{\prime},H_{1},\ldots,H_{k}}(\mathbb{X})$,
hence so is their joining.

∎

######  Remark.

We caution the reader that Lemma 2.4 (iii) is not valid for $\displaystyle
k=1$. In fact, for an ergodic $\displaystyle{\mathbb{Z}}$-system
$\displaystyle\mathbb{X}=(X,\mathcal{B},\mu,T)$ where $\displaystyle T^{2}$ is
not ergodic, we have $\displaystyle
Z_{\mathbb{Z}}(\mathbb{X})=I(\mathbb{Z})\neq
I(2\mathbb{Z})=Z_{2\mathbb{Z}}(\mathbb{X})$. The reason that (iii) fails for
$\displaystyle k=1$ is that the inequality in (5) is no longer valid since the
term $\displaystyle\Bigl{\|}f\cdot
T_{g_{i}g}f\Bigr{\|}^{2^{k-1}}_{\mathbb{X},H_{1},\dots,H_{k-1}}$ is replaced
by $\displaystyle\int_{X}f\cdot T_{g_{i}g}f\,d\mu$, which might be negative.

As an immediate corollary of Lemma 2.4 (ii), we have:

######  Corollary 2.5.

Let $\displaystyle H_{1},\dots,H_{k}$ be subgroups of
$\displaystyle{\mathbb{Z}}^{d}$. If the $\displaystyle H_{i}$-action
$\displaystyle(T_{g})_{g\in H_{i}}$ is ergodic on $\displaystyle\mathbb{X}$
for all $\displaystyle 1\leq i\leq k$, then $\displaystyle
Z_{H_{1},\dots,H_{k}}(\mathbb{X})=Z_{({{\mathbb{Z}}^{d}})^{\times
k}}(\mathbb{X})$.

###  2.3. Structure theorem and nilsystems

Let $\displaystyle X=N/\Gamma$, where $\displaystyle N$ is a ($\displaystyle
k$-step) nilpotent Lie group and $\displaystyle\Gamma$ is a discrete cocompact
subgroup of $\displaystyle N$. Let $\displaystyle\mathcal{B}$ be the Borel
$\displaystyle\sigma$-algebra of $\displaystyle X,$ $\displaystyle\mu$ the
Haar measure on $\displaystyle X,$ and for $\displaystyle
g\in{\mathbb{Z}}^{d},$ let $\displaystyle T_{g}\colon X\to X$ with
$\displaystyle T_{g}x=b_{g}\cdot x$ for some group homomorphism $\displaystyle
g\mapsto b_{g}$ from $\displaystyle{\mathbb{Z}}^{d}$ to $\displaystyle N$. We
say that
$\displaystyle\mathbb{X}=(X,\mathcal{B},\mu,(T_{g})_{g\in{\mathbb{Z}}^{d}})$
is a _( $\displaystyle k$-step) $\displaystyle{\mathbb{Z}}^{d}$-nilsystem_.

An important reason which makes the Host-Kra characteristic factors powerful
is their connection with nilsystems. The following is a slight generalization
of [26, Theorem 3.7] (see [22, Theorem 3.7]), which is a higher dimensional
version of Host-Kra structure theorem ([16]).

######  Theorem 2.6 (Structure theorem).

Let $\displaystyle\mathbb{X}$ be an ergodic
$\displaystyle\mathbb{Z}^{d}$-system. Then $\displaystyle
Z_{(\mathbb{Z}^{d})^{\times k}}(\mathbb{X})$ is an inverse limit of
$\displaystyle(k-1)$-step $\displaystyle\mathbb{Z}^{d}$-nilsystems.

The 1-step Host-Kra nilfactor is the _Kronecker factor_ , which is intimately
related to the spectrum of the system [16]. We say that a
non-$\displaystyle\mu$-a.e. constant function $\displaystyle f\in
L^{\infty}(\mu)$ is an _eigenfunction_ of the
$\displaystyle\mathbb{Z}^{d}$-system
$\displaystyle\mathbb{X}=(X,\mathcal{B},\mu,(T_{g})_{g\in\mathbb{Z}^{d}})$ if
$\displaystyle T_{g}f=\lambda_{g}f$ for all $\displaystyle
g\in\mathbb{Z}^{d},$ where $\displaystyle g\mapsto\lambda_{g}$ is a group
homomorphism from $\displaystyle\mathbb{Z}^{d}$ to
$\displaystyle\mathbb{S}^{1}$. For each $\displaystyle g\in\mathbb{Z}^{d}$, we
say that $\displaystyle\lambda_{g}$ is an _eigenvalue_ of
$\displaystyle\mathbb{X}$. If $\displaystyle(X,\mathcal{B},\mu,T)$ is a
$\displaystyle\mathbb{Z}$-system, we say that a non-$\displaystyle\mu$-a.e.
constant function $\displaystyle f\in L^{\infty}(\mu)$ is an _eigenfunction_
of $\displaystyle T$ if $\displaystyle Tf=\lambda f$ for some
$\displaystyle\lambda\in\mathbb{S}^{1},$ and we say that
$\displaystyle\lambda$ is an _eigenvalue_ of $\displaystyle T$.

The _Kronecker factor_ $\displaystyle\mathcal{K}(\mathbb{X})$ of the
$\displaystyle\mathbb{Z}^{d}$-system
$\displaystyle\mathbb{X}=(X,\mathcal{B},\mu,(T_{g})_{g\in\mathbb{Z}^{d}})$ is
the sub-$\displaystyle\sigma$-algebra of $\displaystyle\mathcal{B}$ that
corresponds to the algebra of functions spanned by the eigenfunctions of
$\displaystyle\mathbb{X}$ in $\displaystyle L^{2}(\mu)$. As a special case of
Theorem 2.6, we have:

######  Lemma 2.7.

For an ergodic $\displaystyle{\mathbb{Z}}^{d}$-system
$\displaystyle\mathbb{X}$, we have that
$\displaystyle\mathcal{K}(\mathbb{X})=Z_{\mathbb{Z}^{d},\mathbb{Z}^{d}}(\mathbb{X})$.

An application of the Kronecker factor is to characterize single averages
along polynomials (for a proof of this result, see Section 2 in [5]):

######  Proposition 2.8.

Let $\displaystyle L\in\mathbb{N}^{\ast},$ $\displaystyle
p\colon\mathbb{Z}^{L}\to\mathbb{Z}$ be a non-constant polynomial,171717 We
warn the reader that this result is only valid for $\displaystyle
p\colon\mathbb{Z}^{L}\to\mathbb{Z}^{d}$ with $\displaystyle d$ equal to 1.
$\displaystyle\mathbb{X}=(X,\mathcal{B},\mu,(T_{g})_{g\in\mathbb{Z}})$ be a
$\displaystyle\mathbb{Z}$-system, and $\displaystyle f\in L^{\infty}(\mu)$. If
$\displaystyle\mathbb{E}(f|Z_{\mathbb{Z},\mathbb{Z}}(\mathbb{X}))=0$, then

$\displaystyle\mathbb{E}_{n\in\mathbb{Z}^{L}}T_{p(n)}f=0.$

We provide an alternative proof of Proposition 2.8 in Section 4 using the
language of this paper.

We conclude this subsection with the following theorem from [20], a
consequence of [20, Theorem B], which we state in a convenient form.

######  Theorem 2.9 (Leibman, [20]).

Let $\displaystyle(X=N/\Gamma,\mathcal{B},\mu,(T_{g})_{g\in{\mathbb{Z}}^{d}})$
be a nilsystem, $\displaystyle x\in X$ and $\displaystyle
p_{i}\colon{\mathbb{Z}}^{L}\to{\mathbb{Z}}^{d}$, $\displaystyle 1\leq i\leq k$
be polynomials. Then the following are equivalent:

  1. (i)

For some $\displaystyle x\in X$, the sequence
$\displaystyle(T_{p_{1}(n)}x,\ldots,T_{p_{k}(n)}x)_{n\in{\mathbb{Z}}^{L}}$ is
dense in $\displaystyle X^{k}$.

  2. (ii)

The sequence
$\displaystyle(T_{p_{1}(n)}y_{1},\ldots,T_{p_{k}(n)}y_{k})_{n\in{\mathbb{Z}}^{L}}$
is dense in $\displaystyle X^{k}$ for all $\displaystyle y_{1},\ldots,y_{k}\in
X$.

  3. (iii)

For any Følner sequence $\displaystyle(I_{N})_{N\in{\mathbb{N}}}$ in
$\displaystyle{\mathbb{Z}}^{L}$, the average $\displaystyle\mathbb{E}_{n\in
I_{N}}f_{1}(T_{p_{1}(n)}x)\cdot\ldots\cdot f_{k}(T_{p_{k}(n)}x)$ converges (in
$\displaystyle L^{2}$ and $\displaystyle\mu$-a.e.) to
$\displaystyle\prod_{i=1}^{k}\int f_{i}d\mu$ as $\displaystyle N\to\infty$.

  4. (iv)

The sequence
$\displaystyle(T_{p_{1}(n)},\ldots,T_{p_{k}(n)})_{n\in{\mathbb{Z}}^{L}}$ is
ergodic for $\displaystyle\mu^{\otimes k}$.

###  2.4. Concatenation theorem

An essential ingredient in our approach is the concatenation theorem
established by Tao and Ziegler (in [24]), which studies the properties of
intersections of different characteristic factors:

######  Theorem 2.10 (Concatenation theorem, [24, Theorem 1.15]).

Let $\displaystyle\mathbb{X}$ be a $\displaystyle{\mathbb{Z}}^{d}$-system,
$\displaystyle k,k^{\prime}\in\mathbb{N}^{\ast}$ and $\displaystyle
H_{1},\dots,H_{k},H^{\prime}_{1},\dots,H^{\prime}_{k^{\prime}}$ subgroups of
$\displaystyle\mathbb{Z}^{d}$. Then

$\displaystyle Z_{H_{1},\dots,H_{k}}\cap
Z_{H^{\prime}_{1},\dots,H^{\prime}_{k^{\prime}}}\subseteq
Z_{(H_{i}+H^{\prime}_{i^{\prime}})_{1\leq i\leq k,1\leq i^{\prime}\leq
k^{\prime}}}.$

As an immediate corollary, we have:

######  Corollary 2.11.

Let $\displaystyle\mathbb{X}$ be a $\displaystyle{\mathbb{Z}}^{d}$-system,
$\displaystyle s,d_{1},\dots,d_{s}\in\mathbb{N}$ and $\displaystyle
H_{i,j},1\leq i\leq s,1\leq j\leq d_{i}$ subgroups of
$\displaystyle{\mathbb{Z}}^{d}$. Then

$\displaystyle\bigcap_{i=1}^{s}Z_{H_{i,1},H_{i,2},\dots,H_{i,d_{i}}}\subseteq
Z_{(H_{1,n_{1}}+H_{2,n_{2}}+\dots+H_{s,n_{s}})_{1\leq n_{i}\leq d_{i},1\leq
i\leq s}}.$

###  2.5. Range of polynomials

In this subsection we state and prove two elementary lemmas regarding the
range of polynomials.

######  Definition.

For
$\displaystyle\textbf{b}=(b_{1},\dots,b_{L})\in(\mathbb{Q}^{d})^{L},b_{i}\in\mathbb{Q}^{d}$,
we define

(7)
$G(\textbf{b})\coloneqq\text{span}_{\mathbb{Q}}\\{b_{1},\dots,b_{L}\\}\cap\mathbb{Z}^{d}.$

Note that $\displaystyle G(\textbf{b})$ can either be seen as a subgroup or a
subspace (over $\displaystyle\mathbb{Z}$) of $\displaystyle\mathbb{Z}^{d}$; we
freely use both.

######  Lemma 2.12.

Let $\displaystyle\mathbb{c}\colon(\mathbb{Z}^{L})^{s}\to(\mathbb{Q}^{d})^{L}$
be a polynomial and let $\displaystyle V$ be a subspace of
$\displaystyle\mathbb{Z}^{d}$ over $\displaystyle{\mathbb{Z}}$. Then the set

$\displaystyle\\{(h_{1},\dots,h_{s})\in(\mathbb{Z}^{L})^{s}\colon
G(\mathbb{c}(h_{1},\dots,h_{s}))\subseteq V\\}$

is either $\displaystyle(\mathbb{Z}^{L})^{s}$ or of (upper) Banach density
$\displaystyle 0$.

###### Proof.

For convenience, denote

$\displaystyle W\coloneqq\\{(h_{1},\dots,h_{s})\in(\mathbb{Z}^{L})^{s}\colon
G(\mathbb{c}(h_{1},\dots,h_{s}))\subseteq V\\},$

where one views $\displaystyle\mathbb{c}$ as the matrix:

$\displaystyle\mathbb{c}(h_{1},\dots,h_{s})=\begin{pmatrix}c_{1,1}(h_{1},\dots,h_{s})&\dots&c_{1,L}(h_{1},\dots,h_{s})\\\
\vdots&\vdots&\vdots\\\
c_{d,1}(h_{1},\dots,h_{s})&\dots&c_{d,L}(h_{1},\dots,h_{s})\end{pmatrix}$

for some polynomials $\displaystyle
c_{i,j}\colon(\mathbb{Z}^{L})^{s}\to\mathbb{Q}$, $\displaystyle 1\leq i\leq
d,1\leq j\leq L$.

We start with the case $\displaystyle V=\\{{\bf 0}\\}$. Let $\displaystyle
W_{i,j}$ be the set of
$\displaystyle(h_{1},\dots,h_{s})\in(\mathbb{Z}^{L})^{s}$ such that
$\displaystyle c_{i,j}(h_{1},\dots,h_{s})=0$. Then $\displaystyle
W=\bigcap_{i=1}^{d}\bigcap_{j=1}^{L}W_{i,j}$ and so it suffices to show that
either each $\displaystyle W_{i,j}$ is $\displaystyle({\mathbb{Z}}^{L})^{s}$
or that some $\displaystyle W_{i,j}$ is of density $\displaystyle 0.$ By
relabelling the variables, we may assume that $\displaystyle L=1$ (and change
$\displaystyle s$ to $\displaystyle Ls$). Hence, it suffices to show that for
a polynomial $\displaystyle c\colon\mathbb{Z}^{s}\to\mathbb{Z}$, the set

$\displaystyle W=\\{(h_{1},\dots,h_{s})\in(\mathbb{Z})^{s}\colon
c(h_{1},\dots,h_{s})=0\\}$

is either $\displaystyle\mathbb{Z}^{s}$ or of density $\displaystyle 0.$

If $\displaystyle s=1$, then either $\displaystyle c\equiv 0$ or
$\displaystyle c(x)=0$ has finitely many roots. So $\displaystyle W$ is either
$\displaystyle\mathbb{Z}$ or of upper Banach density $\displaystyle 0.$
Suppose now that the conclusion holds for some $\displaystyle s\geq 1$, and
assume that $\displaystyle
c(h_{1},\dots,h_{s+1})=\sum_{i=0}^{K}q_{i}(h_{2},\dots,h_{s+1})h^{i}_{1}$ for
some $\displaystyle K\in\mathbb{N}$ and polynomials $\displaystyle
q_{i}\colon\mathbb{Z}^{s}\to\mathbb{Q}$ for all $\displaystyle 0\leq i\leq K$.
Let

$\displaystyle W^{\prime}=\\{(h_{2},\dots,h_{s+1})\in\mathbb{Z}^{s}\colon
q_{i}(h_{2},\dots,h_{s+1})=0,0\leq i\leq K\\}.$

By induction hypothesis, either $\displaystyle W^{\prime}=\mathbb{Z}^{s}$ or
$\displaystyle W^{\prime}$ is of upper Banach density $\displaystyle 0.$ If
$\displaystyle W^{\prime}=\mathbb{Z}^{s}$, then $\displaystyle c\equiv 0$ and
so $\displaystyle W=\mathbb{Z}^{s+1}$. If $\displaystyle W^{\prime}$ is of
upper Banach density 0, then $\displaystyle W\subseteq W_{1}\cup W_{2}$, where
$\displaystyle W_{1}=\mathbb{Z}\times W^{\prime}$ and $\displaystyle
W_{2}=\\{(h_{1},\dots,h_{s+1})\in\mathbb{Z}^{s+1}\colon(h_{2},\dots,h_{s+1})\notin
W^{\prime},c(h_{1},\dots,h_{s+1})=0\\}.$ Since $\displaystyle W^{\prime}$ is
of upper Banach density 0, so is $\displaystyle W_{1}$. On the other hand, for
any $\displaystyle(h_{2},\dots,h_{s+1})\notin W^{\prime}$, $\displaystyle
c(\cdot,h_{2},\dots,h_{s+1})$ is not constant 0 and so has at most
$\displaystyle K$ roots. This implies that $\displaystyle W_{2}$ is of upper
Banach density 0, so $\displaystyle W$ is of density $\displaystyle 0$,
completing the induction.

Now assume that $\displaystyle V\neq\\{{\bf 0}\\}$. Since $\displaystyle V$ is
a subspace of $\displaystyle\mathbb{Z}^{d}$ over $\displaystyle{\mathbb{Z}}$,
under a change of coordinates, we may assume that $\displaystyle
V=\\{0\\}^{\ell}\times\mathbb{Z}^{d-\ell}$ for some $\displaystyle
0\leq\ell\leq d$. If $\displaystyle\ell=0$, then $\displaystyle
V={\mathbb{Z}}^{d}$ and there is nothing to prove. If $\displaystyle\ell>0$,
then by restricting to the first polynomials $\displaystyle c_{i,j},1\leq
i\leq d,1\leq j\leq\ell$, we are reduced to the case $\displaystyle V=\\{{\bf
0}\\}$, finishing the proof. ∎

######  Lemma 2.13.

Let $\displaystyle\mathbb{c}\colon(\mathbb{Z}^{L})^{s}\to(\mathbb{Q}^{d})^{L}$
be a polynomial given by

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

for some $\displaystyle\mathbb{u}(a_{1},\dots,a_{s})\in(\mathbb{Q}^{d})^{L}$
which all but finitely many equal to $\displaystyle 0.$ Then

$\displaystyle\text{span}_{\mathbb{Q}}\\{G(\mathbb{c}(h_{1},\dots,h_{s}))\colon
h_{1},\dots,h_{s}\in\mathbb{Z}^{L}\\}=\text{span}_{\mathbb{Q}}\\{G(\mathbb{u}(a_{1},\dots,a_{s}))\colon
a_{1},\dots,a_{s}\in\mathbb{N}^{L}\\}.$

For the reader’s convenience we first make the statement clear with an
example, with $\displaystyle L=2,$ $\displaystyle s=1$, $\displaystyle d=4,$
and then present the proof. Let $\displaystyle{\bf
c}\colon{\mathbb{Z}}^{2}\to({\mathbb{Z}}^{4})^{2}$ be given by

$\displaystyle{\bf c}(h_{1},h_{2})=\begin{pmatrix}h_{1}&0\\\
-3h_{1}h_{2}&h_{1}\\\ h_{1}^{2}&-h_{2}-2h_{2}^{2}\\\
7h_{1}h_{2}&h_{1}^{2}\end{pmatrix}.$

Denoting $\displaystyle h=(h_{1},h_{2}),$ we have

$\displaystyle\displaystyle{\bf c}(h_{1},h_{2})$
$\displaystyle\displaystyle=h_{1}\begin{pmatrix}1&0\\\ 0&1\\\ 0&0\\\
0&0\end{pmatrix}+h_{2}\begin{pmatrix}0&0\\\ 0&0\\\ 0&-1\\\
0&0\end{pmatrix}+h_{1}h_{2}\begin{pmatrix}0&0\\\ -3&0\\\ 0&0\\\
7&0\end{pmatrix}+h_{1}^{2}\begin{pmatrix}0&0\\\ 0&0\\\ 1&0\\\
0&1\end{pmatrix}+h_{2}^{2}\begin{pmatrix}0&0\\\ 0&0\\\ 0&-2\\\
0&0\end{pmatrix}$ $\displaystyle\displaystyle=h^{(1,0)}\begin{pmatrix}1&0\\\
0&1\\\ 0&0\\\ 0&0\end{pmatrix}+h^{(0,1)}\begin{pmatrix}0&0\\\ 0&0\\\ 0&-1\\\
0&0\end{pmatrix}+h^{(1,1)}\begin{pmatrix}0&0\\\ -3&0\\\ 0&0\\\
7&0\end{pmatrix}+h^{(2,0)}\begin{pmatrix}0&0\\\ 0&0\\\ 1&0\\\
0&1\end{pmatrix}+h^{(0,2)}\begin{pmatrix}0&0\\\ 0&0\\\ 0&-2\\\
0&0\end{pmatrix}$ $\displaystyle\displaystyle=h^{(1,0)}{\bf
u}(1,0)+h^{(0,1)}{\bf u}(0,1)+h^{(1,1)}{\bf u}(1,1)+h^{(2,0)}{\bf
u}(2,0)+h^{(0,2)}{\bf u}(0,2),$

where the $\displaystyle{\bf u}(i,j)$ denote the corresponding matrices from
the previous step.

Lemma 2.13 establishes that the span of the columns of $\displaystyle{\bf
c}(h_{1},h_{2})$ (for all $\displaystyle h_{1},h_{2}\in\mathbb{Z}$) equals to
the span of the columns of the $\displaystyle\mathbb{u}(a_{1},a_{2})$ (for all
$\displaystyle a_{1},a_{2}\in\mathbb{N}$). More explicitly, it states that

$\displaystyle\text{span}_{{\mathbb{Q}}}\left\\{\begin{pmatrix}h_{1}\\\
-3h_{1}h_{2}\\\ h_{1}^{2}\\\ 7h_{1}h_{2}\end{pmatrix},\begin{pmatrix}0\\\
h_{1}\\\ -h_{2}-2h_{2}^{2}\\\
h_{1}^{2}\end{pmatrix}:h_{1},h_{2}\in{\mathbb{Z}}\right\\}$

equals to

$\displaystyle\text{span}_{{\mathbb{Q}}}\left\\{\begin{pmatrix}1\\\ 0\\\ 0\\\
0\end{pmatrix},\begin{pmatrix}0\\\ 1\\\ 0\\\
0\end{pmatrix},\begin{pmatrix}0\\\ 0\\\ -1\\\
0\end{pmatrix},\begin{pmatrix}0\\\ -3\\\ 0\\\
7\end{pmatrix},\begin{pmatrix}0\\\ 0\\\ 1\\\
0\end{pmatrix},\begin{pmatrix}0\\\ 0\\\ 0\\\
1\end{pmatrix},\begin{pmatrix}0\\\ 0\\\ -2\\\ 0\end{pmatrix}\right\\}.$

###### Proof of Lemma 2.13.

We first assume that $\displaystyle L=1$. In this case, we have that

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

for $\displaystyle h_{1},\dots,h_{s}\in{\mathbb{Z}}$ and some $\displaystyle
u(a_{1},\dots,a_{s})\in\mathbb{Q}^{d}$. It suffices to show that

$\displaystyle\text{span}_{\mathbb{Q}}\\{c(h_{1},\dots,h_{s})\colon
h_{1},\dots,h_{s}\in\mathbb{Z}\\}=\text{span}_{\mathbb{Q}}\\{u(a_{1},\dots,a_{s})\colon
a_{1},\dots,a_{s}\in\mathbb{N}\\}.$

Since $\displaystyle c(h_{1},\ldots,h_{s})$ belongs to the
$\displaystyle{\mathbb{Q}}$-span of
$\displaystyle\\{u(a_{1},\ldots,a_{s})\\}_{a_{1},\ldots,a_{s}\in{\mathbb{N}}},$
the inclusion “$\displaystyle\subseteq$” is straightforward. We then show the
“$\displaystyle\supseteq$” direction. When $\displaystyle s=1,$ we have that
$\displaystyle c(h_{1})=\sum_{i=0}^{K}h^{i}_{1}u(i)$ for some $\displaystyle
K\in\mathbb{N}$. Since the matrix $\displaystyle(j^{i})_{0\leq i,j\leq
K},$202020 Recall that we set $\displaystyle 0^{0}\coloneqq 1.$ is (the
transpose of) a Vandermonde matrix, its determinant is non-zero, so each
$\displaystyle u(i)$ is a linear combination of $\displaystyle
c(0),\dots,c(K)$. Therefore, the conclusion holds for $\displaystyle s=1$.

We now suppose that the conclusion holds for some $\displaystyle s\geq 1$ and
we prove it for $\displaystyle s+1$. Write

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

for some polynomials $\displaystyle
v_{i}\colon\mathbb{Z}^{s}\to\mathbb{Q}^{d}$ given by

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

Since the conclusion holds for $\displaystyle s=1,$ we have that for all
$\displaystyle h_{1},\dots,h_{s}\in\mathbb{Z}$ and $\displaystyle
i\in\mathbb{N}$, $\displaystyle
v_{i}(h_{1},\dots,h_{s})\in\text{span}_{\mathbb{Q}}\\{c(h_{1},\ldots,h_{s},h_{s+1}):\;h_{s+1}\in\mathbb{Z}\\}$.
Applying the induction hypothesis for $\displaystyle s$, we have that

$\displaystyle
u(a_{1},\dots,a_{s},i)\in\text{span}_{\mathbb{Q}}\\{v_{i}(h_{1},\dots,h_{s})\colon
h_{1},\dots,h_{s}\in\mathbb{Z}\\}$

for all $\displaystyle a_{1},\dots,a_{s},i\in\mathbb{N},$ hence the conclusion
holds for $\displaystyle s+1$. By induction, the $\displaystyle L=1$ case is
complete.

For the general case, suppose that
$\displaystyle\mathbb{c}(h_{1},\dots,h_{s})=(c_{1}(h_{1},\dots,h_{s}),\dots,c_{L}(h_{1},\dots,h_{s}))$
and $\displaystyle\mathbb{u}(a_{1},\dots,a_{s})$
$\displaystyle=(u_{1}(a_{1},\dots,a_{s}),\dots,u_{L}(a_{1},\dots,a_{s}))$,
where $\displaystyle c_{i}\colon(\mathbb{Z}^{L})^{s}\to\mathbb{Q}^{d}$,
$\displaystyle u_{i}\colon(\mathbb{N}^{L})^{s}\to\mathbb{Q}^{d}$,
$\displaystyle 1\leq i\leq L$. Then

(8)
$\begin{split}c_{i}(h_{1},\dots,h_{s})=\sum_{a_{1},\dots,a_{s}\in\mathbb{N}^{L}}h^{a_{1}}_{1}\dots
h^{a_{s}}_{s}\cdot u_{i}(a_{1},\dots,a_{s})\end{split}$

for all $\displaystyle 1\leq i\leq L$. By definition, one easily checks that

$\displaystyle\text{span}_{\mathbb{Q}}\\{G(\mathbb{c}(h_{1},\dots,h_{s}))\colon
h_{1},\dots,h_{s}\in\mathbb{Z}^{L}\\}=\text{span}_{\mathbb{Q}}\\{c_{i}(h_{1},\dots,h_{s})\colon
h_{1},\dots,h_{s}\in\mathbb{Z}^{L},1\leq i\leq L\\},$

and

$\displaystyle\text{span}_{\mathbb{Q}}\\{G(\mathbb{u}(a_{1},\dots,a_{s}))\colon
a_{1},\dots,a_{s}\in\mathbb{N}^{L}\\}=\text{span}_{\mathbb{Q}}\\{u_{i}(a_{1},\dots,a_{s})\colon
a_{1},\dots,a_{s}\in\mathbb{N}^{L},1\leq i\leq L\\}.\;\;\;\;\;\;\;\;\;$

So, it suffices to show that for every $\displaystyle 1\leq i\leq L$,

$\begin{split}\text{span}_{\mathbb{Q}}\\{c_{i}(h_{1},\dots,h_{s})\colon
h_{1},\dots,h_{s}\in\mathbb{Z}^{L}\\}=\text{span}_{\mathbb{Q}}\\{u_{i}(a_{1},\dots,a_{s})\colon
a_{1},\dots,a_{s}\in\mathbb{N}^{L}\\},\;\;\text{or}\end{split}$ (9)
$\begin{split}\text{span}_{\mathbb{Q}}\\{c_{i}(h)\colon
h\in\mathbb{Z}^{Ls}\\}=\text{span}_{\mathbb{Q}}\\{u_{i}(a)\colon
a\in\mathbb{N}^{Ls}\\},\;\;\;\;\;\;\end{split}$

by viewing $\displaystyle(h_{1},\dots,h_{s})$ and
$\displaystyle(a_{1},\dots,a_{s})$ as the $\displaystyle Ls$-dimensional
vectors $\displaystyle h$ and $\displaystyle a$. Rewriting (8) as

$\begin{split}c_{i}(h)=\sum_{a\in\mathbb{N}^{Ls}}h^{a}\cdot
u_{i}(a),\end{split}$

we can apply the conclusion of the case $\displaystyle L^{\prime}=1$,
$\displaystyle s^{\prime}=Ls$, $\displaystyle d^{\prime}=d$ and $\displaystyle
c_{i}\colon({\mathbb{Z}}^{L^{\prime}})^{s^{\prime}}=({\mathbb{Z}}^{L})^{s}\to(\mathbb{Z}^{d^{\prime}})^{L^{\prime}}={\mathbb{Z}}^{d}$
to show (9). This finishes the proof. ∎

##  3\. Equivalent conditions for $\displaystyle((T_{1}\times\dots\times
T_{d})^{p(n)})_{n\in\mathbb{Z}^{L}}$ being ergodic

In this short section, we provide equivalent conditions of Property (ii) in
Theorem 1.4, _i.e._ , we characterize when
$\displaystyle((T_{1}\times\dots\times T_{d})^{p(n)})_{n\in\mathbb{Z}^{L}}$ is
ergodic for $\displaystyle\mu^{\otimes d}$.

The following lemma is an implication of [12, Lemma 4.18].

######  Lemma 3.1.

Let $\displaystyle\mathbb{X}_{i}=(X_{i},\mathcal{B}_{i},\mu_{i},T_{i}),$
$\displaystyle 1\leq i\leq d$ be $\displaystyle\mathbb{Z}$-systems. The set of
eigenvalues of $\displaystyle T_{1}\times\dots\times T_{d}$ consists of all
numbers of the form $\displaystyle\prod_{i=1}^{d}\lambda_{i}$, where
$\displaystyle\lambda_{i}$ is either $\displaystyle 1$ or an eigenvalue of
$\displaystyle T_{i}$, where at least one $\displaystyle\lambda_{i}$ is an
eigenvalue.

###### Proof.

Suppose first that $\displaystyle\lambda_{i}$ is either 1 or an eigenvalue of
$\displaystyle T_{i}$ and that at least one $\displaystyle\lambda_{i}$ is an
eigenvalue. Then, for all $\displaystyle 1\leq i\leq d$, $\displaystyle
T_{i}f_{i}=\lambda_{i}f_{i}$ for some $\displaystyle f_{i}\in
L^{\infty}(\mu_{i})$, where not all $\displaystyle f_{i}$’s are
$\displaystyle\mu_{i}$-a.e. constant. Then
$\displaystyle(T_{1}\times\dots\times T_{d})(f_{1}\otimes\dots\otimes
f_{d})=\left(\prod_{i=1}^{d}\lambda_{i}\right)(f_{1}\otimes\dots\otimes
f_{d}).$ Since $\displaystyle f_{1}\otimes\dots\otimes f_{d}$ is not
$\displaystyle(\mu_{1}\times\dots\times\mu_{d})$-a.e. constant,
$\displaystyle\prod_{i=1}^{d}\lambda_{i}$ is an eigenvalue of $\displaystyle
T_{1}\times\dots\times T_{d}$.

Conversely, let $\displaystyle\lambda$ be an eigenvalue of $\displaystyle
T_{1}\times\dots\times T_{d}$ with a corresponding eigenfunction
$\displaystyle f$. By [12, Lemma 4.18], $\displaystyle
f=\sum_{n}c_{n}f_{1,n}\otimes\dots\otimes f_{d,n}$, where $\displaystyle
c_{n}\in\mathbb{C}$, $\displaystyle T_{i}f_{i,n}=\lambda_{i,n}f_{i,n}$ for
some $\displaystyle\lambda_{i,n}\in\mathbb{S}^{1}$ with
$\displaystyle\prod_{i=1}^{d}\lambda_{i,n}=\lambda$. Each
$\displaystyle\lambda_{i,n}$ is either 1 or an eigenvalue of $\displaystyle
T_{i}$. Since $\displaystyle f$ is not
$\displaystyle(\mu_{1}\times\dots\times\mu_{d})$-a.e. constant, some
$\displaystyle f_{1,n}\otimes\dots\otimes f_{d,n}$ is also not
$\displaystyle(\mu_{1}\times\dots\times\mu_{d})$-a.e. constant. For such
$\displaystyle n$, at least one of
$\displaystyle\lambda_{1,n},\dots,\lambda_{d,n}$ is an eigenvalue of
$\displaystyle T_{i}$. Note that if $\displaystyle f_{i,n}$ is
$\displaystyle\mu_{i}$-a.e. constant, then $\displaystyle\lambda_{i,n}=1$.
Otherwise $\displaystyle\lambda_{i,n}$ is an eigenvalue for $\displaystyle
T_{i}$, which finishes the proof. ∎

Let $\displaystyle p\colon\mathbb{Z}^{L}\to\mathbb{Z}$ be a polynomial and
$\displaystyle\lambda\in\mathbb{S}^{1}$. We say that $\displaystyle\lambda$ is
_uniform_ for $\displaystyle p$ if
$\displaystyle\mathbb{E}_{n\in\mathbb{Z}^{L}}\lambda^{p(n)}=0$. So,
$\displaystyle\lambda=1$ is not uniform for any integer-valued polynomial,
while by Weyl’s equidistribution theorem, every $\displaystyle\lambda=e^{2\pi
ia}$ for some $\displaystyle a\notin\mathbb{Q}$ is uniform for all integer-
valued polynomials.

The following proposition, which lists conditions equivalent to Property (ii)
of Theorem 1.4, is the main result of the section.

######  Proposition 3.2 (Conditions equivalent to (ii) of Theorem 1.4).

Let $\displaystyle(X,\mathcal{B},\mu,T_{1},\dots,T_{d})$ be a system with
commuting transformations and $\displaystyle
p\colon\mathbb{Z}^{L}\to\mathbb{Z}$ be a polynomial. The following are
equivalent:

  * (i)

$\displaystyle((T_{1}\times\dots\times T_{d})^{p(n)})_{n\in\mathbb{Z}^{L}}$ is
ergodic for $\displaystyle\mu^{\otimes d}$.

  * (ii)

Every eigenvalue of $\displaystyle T_{1}\times\dots\times T_{d}$ is uniform
for $\displaystyle p$.

  * (iii)

For every $\displaystyle 1\leq i\leq d$, if $\displaystyle\lambda_{i}$ is
either 1 or an eigenvalue of $\displaystyle T_{i}$, where at least one
$\displaystyle\lambda_{i}$ is an eigenvalue, then
$\displaystyle\prod_{i=1}^{d}\lambda_{i}$ is uniform for $\displaystyle p$.

###### Proof.

For convenience denote $\displaystyle{\bf
Y}=(Y,\mathcal{D},\nu,T)=(X^{d},\mathcal{B}^{\otimes d},\mu^{\otimes
d},T_{1}\times\dots\times T_{d})$.

(i) $\displaystyle\Rightarrow$ (ii): Suppose that $\displaystyle\lambda$ is an
eigenvalue of $\displaystyle T$. Let $\displaystyle f\in L^{\infty}(\nu)$ be a
non-$\displaystyle\nu$-a.e. constant function such that $\displaystyle
Tf=\lambda f$. By (i),

$\displaystyle
0=\mathbb{E}_{n\in\mathbb{Z}^{L}}T^{p(n)}f=\mathbb{E}_{n\in\mathbb{Z}^{L}}\lambda^{p(n)}f.$

Since $\displaystyle f$ is not $\displaystyle\nu$-a.e. constant,
$\displaystyle\mathbb{E}_{n\in\mathbb{Z}^{L}}\lambda^{p(n)}=0$ and so
$\displaystyle\lambda$ is uniform for $\displaystyle p$.

(ii) $\displaystyle\Rightarrow$ (i): It suffices to show that for all
$\displaystyle f\in L^{\infty}(\nu)$ with $\displaystyle\int_{Y}f\,d\nu=0$, we
have that

$\displaystyle\mathbb{E}_{n\in\mathbb{Z}^{L}}T^{p(n)}f=0.$

By Proposition 2.8, it follows that

$\displaystyle\mathbb{E}_{n\in\mathbb{Z}^{L}}T^{p(n)}f=\mathbb{E}_{n\in\mathbb{Z}^{L}}T^{p(n)}\mathbb{E}(f|Z_{T,T}({\bf{Y}})).$

By Lemma 2.7, we can approximate
$\displaystyle\mathbb{E}(f|Z_{T,T}({\bf{Y}}))$ in $\displaystyle L^{2}(\nu)$
by finite linear combinations of eigenfunctions of $\displaystyle T$. So, we
may assume without loss of generality that
$\displaystyle\mathbb{E}(f|Z_{T,T}({\bf{Y}}))$ itself is an eigenfunction of
$\displaystyle T$ and $\displaystyle
T\mathbb{E}(f|Z_{T,T}({\bf{Y}}))=\lambda\mathbb{E}(f|Z_{T,T}({\bf{Y}}))$.
Since $\displaystyle\lambda$ is uniform for $\displaystyle p$,

$\displaystyle\mathbb{E}_{n\in\mathbb{Z}^{L}}T^{p(n)}\mathbb{E}(f|Z_{T,T}({\bf{Y}}))=\mathbb{E}_{n\in\mathbb{Z}^{L}}\lambda^{p(n)}\mathbb{E}(f|Z_{T,T}({\bf{Y}}))=0$

and we are done.

(ii) $\displaystyle\Leftrightarrow$ (iii): This is a direct corollary of Lemma
3.1. ∎

##  4\. PET induction

This section deals and explains the PET induction scheme, which is one of the
main tools that we use in order to study expressions of the form (1) and
(2).212121 For us, PET is an abbreviation for “Polynomial Exhaustion
Technique” (PET also stands for “Polynomial Ergodic Theorem”). This technique
was introduced by Bergelson (in the now classical [6]) to study multiple
averages for essentially distinct polynomials in weakly mixing systems and
show the joint ergodicity property in that setting. His method used an
induction argument via van der Corput lemma, reformulated in his setting, to
reduce the “complexity” of the family of polynomials.

Following this pivotal work of Bergelson, variations of the initial PET
induction scheme were used to tackle more general cases, as the one in [9] to
deal with multiple, commuting $\displaystyle T_{i}$’s and “nice” families of
polynomials, and in [17] to deal with multiple, commuting, $\displaystyle
T_{i}$’s and “standard” families of multi-variable polynomials, which we
actually follow here too.

The idea is the following: one runs the van der Corput lemma (vdC-operation)
in some family of integer valued functions-sequences satisfying some special
property and gets a family also satisfying the special property but of lower
“complexity”. This allows one to run an inductive argument and arrive at a
base case. In our case the base case is when all the iterates are linear.

Of course, in all the different aforementioned cases, one has to do several
technical variations in the method. In this paper for example, an essential
detail is that whenever we talk about a polynomial with multiple variables, we
always treat the first variable as a special one (see below for more details).
Also, to the best of our knowledge, it is the first time that via the vdC-
operations, while running (the variation of) the PET induction, we track down
the coefficients of the polynomials (see Section 6), which is crucial for our
arguments.

######  Definition.

For a polynomial $\displaystyle
p(n;h_{1},\dots,h_{s})\colon(\mathbb{Z}^{L})^{s+1}\to\mathbb{Z}$, we denote
with $\displaystyle\deg(p)$ _the degree of $\displaystyle p$ with respect to
$\displaystyle n$_ (for example, for $\displaystyle s=1,L=2$, the degree of
$\displaystyle
p(n_{1},n_{2};h_{1,1},h_{1,2})=h_{1,1}h_{1,2}n_{1}^{2}+h_{1,1}^{5}n_{2}$ is
2).

For a polynomial $\displaystyle
p(n;h_{1},\dots,h_{s})=(p_{1}(n;h_{1},\dots,h_{s}),\dots,p_{d}(n;h_{1},\dots,h_{s}))\colon(\mathbb{Z}^{L})^{s+1}\to\mathbb{Z}^{d},$
we let $\displaystyle\deg(p)=\max_{1\leq i\leq d}\deg(p_{i})$ and we say that
$\displaystyle p$ is _essentially constant_ if $\displaystyle
p(n;h_{1},\dots,h_{s})$ is independent of the variable $\displaystyle n$. We
say that the polynomials $\displaystyle
p,q\colon(\mathbb{Z}^{L})^{s+1}\to\mathbb{Z}^{d}$ are _essentially distinct_
if $\displaystyle p-q$ is not essentially constant, and _essentially equal_
otherwise.

Actually, for a tuple $\displaystyle\mathbb{q}=(q_{1},\dots,q_{\ell})$ with
polynomials $\displaystyle
q_{1},\dots,q_{\ell}\colon(\mathbb{Z}^{L})^{s+1}\to\mathbb{Z}^{d}$, we let
$\displaystyle\deg(\mathbb{q})=\max_{1\leq i\leq\ell}\deg(q_{i})$. We say that
$\displaystyle\mathbb{q}$ is _non-degenerate_ if $\displaystyle
q_{1},\dots,q_{\ell}$ are all not essentially constant, and are pairwise
essentially distinct.222222 The separation between using or not bold
characters might look confusing in the beginning, it makes it clearer though
when we use both vectors and vectors of vectors of polynomials.

Fix a $\displaystyle\mathbb{Z}^{d}$-system
$\displaystyle(X,\mathcal{B},\mu,(T_{g})_{g\in\mathbb{Z}^{d}})$. Let
$\displaystyle
q_{1},\dots,q_{\ell}\colon(\mathbb{Z}^{L})^{s+1}\to\mathbb{Z}^{d}$ be
polynomials and $\displaystyle g_{1},\dots,g_{\ell}\colon$ $\displaystyle
X\times(\mathbb{Z}^{L})^{s}\to\mathbb{R}$ be functions such that each
$\displaystyle g_{m}(\cdot;h_{1},\dots,h_{s})$ is an $\displaystyle
L^{\infty}(\mu)$ function bounded by $\displaystyle 1$ for all $\displaystyle
h_{1},\dots,h_{s}\in\mathbb{Z},1\leq m\leq\ell$. For convenience, let
$\displaystyle\mathbb{q}=(q_{1},\dots,q_{\ell})$ and
$\displaystyle\mathbb{g}=(g_{1},\dots,g_{\ell})$. We call $\displaystyle
A=(L,s,\ell,\mathbb{g},\mathbb{q})$ a _PET-tuple_ , and for
$\displaystyle\kappa\in{\mathbb{N}}$ we set

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

We define $\displaystyle\deg(A)=\deg(\mathbb{q})$, and we say that
$\displaystyle A$ is _non-degenerate_ if $\displaystyle\mathbb{q}$ is non-
degenerate. For any $\displaystyle f\in L^{\infty}(\mu)$, we say that
$\displaystyle A=(L,s,\ell,\mathbb{g},\mathbb{q})$ is _standard_ for
$\displaystyle f$ if there exists $\displaystyle 1\leq m\leq\ell$ such that
$\displaystyle\deg(A)=\deg(q_{m})$ and $\displaystyle
g_{m}(x;h_{1},\dots,h_{s})=f(x)$. That is, $\displaystyle f$ appears as one of
the functions in $\displaystyle\mathbb{g}$, only depending on the first
variable, and that the polynomial acting on $\displaystyle f$ is of the
highest degree. We say $\displaystyle A=(L,s,\ell,\mathbb{g},\mathbb{q})$ is
_semi-standard_ for $\displaystyle f$ if there exists $\displaystyle 1\leq
m\leq\ell$ such that $\displaystyle g_{m}(x;h_{1},\dots,h_{s})=f(x),$ which is
similar to being standard, but we do not require the polynomial acting on
$\displaystyle f$ to be of the highest degree.

For each PET-tuple $\displaystyle A=(L,s,\ell,\mathbb{g},\mathbb{q})$ and
polynomial $\displaystyle q\colon(\mathbb{Z}^{L})^{s+1}\to\mathbb{Z}^{d}$, we
define the _vdC-operation_ , $\displaystyle\partial_{q}A$, according to the
following three steps:

Step 1: For all $\displaystyle 1\leq m\leq\ell$, let $\displaystyle
g^{\prime}_{m}=g^{\prime}_{m+\ell}=g_{m},$ and $\displaystyle
q_{1}^{\prime},\ldots,q_{2\ell}^{\prime}\colon(\mathbb{Z}^{L})^{s+2}\to\mathbb{Z}^{d}$
be polynomials defined as

$\displaystyle\displaystyle
q^{\prime}_{m}(n;h_{1},\ldots,h_{s+1})=\left\\{\begin{array}[]{ll}q_{m}(n;h_{1},\ldots,h_{s})-q(n;h_{1},\ldots,h_{s})&\;,1\leq
m\leq\ell\\\
q_{m-\ell}(n+h_{s+1};h_{1},\ldots,h_{s})-q(n;h_{1},\ldots,h_{s})&\;,\ell+1\leq
m\leq 2\ell\end{array}\right.,$

_i.e._ , we subtract the polynomial $\displaystyle q$ from the first
$\displaystyle\ell$ polynomials and for the second $\displaystyle\ell$ ones we
first shift by $\displaystyle h_{s+1}$ about the first variable and then we
subtract $\displaystyle q$.

Step 2: We remove from $\displaystyle
q^{\prime}_{1}(n;h_{1},\dots,h_{s+1}),\dots,q^{\prime}_{2\ell}(n;h_{1},\dots,h_{s+1})$
the polynomials which are essentially constant and the corresponding terms
with those as iterates (this will be justified via the use of the Cauchy-
Schwarz inequality and the fact that the functions $\displaystyle g_{m}$ are
bounded), and then put the non-essentially constant ones in groups
$\displaystyle
J_{i}=\\{q^{\prime\prime}_{i,1},\dots,q^{\prime\prime}_{i,t_{i}}\\},$
$\displaystyle 1\leq i\leq r$ for some $\displaystyle r,$ $\displaystyle
t_{i}\in\mathbb{N}^{\ast}$ such that two polynomials are essentially distinct
if and only if they belong to different groups. We now write $\displaystyle
q^{\prime\prime}_{i,j}(n;h_{1},\dots,h_{s+1})=q^{\prime\prime}_{i,1}(n;h_{1},\dots,h_{s+1})+p^{\prime\prime}_{i,j}(h_{1},\dots,h_{s+1})$
for some polynomial $\displaystyle p^{\prime\prime}_{i,j}$ for all
$\displaystyle 1\leq j\leq t_{i},$ $\displaystyle 1\leq i\leq r$. For
convenience, we also relabel $\displaystyle
g^{\prime}_{1},\dots,g^{\prime}_{2\ell}$ accordingly as $\displaystyle
g^{\prime\prime}_{i,j}$ for all $\displaystyle 1\leq j\leq t_{i},$
$\displaystyle 1\leq i\leq r$.

Step 3: For all $\displaystyle 1\leq i\leq r$, let $\displaystyle
q^{\ast}_{i}=q^{\prime\prime}_{i,1}$ and

$\displaystyle
g^{\ast}_{i}(x;h_{1},\dots,h_{s+1})=g^{\prime\prime}_{i,1}(x;h_{1},\dots,h_{s+1})\prod^{t_{i}}_{j=2}T_{p^{\prime\prime}_{i,j}(h_{1},\dots,h_{s+1})}g^{\prime\prime}_{i,j}(x;h_{1},\dots,h_{s+1}).$

Set $\displaystyle\mathbb{q}^{\ast}=(q^{\ast}_{1},\dots,q^{\ast}_{r})$,
$\displaystyle\mathbb{g}^{\ast}=(g^{\ast}_{1},\dots,g^{\ast}_{r})$ and let
this new PET-tuple be
$\displaystyle\partial_{q}A=(L,s+1,r,\mathbb{g}^{\ast},\mathbb{q}^{\ast})$.
232323 Here we abuse the notation by writing $\displaystyle\partial_{q}A$ to
denote any of such operations obtained from Step 1 to 3. Strictly speaking,
$\displaystyle\partial_{q}A$ is not uniquely defined as the order of grouping
of $\displaystyle q^{\prime}_{1},\dots,q^{\prime}_{2\ell}$ in Step 2 is
ambiguous. However, this is done without loss of generality, since the order
does not affect the value of $\displaystyle S(\partial_{q}A,\cdot)$.

In practice, the polynomial $\displaystyle q$ is some of the initial
polynomials $\displaystyle q_{1},\ldots,q_{\ell}.$ Therefore, if
$\displaystyle q=q_{t}$ for some $\displaystyle 1\leq t\leq\ell$, we write
$\displaystyle\partial_{t}A$ instead of $\displaystyle\partial_{q_{t}}A$ to
lighten the notation.

We will use the previous notation and quantifiers for the vdC-operation from
now on.

The following important proposition informs us that, modulo some power and
some constant which are unimportant for our purpose, the value of
$\displaystyle S(\cdot,\cdot)$ grows by using the vdC-operation described
above.

######  Proposition 4.1.

Let $\displaystyle(X,\mathcal{B},\mu,(T_{g})_{g\in\mathbb{Z}^{d}})$ be a
$\displaystyle\mathbb{Z}^{d}$-system, $\displaystyle
A=(L,s,\ell,\mathbb{g},\mathbb{q})$ a PET-tuple, and $\displaystyle
q\colon(\mathbb{Z}^{L})^{s+1}\to\mathbb{Z}^{d}$ a polynomial. Then
$\displaystyle\partial_{q}A$ is non-degenerate and $\displaystyle
S(A,2\kappa)\leq 4^{\kappa}S(\partial_{q}A,\kappa)$ for every
$\displaystyle\kappa\in{\mathbb{N}}$.

###### Proof.

Since in Step 2 of the vdC-operation, essentially constant polynomials are
removed and polynomials which are essentially the same are grouped together,
we have that $\displaystyle\partial_{q}A$ is non-degenerate.

On the other hand, we have that $\displaystyle S(A,2\kappa)$ equals to

$\displaystyle\displaystyle\overline{\mathbb{E}}^{\square}_{h_{1},\dots,h_{s}\in\mathbb{Z}^{L}}\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\
\text{ F\o lner
seq.}\end{subarray}}\varlimsup_{N\to\infty}\Bigl{\|}\mathbb{E}_{n\in
I_{N}}\prod_{m=1}^{\ell}T_{q_{m}(n;h_{1},\dots,h_{s})}g_{m}(x;h_{1},\dots,h_{s})\Bigr{\|}^{2\kappa}_{L^{2}(\mu)}$
$\displaystyle\displaystyle\leq$ $\displaystyle\displaystyle
4^{\kappa}\overline{\mathbb{E}}^{\square}_{h_{1},\dots,h_{s+1}\in\mathbb{Z}^{L}}\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\
\text{ F\o lner
seq.}\end{subarray}}\varlimsup_{N\to\infty}\left|\mathbb{E}_{n\in
I_{N}}\left\langle\prod_{m=1}^{l}T_{q_{m}(n;h_{1},\dots,h_{s})}g_{m}(x;h_{1},\dots,h_{s}),\right.\right.\;\;\;\text{(by
Lemma~{}\ref{lemma:iteratedVDC})}$
$\displaystyle\displaystyle\left.\left.\hskip
122.34692pt\prod_{m=1}^{l}T_{q_{m}(n+h_{s+1};h_{1},\dots,h_{s})}g_{m}(x;h_{1},\dots,h_{s})\right\rangle\right|^{\kappa}$
$\displaystyle\displaystyle=$ $\displaystyle\displaystyle
4^{\kappa}\overline{\mathbb{E}}^{\square}_{h_{1},\dots,h_{s+1}\in\mathbb{Z}^{L}}\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\
\text{ F\o lner
seq.}\end{subarray}}\varlimsup_{N\to\infty}\left|\mathbb{E}_{n\in
I_{N}}\left\langle\prod_{m=1}^{l}T_{{q}^{\prime}_{m}(n;h_{1},\dots,h_{s},h_{s+1})}g^{\prime}_{m}(x;h_{1},\dots,h_{s}),\right.\right.\;$
$\displaystyle\displaystyle\left.\left.\hskip
122.34692pt\prod_{m=1}^{l}T_{{q}^{\prime}_{m+l}(n;h_{1},\dots,h_{s},h_{s+1})}g^{\prime}_{m}(x;h_{1},\dots,h_{s})\right\rangle\right|^{\kappa}\;\;\;\;\;\;\;\;\;\;\;\;\text{(end
of Step 1)}$ $\displaystyle\displaystyle\leq$ $\displaystyle\displaystyle
4^{\kappa}\overline{\mathbb{E}}^{\square}_{h_{1},\dots,h_{s+1}\in\mathbb{Z}^{L}}\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\
\text{ F\o lner
seq.}\end{subarray}}\varlimsup_{N\to\infty}\Bigl{\|}\mathbb{E}_{n\in
I_{N}}\prod_{i=1}^{r}T_{q^{\prime\prime}_{i,1}(n;h_{1},\dots,h_{s+1})}\big{(}g^{\prime\prime}_{i,1}(x;h_{1},\dots,h_{s+1})\cdot$
$\displaystyle\displaystyle\hskip
122.34692pt\prod_{j=2}^{t_{i}}T_{p^{\prime\prime}_{i,j}(h_{1},\dots,h_{s+1})}g^{\prime\prime}_{i,j}(x;h_{1},\dots,h_{s+1})\big{)}\Bigr{\|}_{L^{2}(\mu)}^{\kappa}\;\;\;\;\;\;\;\;\;\text{
\stackanchor{(Cauchy-Schwarz}{and Step 2)} }$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle
4^{\kappa}\overline{\mathbb{E}}^{\square}_{h_{1},\dots,h_{s+1}\in\mathbb{Z}^{L}}\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\
\text{ F\o lner
seq.}\end{subarray}}\varlimsup_{N\to\infty}\Bigl{\|}\mathbb{E}_{n\in
I_{N}}\prod_{i=1}^{r}T_{q^{\ast}_{i}(n;h_{1},\dots,h_{s+1})}g^{\ast}_{i}(x;h_{1},\dots,h_{s+1})\Bigr{\|}_{L^{2}(\mu)}^{\kappa}\;\;\;\;\;\text{
(Step 3)},$

which is $\displaystyle 4^{\kappa}S(\partial_{q}A,\kappa),$ completing the
proof. ∎

The following theorem shows that when we start with a PET-tuple which is
standard for a function, then after finitely many vdC-operations, we arrive at
a new PET-tuple of degree 1 which is still standard for the same function.
This is useful because by [17, Proposition 3.1], whenever we have an average
with linear iterates, we can bound the limsup of the norm of the average by
some Host-Kra seminorm of the functions. We caution the reader that in our
method, we alternate this standard procedure and instead of deriving to linear
iterates for “some functions”, we run the PET induction multiple times to
arrive at linear iterates isolating “each function” separately.

######  Theorem 4.2.

Let $\displaystyle(X,\mathcal{B},\mu,(T_{g})_{g\in\mathbb{Z}^{d}})$ be a
$\displaystyle\mathbb{Z}^{d}$-system and $\displaystyle f\in L^{\infty}(\mu)$.
If $\displaystyle A$ is a non-degenerate PET-tuple which is standard for
$\displaystyle f$, then there exist
$\displaystyle\rho_{1},\dots,\rho_{t}\in\mathbb{N}^{\ast},$ for some
$\displaystyle t\in\mathbb{N},$ such that
$\displaystyle\partial_{\rho_{t}}\dots\partial_{\rho_{1}}A$ is a non-
degenerate PET-tuple which is standard for $\displaystyle f$ with
$\displaystyle\deg(\partial_{\rho_{t}}\dots\partial_{\rho_{1}}A)=1$.

As an example to demonstrate how the method works, we present some
computations for our Example 1.

First part of computations for Example 1: For a
$\displaystyle\mathbb{Z}^{2}$-system
$\displaystyle(X,\mathcal{B},\mu,(T_{g})_{g\in\mathbb{Z}^{2}})$ and
$\displaystyle f_{1},f_{2}\in L^{\infty}(\mu),$ the PET-tuple of Example 1 is

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

where $\displaystyle p_{1}(n)=(n^{2}+n,0)=(n^{2}+n)e_{1},$ $\displaystyle
p_{2}(n)=(0,n^{2})=n^{2}e_{2},$ for $\displaystyle e_{1}=(1,0)$ and
$\displaystyle e_{2}=(0,1).$ For $\displaystyle i=1$ and 2, we explain how to
find a sequence of vdC-operations to reduce $\displaystyle A$ into a non-
degenerate PET-tuple of degree 1 which is standard for $\displaystyle f_{i}$.

We first isolate the function $\displaystyle f_{1}.$ Setting $\displaystyle
e=(1,-1),$ we have
$\displaystyle\partial_{2}A=(1,1,3,(f_{1},f_{1},f_{2}),\mathbb{p}_{1}),$ where
the tuple $\displaystyle\mathbb{p}_{1}$ essentially equals to

$\displaystyle(n^{2}e+ne_{1},n^{2}e+(2h_{1}+1)ne_{1},2h_{1}ne_{2})$

(one term is removed because it is essentially constant and so
$\displaystyle\ell=3$). Then
$\displaystyle\partial_{3}\partial_{2}A=(1,2,4,(f_{1},f_{1},$ $\displaystyle
f_{1},f_{1}),\mathbb{p}_{2}),$ where the tuple $\displaystyle\mathbb{p}_{2}$
essentially equals to

$\displaystyle((n^{2}+2h_{1}n)e+(1-2h_{1})ne_{1},(n^{2}+2h_{1}n)e+ne_{1},(n^{2}+2(h_{1}+h_{2})n)e+(1-2h_{1})ne_{1},(n^{2}+2(h_{1}+h_{2})n)e+ne_{1})$
(two terms are removed because they are essentially constant and so
$\displaystyle\ell=4$). Finally
$\displaystyle\partial_{2}\partial_{3}\partial_{2}A=(1,3,7,(f_{1},\dots,f_{1}),\mathbb{p}_{3}),$
where the tuple $\displaystyle\mathbb{p}_{3}$ essentially equals to

$\displaystyle(-2h_{1}ne_{1},2h_{2}ne-2h_{1}e_{1},2h_{2}ne,2h_{3}ne-2h_{1}ne_{1},2h_{3}ne,2(h_{2}+h_{3})ne-2h_{1}ne_{1},2(h_{2}+h_{3})ne)$
(one term is removed because it is essentially constant and so
$\displaystyle\ell=7$). We have that
$\displaystyle\partial_{2}\partial_{3}\partial_{2}A$ is non-degenerate and
standard for $\displaystyle f_{1}$, and
$\displaystyle\deg(\partial_{2}\partial_{3}\partial_{2}A)=1$.

We continue by isolating $\displaystyle f_{2}$. Note that
$\displaystyle\partial_{1}A=(1,1,3,(f_{2},f_{1},f_{2}),\mathbb{p}_{1}),$ where
the tuple $\displaystyle\mathbb{p}_{1}$ essentially equals to

$\displaystyle(-n^{2}e-ne_{1},2h_{1}ne_{1},-n^{2}e-ne_{1}+2h_{1}ne_{2})$

(one term is removed for it is essentially constant and so
$\displaystyle\ell=3$). Then
$\displaystyle\partial_{2}\partial_{1}A=(1,2,4,(f_{2},f_{2},f_{2},$
$\displaystyle f_{2}),\mathbb{p}_{2}),$ where the tuple
$\displaystyle\mathbb{p}_{2}$ essentially equals to

$\displaystyle(-n^{2}e-(2h_{1}+1)ne_{1},-(n^{2}+2h_{1}n)e-ne_{1},-(n^{2}+2h_{2}n)e-(2h_{1}+1)ne_{1},-(n^{2}+2(h_{1}+h_{2})n)e-ne_{1})$

(two terms are removed because they are essentially constant and so
$\displaystyle\ell=4$). Finally
$\displaystyle\partial_{1}\partial_{2}\partial_{1}A=(1,3,7,(f_{2},\dots,f_{2}),\mathbb{p}_{3}),$
where the tuple $\displaystyle\mathbb{p}_{3}$ essentially equals to

$\displaystyle(2h_{1}ne_{2},-2h_{2}ne,-2h_{2}ne+2h_{1}ne_{2},-2h_{3}ne,-2h_{3}ne+2h_{1}ne_{2},-2(h_{2}+h_{3})ne,-2(h_{2}+h_{3})ne+2h_{1}ne_{2})$
(one term is removed because it is essentially constant and so
$\displaystyle\ell=7$). We have that
$\displaystyle\partial_{1}\partial_{2}\partial_{1}A$ is non-degenerate and
standard for $\displaystyle f_{2}$, and
$\displaystyle\deg(\partial_{1}\partial_{2}\partial_{1}A)=1$.

###### Proof of Theorem 4.2.

We follow the ideas of the PET induction in [17] and [21].

If $\displaystyle\deg(A)=1,$ there is nothing to prove. So, we assume that
$\displaystyle\deg(A)\geq 2,$ $\displaystyle
A=(L,s,\ell,\mathbb{g}=(g_{1},\dots,g_{\ell}),\mathbb{q}=(q_{1},\dots,q_{\ell}))$,
with $\displaystyle q_{i}=(q_{i,1},\dots,q_{i,d}),$ $\displaystyle 1\leq
i\leq\ell$, where each $\displaystyle q_{i,j}$ is a polynomial from
$\displaystyle\mathbb{Z}^{s+1}$ to $\displaystyle\mathbb{Z}$. Recall that
$\displaystyle\deg(q_{i})=\max_{1\leq j\leq d}\deg(q_{i,j})$. In this proof,
we are thinking of $\displaystyle\mathbb{q}$ as an $\displaystyle\ell\times d$
matrix $\displaystyle(q_{i,j})_{1\leq i\leq\ell,1\leq j\leq d}$ with
polynomial entries.

We say that $\displaystyle p,q\colon(\mathbb{Z}^{L})^{s+1}\to\mathbb{Z}$ are
_equivalent_ , and we write that $\displaystyle p\sim q$, if
$\displaystyle\deg(p)=\deg(q)$ and $\displaystyle\deg(p-q)<\deg(p)$;
otherwise, we write $\displaystyle p\nsim q$. It is not hard to see that
“$\displaystyle\sim$” defines an equivalence relation. Suppose that
$\displaystyle\deg(\mathbb{q})\leq D$. We define the _column weight_ of the
column $\displaystyle j$ to be the vector $\displaystyle
w_{j}(\mathbb{q})=(w_{1,j}(\mathbb{q}),\dots,w_{D,j}(\mathbb{q}))$, where each
$\displaystyle w_{k,j}(\mathbb{q})$ is equal to the number of equivalent
classes in $\displaystyle\mathbb{q}$ of degree $\displaystyle k$ in the column
$\displaystyle j$ (_i.e._ , among $\displaystyle q_{1,j},\dots,q_{\ell,j}$).
For two column weights $\displaystyle\mathbb{v}=(v_{1},\dots,v_{D})$ and
$\displaystyle\mathbb{v}^{\prime}=(v^{\prime}_{1},\dots,v^{\prime}_{D})$, we
say that $\displaystyle\mathbb{v}<\mathbb{v}^{\prime}$ if there exists
$\displaystyle 1\leq k\leq D$ such that $\displaystyle v_{k}<v^{\prime}_{k}$
and $\displaystyle v_{k^{\prime}}=v^{\prime}_{k^{\prime}}$ for all
$\displaystyle k^{\prime}>k$ (notice that we start comparing them from the
last coordinate because this is the one associated to the highest degree).
Then, the set of weights and the set of column degrees are well ordered sets.
Putting this information about $\displaystyle\mathbb{q}$ in rows, we get the
$\displaystyle D\times d$ matrix $\displaystyle
w_{\mathbb{q}}=[w_{1}(\mathbb{q}),\dots,w_{d}(\mathbb{q})]$ which we call the
subweigth of $\displaystyle\mathbb{q}$.

Given a matrix $\displaystyle M$ (with polynomial entries), we define its
k-reduction, denoted by $\displaystyle R_{k}(M),$ to be the submatrix of
$\displaystyle M$ obtained by only considering the rows whose first
$\displaystyle k$ elements are $\displaystyle 0,$ after discarding these
$\displaystyle 0$’s. For instance, for the matrix

$\displaystyle M=\begin{pmatrix}0&p_{1}&p_{2}&p_{3}\\\ p_{4}&0&0&p_{5}\\\
0&0&p_{6}&p_{7}\\\ 0&0&0&p_{8}\end{pmatrix}$

where $\displaystyle p_{1},\ldots,p_{8}$ are non-zero polynomials, its
$\displaystyle i$-reduction for $\displaystyle i=1,2,3,4$ is
$\displaystyle\begin{pmatrix}p_{1}&p_{2}&p_{3}\\\ 0&p_{6}&p_{7}\\\
0&0&p_{8}\end{pmatrix}$, $\displaystyle\begin{pmatrix}p_{6}&p_{7}\\\
0&p_{8}\end{pmatrix}$ , $\displaystyle\begin{pmatrix}p_{8}\end{pmatrix}$ and
$\displaystyle\emptyset$ respectively. By convention, the 0-reduction
$\displaystyle R_{0}(M)$ is $\displaystyle M$ itself and the $\displaystyle
k$-reduction for $\displaystyle k\geq\ell$ is $\displaystyle\emptyset$.

We now define an order associated to matrices. The weight of a matrix
$\displaystyle\mathbb{q}$ with polynomial entries, denoted by $\displaystyle
W(\mathbb{q})$, is the vector of the matrices
$\displaystyle(w(R_{0}(\mathbb{q})),w(R_{1}(\mathbb{q})),\ldots,w(R_{\ell-1}(\mathbb{q})))$,
where $\displaystyle\ell$ is the number of columns of
$\displaystyle\mathbb{q}$. Given two polynomial matrices
$\displaystyle\mathbb{q}$ and $\displaystyle\mathbb{q}^{\prime}$,
$\displaystyle\deg(\mathbb{q}),\deg(\mathbb{q}^{\prime})\leq D$, we say that
$\displaystyle W(\mathbb{q}^{\prime})<W(\mathbb{q})$ if there exist
$\displaystyle 1\leq J,K\leq\ell$ such that

$\displaystyle
w_{j}(R_{k}(\mathbb{q}))=w_{j}(R_{k}(\mathbb{q}^{\prime}))\text{ for all
}j<J\text{ and all }k=0,\ldots,\ell-1;$

and

$\displaystyle
w_{J}(R_{k}(\mathbb{q}))=w_{J}(R_{k}(\mathbb{q}^{\prime}))\text{ for all
}k=0,\ldots,K-1\text{ and
}w_{J}(R_{K}(\mathbb{q}))<w_{J}(R_{K}(\mathbb{q}^{\prime})).$

Under this order, the set of weights of matrices is well-ordered. For a PET-
tuple $\displaystyle A=(L,s,\ell,\mathbb{g},\mathbb{q})$, we define
$\displaystyle W(A)=W(\mathbb{q})$ to be the _weight_ of $\displaystyle A$.

Claim: Let $\displaystyle A$ be a non-degenerate PET-tuple which is standard
for $\displaystyle f$ with $\displaystyle\deg(A)\geq 2$. There exists
$\displaystyle 1\leq\rho\leq\ell$ such that $\displaystyle\partial_{\rho}A$ is
non-degenerate and standard for $\displaystyle f$ with $\displaystyle
W(\partial_{\rho}A)<W(A)$.

We first finish the proof of the theorem assuming that the claim holds. Let
$\displaystyle A$ be a non-degenerate PET-tuple which is standard for
$\displaystyle f$ and $\displaystyle\deg(A)\geq 2$. After using the claim
finitely many steps, the decreasing chain $\displaystyle
W(A)>W(\partial_{\rho_{1}}A)>W(\partial_{\rho_{2}}\partial_{\rho_{1}}A)>\dots$
will eventually terminate, so we will end up with a non-degenerate PET-tuple
$\displaystyle\partial_{\rho_{t}}\dots\partial_{\rho_{1}}A$ which is standard
for $\displaystyle f$, with
$\displaystyle\deg(\partial_{\rho_{t}}\dots\partial_{\rho_{1}}A)=1$. This
finishes the proof.

So it suffices to prove the claim. Relabeling if necessary, we may assume
without loss of generality that $\displaystyle g_{1}=f$ and
$\displaystyle\deg(q_{1,1})=\deg(A)\geq 2$. Let $\displaystyle
j_{0}\in\\{0,\ldots,\ell\\}$ be the smallest integer such that $\displaystyle
R_{j_{0}+1}(\mathbb{q})=\emptyset$. We choose $\displaystyle
1\leq\rho\leq\ell$ in the following way:

  1. (i)

Case that $\displaystyle j_{0}=0$. This case has three sub-cases.

    1. (a)

If some $\displaystyle q_{i,1}\nsim q_{1,1}$, then let $\displaystyle\rho$ be
the smallest integer such that $\displaystyle q_{\rho,1}\nsim q_{1,1}$.

In this case, since $\displaystyle q_{\rho,1}\nsim q_{1,1}$ and $\displaystyle
A$ is standard for $\displaystyle f$, $\displaystyle\partial_{\rho}A$ is
standard for $\displaystyle f$. Moreover, $\displaystyle
w_{D,1}(\partial_{\rho}A)=w_{D,1}(A)-1$ and so $\displaystyle
W(\partial_{\rho}A)<W(A)$.

    2. (b)

If all $\displaystyle q_{1,1},\dots,q_{\ell,1}$ are equivalent and there exist
$\displaystyle 2\leq i\leq\ell,$ $\displaystyle 1\leq j\leq d$ such that
$\displaystyle q_{i,j}\nsim q_{1,j}$, and either $\displaystyle\deg(q_{i,j})$
or $\displaystyle\deg(q_{1,j})$ equals to $\displaystyle\deg(\mathbb{q})$,
then let $\displaystyle\rho$ be the smallest integer such that there exists
$\displaystyle 1\leq j\leq d$ with $\displaystyle q_{\rho,j}\nsim q_{1,j}$,
and either $\displaystyle\deg(q_{\rho,j})$ or $\displaystyle\deg(q_{1,j})$
equals to $\displaystyle\deg(\mathbb{q})$. In this case, since $\displaystyle
q_{\rho,j}$ is not equivalent to $\displaystyle q_{1,j}$, and either
$\displaystyle\deg(q_{\rho,j})$ or $\displaystyle\deg(q_{1,j})$ equals to
$\displaystyle\deg(\mathbb{q})$, $\displaystyle\partial_{\rho}A$ is standard
for $\displaystyle f$. Moreover, $\displaystyle
w_{D,1}(\partial_{\rho}A)=0<w_{D,1}(A)$ and so $\displaystyle
W(\partial_{\rho}A)<W(A)$.

    3. (c)

If all $\displaystyle q_{1,1},\dots,q_{\ell,1}$ are equivalent, and for all
$\displaystyle 1\leq j\leq d$, either $\displaystyle\deg(q_{i,j})$ is
$\displaystyle\deg(q_{1,j})$ for all $\displaystyle 1\leq i\leq\ell$ or
$\displaystyle\deg(q_{i,j})<\deg(\mathbb{q})$ for all $\displaystyle 1\leq
i\leq\ell$, then let $\displaystyle\rho=\ell+1$.242424 We leave it to the
interested reader to check that (a), (b) and (c) cover all the possibilities
in Case (i).

In this case, $\displaystyle\deg(\partial_{\rho}A)<\deg(A)$. Since
$\displaystyle\deg(q_{1,1})\geq 2$, we have that

$\displaystyle\deg(q_{1,1}(n,h_{1},\dots,h_{s})-q_{1,1}(n+h_{s+1},h_{1},\dots,h_{s}))=\deg(q_{1,1})-1=\deg(\partial_{\rho}A)\geq
1.$

So $\displaystyle\partial_{\rho}A$ is standard for $\displaystyle f$.
Moreover, $\displaystyle w_{D,1}(\partial_{\rho}A)=0<w_{D,1}(A)$ and so
$\displaystyle W(\partial_{\rho}A)<W(A)$.

  2. (ii)

Case that $\displaystyle j_{0}>0$. Consider the reduction $\displaystyle
R_{j_{0}}(\mathbb{q})$ of the matrix $\displaystyle\mathbb{q}$.

    1. (a)

Suppose that an entry of the first column of $\displaystyle
R_{j_{0}}(\mathbb{q})$ (which is of course an entry of the $\displaystyle
j_{0}+1$ column of $\displaystyle\mathbb{q}$) is not equivalent to any other
entry of the first column of $\displaystyle R_{j_{0}}(\mathbb{q})$. Among such
entries, let $\displaystyle\rho$ be the smallest index such that
$\displaystyle q_{\rho,j_{0}+1}$ has minimal degree.

In this case, we have that $\displaystyle\partial_{\rho}A$ is standard for
$\displaystyle f$. Moreover,

$\displaystyle
w_{\deg(q_{\rho,j_{0}+1}),1}(\partial_{\rho}^{j_{0}}A)>w_{\deg(q_{\rho,j_{0}+1}),j_{0}}(R_{j_{0}}(\mathbb{q})),$

where $\displaystyle\partial_{\rho}^{k}=\partial_{\rho}\ldots\partial_{\rho}$
($\displaystyle k$ times). One can check that this implies that $\displaystyle
W(\partial_{\rho}A)<W(A)$.

    2. (b)

Suppose all entries in the first column of $\displaystyle
R_{j_{0}}(\mathbb{q})$ are equivalent. Then let $\displaystyle\rho$ be such
that $\displaystyle q_{\rho,j_{0}+1}$ corresponds to the first entry of the
first column of $\displaystyle R_{j_{0}}(\mathbb{q})$.

In this case, $\displaystyle\partial_{\rho}A$ is standard for $\displaystyle
f$. Moreover,

$\displaystyle
w_{\deg(q_{\rho,j_{0}+1}),1}(\partial_{\rho}^{j_{0}}A)>w_{\deg(q_{\rho,j_{0}+1}),j_{0}}(R_{j_{0}}(\mathbb{q})).$

One can check that this fact implies that $\displaystyle
W(\partial_{\rho}A)<W(A)$.

This proves the claim and completes the proof. ∎

We now provide a proof of Proposition 2.8.

###### Proof of Proposition 2.8.

Let $\displaystyle A=(L,0,1,\\{f\\},\\{p\\})$. It suffices to show that
$\displaystyle S(A,\kappa)=0$ for some $\displaystyle\kappa\in{\mathbb{N}}$,
assuming that
$\displaystyle\mathbb{E}(f|Z_{{\mathbb{Z}},{\mathbb{Z}}}(\mathbb{X}))=0$. For
any $\displaystyle s\in\mathbb{N}^{\ast}$ and function $\displaystyle
u\colon(\mathbb{Z}^{L})^{s}\to\mathbb{Z}$, let $\displaystyle\Delta
u\colon(\mathbb{Z}^{L})^{s+1}\to\mathbb{Z}$ be the function
$\displaystyle\Delta
u(x_{1},\dots,x_{s+1})=u(x_{1}+x_{s+1},\dots,x_{s})-u(x_{1},\dots,x_{s})$ and
$\displaystyle\Delta^{k}u=(\Delta\circ\dots\circ\Delta)u$ ($\displaystyle k$
times).

If $\displaystyle\deg(p)>1,$ then it is easy to verify that
$\displaystyle\partial_{1}A=(L,1,1,\\{f\\},\\{\Delta p\\})$. By induction,
$\displaystyle\partial^{k}_{1}A=(L,k,1,\\{f\\},\\{\Delta^{k}p\\})$ for all
$\displaystyle k<\deg(p)$. By Proposition 4.1, we have that $\displaystyle
S(A,2^{K})\leq 4^{2^{K}-1}S(\partial_{1}^{K}A,1)$, where $\displaystyle
K=\deg(p)-1$. It is easy to see that $\displaystyle\deg(\Delta
p)=\deg(p)-1,$252525 Recall that “$\displaystyle\deg$” only “sees” the first
variable. and so $\displaystyle\deg(\Delta^{K}p)=1$. We may then assume that
$\displaystyle\Delta^{K}p(n,h_{1},\dots,h_{K})=c(h_{1},\dots,h_{K})\cdot
n+c^{\prime}(h_{1},\dots,h_{K})$ for some polynomials $\displaystyle
c(h_{1},\dots,h_{K})\in\mathbb{Z}^{L},c^{\prime}(h_{1},\dots,h_{K})\in\mathbb{Z}$
of $\displaystyle h_{1},\ldots,h_{K}$ with $\displaystyle c$ not being the
constant zero vector. By Theorem 2.3,

(10)
$\overline{\mathbb{E}}_{n\in\mathbb{Z}^{L}}T_{\Delta^{K}p(n,h_{1},\dots,h_{K})}f=T_{c^{\prime}(h_{1},\ldots,h_{K})}\mathbb{E}(f|\mathcal{I}(G(c(h_{1},\dots,h_{K})))).$

If $\displaystyle c(h_{1},\dots,h_{K})\neq 0$, then

$\displaystyle\mathcal{I}(G(c(h_{1},\dots,h_{K})))=Z_{G(c(h_{1},\dots,h_{K}))}\subseteq
Z_{\mathbb{Z},G(c(h_{1},\dots,h_{K}))}=Z_{\mathbb{Z},\mathbb{Z}},$

where in the last equality we used Lemma 2.4 (iv), since $\displaystyle
G(c(h_{1},\dots,h_{K}))$ is a finite index subgroup of
$\displaystyle{\mathbb{Z}}$. By Lemma 2.12, the set of
$\displaystyle(h_{1},\dots,h_{K})\in(\mathbb{Z}^{L})^{K}$ such that
$\displaystyle c(h_{1},\dots,h_{K})=0$ is of upper Banach density
$\displaystyle 0,$ so

$\displaystyle\displaystyle S(\partial_{1}^{K}A,1)=$
$\displaystyle\displaystyle\overline{\mathbb{E}}^{\square}_{h_{1},\dots,h_{K}\in{\mathbb{Z}}^{L}}\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\
\text{ F\o lner
seq.}\end{subarray}}\varlimsup_{N\to\infty}\Bigl{\|}\mathbb{E}_{n\in
I_{N}}T_{\Delta^{K}p(n,h_{1},\dots,h_{K})}f\Bigr{\|}_{L^{2}(\mu)}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\overline{\mathbb{E}}^{\square}_{h_{1},\dots,h_{K}\in{\mathbb{Z}}^{L}}\sup_{\begin{subarray}{c}(I_{N})_{N\in\mathbb{N}}\\\
\text{ F\o lner
seq.}\end{subarray}}\varlimsup_{N\to\infty}\Bigl{\|}\mathbb{E}_{n\in
I_{N}}T_{\Delta^{K}p(n,h_{1},\dots,h_{K})}\mathbb{E}(f|Z_{\mathbb{Z},\mathbb{Z}})\Bigr{\|}_{L^{2}(\mu)}=0.$

This implies that $\displaystyle S(A,2^{K})=0$, which finishes the proof. ∎

##  5\. Characterizing multiple averages along polynomials

In this section we state Theorem 5.1, the stronger form of Theorem 1.1, which
is the main contribution of this work. Its validity implies (see below) both
Theorems 1.3 and 1.4, our main joint ergodicity results.

###  5.1. Characteristic factors for multiple averages

Recall that a family of (integer valued) polynomials $\displaystyle
p_{1},\dots,p_{k}\colon\mathbb{Z}^{L}$ $\displaystyle\to\mathbb{Z}^{d}$ is
non-degenerate if $\displaystyle p_{i},p_{i}-p_{j}$ are not essentially
constant for all $\displaystyle 1\leq i,j\leq k$, $\displaystyle i\neq j$. The
following theorem states that in order to study multiple averages along
polynomials, it suffices to assume that all the functions $\displaystyle
f_{i}$ are measurable with respect to certain Host-Kra characteristic factors:

######  Theorem 5.1 (Characteristic factors for multiple averages along
polynomials).

Let $\displaystyle d,k,L\in\mathbb{N}^{\ast}$ and $\displaystyle
p_{1},\dots,p_{k}\colon\mathbb{Z}^{L}\to\mathbb{Z}^{d}$ be a non-degenerate
family of polynomials of degree at most $\displaystyle K$. Suppose that
$\displaystyle p_{i}(n)=\sum_{v\in\mathbb{N}^{L},|v|\leq K}b_{i,v}n^{v}$ for
some $\displaystyle b_{i,v}\in\mathbb{Q}^{d}$. Let $\displaystyle
R\subseteq\mathbb{Q}^{d}$ be the set

$\displaystyle R\coloneqq\bigcup_{v\in\mathbb{N}^{L},0<|v|\leq
K}\\{b_{i,v},b_{i,v}-b_{i^{\prime},v}\colon 1\leq i,i^{\prime}\leq
k\\}\backslash\\{{\bf 0}\\}.$

Let $\displaystyle(X,\mathcal{B},\mu,(T_{g})_{g\in\mathbb{Z}^{d}})$ be a
$\displaystyle\mathbb{Z}^{d}$-system. For every $\displaystyle
f_{1},\dots,f_{k}\in L^{\infty}(\mu)$, we have that

(11) $\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
$\displaystyle\mathbb{E}(f_{i}|Z_{\\{G(r)^{\times\infty}\\}_{r\in R}})=0$ for
some $\displaystyle 1\leq i\leq k$.}$

In particular, if $\displaystyle(T_{g})_{g\in G(r)}$ is ergodic for
$\displaystyle\mu$ for all $\displaystyle r\in R$, then for every
$\displaystyle f_{1},\dots,f_{k}\in L^{\infty}(\mu)$,

(12) $\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
$\displaystyle\mathbb{E}(f_{i}|Z_{(\mathbb{Z}^{d})^{\times\infty}})=0$ for
some $\displaystyle 1\leq i\leq k$.}$

######  Remark.

The following weaker form of (11) in Theorem 5.1 can be derived by the results
of [17]:

$\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
$\displaystyle\mathbb{E}(f_{i}|Z_{\\{G(r)^{\times\infty}\\}_{r\in{\mathbb{Z}}^{d}\setminus\\{\bf
0\\}}})=0$ for some $\displaystyle 1\leq i\leq k$.}$

Hence, (12) holds if $\displaystyle T_{g}$ is assumed to be ergodic for
$\displaystyle\mu$ for all $\displaystyle
g\in\mathbb{Z}^{d}\backslash\\{{\bf{0}}\\}$. Theorem 5.1 improves the result
of [17] since one only needs to require finitely many $\displaystyle T_{g}$’s
to be ergodic (_i.e._ , the generators of $\displaystyle G(r),r\in R$) in
order to deduce (12).

On the other hand, it is worth noting that (11) has room for improvement
(meaning that it is possible for one to replace the factor $\displaystyle
Z_{\\{G(r)^{\times\infty}\\}_{r\in R}}$ of (11) with smaller ones), as we
shall see in the examples below. Actually, we do have a stronger version of
(11) (see the proof of Theorem 5.1), but (11) already captures the essence of
our result as it is stated here.

Another important example of polynomial averages is the following, for which
we actually characterize its convergence to the “expected” limit, where all
the transformations have the same polynomial iterate.

######  Example 2.

Let $\displaystyle(X,\mathcal{B},\mu,T_{1},\dots,T_{d})$ be a system with
commuting transformations. One should think of $\displaystyle
T_{1},\dots,T_{d}$ as a $\displaystyle\mathbb{Z}^{d}$-action
$\displaystyle(S_{g})_{g\in\mathbb{Z}^{d}}$ with $\displaystyle
T_{i}=S_{e_{i}}$, where we recall that $\displaystyle e_{i}\in\mathbb{Z}^{d}$
denotes the vector whose $\displaystyle i$th entry is 1 and all other entries
are $\displaystyle 0$’s. Let $\displaystyle
p_{1},\dots,p_{d}\colon\mathbb{Z}\to\mathbb{Z}^{d}$ be polynomials given by
$\displaystyle p_{i}(n)=p(n)e_{i}$ for some polynomial $\displaystyle
p\colon\mathbb{Z}\to\mathbb{Z}$. By Theorem 5.1, we have that

(13) $\mathbb{E}_{n\in{\mathbb{Z}}}T_{1}^{p(n)}f_{1}\cdot\ldots\cdot
T_{d}^{p(n)}f_{d}=0\text{ if
$\displaystyle\mathbb{E}(f_{i}|Z_{\\{G(r)^{\times\infty}\\}_{r\in R}})=0$ for
some $\displaystyle 1\leq i\leq d$,}$

where $\displaystyle R=\\{T_{i},T_{i}T^{-1}_{j}\colon 1\leq i,j\leq d,i\neq
j\\}$. We remark that $\displaystyle Z_{\\{G(r)^{\times\infty}\\}_{r\in R}}$
is not necessarily the smallest factor with this property. For example, if
$\displaystyle p(n)=n$, then (13) is a weaker form of Proposition 6.1 (or [15,
Proposition 1]).

Continuation of Example 1. Recall the $\displaystyle\mathbb{Z}^{2}$-system
$\displaystyle\mathbb{X}$ with two commuting transformations $\displaystyle
T_{1},T_{2}$ and $\displaystyle p_{1},p_{2}\colon\mathbb{Z}\to\mathbb{Z}^{2}$
polynomials given by $\displaystyle p_{1}(n)=(n^{2}+n,0)$ and $\displaystyle
p_{2}(n)=(0,n^{2})$. By Theorem 5.1, we have that

(14) $\begin{split}\quad\mathbb{E}_{n\in{\mathbb{Z}}}T_{1}^{n^{2}+n}f_{1}\cdot
T_{2}^{n^{2}}f_{2}=0\text{ if
$\displaystyle\mathbb{E}(f_{i}|Z_{\\{G(r)^{\times\infty}\\}_{r\in R}})=0$ for
$\displaystyle i=1$ or 2,}\end{split}$

where $\displaystyle R=\\{T_{1},T_{2},T_{1}T^{-1}_{2}\\}$. Again
$\displaystyle Z_{\\{G(r)^{\times\infty}\\}_{r\in R}}$ is not the smallest
factor with this property (later, in equality (39), we will obtain an
improvement of (14)).

It is an interesting, in general open (and definitely hard), question to ask
what are the smallest factors $\displaystyle Z_{1},\dots,Z_{k}$ of
$\displaystyle\mathbb{X}$ such that for every $\displaystyle
f_{1},\dots,f_{k}\in L^{\infty}(\mu)$,

$\mathbb{E}_{n\in{\mathbb{Z}}}T_{p_{1}(n)}f_{1}\cdot\ldots\cdot
T_{p_{k}(n)}f_{k}=0\text{ if $\displaystyle\mathbb{E}(f_{i}|Z_{i})=0$ for some
$\displaystyle 1\leq i\leq k$.}$

###  5.2. Proofs of the joint ergodicity results assuming Theorem 5.1

In this subsection we explain how to derive our main joint ergodicity results,
Theorems 1.3 and 1.4, assuming the validity of Theorem 5.1.

###### Proof of Theorem 1.3 assuming Theorem 5.1.

Let $\displaystyle R$ be defined as in Theorem 5.1. Since $\displaystyle
T_{g}$ is ergodic for all $\displaystyle g\in R$, by Theorem 5.1, we may
assume without loss of generality that all $\displaystyle f_{1},\dots,f_{k}$
are measurable with respect to $\displaystyle
Z_{(\mathbb{Z}^{d})^{\times\infty}}(\mathbb{X})$ (note that conditions (i) and
(ii) remain valid when passing to a factor system). By $\displaystyle
L^{1}(\mu)$-approximation, we may assume without loss of generality that all
$\displaystyle f_{1},\dots,f_{k}$ are measurable with respect to
$\displaystyle Z_{(\mathbb{Z}^{d})^{\times M}}(\mathbb{X})$ for some
$\displaystyle M\in\mathbb{N}$. By Theorem 2.6 and again by $\displaystyle
L^{1}(\mu)$-approximation, we may further assume without loss of generality
that all $\displaystyle f_{1},\dots,f_{k}$ are measurable with respect a
factor of $\displaystyle\mathbb{X}$ which is isomorphic to an
$\displaystyle(M-1)$-step $\displaystyle\mathbb{Z}^{d}$-nilsystem.

So, we may assume that $\displaystyle\mathbb{X}$ is itself an
$\displaystyle(M-1)$-step $\displaystyle\mathbb{Z}^{d}$-nilsystem. Since
$\displaystyle(T_{p_{1}(n)}\times\dots\times
T_{p_{k}(n)})_{n\in{\mathbb{Z}}^{L}}$ is ergodic for
$\displaystyle\mu^{\otimes k}$, we have that

(15) $\mathbb{E}_{n\in\mathbb{Z}^{L}}T_{p_{1}(n)}f_{1}(x_{1})\cdot\ldots\cdot
T_{p_{k}(n)}f_{k}(x_{k})=\int_{X}f_{1}\,d\mu\cdot\ldots\cdot\int_{X}f_{k}\,d\mu$

for $\displaystyle\mu^{\otimes k}$-a.e. $\displaystyle(x_{1},\dots,x_{k})\in
X^{k}$. By Theorem 2.9, (15) holds for all
$\displaystyle(x_{1},\dots,x_{k})\in X^{k}$ and so in particular

(16) $\mathbb{E}_{n\in\mathbb{Z}^{L}}T_{p_{1}(n)}f_{1}(x)\cdot\ldots\cdot
T_{p_{k}(n)}f_{k}(x)=\int_{X}f_{1}\,d\mu\cdot\ldots\cdot\int_{X}f_{k}\,d\mu$

for all $\displaystyle x\in X$ (as a pointwise limit). By the dominated
convergence theorem, (16) also holds as an $\displaystyle L^{2}(\mu)$-limit,
which finishes the proof. ∎

Before we proceed with the proof of Theorem 1.4, we need the following lemma
and proposition:

######  Lemma 5.2.

Let $\displaystyle(X,\mathcal{B},\mu,T_{1},\dots,T_{d})$ be a system with
commuting transformations. Then in the product space
$\displaystyle(X^{d},\mathcal{B}^{d},\mu^{\otimes d})$, the
$\displaystyle\sigma$-algebra of $\displaystyle T_{1}\times\cdots\times
T_{d}$-invariant sets is measurable with respect to
$\displaystyle\bigotimes_{i=1}^{d}Z_{T_{i},T_{i}}$.

###### Proof.

It suffices to show that $\displaystyle\mathbb{E}(f_{1}\otimes\cdots\otimes
f_{d}|\mathcal{I}(T_{1}\times\cdots\times T_{d}))=0$ whenever
$\displaystyle\mathbb{E}(f_{i}|Z_{T_{i},T_{i}})=0$ for some $\displaystyle
1\leq i\leq d$. Without loss of generality, we may assume that all functions
are bounded by $\displaystyle 1$ in $\displaystyle L^{\infty}(\mu)$ and that
$\displaystyle\mathbb{E}(f_{1}|Z_{T_{1},T_{1}})=0$ (or equivalently
$\displaystyle\|f_{1}\|_{T_{1},T_{1}}=0$). By Lemma 2.1 and Jensen’s
inequality, setting $\displaystyle a(n)=T_{1}^{n}f_{1}\otimes\cdots\otimes
T_{d}^{n}f_{d},$ we have that

$\displaystyle\displaystyle\|\mathbb{E}(f_{1}\otimes\cdots\otimes
f_{d}|\mathcal{I}(T_{1}\times\cdots\times T_{d}))\|^{4}_{L^{2}(\mu^{\otimes
d})}$
$\displaystyle\displaystyle=\|\mathbb{E}_{n\in{\mathbb{Z}}}(T_{1}\times\cdots\times
T_{d})^{n}f_{1}\otimes\cdots\otimes f_{d}\|^{4}_{L^{2}(\mu^{\otimes d})}$
$\displaystyle\displaystyle\leq\left(4\mathbb{E}^{\square}_{h\in{\mathbb{Z}}}\mathbb{E}_{n\in{\mathbb{Z}}}\langle
a(n),a(n+h)\rangle\right)^{2}$
$\displaystyle\displaystyle=16\mathbb{E}^{\square}_{h\in{\mathbb{Z}}}\mathbb{E}_{n\in{\mathbb{Z}}}\langle
f_{1}\otimes\cdots\otimes f_{d},T_{1}^{h}f_{1}\otimes\cdots\otimes
T_{d}^{h}f_{d}\rangle^{2}$ $\displaystyle\displaystyle\leq
16\mathbb{E}^{\square}_{h\in{\mathbb{Z}}}\left|\int f_{1}\cdot
T_{1}^{h}f_{1}d\mu\right|^{2}$ $\displaystyle\displaystyle\leq
16\mathbb{E}^{\square}_{h\in{\mathbb{Z}}}\left|\int\mathbb{E}(f_{1}\cdot
T_{1}^{h}f_{1}|\mathcal{I}(T_{1}))d\mu\right|^{2}$
$\displaystyle\displaystyle\leq
16\mathbb{E}^{\square}_{h\in{\mathbb{Z}}}\left\|\mathbb{E}(f_{1}\cdot
T_{1}^{h}f_{1}|\mathcal{I}(T_{1}))\right\|_{L^{2}(\mu)}^{2}$
$\displaystyle\displaystyle=16\|f_{1}\|_{T_{1},T_{1}}^{4},$

where the last line follows, for instance, from Lemma 2.4 (iii). This finishes
the proof. ∎

######  Proposition 5.3.

Let $\displaystyle d,L\in\mathbb{N}^{\ast},$ $\displaystyle
p\colon\mathbb{Z}^{L}\to\mathbb{Z}$ a polynomial and
$\displaystyle(X,\mathcal{B},\mu,T_{1},\dots,T_{d})$ a system with commuting
transformations such that
$\displaystyle(T_{1}^{p(n)},\dots,T_{d}^{p(n)})_{n\in\mathbb{Z}^{L}}$ is
jointly ergodic for $\displaystyle\mu$. Then

  * (i)

$\displaystyle((T_{i}T_{j}^{-1})^{p(n)})_{n\in\mathbb{Z}^{L}}$ is ergodic for
$\displaystyle\mu$ for all $\displaystyle 1\leq i,j\leq d,i\neq j$; and

  * (ii)

$\displaystyle(T_{1}^{p(n)}\times\dots\times
T_{d}^{p(n)})_{n\in\mathbb{Z}^{L}}$ is ergodic for $\displaystyle\mu^{\otimes
d}$.

###### Proof.

The idea of the proof for Part (i) is similar to [4, Proposition 2.1]. Since
the language we use is different, we present the proof for completeness.

By assumption,

(17) $\mathbb{E}_{n\in\mathbb{Z}^{L}}T^{p(n)}_{1}f_{1}\cdot\ldots\cdot
T^{p(n)}_{d}f_{d}=\int_{X}f_{1}\,d\mu\cdot\ldots\cdot\int_{X}f_{d}\,d\mu$

for all $\displaystyle f_{1},\dots,f_{d}\in L^{\infty}(\mu)$. Suppose first
that (i) fails. We may assume without loss of generality that
$\displaystyle((T_{1}T^{-1}_{2})^{p(n)})_{n\in\mathbb{Z}^{L}}$ is not ergodic
for $\displaystyle\mu$. So there exist $\displaystyle g\in L^{\infty}(\mu)$
not $\displaystyle\mu$-a.e. equal to a constant function and a function
$\displaystyle g^{\prime}\in L^{2}(\mu)$ such that

$\displaystyle
g^{\prime}\coloneqq\mathbb{E}_{n\in\mathbb{Z}^{L}}(T_{1}T^{-1}_{2})^{p(n)}g\neq\int_{X}g\,d\mu$

(the existence of the limit can be seen by Furstenberg’s classical spectral-
theorem approach, or even by a known “non-spectral” approach due to
Bergelson). Then $\displaystyle\int_{X}g^{\prime}\,d\mu=\int_{X}g\,d\mu$. Note
that $\displaystyle g^{\prime}$ cannot be $\displaystyle\mu$-a.e. equal to a
constant. Letting $\displaystyle f_{1}=g,f_{2}=g^{\prime}$ and $\displaystyle
f_{0}=f_{3}=\dots=f_{d}\equiv 1$, we have that

$\displaystyle\displaystyle\int_{X}f_{0}\cdot\mathbb{E}_{n\in\mathbb{Z}^{L}}T^{p(n)}_{1}f_{1}\cdot\ldots\cdot
T^{p(n)}_{d}f_{d}\,d\mu$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\mathbb{E}_{n\in\mathbb{Z}^{L}}\int_{X}T^{p(n)}_{1}g\cdot
T^{p(n)}_{2}g^{\prime}\,d\mu$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\mathbb{E}_{n\in\mathbb{Z}^{L}}\int_{X}(T_{1}T^{-1}_{2})^{p(n)}g\cdot
g^{\prime}\,d\mu$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\int_{X}\mathbb{E}_{n\in\mathbb{Z}^{L}}(T_{1}T^{-1}_{2})^{p(n)}g\cdot
g^{\prime}\,d\mu$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\int_{X}g^{\prime
2}\,d\mu>\left(\int_{X}g^{\prime}\,d\mu\right)^{2}=\left(\int_{X}g\,d\mu\right)^{2}$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\int_{X}f_{1}\,d\mu\cdot\ldots\cdot\int_{X}f_{d}\,d\mu,$

a contradiction to (17), proving (i).

To show (ii), it suffices to show that for all $\displaystyle
f_{1},\dots,f_{d}\in L^{\infty}(\mu)$ with
$\displaystyle\prod_{i=1}^{d}\int_{X}f_{i}\,d\mu=0$, we have that

(18)
$\displaystyle\displaystyle\mathbb{E}_{n\in\mathbb{Z}^{L}}\bigotimes_{i=1}^{d}T_{i}^{p(n)}f_{i}=\mathbb{E}_{n\in\mathbb{Z}^{L}}(T_{1}\times\dots\times
T_{d})^{p(n)}\bigotimes_{i=1}^{d}f_{i}=0.$

We first claim that

$\displaystyle\displaystyle\mathbb{E}_{n\in\mathbb{Z}^{L}}\bigotimes_{i=1}^{d}T_{i}^{p(n)}f_{i}=0\text{
if
$\displaystyle\mathbb{E}(f_{i}|Z_{\mathbb{Z}^{d},\mathbb{Z}^{d}}(\mathbb{X}))=0$
for some $\displaystyle 1\leq i\leq d$.}$

We apply the proof of Proposition 2.8 to the
$\displaystyle{\mathbb{Z}}$-system
$\displaystyle(X^{d},\mathcal{B}^{d},\mu^{\otimes d},T_{1}\times\dots\times
T_{d})$. Suppose that
$\displaystyle\mathbb{E}(f_{i}|Z_{\mathbb{Z}^{d},\mathbb{Z}^{d}}(\mathbb{X}))=0$
for some $\displaystyle 1\leq i\leq d$. By Theorem 2.3 and (10) in the proof
of Proposition 2.8, it suffices to show that the set of
$\displaystyle(h_{1},\dots,h_{K})\in({\mathbb{Z}}^{L})^{K}$ such that

(19)
$\mathbb{E}\Bigl{(}\bigotimes_{i=1}^{d}f_{i}|\mathcal{I}(G(c(h_{1},\ldots,h_{K}))\Bigr{)}=0$

is of density 1, where $\displaystyle
c\colon({\mathbb{Z}}^{L})^{K}\to{\mathbb{Z}}$ is a non-constant polynomial. If
$\displaystyle c(h_{1},\dots,h_{K})\neq 0$, then
$\displaystyle\mathcal{I}(G(c(h_{1},\ldots,h_{K}))$ is the
sub-$\displaystyle\sigma$-algebra of $\displaystyle\mathcal{B}^{d}$ consists
of the $\displaystyle(T_{1}\times\dots\times
T_{d})^{c(h_{1},\dots,h_{K})}$-invariant sets. By Lemma 5.2,

$\displaystyle\mathcal{I}(G(c(h_{1},\ldots,h_{K}))\subseteq\bigotimes_{i=1}^{d}Z_{T^{c(h_{1},\ldots,h_{K})}_{i},T^{c(h_{1},\ldots,h_{K})}_{i}}=\bigotimes_{i=1}^{d}Z_{T_{i},T_{i}},$

where we used Lemma 2.4 (iv) in the last equality. On the other hand, by (17),
we have that $\displaystyle(T^{p(n)}_{i})_{n\in\mathbb{Z}^{L}}$ is ergodic for
$\displaystyle\mu$ for all $\displaystyle 1\leq i\leq d$, which implies that
$\displaystyle T_{i}$ is ergodic for $\displaystyle\mu$. By Lemma 2.4 (ii), we
have that

$\displaystyle\mathcal{I}(G(c(h_{1},\ldots,h_{K}))\subseteq\bigotimes_{i=1}^{d}Z_{T_{i},T_{i}}=\bigotimes_{i=1}^{d}Z_{{\mathbb{Z}}^{d},{\mathbb{Z}}^{d}}.$

Since
$\displaystyle\mathbb{E}(f_{i}|Z_{\mathbb{Z}^{d},\mathbb{Z}^{d}}(\mathbb{X}))=0$,
we have that
$\displaystyle\mathbb{E}\Bigl{(}\bigotimes_{i=1}^{d}f_{i}\Big{|}\bigotimes_{i=1}^{d}Z_{{\mathbb{Z}}^{d},{\mathbb{Z}}^{d}}\Bigr{)}=\bigotimes_{i=1}^{d}\mathbb{E}(f_{i}|Z_{{\mathbb{Z}}^{d},{\mathbb{Z}}^{d}})=0$,
and so (19) holds whenever $\displaystyle c(h_{1},\ldots,h_{K})\neq 0$. By
Proposition 2.8, such tuples $\displaystyle(h_{1},\ldots,h_{K})$ are of
density 1. This proves the claim.

By the claim, it now suffices to prove (18) under the assumption that all
$\displaystyle f_{i}$ are measurable with respect to $\displaystyle
Z_{{\mathbb{Z}}^{d},{\mathbb{Z}}^{d}}$. By Lemma 2.7, we can approximate each
$\displaystyle f_{i}$ in $\displaystyle L^{2}(\mu)$ by eigenfunctions of
$\displaystyle\mathbb{X}$. By multi-linearity, we may assume without loss of
generality that each $\displaystyle f_{i}$ is a non-constant eigenfunction of
$\displaystyle\mathbb{X}$ given by $\displaystyle
T_{g}f_{i}=\lambda_{i}(g)f_{i}$ for all $\displaystyle g\in\mathbb{Z}^{d}$ for
some group homomorphism
$\displaystyle\lambda_{i}\colon\mathbb{Z}^{d}\to\mathbb{S}^{1}$ and that
$\displaystyle f_{i}(x)\neq 0$ $\displaystyle\mu$-a.e $\displaystyle x\in X$.
Then by (17),

$\displaystyle
0=\prod_{i=1}^{d}\int_{X}f_{i}d\mu=\mathbb{E}_{n\in\mathbb{Z}^{L}}\prod_{i=1}^{d}T_{i}^{p(n)}f_{i}=\Bigl{(}\mathbb{E}_{n\in\mathbb{Z}^{L}}\prod_{i=1}^{d}\lambda_{i}(p(n)e_{i})\Bigr{)}\prod_{i=1}^{d}f_{i}.$

This implies that
$\displaystyle\mathbb{E}_{n\in\mathbb{Z}^{L}}\prod_{i=1}^{d}\lambda_{i}(p(n)e_{i})=0$.
So,

$\displaystyle\mathbb{E}_{n\in\mathbb{Z}^{L}}\bigotimes_{i=1}^{d}T_{i}^{p(n)}f_{i}=\Bigl{(}\mathbb{E}_{n\in\mathbb{Z}^{L}}\prod_{i=1}^{d}\lambda_{i}(p(n)e_{i})\Bigr{)}\bigotimes_{i=1}^{d}f_{i}=0.$

This proves (ii) and finishes the proof. ∎

###### Proof of Theorem 1.4 assuming Theorem 5.1.

We first prove the “if” part. We want to show that

(20) $\mathbb{E}_{n\in\mathbb{Z}^{L}}T^{p(n)}_{1}f_{1}\cdot\ldots\cdot
T^{p(n)}_{d}f_{d}=\int_{X}f_{1}\,d\mu\cdot\ldots\cdot\int_{X}f_{d}\,d\mu$

for all $\displaystyle f_{1},\dots,f_{d}\in L^{\infty}(\mu)$.

Regard $\displaystyle T_{i}$ as $\displaystyle T_{e_{i}}$ and let
$\displaystyle p_{1},\dots,p_{d}\colon\mathbb{Z}^{L}\to\mathbb{Z}^{d}$ be
polynomials given by $\displaystyle p_{i}(n)=p(n)e_{i}$. In this case, there
exists $\displaystyle q\in\mathbb{Q}\backslash\\{0\\}$ such that the set
$\displaystyle R$ defined in Theorem 5.1 is $\displaystyle
R=\\{qe_{i},q(e_{i}-e_{j})\colon 1\leq i,j\leq d,i\neq j\\}$. By assumption
(i), all the $\displaystyle T_{i}T^{-1}_{j}$’s (or $\displaystyle
T_{e_{i}-e_{j}}$’s), $\displaystyle i\neq j$ are ergodic for
$\displaystyle\mu$, and so $\displaystyle(T_{g})_{g\in
G(q(e_{i}-e_{j}))}=(T_{g})_{g\in G(e_{i}-e_{j})}$ is ergodic for
$\displaystyle\mu$. By assumption (ii),
$\displaystyle(T_{i}^{p(n)})_{n\in\mathbb{Z}^{L}}$ is ergodic for
$\displaystyle\mu$ for all $\displaystyle 1\leq i\leq d$, which implies that
$\displaystyle T_{i}$ (or $\displaystyle T_{e_{i}}$) is ergodic for
$\displaystyle\mu$. So $\displaystyle(T_{g})_{g\in G(qe_{i})}=(T_{g})_{g\in
G(e_{i})}$ is ergodic for $\displaystyle\mu$. Thus the assumptions of Theorem
5.1 are fulfilled.

By Theorem 5.1, we may assume without loss of generality that
$\displaystyle\mathbb{X}=Z_{(\mathbb{Z}^{d})^{\times}}(\mathbb{X})$ (note that
conditions (i) and (ii) remain valid when passing to a factor system). Since
$\displaystyle(T_{1}^{p(n)}\times\dots\times
T_{d}^{p_{d}(n)})_{n\in\mathbb{Z}^{L}}$ is ergodic for
$\displaystyle\mu^{\otimes d}$, a similar argument, using Theorem 2.9, as in
the proof of Theorem 1.3, yields the “if” part of this theorem.

To prove the “only if” part, assume that (20) holds for all $\displaystyle
f_{1},\dots,f_{d}\in L^{\infty}(\mu)$. If $\displaystyle T_{i}T^{-1}_{j}$ is
not ergodic for some $\displaystyle 1\leq i,j\leq d,i\neq j$, then there
exists $\displaystyle g\in L^{\infty}(\mu)$ which is not
$\displaystyle\mu$-a.e. equal to a constant such that $\displaystyle
T_{i}g=T_{j}g$. So

$\displaystyle\mathbb{E}_{n\in\mathbb{Z}^{L}}(T_{i}T^{-1}_{j})^{p(n)}g=\mathbb{E}_{n\in\mathbb{Z}^{L}}g=g\neq\int_{X}gd\mu,$

which implies that
$\displaystyle((T_{i}T^{-1}_{j})^{p(n)})_{n\in\mathbb{Z}^{L}}$ is not ergodic
for $\displaystyle\mu$, a contradiction to (i) in Proposition 5.3. This proofs
(i).

Since (20) holds, (ii) follows directly from the statement (ii) of Proposition
5.3 and the proof is complete. ∎

###  5.3. Ingredients to proving Theorem 5.1

The rest of the paper is devoted to the proof of Theorem 5.1. In order to keep
track of the coefficients of the polynomials after the iterated van der Corput
operations, we introduce the following definition:

######  Definition.

Let $\displaystyle d\in\mathbb{N}^{\ast}$ and $\displaystyle V$ denote the
collection of all finite subsets
$\displaystyle\\{u_{1},\dots,u_{k}\\}\subseteq\mathbb{Q}^{d}$ containing the
zero vector $\displaystyle{\bf 0}$. For $\displaystyle
R_{1}=\\{u_{1},\dots,u_{k}\\}\in V$ and $\displaystyle
R_{2}\subseteq\mathbb{Q}^{d}$, we say that $\displaystyle R_{1}$ is
_equivalent_ to $\displaystyle R_{2}$ (denoted as $\displaystyle R_{1}\sim
R_{2}$) if there exists $\displaystyle 1\leq i\leq k$ such that $\displaystyle
R_{2}=\\{-ru_{i},r(u_{j}-u_{i})\colon 1\leq j\leq k\\}$ for some
$\displaystyle r\in\mathbb{Q}\backslash\\{0\\}.$ Note that $\displaystyle
R_{1}\sim R_{2}$ implies that $\displaystyle R_{1}$ and $\displaystyle R_{2}$
have the same cardinality.272727 Note that $\displaystyle{\bf 0}\in R_{2}$ as
$\displaystyle r(u_{j}-u_{i})={\bf 0}$ for $\displaystyle j=i.$

######  Lemma 5.4.

The relation $\displaystyle\sim$ is an equivalence relation on $\displaystyle
V$.

###### Proof.

If $\displaystyle R_{1}=\\{u_{1},\dots,u_{k}\\}$ and $\displaystyle u_{i}=0$,
then $\displaystyle R_{1}=\\{-ru_{i},r(u_{j}-u_{i})\colon 1\leq j\leq k\\}$
for $\displaystyle r=1,$ and so $\displaystyle R_{1}\sim R_{1}$. Suppose that
$\displaystyle R_{1}\sim R_{2}$. We may write $\displaystyle
R_{1}=\\{u_{1},\dots,u_{k}\\}$ and $\displaystyle
R_{2}=\\{v_{1},\dots,v_{k}\\}$, where $\displaystyle v_{i}=-ru_{i}$ and
$\displaystyle v_{j}=r(u_{j}-u_{i})$ for all $\displaystyle 1\leq j\leq
k,j\neq i$ for some $\displaystyle 1\leq i\leq k$. It follows that
$\displaystyle u_{i}=-(1/r)v_{i}$ and $\displaystyle u_{j}=(1/r)(v_{j}-v_{i})$
which means $\displaystyle R_{2}\sim R_{1}$.