in which the eigenfunctions $u_{{\underline{h}},{\underline{h}}^{\prime}}$ are
shown to satisfy extra properties.

For the proof of Proposition 5.7, we need the following version of the Gowers-
Cauchy-Schwarz inequality. Its proof proceeds identically to the single-
transformation version given in [15, Lemma 4.3].

######  Lemma 5.8 (Twisted Gowers-Cauchy-Schwarz inequality).

Let $(X,{\mathcal{X}},\mu,T_{1},\ldots,T_{\ell})$ be a system,
$s\in{\mathbb{N}}$,
${\textbf{b}}_{1},\ldots,{\textbf{b}}_{s}\in{\mathbb{Z}}^{\ell}$ be vectors,
and the functions
$(f_{\underline{\epsilon}})_{{\underline{\epsilon}}\in\\{0,1\\}^{s}},(u_{\underline{h}})_{{\underline{h}}\in{\mathbb{N}}^{s}}\subseteq
L^{\infty}(\mu)$ be 1-bounded. Then for every $H\in{\mathbb{N}}$, we have

$\displaystyle\Big{|}{\mathbb{E}}_{{\underline{h}}\in[H]^{s}}\,\int\prod_{{\underline{\epsilon}}\in\\{0,1\\}^{s}}T^{{\textbf{b}}_{1}\epsilon_{1}h_{1}+\cdots+{\textbf{b}}_{s}\epsilon_{s}h_{s}}f_{\underline{\epsilon}}\cdot
u_{{\underline{h}}}\,d\mu\Big{|}^{2^{s}}$
$\displaystyle\leq{\mathbb{E}}_{{\underline{h}},{\underline{h}}^{\prime}\in[H]^{s}}\,\int\Delta_{{\textbf{b}}_{1},\ldots,{\textbf{b}}_{s};{\underline{h}}-{\underline{h}}^{\prime}}f_{\underline{1}}\cdot
T^{-({\textbf{b}}_{1}h_{1}^{\prime}+\cdots+{\textbf{b}}_{s}h^{\prime}_{s})}\Big{(}\prod_{{\underline{\epsilon}}\in\\{0,1\\}^{s}}\mathcal{C}^{|{\underline{\epsilon}}|}u_{{\underline{h}}^{\underline{\epsilon}}}\Big{)}\,d\mu.$

###### Proof of Proposition 5.7.

Using the assumption (54), we use respectively the identities (34), (37), and
(33), to deduce that

$\displaystyle\lim_{H\to\infty}{\mathbb{E}}_{{\underline{h}}\in[H]^{s}}\int\Delta_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}};{\underline{h}}}f\cdot
u_{\underline{h}}\,d\mu>0$

for 1-bounded functions $u_{\underline{h}}$,
${\underline{h}}\in{\mathbb{N}}^{s}$, which we can take to be:

  * •

$u_{\underline{h}}:=\overline{{\mathbb{E}}(\Delta_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}};{\underline{h}}}f|{\mathcal{I}}(T^{{\textbf{b}}_{s+1}}))}$
if $s^{\prime}=1$;

  * •

$u_{\underline{h}}\in{\mathcal{E}}(T^{{\textbf{b}}_{s+1}})$ if $s^{\prime}=2$;

or

  * •

$u_{\underline{h}}:={\mathcal{D}}_{s^{\prime},T^{{\textbf{b}}_{s+1}}}(\Delta_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}};{\underline{h}}^{\underline{\epsilon}}}f)$
for $s^{\prime}\in{\mathbb{N}}$.

From the fact that $f=\lim_{k\to\infty}{\mathbb{E}}_{n\in[N_{k}]}\,f_{n,k}$,
where the averages converge weakly, we deduce that

$\liminf_{H\to\infty}\lim_{k\to\infty}{\mathbb{E}}_{n\in[N_{k}]}\,{\mathbb{E}}_{{\underline{h}}\in[H]^{s}}\,\int\prod_{{\underline{\epsilon}}\in\\{0,1\\}^{s}\setminus\\{\underline{1}\\}}T^{{\textbf{b}}_{1}\epsilon_{1}h_{1}+\cdots+{\textbf{b}}_{s}\epsilon_{s}h_{s}}{\mathcal{C}}^{|{\underline{\epsilon}}|}f\cdot
T^{{\textbf{b}}_{1}h_{1}+\cdots+{\textbf{b}}_{s}h_{s}}f_{n,k}\cdot
u_{\underline{h}}\,d\mu>0.$

For fixed $k,n,H\in{\mathbb{N}}$, we apply Lemma 5.8 with
$f_{\underline{\epsilon}}:={\mathcal{C}}^{|{\underline{\epsilon}}|}f$ for
${\underline{\epsilon}}\in\\{0,1\\}^{s}\setminus\\{1\\}$ and
$f_{\underline{1}}:=f_{n,k}$, obtaining

$\liminf_{H\to\infty}\limsup_{k\to\infty}{\mathbb{E}}_{n\in[N_{k}]}\,{\mathbb{E}}_{{\underline{h}},{\underline{h}}^{\prime}\in[H]^{s}}\,\int\Delta_{{\textbf{b}}_{1},\ldots,{\textbf{b}}_{s};{\underline{h}}-{\underline{h}}^{\prime}}f_{n,k}\cdot
u_{{\underline{h}},{\underline{h}}^{\prime}}\,d\mu>0,$

where for ${\underline{h}},{\underline{h}}^{\prime}\in{\mathbb{N}}^{s}$ we
have

$\displaystyle
u_{{\underline{h}},{\underline{h}}^{\prime}}:=T^{-({\textbf{b}}_{1}h_{1}^{\prime}+\cdots+{\textbf{b}}_{s}h^{\prime}_{s})}\Big{(}\prod_{{\underline{\epsilon}}\in\\{0,1\\}^{s}}\mathcal{C}^{|{\underline{\epsilon}}|}u_{{\underline{h}}^{\underline{\epsilon}}}\Big{)}.$

If the functions $u_{\underline{h}}$ are $T^{{\textbf{b}}_{s+1}}$-invariant,
then so are $u_{{\underline{h}},{\underline{h}}^{\prime}}$. If
$u_{\underline{h}}\in{\mathcal{E}}(T^{{\textbf{b}}_{s+1}})$, then
$u_{{\underline{h}},{\underline{h}}^{\prime}}\in{\mathcal{E}}(T^{{\textbf{b}}_{s+1}})$
as well. Lastly, if
$u_{{\underline{h}}}:={\mathcal{D}}_{s^{\prime},T^{{\textbf{b}}_{s+1}}}(\Delta_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}};{\underline{h}}}f)$,
then

$\displaystyle u_{{\underline{h}},{\underline{h}}^{\prime}}$
$\displaystyle=\prod_{{\underline{\epsilon}}\in\\{0,1\\}^{s^{\prime}}}\mathcal{C}^{|{\underline{\epsilon}}|}T^{-({\textbf{b}}_{1}h_{1}^{\prime}+\cdots+{\textbf{b}}_{s}h^{\prime}_{s})}{\mathcal{D}}_{s^{\prime},T^{{\textbf{b}}_{s+1}}}(\Delta_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}};{\underline{h}}^{\underline{\epsilon}}}f)$
$\displaystyle=\prod_{{\underline{\epsilon}}\in\\{0,1\\}^{s^{\prime}}}{\mathcal{D}}_{s^{\prime},T^{{\textbf{b}}_{s+1}}}\Big{(}\mathcal{C}^{|{\underline{\epsilon}}|}T^{-({\textbf{b}}_{1}h_{1}^{\prime}+\cdots+{\textbf{b}}_{s}h^{\prime}_{s})}\Delta_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}};{\underline{h}}^{\underline{\epsilon}}}f\Big{)}.$

The result follows from the fact that the limsup of a sum is at most the sum
of the limsups. ∎

###  5.6. Removing low complexity functions

In the final stage of the degree lowering argument in Section 7, we need the
following result which roughly speaking states that if a positivity property
holds when the average of
$\Delta_{{\textbf{b}}_{1},\ldots,{\textbf{b}}_{s};{\underline{h}}}f$ is
twisted by a product of low complexity functions, then these functions can be
removed. Lemma 5.9 is a generalisation of [15, Lemma 3.4]; although its
statement differs somewhat from the result in [15], where we just assume that
the vectors ${\textbf{b}}_{1}=\cdots={\textbf{b}}_{s}$ are the same and
$b_{j,{\underline{h}},H}$ are sequences of numbers rather than functions, the
proof requires only marginal changes so we omit it.

######  Lemma 5.9 (Removing low-complexity functions).

Let $\ell,s\in{\mathbb{N}}$, $(X,{\mathcal{X}},\mu,T_{1},\ldots,T_{\ell})$ be
a system, $f\in L^{\infty}(\mu)$ be a function and
${\textbf{b}}_{1},\ldots,{\textbf{b}}_{s}\in{\mathbb{Z}}^{\ell}$ be vectors.
For $j\in[s]$, $H\in{\mathbb{N}}$ and ${\underline{h}}\in[H]^{s}$, let
$b_{j,{\underline{h}},H}\in L^{\infty}(\mu)$ be 1-bounded functions such that
the sequence of functions ${\underline{h}}\mapsto b_{j,{\underline{h}},H}$
does not depend on the variable $h_{j}$. If

(55)
$\displaystyle\limsup_{N\to\infty}\left\|{\mathbb{E}}_{{\underline{h}}\in[H]^{s}}\,\prod_{j=1}^{s}b_{j,{\underline{h}},H}\cdot\Delta_{{\textbf{b}}_{1},\ldots,{\textbf{b}}_{s};{\underline{h}}}f\right\|_{L^{2}(\mu)}>0,$

then
$\lvert\\!|\\!|f|\\!|\\!\rvert_{{\textbf{b}}_{1},\ldots,{\textbf{b}}_{s}}>0$.

We will apply Lemma 5.9 in Section 7.4 with
${\textbf{b}}_{1}=\cdots={\textbf{b}}_{s-1}:={\mathbf{e}}_{m}$ and
${\textbf{b}}_{s}:=r{\mathbf{e}}_{m}$ for some $r\in{\mathbb{N}}$.

More generally, we could replace
$\Delta_{{\textbf{b}}_{1},\ldots,{\textbf{b}}_{s};{\underline{h}}}f$ in (55)
by
$\prod_{{\underline{\epsilon}}\in\\{0,1\\}^{s}}T^{{\textbf{b}}_{1}\epsilon_{1}h_{1}+\cdots+{\textbf{b}}_{s}\epsilon_{s}h_{s}}f_{\underline{\epsilon}}$
for 1-bounded functions $f_{\underline{\epsilon}}$, in which case the
conclusion
$\lvert\\!|\\!|f_{\underline{1}}|\\!|\\!\rvert_{{\textbf{b}}_{1},\ldots,{\textbf{b}}_{s}}>0$
would follow. From this standpoint, Lemma 5.9 can be viewed as yet another
refinement of the Gowers-Cauchy-Schwarz inequality (28). We do not need this
strengthening for our arguments, though, therefore we present the result in a
simpler version that actually gets applied.

##  6\. Proof of Theorem 2.2 and 2.6 \- Reduction to a degree lowering
property

In this section we prove Theorem 2.6, which as we remarked before, implies
Theorem 2.2. We omit the proof of Theorem 2.5 since it is similar, in fact
simpler, as it does not require the use of Proposition 4.5.

###  6.1. Preparation for induction

In order to prove Theorem 2.6, it will be more convenient to prove the
following statement.

######  Proposition 6.1.

Let $a_{1},\ldots,a_{\ell}\colon{\mathbb{N}}\to{\mathbb{Z}}$ be good for
seminorm control and irrational equidistribution for the system
$(X,{\mathcal{X}},\mu,T_{1},\ldots,T_{\ell})$, and suppose that equation (8)
holds when all but one of the functions are rational $T_{j}$-eigenfunctions.
Then for every $m\in\\{0,1,\ldots,\ell\\}$, the following property holds:

$(P_{m})$ If $(X,{\mathcal{X}},\mu,T_{1},\ldots,T_{\ell})$ is a system and
$f_{1},\ldots,f_{\ell}\in L^{\infty}(\mu)$ are such that
$f_{j}\in{\mathcal{E}}(T_{j})$ for $j=m+1,\ldots,\ell$, then

(56) $\lim_{N\to\infty}{\mathbb{E}}_{n\in[N]}\,T_{1}^{a_{1}(n)}f_{1}\cdots
T_{\ell}^{a_{\ell}(n)}f_{\ell}=0$

in $L^{2}(\mu)$ if ${\mathbb{E}}(f_{j}|{{\mathcal{K}}_{\text{\rm
rat}}}(T_{j}))=0$ for some $j\in[\ell]$.

Note that the implication (i)$\implies$(ii) of Theorem 2.6 follows immediately
from the case $m=\ell$. In what follows, we are going to show that property
$(P_{m})$ holds by (finite) induction on $m\in\\{0,1,\ldots,\ell\\}$. Since
the collection of sequences $a_{1},\ldots,a_{\ell}$ is good for irrational
equidistribution, we get by Lemma 2.4 (iii) that property $(P_{0})$ holds.

So we fix $m\in[\ell]$ and assume that property $(P_{m-1})$ holds. We are
going to show that property $(P_{m})$ holds. We briefly sketch how $(P_{m})$
is derived from $(P_{m-1})$ before delving into the full proof. Suppose that
(56) fails. Then it also fails when $f_{m}$ is replaced by a more structured
function $\tilde{f}_{m}$ defined as in (49). The assumption that the sequences
$a_{1},\ldots,a_{m}$ are good for seminorm control implies that
$\lvert\\!|\\!|\tilde{f}_{m}|\\!|\\!\rvert_{s+1,T_{m}}>0$ for some
$s\in{\mathbb{N}}$ (Proposition 6.3). Then the degree lowering argument shows
that $\lvert\\!|\\!|\tilde{f}_{m}|\\!|\\!\rvert_{s,T_{m}}>0$ as long as $s\geq
2$. This result, stated as Proposition 6.4, is the most technically engaging
part of the proof, and its proof is carried out in detail in Section 7.
Iterating, we deduce that
$\lvert\\!|\\!|\tilde{f}_{m}|\\!|\\!\rvert_{2,T_{m}}>0$. We then derive the
property $(P_{m})$ using this fact, the definition of $\tilde{f}_{m}$ and the
property $(P_{m-1})$.

###  6.2. Property $(P_{m-1})$ for the ergodic components

We will deduce from property $(P_{m-1})$ and Corollary 5.5 (ii) the following
result that is better suited for our purposes since it provides information
for the ergodic components of the system (with respect to the transformation
$T_{m}$). It will be used in Section 7.2 while carrying out the degree
lowering argument.

######  Proposition 6.2.

Let $(X,{\mathcal{X}},\mu,T_{1},\ldots,T_{\ell})$ be a system and for some
$m\in[\ell]$ let $\mu=\int\mu_{m,x}\,d\mu$ be the ergodic decomposition with
respect to the transformation $T_{m}$. Let also
$a_{1},\ldots,a_{\ell}\colon{\mathbb{N}}\to{\mathbb{Z}}$ be sequences and
${\mathcal{F}}$ be a countable subset of $L^{\infty}(\mu)$. Then every
sequence $N_{k}\to\infty$ has a subsequence $N_{k}^{\prime}\to\infty$ such
that for some $L_{K}\to\infty$, the following properties hold:

  1. (i)

For $\mu$-a.e. $x\in X$, the limit

(57)
$\lim_{K\to\infty}{\mathbb{E}}_{k\in[L_{K}]}{\mathbb{E}}_{n\in[N_{k}^{\prime}]}\,T_{1}^{a_{1}(n)}f_{1}\cdots
T_{\ell}^{a_{\ell}(n)}f_{\ell}$

exists in $L^{2}(\mu_{m,x})$ for all $f_{j}\in{\mathcal{F}}$, $j\in[\ell]$,
$j\neq m$, and $f_{m}\in L^{\infty}(\mu_{m,x})$.

  2. (ii)

If furthermore property $(P_{m-1})$ of Proposition 6.1 holds, then for
$\mu$-a.e. $x\in X$, the limit in (57) is $0$ in $L^{2}(\mu_{m,x})$ for all
$f_{j}\in{\mathcal{F}}$, $j\in[\ell]$, $j\neq m$, with
$f_{j}\in{\mathcal{E}}(T_{j})$ for $j=m+1,\ldots,\ell$, and
$f_{m}\in{\mathcal{E}}(T_{m},\mu_{m,x})$ that satisfies
${\mathbb{E}}(f_{m}|{{\mathcal{K}}_{\text{\rm rat}}}(T_{m},\mu_{m,x}))=0$.

######  Remark.

It is important that the subsequence $N_{k}^{\prime}$ and $L_{K}$ can be
chosen independently of the functions $f_{m}\in{\mathcal{E}}(T_{m},\mu_{m,x})$
with ${\mathbb{E}}(f_{m}|{{\mathcal{K}}_{\text{\rm rat}}}(T_{m},\mu_{m,x}))=0$
(as $x$ varies this is an uncountable collection of functions so we do not get
this independence for free). Note also that for $j\neq m$ the transformation
$T_{j}$ may not preserve the measures $\mu_{m,x}$ and this makes it difficult
to prove this result by working on every ergodic component separately.

###### Proof.

Let $N_{k}\to\infty$ be a sequence. The existence of a subsequence
$N_{k}^{\prime}\to\infty$ of $N_{k}$ and $L_{K}\to\infty$ so that property (i)
is satisfied follows from Corollary 5.5 (ii). We claim that this
$N_{k}^{\prime}$ and $L_{K}\to\infty$ also satisfies property $(ii)$.

Let $\\{\chi_{l}\colon l\in{\mathbb{N}}\\}$, be a relative orthonormal basis
of nonergodic eigenfunctions for ${\mathcal{Z}}_{1}(\mu,T_{m})$ given by
Proposition 4.1.151515The orthonormality is not needed for this argument, only
their density properties will be used. Then $\chi_{l}\in{\mathcal{E}}(T_{m})$
and $\chi_{l}-{\mathbb{E}}(\chi_{l}|{{\mathcal{K}}_{\text{\rm
rat}}}(T_{m},\mu))\in{\mathcal{E}}(T_{m})$, and this difference is orthogonal
to ${{\mathcal{K}}_{\text{\rm rat}}}(T_{m},\mu)$. We deduce from Property
$(P_{m-1})$ that

$\lim_{N\to\infty}\left\|{\mathbb{E}}_{n\in[N]}\,T_{m}^{a_{m}(n)}\big{(}\chi_{l}-{\mathbb{E}}(\chi_{l}|{{\mathcal{K}}_{\text{\rm
rat}}}(T_{m},\mu))\big{)}\cdot\prod_{j\in[\ell],j\neq
m}T_{j}^{a_{j}(n)}f_{j}\right\|_{L^{2}(\mu)}=0$

for every $l\in{\mathbb{N}}$ and $f_{j}\in{\mathcal{F}}$, $j\in[\ell]$, $j\neq
m$, with $f_{j}\in{\mathcal{E}}(T_{j})$ for $j=m+1,\ldots,\ell$. Since linear
combinations of functions of the form
$\\{\psi\cdot\chi_{l}\colon\,l\in{\mathbb{N}},\psi\in{\mathcal{I}}(T_{m})\\}$
are dense in $L^{2}({\mathcal{Z}}_{1}(T_{m},\mu))$, using linearity we deduce
that

$\lim_{N\to\infty}\left\|{\mathbb{E}}_{n\in[N]}\,T_{m}^{a_{m}(n)}\big{(}f_{m}-{\mathbb{E}}(f_{m}|{{\mathcal{K}}_{\text{\rm
rat}}}(T_{m},\mu))\big{)}\cdot\prod_{j\in[\ell],j\neq
m}T_{j}^{a_{j}(n)}f_{j}\right\|_{L^{2}(\mu)}=0$

for all $f_{m}\in L^{\infty}({\mathcal{Z}}_{1}(T_{m},\mu))$ and
$f_{j}\in{\mathcal{F}}$, $j\in[\ell]$, $j\neq m$, with
$f_{j}\in{\mathcal{E}}(T_{j})$ for $j=m+1,\ldots,\ell$. By Lemma 3.3 there
exists a countable set of functions ${\mathcal{A}}$ in
$L^{\infty}({\mathcal{Z}}_{1}(T_{m},\mu))$ such that

(58) $\text{ for }\mu\text{-a.e. }x\in X\text{ the set }{\mathcal{A}}\text{ is
dense in }L^{2}({\mathcal{Z}}_{1}(T,\mu_{m,x})).$

The mean convergence property for $N_{k}^{\prime}$ and $L_{K}\to\infty$ gives
that for $\mu$-a.e. $x\in X$, we have

(59)
$\lim_{K\to\infty}\left\|{\mathbb{E}}_{k\in[L_{K}]}{\mathbb{E}}_{n\in[N_{k}^{\prime}]}\,T_{m}^{a_{m}(n)}\big{(}f_{m}-{\mathbb{E}}(f_{m}|{{\mathcal{K}}_{\text{\rm
rat}}}(T_{m},\mu))\big{)}\cdot\prod_{j\in[\ell],j\neq
m}T_{j}^{a_{j}(n)}f_{j}\right\|_{L^{2}(\mu_{m,x})}=0$

for all $f_{m}\in{\mathcal{A}}$ and $f_{j}\in{\mathcal{F}}$, $j\in[\ell]$,
$j\neq m$, with $f_{j}\in{\mathcal{E}}(T_{j})$ for $j=m+1,\ldots,\ell$. Recall
that for every system $(X,{\mathcal{X}},\mu,T)$ with ergodic decomposition
$\mu=\int\mu_{x}\,d\mu$ and $f\in L^{2}(\mu)$, we have
${\mathbb{E}}(f|{{\mathcal{K}}_{\text{\rm
rat}}}(T,\mu))={\mathbb{E}}(f|{{\mathcal{K}}_{\text{\rm rat}}}(T,\mu_{x}))$
for $\mu$-a.e. $x\in X$.161616One way to see this is to note that by the
pointwise ergodic theorem (applied first for the system
$(X,{\mathcal{X}},\mu,T)$ and then for the systems
$(X,{\mathcal{X}},\mu_{x},T)$) , we have for $\mu$-a.e. $x^{\prime}\in X$ that
${\mathbb{E}}(f|{{\mathcal{K}}_{\text{\rm
rat}}}(T,\mu))(x^{\prime})=\lim_{r\to\infty}\lim_{N\to\infty}{\mathbb{E}}_{n\in[N]}f(T^{r!n}x^{\prime})={\mathbb{E}}(f|{{\mathcal{K}}_{\text{\rm
rat}}}(T,\mu_{x}))(x^{\prime})$ for $\mu$-a.e. $x\in X$. As a consequence, in
(59) we can replace the measure $\mu$ with $\mu_{m,x}$ in the conditional
expectation, and we get for $\mu$-a.e. $x\in X$ that

(60)
$\lim_{K\to\infty}\left\|{\mathbb{E}}_{k\in[L_{K}]}{\mathbb{E}}_{n\in[N_{k}^{\prime}]}\,T_{m}^{a_{m}(n)}\big{(}f_{m}-{\mathbb{E}}(f_{m}|{{\mathcal{K}}_{\text{\rm
rat}}}(T_{m},\mu_{m,x}))\big{)}\cdot\prod_{j\in[\ell],j\neq
m}T_{j}^{a_{j}(n)}f_{j}\right\|_{L^{2}(\mu_{m,x})}=0$

for all $f_{m}\in{\mathcal{A}}$ and $f_{j}\in{\mathcal{F}}$, $j\in[\ell]$,
$j\neq m$, with $f_{j}\in{\mathcal{E}}(T_{j})$ for $j=m+1,\ldots,\ell$. Using
(58) and an approximation argument in $L^{2}(\mu_{m,x})$, we deduce that for
$\mu$-a.e. $x\in X$, we have

(61)
$\lim_{K\to\infty}\left\|{\mathbb{E}}_{k\in[L_{K}]}{\mathbb{E}}_{n\in[N_{k}^{\prime}]}\,T_{m}^{a_{m}(n)}\big{(}f_{m}-{\mathbb{E}}(f_{m}|{{\mathcal{K}}_{\text{\rm
rat}}}(T_{m},\mu_{m,x}))\big{)}\cdot\prod_{j\in[\ell],j\neq
m}T_{j}^{a_{j}(n)}f_{j}\right\|_{L^{2}(\mu_{m,x})}=0$

for all $f_{m}\in L^{\infty}({\mathcal{Z}}_{1}(T_{m},\mu_{x}))$ and
$f_{j}\in{\mathcal{F}}$, $j\in[\ell]$, $j\neq m$, with
$f_{j}\in{\mathcal{E}}(T_{j})$ for $j=m+1,\ldots,\ell$. Since
${\mathcal{E}}(T_{m},\mu_{m,x})\subset{\mathcal{Z}}_{1}(T_{m},\mu_{m,x})$,
this immediately implies the asserted statement. ∎

###  6.3. Replacing $f_{m}$ with an averaged function

Using Proposition 5.6, we deduce the following result.

######  Proposition 6.3.

Let $a_{1},\ldots,a_{\ell}\colon{\mathbb{N}}\to{\mathbb{Z}}$ be sequences that
are good for seminorm control for the system
$(X,{\mathcal{X}},\mu,T_{1},\ldots,T_{\ell})$. Then there exists
$s\in{\mathbb{N}}$ such that the following holds: if $m\in[\ell]$ and (47)
holds for some $f_{1},\ldots,f_{\ell}\in L^{\infty}(\mu)$ with
$f_{j}\in{\mathcal{E}}(T_{j})$ for $j=m+1,\ldots,\ell$, then

(62) $\lvert\\!|\\!|\tilde{f}_{m}|\\!|\\!\rvert_{s+1,T_{m}}>0$

where $\tilde{f}_{m}$ is given by (49) (as a weak limit) and satisfies (50).

###  6.4. Reduction to a degree lowering property

Our new goal is to establish the following statement:

######  Proposition 6.4.

Let $a_{1},\ldots,a_{\ell}\colon{\mathbb{N}}\to{\mathbb{Z}}$ be good for
seminorm control and irrational equidistribution for the system
$(X,{\mathcal{X}},\mu,T_{1},\ldots,T_{\ell})$. Suppose that property
$(P_{m-1})$ of Proposition 6.1 holds for some $m\in[\ell]$ and $\tilde{f}_{m}$
is given by (49) and satisfies (50) where all functions involved are
$1$-bounded and $f_{j}\in{\mathcal{E}}(T_{j})$ for $j=m+1,\ldots,\ell$.
Lastly, suppose that $\lvert\\!|\\!|\tilde{f}_{m}|\\!|\\!\rvert_{s+1,T_{m}}>0$
for some $s\geq 2$. Then
$\lvert\\!|\\!|\tilde{f}_{m}|\\!|\\!\rvert_{s,T_{m}}>0$.

Before proceeding to the proof of Proposition 6.4 we show how it can be used
to prove Proposition 6.1 (and hence Theorem 2.2).

###### Proof of Proposition 6.1 assuming Proposition 6.4.

Since $a_{1},\ldots,a_{\ell}$ are good for irrational equidistribution we get
by Lemma 2.4 (ii) that property $P_{0}$ holds. So we fix $m\in[\ell]$ and we
assume that property $(P_{m-1})$ holds. Our goal is to show that property
$(P_{m})$ also holds.

Arguing by contradiction, suppose that (56) does not hold, or equivalently,
that (47) holds for some $f_{1},\ldots,f_{\ell}\in L^{\infty}(\mu)$ where
$f_{j}\in{\mathcal{E}}(T_{j})$ for $j=m+1,\ldots,\ell$. It suffices to show
that then ${\mathbb{E}}(f_{j}|{\mathcal{K}}_{rat}(T_{j}))\neq 0$ for
$j\in[\ell]$.

Since the sequences $a_{1},\ldots,a_{\ell}$ are good for seminorm control, by
combining Proposition 6.3 and (47), we deduce that if $\tilde{f}_{m}$ is as in
(49), then $\lvert\\!|\\!|\tilde{f}_{m}|\\!|\\!\rvert_{s+1,T_{m}}>0$ for some
$s\in{\mathbb{N}}$. Iterating the conclusion of Proposition 6.4 exactly $s-1$
times, we get $\lvert\\!|\\!|\tilde{f}_{m}|\\!|\\!\rvert_{2,T_{m}}>0$. This
implies using Proposition 4.3 that

$\int\tilde{f}_{m}\cdot\chi_{m}\,d\mu>0$

for some $\chi_{m}\in\mathcal{E}(T_{m})$. We use the definition (49) of
$\tilde{f}_{m}$, compose with $T_{m}^{a_{m}(n)}$, use that the transformations
$T_{1},\ldots,T_{\ell}$ commute, and then the Cauchy-Schwarz inequality. We
deduce that

$\limsup_{k\to\infty}\left\|{\mathbb{E}}_{n\in[N_{k}]}\,T_{m}^{a_{m}(n)}\chi_{m}\prod_{j\in[\ell],j\neq
m}T_{j}^{a_{j}(n)}f_{j}\right\|_{L^{2}(\mu)}>0.$

Since $\chi_{m}\in{\mathcal{E}}(T_{m})$ and $f_{j}\in{\mathcal{E}}(T_{j})$,
$j=m+1,\ldots,\ell$, property $(P_{m-1})$ applies and gives that
${\mathbb{E}}(f_{j}|{\mathcal{K}}_{rat}(T_{j}))\neq 0$ for $j\in[\ell]$,
$j\neq m$.

It remains to show that ${\mathbb{E}}(f_{m}|{{\mathcal{K}}_{\text{\rm
rat}}}(T_{m}))\neq 0$. To do this, we first note that we have established the
$L^{2}(\mu)$-identity (56) for all $f_{1},\ldots,f_{\ell}\in L^{\infty}(\mu)$
that satisfy ${\mathbb{E}}(f_{j}|{{\mathcal{K}}_{\text{\rm rat}}}(T_{j}))=0$
for some $j\in[\ell]$, $j\neq m$ and $f_{j}\in{\mathcal{E}}(T_{j})$ for
$j=m+1,\ldots,\ell$. We claim that under the same restrictions on
$f_{1},\ldots,f_{\ell}$, we have the identity

(63)
$\lim_{N\to\infty}\left\|{\mathbb{E}}_{n\in[N]}\,\prod_{j\in[\ell]}T_{j}^{a_{j}(n)}f_{j}-{\mathbb{E}}_{n\in[N]}\,T_{m}^{a_{m}(n)}f_{m}\prod_{j\in[\ell],j\neq
m}T_{j}^{a_{j}(n)}{\mathbb{E}}(f_{j}|{{\mathcal{K}}_{\text{\rm
rat}}}(T_{j}))\right\|_{L^{2}(\mu)}=0.$

To see this, we first note that if $(X,{\mathcal{X}},\mu,T)$ is a system and
$f\in{\mathcal{E}}(T)$, then by Lemma 4.6 we have
$f-{\mathbb{E}}(f|{{\mathcal{K}}_{\text{\rm rat}}}(T))\in{\mathcal{E}}(T)$ and
trivially ${\mathbb{E}}\big{(}f-{\mathbb{E}}(f|{{\mathcal{K}}_{\text{\rm
rat}}}(T))|{{\mathcal{K}}_{\text{\rm rat}}}(T)\big{)}=0$. Hence, identity (56)
implies that (63) holds if for $j\in[\ell],j\neq m$, we replace at least one
of the functions $f_{j}\in L^{\infty}(\mu)$ with
$f_{j}-{\mathbb{E}}(f_{j}|{{\mathcal{K}}_{\text{\rm rat}}}(T))$. Using this
property, writing $f_{j}={\mathbb{E}}(f_{j}|{{\mathcal{K}}_{\text{\rm
rat}}}(T_{j}))+(f_{j}-{\mathbb{E}}(f_{j}|{{\mathcal{K}}_{\text{\rm
rat}}}(T_{j})))$ for $j\neq m$, and expanding the product, we deduce that (63)
holds. Combining (47) and (63) we get that (47) holds for some $f_{j}\in
L^{\infty}(\mu)$ with $f_{j}\in{{\mathcal{K}}_{\text{\rm rat}}}(T_{j})$ for
$j\in[\ell]$, $j\neq m$.

Using $L^{2}(\mu)$-approximation and linearity, we conclude that (47) holds
for some rational $T_{j}$-eigenfunctions $f_{j}$, $j\in[\ell]$, $j\neq m$. Our
assumption that (63) holds when all but one of the functions are
$T_{j}$-rational eigenfunctions implies that
${\mathbb{E}}(f_{m}|{{\mathcal{K}}_{\text{\rm rat}}}(T_{m}))\neq 0$,
completing the proof. ∎

We remark that in order to prove Theorem 2.5 (which implies Theorem 2.1),
there is a small difference in the last step where we want to prove the result
for $\ell=1$, since this case is not covered by our assumption as was the case
with Theorem 2.6 . The case $\ell=1$ follows from the $\ell=1$ case of the
corresponding result in [15, Theorem 1.1] (and is done by a further degree
lowering argument).

##  7\. Proof of Theorems 2.2 and 2.6 \- Proof of the degree lowering property

The goal of this subsection is to complete the proof of Theorem 2.6 by proving
Proposition 6.4. The proof goes as follows. Starting with the assumption
$\lvert\\!|\\!|\tilde{f}_{m}|\\!|\\!\rvert_{s+1,T_{m}}>0$, we deduce that
$\lvert\\!|\\!|\tilde{f}_{m}|\\!|\\!\rvert_{s+1,T_{m},\mu_{m,x}}>0$ on a set
$A$ of positive measure. We then use Proposition 4.5 to deduce that for $x\in
A$ and a positive lower density set of values
${\underline{h}}\in{\mathbb{N}}^{s-1}$, the multiplicative derivative
$\Delta_{s-1,T_{m};{\underline{h}}}\tilde{f}_{m}$ correlates with an
eigenfunction $\chi_{m,{\underline{h}},x}$ whose eigenvalue is either
irrational or rational with denominator bounded by $r_{x}$. Using a version of
the dual-difference interchange argument and invoking property $(P_{m-1})$ (in
the version given by Proposition 6.2), we deduce that for many values
${\underline{h}}$, the eigenfunction $\chi_{m,{\underline{h}},x}$ is a product
of low complexity functions and a term invariant under $T_{m}^{r_{x}!}$. We
then use Lemma 5.9 to conclude that
$\lvert\\!|\\!|\tilde{f}_{m}|\\!|\\!\rvert_{s,T_{m},\mu_{x}}>0$ for $x\in A$,
and the claim $\lvert\\!|\\!|\tilde{f}_{m}|\\!|\\!\rvert_{s,T_{m}}>0$ follows
from the formula (23) and the fact that $A$ has positive measure.

###  7.1. Ergodic decomposition and inverse theorem

Let

$\mu=\int\mu_{m,x}\,d\mu$

be the ergodic decomposition of $\mu$ with respect to the transformation
$T_{m}$. Then the measures $\mu_{m,x}$ are $T_{m}$-invariant (but not
necessarily $T_{j}$-invariant for $j\neq m$) and the systems
$(X,{\mathcal{X}},\mu_{m,x},T_{m})$ are ergodic for $\mu$-a.e. $x\in X$.

Since $\lvert\\!|\\!|\tilde{f}_{m}|\\!|\\!\rvert_{s+1,T_{m}}>0$, there exists
a $\mu$-measurable $T$-invariant set $A$ with $\mu(A)>0$ such that for every
$x\in A$, we have

$\lvert\\!|\\!|\tilde{f}_{m}|\\!|\\!\rvert_{s+1,T_{m},\mu_{m,x}}>0.$

Using Proposition 4.5, we get that for every $x\in A$, there exist
$r_{x}=r_{\mu_{m,x}}\in{\mathbb{N}}$ and
$\chi_{m,{\underline{h}},x}=\chi_{m,{\underline{h}},\mu_{m,x}}\in{\mathcal{E}}(T_{m},\mu_{m,x})$
such that171717It is important that $r_{x}$ does not depend on
${\underline{h}},{\underline{h}}^{\prime}\in{\mathbb{N}}$ and
$r_{x},\chi_{m,{\underline{h}},{\underline{h}}^{\prime},x}$ depend only on the
measure $\mu_{m,x}$ and not on $x$ itself.

(64)
${\mathbb{E}}(\chi_{m,{\underline{h}},{\underline{h}}^{\prime},x}|{{\mathcal{K}}_{\text{\rm
rat}}}(T_{m},\mu_{m,x}))={\mathbb{E}}(\chi_{m,{\underline{h}},{\underline{h}}^{\prime},x}|{\mathcal{K}}_{r_{x}}(T_{m},\mu_{m,x}))$

for all ${\underline{h}},{\underline{h}}^{\prime}\in{\mathbb{N}}^{s-1}$ and
$x\in X$, where $\chi_{m,{\underline{h}},{\underline{h}}^{\prime},x}$ are as
in (70) below and

(65)
$\liminf_{H\to\infty}{\mathbb{E}}_{{\underline{h}}\in[H]^{s-1}}\int\Delta_{s-1,T_{m};{\underline{h}}}\tilde{f}_{m}\cdot\chi_{m,{\underline{h}},x}\,d\mu_{m,x}>0.$

We deduce that there exist $a=a_{\mu_{m,x}}>0$ and a subset
$\Lambda=\Lambda_{\mu_{m,x}}$ of ${\mathbb{N}}^{s-1}$ with positive lower
density, such that

(66)
$\int\Delta_{s-1,T_{m};{\underline{h}}}\tilde{f}_{m}\cdot\chi_{m,{\underline{h}},x}\,d\mu_{m,x}>a$

for all ${\underline{h}}\in\Lambda$.

###  7.2. Passing information to the ergodic components - Choice of
$(N_{k}^{\prime})$

Recall that we work under the assumption that property $(P_{m-1})$ of
Proposition 6.1 holds for some $m\in[\ell]$. Let $N_{k}$ be a sequence that
defines $\tilde{f}_{m}$ as in (49), so that

$\displaystyle\tilde{f}_{m}=\lim_{k\to\infty}{\mathbb{E}}_{n\in[N_{k}]}\,f_{n,k}$

weakly, and

(67) $f_{n,k}:=T_{m}^{-a_{m}(n)}\bar{g}_{k}\cdot\prod_{j\in[\ell],j\neq
m}T_{j}^{a_{j}(n)}T_{m}^{-a_{m}(n)}\bar{f}_{j}.$

We consider the following countable collection of $1$-bounded functions in
$L^{\infty}(\mu)$:

${\mathcal{F}}:=\\{\Delta_{s-1,T_{m};{\underline{h}}}f_{j}:\,j\in[\ell],\
j\neq m,\ {\underline{h}}\in{\mathbb{Z}}^{s-1}\\}.$

We claim that we can pick a subsequence $N_{k}^{\prime}\to\infty$ of $N_{k}$
and $L_{K}\to\infty$ such that the following properties hold:

  1. (i)

For $\mu$-a.e. $x\in X$, we have191919In order to deduce this property weak
convergence in $L^{2}(\mu)$ does not suffice, we need mean convergence, which
is one of the reasons why we use the extra averaging over $k$ that enables to
deduce mean convergence. There are also several other places in the argument
when mean convergence is needed.

(68)
$\tilde{f}_{m}=\lim_{K\to\infty}{\mathbb{E}}_{k\in[L_{K}]}{\mathbb{E}}_{n\in[N_{k}^{\prime}]}\,f_{n,k}\
\text{ in }L^{2}(\mu_{m,x}).$

  2. (ii)

For $\mu$-a.e. $x\in X$, the limit

$\lim_{K\to\infty}{\mathbb{E}}_{k\in[L_{K}]}{\mathbb{E}}_{n\in[N_{k}^{\prime}]}\,T_{1}^{a_{1}(n)}f^{\prime}_{1}\cdots
T_{\ell}^{a_{\ell}(n)}f^{\prime}_{\ell}\,\text{ exists in }L^{2}(\mu_{m,x})$

for all $f^{\prime}_{j}\in{\mathcal{F}}$, $j\in[\ell]$, $j\neq m$, and
$f^{\prime}_{m}\in L^{\infty}(\mu_{m,x})$, and it is $0$ if in addition we
have $f^{\prime}_{j}\in{\mathcal{E}}(T_{j})$ for $j=m+1,\ldots,\ell$ and
$f^{\prime}_{m}\in{\mathcal{E}}(T_{m},\mu_{m,x})$ satisfies
${\mathbb{E}}(f^{\prime}_{m}|{{\mathcal{K}}_{\text{\rm
rat}}}(T_{m},\mu_{m,x}))=0$.

To see this, we first use both parts of Proposition 6.2 to pick a subsequence
$N_{k}^{\prime}\to\infty$ of $N_{k}$ and $L_{K}\to\infty$, such that property
(ii) holds. Using Proposition 5.6, and specifically the identity (50), we
obtain (68) but with convergence taking place in $L^{2}(\mu)$ rather than
$L^{2}(\mu_{m,x})$. By passing to a subsequence $L_{K}^{\prime}\to\infty$ of
$L_{K}$ we get (as in the proof of Corollary 5.5 (ii)) that property (i) holds
(now with convergence taking place in $L^{2}(\mu_{m,x})$) with
$L_{K}^{\prime}$ in place of $L_{K}$. Since $L_{K}^{\prime}$ is a subsequence
of $L_{K}$, then also property (ii) holds with $L_{K}^{\prime}$ in place of
$L_{K}$. After renaming $L_{K}^{\prime}$ as $L_{K}$ we get both claimed
properties.

###  7.3. Dual-difference interchange and use of $(P_{m-1})$

Combining the identity

$\Delta_{s-1,T_{m};{\underline{h}}}\tilde{f}_{m}=\prod_{{\underline{\epsilon}}\in\\{0,1\\}^{s}}\mathcal{C}^{|{\underline{\epsilon}}|}T_{m}^{{\underline{\epsilon}}\cdot\underline{h}}\tilde{f}_{m}$

with (66) and (68), we deduce that for $\mu$-a.e. $x\in A$, we have

$\liminf_{H\to\infty}\lim_{K\to\infty}{\mathbb{E}}_{k\in[L_{K}]}{\mathbb{E}}_{n\in[N^{\prime}_{k}]}\\\
{\mathbb{E}}_{{\underline{h}}\in[H]^{s-1}}\,\int\prod_{{\underline{\epsilon}}\in\\{0,1\\}^{s}\setminus\\{\underline{1}\\}}\mathcal{C}^{|{\underline{\epsilon}}|}T_{m}^{{\underline{\epsilon}}\cdot\underline{h}}\tilde{f}_{m}\cdot
T_{m}^{h_{1}+\cdots+h_{s-1}}f_{n,k}\cdot\chi_{m,{\underline{h}},x}\cdot{\bf
1}_{\Lambda}(\underline{h})\,d\mu_{m,x}>0.$

We will use Lemma 5.8 with $s-1$ in place of $s$, for $T:=T_{m}$,
$\mu:=\mu_{m,x}$, $f_{\underline{1}}:=f_{n,k}$,
$f_{\underline{\epsilon}}:=\mathcal{C}^{|{\underline{\epsilon}}|}\tilde{f}_{m}$
for ${\underline{\epsilon}}\in\\{0,1\\}^{s-1}\setminus\underline{1}$, and
$g_{\underline{h}}:=\chi_{m,{\underline{h}},x}\cdot{\bf
1}_{\Lambda}(\underline{h})$, in order to upper bound the average over
${\underline{h}},{\underline{h}}^{\prime}$. We deduce that202020We cannot
claim that the limit as $K\to\infty$ below exists because $f_{n,k}$ depend on
the functions $g_{k}$ and we have not bothered to treat mean convergence in
such cases. This is not going to create any trouble for us since the functions
$g_{k}$ will be eliminated on the next step.

$\liminf_{H\to\infty}\limsup_{K\to\infty}{\mathbb{E}}_{k\in[L_{K}]}{\mathbb{E}}_{n\in[N^{\prime}_{k}]}\\\
{\mathbb{E}}_{{\underline{h}},{\underline{h}}^{\prime}\in[H]^{s-1}}\,\int\Delta_{s-1,T_{m};{\underline{h}}-{\underline{h}}^{\prime}}f_{n,k}\cdot
T_{m}^{-|{\underline{h}}|}\Big{(}\prod_{{\underline{\epsilon}}\in\\{0,1\\}^{s-1}}\big{(}\mathcal{C}^{|{\underline{\epsilon}}|}\chi_{m,{\underline{h}}^{\underline{\epsilon}},x}\cdot{\bf
1}_{\Lambda}({\underline{h}}^{\underline{\epsilon}})\big{)}\Big{)}\,d\mu_{m,x}>0.$

Let

(69)
$f_{j,{\underline{h}},{\underline{h}}^{\prime}}:=\Delta_{s-1,T_{m};{\underline{h}}-{\underline{h}}^{\prime}}f_{j}\quad\textrm{for}\quad
j\in[\ell],\,j\neq m,\,{\underline{h}},{\underline{h}}^{\prime}\in[H]^{s-1}.$

Note that since $T_{j},T_{m}$ commute and $f_{j}\in{\mathcal{E}}(T_{j},\mu)$
for $j=m+1,\ldots,\ell$, we have that
$f_{j,{\underline{h}},{\underline{h}}^{\prime}}\in{\mathcal{E}}(T_{j},\mu)$.
For all ${\underline{h}},{\underline{h}}^{\prime}\in{\mathbb{N}}^{s-1}$ and
$x\in A$, let also

(70)
$\chi_{m,{\underline{h}},{\underline{h}}^{\prime},x}:=\prod_{{\underline{\epsilon}}\in\\{0,1\\}^{s-1}}\mathcal{C}^{|{\underline{\epsilon}}|}\chi_{m,{\underline{h}}^{\underline{\epsilon}},x}\in{\mathcal{E}}(T_{m},\mu_{m,x}),$

and $\Lambda^{\prime}=\Lambda^{\prime}_{\mu_{m,x}}$ be defined by

$\Lambda^{\prime}:=\\{({\underline{h}},{\underline{h}}^{\prime})\in{\mathbb{N}}^{2(s-1)}\colon{\underline{h}}^{\underline{\epsilon}}\in\Lambda\text{
for all }{\underline{\epsilon}}\in\\{0,1\\}^{s-1}\\}.$

Using the previous facts and the form of $f_{n,k}$ from (67), we deduce after
composing with $T_{m}^{a_{m}(n)}$ and applying the Cauchy-Schwarz inequality
that for $\mu$-a.e. $x\in A$, we have

$\liminf_{H\to\infty}{\mathbb{E}}_{{\underline{h}},{\underline{h}}^{\prime}\in[H]^{s-1}}{\bf
1}_{\Lambda^{\prime}}(h,h^{\prime})\\\
\lim_{K\to\infty}\left\|{\mathbb{E}}_{k\in[L_{K}]}{\mathbb{E}}_{n\in[N^{\prime}_{k}]}\,T_{m}^{a_{m}(n)}\chi_{m,{\underline{h}},{\underline{h}}^{\prime},x}\prod_{j\in[\ell],j\neq
m}T_{j}^{a_{j}(n)}f_{j,{\underline{h}},{\underline{h}}^{\prime}}\right\|_{L^{2}(\mu_{m,x})}>0,$

where the limit $\lim_{K\to\infty}$ exists for all
${\underline{h}},{\underline{h}}^{\prime}\in{\mathbb{N}}^{s-1}$ by property
(ii) of Section 7.2, the definition of the functions
$f_{j,{\underline{h}},{\underline{h}}^{\prime}}$ in (69), and the choice of
${\mathcal{F}}$. Since ${\bf
1}_{\Lambda}({\underline{h}},{\underline{h}}^{\prime})\leq{\bf
1}_{\Lambda}({\underline{h}})$ and $\Lambda$ has positive lower density, we
deduce that for $\mu$-a.e. $x\in A$, we have

$\liminf_{H\to\infty}{\mathbb{E}}_{{\underline{h}}^{\prime}\in[H]^{s-1}}{\mathbb{E}}_{{\underline{h}}\in\Lambda\cap[H]^{s-1}}\\\
\lim_{K\to\infty}\left\|{\mathbb{E}}_{k\in[L_{K}]}{\mathbb{E}}_{n\in[N_{k}^{\prime}]}\,T_{m}^{a_{m}(n)}\chi_{m,{\underline{h}},{\underline{h}}^{\prime},x}\prod_{j\in[\ell],j\neq
m}T_{j}^{a_{j}(n)}f_{j,{\underline{h}},{\underline{h}}^{\prime}}\right\|_{L^{2}(\mu_{m,x})}>0.$

Using property (ii) of Section 7.2 (indirectly we use property $(P_{m-1})$
here) we get that for $\mu$-a.e. $x\in A$, the limits $\lim_{K\to\infty}$
below exist for all
${\underline{h}},{\underline{h}}^{\prime}\in{\mathbb{N}}^{s-1}$ and the
following holds222222The existence of the limits $\lim_{K\to\infty}$ is also
crucial here, we would run into trouble if we had $\limsup_{K\to\infty}$
before since on the next step we could only apply the result about
characteristic factors on some fixed well chosen subsequence of $L_{K}$.

$\liminf_{H\to\infty}{\mathbb{E}}_{{\underline{h}}^{\prime}\in[H]^{s-1}}{\mathbb{E}}_{{\underline{h}}\in\Lambda\cap[H]^{s-1}}\\\
\lim_{K\to\infty}\left\|{\mathbb{E}}_{k\in[L_{K}]}{\mathbb{E}}_{n\in[N_{k}^{\prime}]}\,T_{m}^{a_{m}(n)}{\mathbb{E}}(\chi_{m,{\underline{h}},{\underline{h}}^{\prime},x}|{{\mathcal{K}}_{\text{\rm
rat}}}(T_{m},\mu_{m,x}))\prod_{j\in[\ell],j\neq
m}T_{j}^{a_{j}(n)}f_{j,{\underline{h}},{\underline{h}}^{\prime}}\right\|_{L^{2}(\mu_{m,x})}>0.$

Using the pigeonhole principle, we pick for $\mu$-a.e. $x\in A$ and all
$H\in{\mathbb{N}}$ an element
${\underline{h}}^{\prime}_{H}={\underline{h}}^{\prime}_{H,x}\in[H]^{s-1}$ for
which

$\liminf_{H\to\infty}{\mathbb{E}}_{{\underline{h}}\in\Lambda\cap[H]^{s-1}}\\\
\lim_{K\to\infty}\left\|{\mathbb{E}}_{k\in[L_{K}]}{\mathbb{E}}_{n\in[N_{k}^{\prime}]}\,T_{m}^{a_{m}(n)}{\mathbb{E}}(\chi_{m,{\underline{h}},{\underline{h}}^{\prime}_{H},x}|{{\mathcal{K}}_{\text{\rm
rat}}}(T_{m},\mu_{m,x}))\prod_{j\in[\ell],j\neq
m}T_{j}^{a_{j}(n)}f_{i,{\underline{h}},{\underline{h}}^{\prime}_{H}}\right\|_{L^{2}(\mu_{m,x})}>0.$

Recall that by (64), we have
${\mathbb{E}}(\chi_{m,{\underline{h}},{\underline{h}}^{\prime},x}|{{\mathcal{K}}_{\text{\rm
rat}}}(T_{m},\mu_{m,x}))={\mathbb{E}}(\chi_{m,{\underline{h}},{\underline{h}}^{\prime},x}|{\mathcal{K}}_{r_{x}}(T_{m},\mu_{m,x}))$
for all ${\underline{h}},{\underline{h}}^{\prime}\in{\mathbb{N}}^{s-1}$ and
$x\in A$. Since
$\chi_{m,{\underline{h}},{\underline{h}}^{\prime},x}\in{\mathcal{E}}(T_{m},\mu_{m,x})$
and (ergodic) eigenfunctions with different eigenvalues are orthogonal, we
deduce that
${\mathbb{E}}(\chi_{m,{\underline{h}},{\underline{h}}^{\prime}_{H},x}|{{\mathcal{K}}_{\text{\rm
rat}}}(T_{m},\mu_{m,x}))=0$ unless the $T_{m}$-eigenfunction
$\chi_{m,{\underline{h}},{\underline{h}}^{\prime}_{H},x}$ is
$T^{r_{x}}$-invariant. We conclude from this that for $\mu$-a.e. $x\in A$ and
every $H\in{\mathbb{N}}$, there exist sets
$\Lambda_{H}:=\Lambda_{H,\mu_{m,x}}\subset\Lambda\cap[H]^{s-1}$ with

(71) $\liminf_{H\to\infty}\frac{|\Lambda_{H}|}{H^{s-1}}>0$

and such that
$T_{m}^{r_{x}}\chi_{m,{\underline{h}},{\underline{h}}^{\prime}_{H},x}=\chi_{m,{\underline{h}},{\underline{h}}^{\prime}_{H},x}$
holds $\mu_{m,x}$-a.e. for all ${\underline{h}}\in\Lambda_{H}$ and
$H\in{\mathbb{N}}$.

###  7.4. Extracting a lower bound on
$\lvert\\!|\\!|\tilde{f}_{m}|\\!|\\!\rvert_{s,T_{m}}$

We now combine all the statements to deduce the claim
$\lvert\\!|\\!|\tilde{f}_{m}|\\!|\\!\rvert_{s,T_{m}}>0$. From the definition
of $\chi_{m,{\underline{h}},{\underline{h}}^{\prime}_{H},x}$ it follows that

(72)
$\displaystyle\chi_{m,{\underline{h}},x}=\prod_{{{\underline{\epsilon}}}\in\\{0,1\\}^{s-1}\setminus\underline{0}}{\mathcal{C}}^{|{\underline{\epsilon}}|+1}\chi_{m,{\underline{h}}_{H}^{\underline{\epsilon}},x}\cdot\chi_{m,{\underline{h}},{\underline{h}}^{\prime}_{H},x}$

for every ${\underline{h}}\in{\mathbb{N}}^{s-1}$ and $H\in{\mathbb{N}}$, where
${\underline{h}}^{\underline{\epsilon}}_{H}=(h^{\epsilon_{1}}_{1,H},\ldots,h^{\epsilon_{s-1}}_{s-1,H})$
and

$\displaystyle h^{\epsilon_{j}}_{j,H}:=\begin{cases}h_{j},\;&\epsilon_{j}=0\\\
h^{\prime}_{j,H},\;&\epsilon_{j}=1.\end{cases}$

For each $H\in{\mathbb{N}}$, ${\underline{h}}\in{\mathbb{N}}^{s-1}$, and
$j\in[s-1]$, we set

(73)
$\displaystyle\tilde{\chi}_{j,{\underline{h}},H,x}:=\prod\limits_{\begin{subarray}{c}{\underline{\epsilon}}\in\\{0,1\\}^{s-1},\epsilon_{j}=1,\\\
\epsilon_{1}=\ldots=\epsilon_{j-1}=0\end{subarray}}{\mathcal{C}}^{|{\underline{\epsilon}}|+1}\chi_{m,{\underline{h}}_{H}^{\underline{\epsilon}},x},$

so that

(74)
$\displaystyle\chi_{m,{\underline{h}},x}=\prod_{j\in[s-1]}\tilde{\chi}_{j,{\underline{h}},H,x}\cdot\chi_{m,{\underline{h}},{\underline{h}}^{\prime}_{H},x}$

for all $H\in{\mathbb{N}}$ and ${\underline{h}}\in N^{s-1}$. The definition
(73) implies that the functions $\tilde{\chi}_{j,{\underline{h}},H,x}$ are
independent of the variable $h_{j}$. Consequently, for $\mu$-a.e. $x\in A$ and
every ${\underline{h}}\in\Lambda_{H}$ and $H\in{\mathbb{N}}$, the
eigenfunction $\chi_{m,{\underline{h}},x}$ can be expressed as a product of
$s-1$ functions
$\tilde{\chi}_{1,{\underline{h}},H,x},\ldots,\tilde{\chi}_{s-1,{\underline{h}},H,x}$,
each of which depends on at most $s-2$ entries of ${\underline{h}}$, and the
function ${\chi}_{m,{\underline{h}},{\underline{h}}^{\prime}_{H},x}$, which is
$T_{m}^{r_{x}}$-invariant $\mu_{m,x}$-almost everywhere. Hence, for many
values ${\underline{h}}$, the eigenfunctions $\chi_{m,{\underline{h}},x}$ are
products of low complexity functions and a function invariant under
$T_{m}^{r_{x}}$. This fact will prove crucial in obtaining the claimed result.

From the estimate (66) and the inclusion $\Lambda_{H}\subseteq\Lambda$, we
infer that

$\displaystyle\int\Delta_{s-1,T_{m};{\underline{h}}}\tilde{f}_{m}\cdot\chi_{m,{\underline{h}},x}\,d\mu_{m,x}>a$

for all ${\underline{h}}\in\Lambda_{H}$ and $H\in{\mathbb{N}}$, and so

(75)
$\displaystyle\liminf_{H\to\infty}{\mathbb{E}}_{{\underline{h}}\in\Lambda_{H}}\int\Delta_{s-1,T_{m};{\underline{h}}}\tilde{f}_{m}\cdot\prod_{j\in[s-1]}\tilde{\chi}_{j,{\underline{h}},H,x}\cdot\chi_{m,{\underline{h}},{\underline{h}}^{\prime}_{H},x}\,d\mu_{m,x}>0.$

At this step, we want to use the invariance property of
$\chi_{m,{\underline{h}},{\underline{h}}^{\prime}_{H},x}$ to get rid of this
function. Letting $r=r_{x}$ for simplicity, we compose each integral with
$T_{m}^{rh^{\prime}_{s}}$, introduce extra averaging over
$h^{\prime}_{s}\in[H]$, and recall that
$T_{m}^{rh^{\prime}_{s}}\chi_{m,{\underline{h}},{\underline{h}}^{\prime}_{H},x}=\chi_{m,{\underline{h}},{\underline{h}}^{\prime}_{H},x}$
for $\mu_{m,x}$-a.e. $x\in X$, so that

(76)
$\displaystyle\liminf_{H\to\infty}{\mathbb{E}}_{{\underline{h}}\in\Lambda_{H}}\int\chi_{m,{\underline{h}},{\underline{h}}^{\prime}_{H},x}\cdot{\mathbb{E}}_{h^{\prime}_{s}\in[H]}\Delta_{s-1,T_{m};{\underline{h}}}\mathopen{}\left(T_{m}^{rh^{\prime}_{s}}\tilde{f}_{m}\mathclose{}\right)\cdot\prod_{j\in[s-1]}T_{m}^{rh^{\prime}_{s}}\tilde{\chi}_{j,{\underline{h}},H,x}\,d\mu_{m,x}>0.$

Applying the triangle inequality inside the integral, extending the summation
by positivity from $\Lambda_{H}$ to all of $[H]^{s-1}$ and invoking the
estimate (71) as well as the fact that $\Lambda$ has positive lower density,
we conclude that

(77)
$\displaystyle\liminf_{H\to\infty}{\mathbb{E}}_{{\underline{h}}\in[H]^{s-1}}\int\Big{|}{\mathbb{E}}_{h^{\prime}_{s}\in[H]}\Delta_{s-1,T_{m};{\underline{h}}}\mathopen{}\left(T_{m}^{rh^{\prime}_{s}}\tilde{f}_{m}\mathclose{}\right)\cdot\prod_{j\in[s-1]}T_{m}^{rh^{\prime}_{s}}\tilde{\chi}_{j,{\underline{h}},H,x}\Big{|}\,d\mu_{m,x}>0.$

We subsequently apply the Cauchy-Schwarz inequality and Lemma 5.1
(specifically, its first part (44)) to (77), so that

$\liminf_{H\to\infty}\Big{|}{\mathbb{E}}_{({\underline{h}},h_{s},h_{s}^{\prime})\in[H]^{s+1}}\,{\bf
1}_{[H]}(h_{s}+h_{s}^{\prime})\\\
\int\Delta_{s,T_{m};({\underline{h}},rh_{s})}\mathopen{}\left(T_{m}^{rh_{s}^{\prime}}\tilde{f}_{m}\mathclose{}\right)\cdot\prod_{j\in[s-1]}\Delta_{1,T_{m};rh_{s}}\mathopen{}\left(T_{m}^{rh_{s}^{\prime}}\tilde{\chi}_{j,{\underline{h}},H,x}\mathclose{}\right)\,d\mu_{m,x}\Big{|}>0.$

Composing each integral with $T_{m}^{-rh_{s}^{\prime}}$, we arrive at the
inequality

$\liminf_{H\to\infty}\Big{|}{\mathbb{E}}_{({\underline{h}},h_{s},h_{s}^{\prime})\in[H]^{s+1}}\,{\bf
1}_{[H]}(h_{s}+h_{s}^{\prime})\\\
\int\Delta_{s,T_{m};({\underline{h}},rh_{s})}\tilde{f}_{m}\cdot\prod_{j\in[s-1]}\Delta_{1,T_{m};rh_{s}}\tilde{\chi}_{j,{\underline{h}},H,x}\,d\mu_{m,x}\Big{|}>0.$

Using the pigeonhole principle, we pick for each $H\in{\mathbb{N}}$ an element
$h^{\prime}_{s}=h^{\prime}_{s,H}\in[H]$ that maximises the average

$\displaystyle\Big{|}{\mathbb{E}}_{({\underline{h}},h_{s})\in[H]^{s}}\,{\bf
1}_{[H]}(h_{s}+h_{s}^{\prime})\int\Delta_{s,T_{m};({\underline{h}},rh_{s})}\tilde{f}_{m}\cdot\prod_{j\in[s-1]}\Delta_{1,T_{m};rh_{s}}\tilde{\chi}_{j,{\underline{h}},H,x}\,d\mu_{m,x}\Big{|}$

so that

(78)
$\displaystyle\liminf_{H\to\infty}\Big{|}{\mathbb{E}}_{({\underline{h}},h_{s})\in[H]^{s}}\,{\bf
1}_{[H]}(h_{s}+h_{s,H}^{\prime})\int\Delta_{s,T_{m};({\underline{h}},rh_{s})}\tilde{f}_{m}\cdot\prod_{j\in[s-1]}\Delta_{1,T_{m};rh_{s}}\tilde{\chi}_{j,{\underline{h}},H,x}\,d\mu_{m,x}\Big{|}>0.$

Letting

$\displaystyle
b_{j,{\underline{h}},h_{s},H,x}:=\begin{cases}\Delta_{1,T_{m};rh_{s}}\tilde{\chi}_{j,{\underline{h}},H,x}\cdot{\bf
1}_{[H]}(h_{s}+h^{\prime}_{s,H}),\;&j=1\\\
\Delta_{1,T_{m};rh_{s}}\tilde{\chi}_{j,{\underline{h}},H,x},\;&2\leq j\leq
s-1\\\ 1,\;&j=s,\end{cases}$

we can express (78) as

(79)
$\displaystyle\liminf_{H\to\infty}\Big{|}{\mathbb{E}}_{({\underline{h}},h_{s})\in[H]^{s}}\int\Delta_{s,T_{m};({\underline{h}},rh_{s})}\tilde{f}_{m}\cdot\prod_{j\in[s]}b_{j,{\underline{h}},h_{s},H,x}\,d\mu_{m,x}\Big{|}>0.$

The crucial property is that each function $b_{j,{\underline{h}},h_{s},H,x}$
is independent of the variable $h_{j}$. Noting that
$\Delta_{s,T_{m};({\underline{h}},rh_{s})}f=\Delta_{{\mathbf{e}}_{m}^{\times(s-1)},{\mathbf{e}}_{m}^{r};({\underline{h}},h_{s})}f$
and applying Lemmas 5.9 and 3.1 to (79), we deduce that
$\lvert\\!|\\!|\tilde{f}_{m}|\\!|\\!\rvert_{s,T_{m},\mu_{m,x}}>0$ for
$\mu$-a.e. $x\in A$. Using this bound and the fact that $A$ has positive
measure, we conclude from (23) that
$\lvert\\!|\\!|\tilde{f}_{m}|\\!|\\!\rvert_{s,T_{m},\mu}>0$, as required. This
completes the proof of Proposition 6.4.

##  8\. Seminorm control and smoothening for pairwise independent polynomials

In this section, we prove Theorem 2.8. The converse direction is standard and
easy to prove, so we focus on the forward direction: if the polynomials
$p_{1},\ldots,p_{\ell}\in{\mathbb{Z}}[n]$ are pairwise independent, then some
Host-Kra factors are characteristic for ergodic averages with commuting
transformations and iterates $p_{1}(n),\ldots,p_{\ell}(n)$.

If $T_{1}=\cdots=T_{\ell}$, then only essential distinctness of the
polynomials is needed, and the claimed result was established in [23, 26]
following a variant of the standard PET argument in [2]. The problem is that
when we deal with multiple transformations, a PET argument alone can only give
a control of an average

(80) $\displaystyle{\mathbb{E}}_{n\in[N]}\,T_{1}^{p_{1}(n)}f_{1}\cdots
T_{\ell}^{p_{\ell}(n)}f_{\ell}$

by box seminorms
$\lvert\\!|\\!|\cdot|\\!|\\!\rvert_{{\textbf{b}}_{1},\ldots,{\textbf{b}}_{s+1}}$
with potentially distinct vectors
${\textbf{b}}_{1},\ldots,{\textbf{b}}_{s+1}$. For instance, for the average

(81) $\displaystyle{\mathbb{E}}_{n\in[N]}\,T_{1}^{n^{2}}f_{1}\cdot
T_{2}^{n^{2}+n}f_{2},$

the usual PET argument gives control by the box seminorm
$\lvert\\!|\\!|f_{2}|\\!|\\!\rvert_{{\mathbf{e}}_{2}^{\times
s_{1}},({\mathbf{e}}_{2}-{\mathbf{e}}_{1})^{\times s_{2}}}$ for some
$s_{1},s_{2}>0$. This is not good enough for applications such as Theorem
2.10. The relevant factors for box seminorms are more complicated than the
Host-Kra factors for a single transformation, and we do not have a comparable
structure theory for them. Neither can we obtain Theorem 2.10 by performing
the degree lowering argument directly with box seminorms without running into
all sorts of difficulties. Therefore it is essential for our applications to
strengthen the control by box seminorms into a control by Gowers-Host-Kra
seminorms. This is achieved step-by-step. We first pass from controlling (80)
by a seminorm
$\lvert\\!|\\!|f_{\ell}|\\!|\\!\rvert_{{\textbf{b}}_{1},\ldots,{\textbf{b}}_{s+1}}$
to a control by
$\lvert\\!|\\!|f_{\ell}|\\!|\\!\rvert_{{\textbf{b}}_{1},\ldots,{\textbf{b}}_{s},{\mathbf{e}}_{\ell}^{\times
s^{\prime}}}$ for some (possibly large) value $s^{\prime}\in{\mathbb{N}}$.
Then we repeat the argument $s$ more times to control (80) by the seminorm
$\lvert\\!|\\!|f_{\ell}|\\!|\\!\rvert_{{\mathbf{e}}_{\ell}^{s^{\prime\prime}}}=\lvert\\!|\\!|f_{\ell}|\\!|\\!\rvert_{s^{\prime\prime},T_{\ell}}$
for some $s^{\prime\prime}\in{\mathbb{N}}$. Finally, we use an approximation
argument based on Proposition 3.2 and induction to obtain a control of (80) by
Gowers-Host-Kra seminorms of the other terms.

As mentioned above, the starting point is to obtain a control of the average
(80) by some box seminorm. For this purpose, we use the following result of
Donoso, Ferré-Moragues, Koutsogiannis, and Sun, reformulated in a language
that better suits our needs.

######  Proposition 8.1 ([9, Theorem 2.5]).

Let $d,\ell\in{\mathbb{N}}$,
$\textbf{p}_{1},\ldots,\textbf{p}_{\ell}\in{\mathbb{Z}}[n]^{\ell}$ be
essentially distinct polynomials that have the form
$\textbf{p}_{j}(n)=\sum_{i=0}^{d}{\textbf{a}}_{ji}\,n^{i}$, $j\in[\ell]$, and
$(X,{\mathcal{X}},\mu,T_{1},\ldots,T_{\ell})$ be a system. Let also
$\textbf{p}_{0}:=0$ and

$d_{\ell j}:=\deg(\textbf{p}_{\ell}-\textbf{p}_{j}),\quad j=0,\ldots,\ell-1.$

Then there exists $s\in{\mathbb{N}}$ (depending only on $d,\ell$) and vectors

(82)
$\displaystyle{\textbf{b}}_{1},\ldots,{\textbf{b}}_{s+1}\in\\{{\textbf{a}}_{\ell
d_{\ell j}}-{\textbf{a}}_{jd_{\ell j}}\colon\;j=0,\ldots,\ell-1\\},$

independent of the system, such that for all 1-bounded functions
$f_{1},\ldots,f_{\ell}\in L^{\infty}(\mu)$ we have

$\displaystyle\lim_{N\to\infty}\left\|{\mathbb{E}}_{n\in[N]}\prod_{j\in[\ell]}T^{\textbf{p}_{j}(n)}f_{j}\right\|_{L^{2}(\mu)}=0$

whenever
$\lvert\\!|\\!|f_{\ell}|\\!|\\!\rvert_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s+1}}}=0$.

######  Remark.

This extends the results about characteristic factors from [23, 26] that cover
the case $T_{1}=\cdots=T_{\ell}$.

The vectors ${\textbf{b}}_{1},\ldots,{\textbf{b}}_{s+1}$ appearing in (82) are
precisely the leading coefficients of the polynomials
$\textbf{p}_{\ell},\textbf{p}_{\ell}-\textbf{p}_{1},\ldots,\textbf{p}_{\ell}-\textbf{p}_{\ell-1}$;
in particular, if $\textbf{p}_{\ell}$ is essentially distinct from
$\textbf{p}_{1},\ldots,\textbf{p}_{\ell-1}$,232323This means that
$\textbf{p}_{\ell}-\textbf{p}_{j}$ is nonconstant $j\in[\ell-1]$, which under
our standing assumption that all polynomials have zero contant term is
equivalent to distinctness. then the vectors
${\textbf{b}}_{1},\ldots,{\textbf{b}}_{s+1}$ are nonzero. For instance, taking
$\textbf{p}_{1}(n):=(n^{2},0)=n^{2}{\mathbf{e}}_{1}$ and
$\textbf{p}_{2}(n):=(0,n^{2}+n)=(n^{2}+n){\mathbf{e}}_{2}$ as in the average
(81), the vectors ${\textbf{b}}_{1},\ldots,{\textbf{b}}_{s+1}$ would come from
the set $\\{{\mathbf{e}}_{2},{\mathbf{e}}_{2}-{\mathbf{e}}_{1}\\}$.

Passing from a control by
$\lvert\\!|\\!|f_{\ell}|\\!|\\!\rvert_{{\textbf{b}}_{1},\ldots,{\textbf{b}}_{s+1}}$
to a control by
$\lvert\\!|\\!|f_{\ell}|\\!|\\!\rvert_{{\textbf{b}}_{1},\ldots,{\textbf{b}}_{s},{\mathbf{e}}_{\ell}^{\times
s^{\prime}}}$ follows a two-step “ping-pong” strategy. Using the control by
$\lvert\\!|\\!|f_{\ell}|\\!|\\!\rvert_{{\textbf{b}}_{1},\ldots,{\textbf{b}}_{s+1}}$,
we first pass to an auxiliary control by some seminorm
$\lvert\\!|\\!|f_{i}|\\!|\\!\rvert_{{\textbf{b}}_{1},\ldots,{\textbf{b}}_{s},{\mathbf{e}}_{i}^{\times
s_{1}}}$ for some $i\in[\ell-1]$, and then we use this auxiliary control to go
back and control the average (80) by
$\lvert\\!|\\!|f_{\ell}|\\!|\\!\rvert_{{\textbf{b}}_{1},\ldots,{\textbf{b}}_{s},{\mathbf{e}}_{\ell}^{\times
s^{\prime}}}$. We call the two steps outlined above _ping_ and _pong_.

###  8.1. Seminorm smoothening for the family $\\{n,n^{2},n^{2}+n\\}$

To better illustrate our techniques and motivate some arguments made in the
general case, we first prove the following seminorm estimate for the pairwise
independent family $n,n^{2},n^{2}+n$.

######  Proposition 8.2.

There exists $s\in{\mathbb{N}}$ with the following property: for every system
$(X,{\mathcal{X}},\mu,T_{1},T_{2},T_{3})$ and 1-bounded functions
$f_{1},f_{2},f_{3}\in L^{\infty}(\mu)$, the average

(83) $\displaystyle{\mathbb{E}}_{n\in[N]}\,T_{1}^{n}f_{1}\cdot
T_{2}^{n^{2}}f_{2}\cdot T_{3}^{n^{2}+n}f_{3}$

converges to 0 in $L^{2}(\mu)$ whenever
$\lvert\\!|\\!|f_{3}|\\!|\\!\rvert_{s,T_{3}}=0$.

By Proposition 8.1, there exist vectors
${\textbf{b}}_{1},\ldots,{\textbf{b}}_{s+1}\in\\{{\mathbf{e}}_{3},{\mathbf{e}}_{3}-{\mathbf{e}}_{2}\\}$
with the property that if
$\lvert\\!|\\!|f_{3}|\\!|\\!\rvert_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s+1}}}=0$,
then the limit of (83) is 0. The key then is the following seminorm
smoothening argument, which allows to replace the vectors
${\textbf{b}}_{1},\ldots,{\textbf{b}}_{s+1}$ by (possibly many copies of)
${\mathbf{e}}_{3}$.

######  Proposition 8.3 (Seminorm smoothening for $n,n^{2},n^{2}+n$).

Let $s\in{\mathbb{N}}$,
${{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s+1}}\in\\{{\mathbf{e}}_{3},{\mathbf{e}}_{3}-{\mathbf{e}}_{2}\\}$
and $(X,{\mathcal{X}},\mu,T_{1},T_{2},T_{3})$ be a system. Suppose that for
all 1-bounded functions $f_{1},f_{2},f_{3}\in L^{\infty}(\mu)$, the equality
$\lvert\\!|\\!|f_{3}|\\!|\\!\rvert_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s+1}}}=0$
implies that

(84)
$\displaystyle\lim_{N\to\infty}{\mathbb{E}}_{n\in[N]}\,T_{1}^{n}f_{1}\cdot
T_{2}^{n^{2}}f_{2}\cdot T_{3}^{n^{2}+n}f_{3}=0.$

Then (84) also holds under the assumption
$\lvert\\!|\\!|f_{3}|\\!|\\!\rvert_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}},{\mathbf{e}}_{3}^{\times
s^{\prime}}}=0$ for some $s^{\prime}\in{\mathbb{N}}$. Moreover, the value
$s^{\prime}$ is independent of the system
$(X,{\mathcal{X}},\mu,T_{1},T_{2},T_{3})$ or the functions
$f_{1},f_{2},f_{3}$.

An iterative application of Proposition 8.3 gives Proposition 8.2.

The content of Proposition 8.3 is that we can “smoothen out” the seminorm,
gradually replacing the potentially distinct vectors
${\textbf{b}}_{1},\ldots,{\textbf{b}}_{s+1}$ by (possibly many copies of) the
same vector ${\mathbf{e}}_{3}$ which corresponds to the transformation $T_{3}$
acting on $f_{3}$.

###### Proof of Proposition 8.3.

If ${\textbf{b}}_{s+1}={\mathbf{e}}_{3}$, there is nothing to prove, so we
assume that ${\textbf{b}}_{s+1}={\mathbf{e}}_{3}-{\mathbf{e}}_{2}$.

Suppose that

(85)
$\displaystyle\lim_{N\to\infty}\left\|{\mathbb{E}}_{n\in[N]}\,T_{1}^{n}f_{1}\cdot
T_{2}^{n^{2}}f_{2}\cdot T_{3}^{n^{2}+n}f_{3}\right\|_{L^{2}(\mu)}>0$

(the limit in (85) and similar limits below exist by [35]). Our argument,
which we call a “ping-pong” argument, proceeds in two steps. In the first
(ping) step, we prove the auxiliary result that

(86)
$\displaystyle\lvert\\!|\\!|f_{2}|\\!|\\!\rvert_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}},{\mathbf{e}}_{2}^{\times
s_{1}}}>0$

for some $s_{1}\in{\mathbb{N}}$. This is achieved by reducing the problem of
getting a seminorm control for the original average (83) to a similar problem
but for an average of the form

(87) $\displaystyle{\mathbb{E}}_{n\in[N]}\,T_{1}^{n}g_{1}\cdot
T_{2}^{n^{2}}g_{2}\cdot T_{2}^{n^{2}+n}g_{3}.$

This average is simpler (in a suitable sense to be explained later) than the
original average, and obtaining seminorm estimates for (87) can be
accomplished by invoking directly Proposition 8.1. In the second (pong) step,
we use the auxiliary estimate (86) to replace the function $f_{2}$ in the
original average by a dual function. The claim
$\lvert\\!|\\!|f_{3}|\\!|\\!\rvert_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}},{\mathbf{e}}_{3}^{\times
s^{\prime}}}>0$ then follows by invoking Proposition B.1, a corollary of
Proposition 8.1, to handle averages of the form

(88)
$\displaystyle{\mathbb{E}}_{n\in[N]}\,T_{1}^{n}g_{1}\cdot\prod_{j=1}^{L}{\mathcal{D}}_{j}(n^{2})\cdot
T_{3}^{n^{2}+n}g_{3},$

where
${\mathcal{D}}_{1},\ldots,{\mathcal{D}}_{L}\in{\mathfrak{D}}_{s^{\prime}}$ are
sequences coming from dual functions (see Section 3.3).

Step 1 (ping): Obtaining control by a seminorm of $f_{2}$.

From (85) and Proposition 5.6 we deduce that

$\displaystyle\lim_{N\to\infty}\left\|{\mathbb{E}}_{n\in[N]}\,T_{1}^{n}f_{1}\cdot
T_{2}^{n^{2}}f_{2}\cdot T_{3}^{n^{2}+n}\tilde{f}_{3}\right\|_{L^{2}(\mu)}>0$

for some function

$\tilde{f}_{3}:=\lim_{k\to\infty}{\mathbb{E}}_{n\in[N_{k}]}\,T_{3}^{-(n^{2}+n)}g_{k}\cdot
T_{3}^{-(n^{2}+n)}T_{1}^{n}\overline{f_{1}}\cdot
T_{3}^{-(n^{2}+n)}T_{2}^{n^{2}}\overline{f_{2}},$

where the limit is a weak limit. Then our assumption gives

$\lvert\\!|\\!|\tilde{f}_{3}|\\!|\\!\rvert_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s+1}}}>0.$

Subsequently, Proposition 5.7 implies that

$\liminf_{H\to\infty}{\mathbb{E}}_{{\underline{h}},{\underline{h}}^{\prime}\in[H]^{s}}\\\
\lim_{N\to\infty}\left\|{\mathbb{E}}_{n\in[N]}\,T_{1}^{n}(\Delta_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}};{\underline{h}}-{\underline{h}}^{\prime}}f_{1})\cdot
T_{2}^{n^{2}}(\Delta_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}};{\underline{h}}-{\underline{h}}^{\prime}}f_{2})\cdot
T_{3}^{n^{2}+n}u_{{\underline{h}},{\underline{h}}^{\prime}}\right\|_{L^{2}(\mu)}>0$

for some 1-bounded functions $u_{{\underline{h}},{\underline{h}}^{\prime}}$,
${\underline{h}},{\underline{h}}^{\prime}\in{\mathbb{N}}^{s}$, invariant under
$T^{{\textbf{b}}_{s+1}}$. This invariance property and the fact that
$T^{{\textbf{b}}_{s+1}}=T_{3}T_{2}^{-1}$ imply that

(89)
$T_{3}u_{{\underline{h}},{\underline{h}}^{\prime}}=T_{2}u_{{\underline{h}},{\underline{h}}^{\prime}},$

and so

$\liminf_{H\to\infty}{\mathbb{E}}_{{\underline{h}},{\underline{h}}^{\prime}\in[H]^{s}}\\\
\lim_{N\to\infty}\left\|{\mathbb{E}}_{n\in[N]}\,T_{1}^{n}(\Delta_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}};{\underline{h}}-{\underline{h}}^{\prime}}f_{1})\cdot
T_{2}^{n^{2}}(\Delta_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}};{\underline{h}}-{\underline{h}}^{\prime}}f_{2})\cdot
T_{2}^{n^{2}+n}u_{{\underline{h}},{\underline{h}}^{\prime}}\right\|_{L^{2}(\mu)}>0.$

Thus, we have replaced $T_{3}$ in the average by $T_{2}$. We note now that all
the terms with highest-degree polynomials involve the same transformation
$T_{2}$. Then there exist $\varepsilon>0$ and a subset
$B\subseteq{\mathbb{N}}^{2s}$ of positive lower density such that

$\displaystyle\lim_{N\to\infty}\left\|{\mathbb{E}}_{n\in[N]}\,T_{1}^{n}(\Delta_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}};{\underline{h}}-{\underline{h}}^{\prime}}f_{1})\cdot
T_{2}^{n^{2}}(\Delta_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}};{\underline{h}}-{\underline{h}}^{\prime}}f_{2})\cdot
T_{2}^{n^{2}+n}u_{{\underline{h}},{\underline{h}}^{\prime}}\right\|_{L^{2}(\mu)}>\varepsilon$

for all $({\underline{h}},{\underline{h}}^{\prime})\in B$. These averages take
the form

$\displaystyle{\mathbb{E}}_{n\in[N]}\,T_{1}^{n}g_{1}\cdot
T_{2}^{n^{2}}g_{2}\cdot T_{2}^{n^{2}+n}g_{3},$

and we know from Proposition 8.1 that they are controlled by
$\lvert\\!|\\!|g_{2}|\\!|\\!\rvert_{s_{1},T_{2}}$ for some
$s_{1}\in{\mathbb{N}}$. Using this fact and Proposition A.1, we deduce that
for all $({\underline{h}},{\underline{h}}^{\prime})\in B$, we have the lower
bound

$\displaystyle\lvert\\!|\\!|\Delta_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}};{\underline{h}}-{\underline{h}}^{\prime}}f_{2}|\\!|\\!\rvert_{s_{1},T_{2}}>\delta$

for some $\delta>0$ independent of
$({\underline{h}},{\underline{h}}^{\prime})$. Consequently,

(90)
$\displaystyle\liminf_{H\to\infty}{\mathbb{E}}_{{\underline{h}},{\underline{h}}^{\prime}\in[H]^{s}}\lvert\\!|\\!|\Delta_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}};{\underline{h}}-{\underline{h}}^{\prime}}f_{2}|\\!|\\!\rvert_{s_{1},T_{2}}>0.$

Together with Lemma 5.2, the inductive formula for seminorms (27) and Hölder
inequality, the inequality (90) implies that

$\lvert\\!|\\!|f_{2}|\\!|\\!\rvert_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}},{\mathbf{e}}_{2}^{\times
s_{1}}}>0.$

Hence, the seminorm
$\lvert\\!|\\!|f_{2}|\\!|\\!\rvert_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}},{\mathbf{e}}_{2}^{\times
s_{1}}}$ controls the average (83).

Starting with a control of (83) by a seminorm of $f_{3}$, we have arrived at a
control by a seminorm of $f_{2}$. Since among the vectors
${\textbf{b}}_{1},\ldots,{\textbf{b}}_{s}$ we may find the vector
${\mathbf{e}}_{3}$ corresponding to the transformation $T_{3}$ and not
involving $T_{2}$ in any way, this seminorm control is not particularly useful
as an independent result. However, as we explain shortly, it proves to be of
importance as an intermediate result, used for obtaining our claimed control
of (83) by
$\lvert\\!|\\!|f_{3}|\\!|\\!\rvert_{{\textbf{b}}_{1},\ldots,{\textbf{b}}_{s},{\mathbf{e}}_{3}^{\times
s^{\prime}}}$.

Step 2 (pong): Obtaining control by a seminorm of $f_{3}$.

To get the claim that
$\lvert\\!|\\!|f_{3}|\\!|\\!\rvert_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}},{\mathbf{e}}_{3}^{\times
s^{\prime}}}$ controls the average for some $s^{\prime}\in{\mathbb{N}}$, we
repeat the procedure once more with $f_{2}$ in place of $f_{3}$. From (85) and
Proposition 5.6, it follows that

$\displaystyle\lim_{N\to\infty}\left\|{\mathbb{E}}_{n\in[N]}\,T_{1}^{n}f_{1}\cdot
T_{2}^{n^{2}}\tilde{f}_{2}\cdot T_{3}^{n^{2}+n}f_{3}\right\|_{L^{2}(\mu)}>0$

for some function

$\tilde{f}_{2}:=\lim_{k\to\infty}{\mathbb{E}}_{n\in[N_{k}]}\,T_{2}^{-n^{2}}g_{k}\cdot
T_{2}^{-n^{2}}T_{1}^{n}\overline{f}_{1}\cdot
T_{2}^{-n^{2}}T_{3}^{n^{2}+n}\overline{f}_{3},$

where the limit is a weak limit. Then the previous result gives

$\lvert\\!|\\!|\tilde{f}_{2}|\\!|\\!\rvert_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}},{\mathbf{e}}_{2}^{\times
s_{1}}}>0.$

Previously, we applied Proposition 5.7 (i) to get rid of
${{\textbf{b}}_{s+1}}$. This time, our goal is to get rid of
${\mathbf{e}}_{2}^{\times s_{1}}$. We apply Proposition 5.7 again, but this
time we invoke its part (iii) since we want to get rid of all the $s_{1}$
vectors ${\mathbf{e}}_{2}$ at once242424If we invoked the first part of
Proposition 5.7 like in the ping step, we would have to deal with difference
functions
$\Delta_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}},{\mathbf{e}}_{2}^{\times(s_{1}-1)};{\underline{h}}-{\underline{h}}^{\prime}}f_{j}$.
Then we could only control the original average in terms of the seminorm
$\lvert\\!|\\!|f_{3}|\\!|\\!\rvert_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}},{\mathbf{e}}_{2}^{\times(s_{1}-1)},{\mathbf{e}}_{3}^{\times
s^{\prime}}}$ which is less useful because of the presence of vectors
${\mathbf{e}}_{2}$.. This gives

$\liminf_{H\to\infty}{\mathbb{E}}_{{\underline{h}},{\underline{h}}^{\prime}\in[H]^{s}}\\\
\lim_{N\to\infty}\left\|{\mathbb{E}}_{n\in[N]}\,T_{1}^{n}(\Delta_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}};{\underline{h}}-{\underline{h}}^{\prime}}f_{1})\cdot{\mathcal{D}}_{{\underline{h}},{\underline{h}}^{\prime}}(n^{2})\cdot
T_{3}^{n^{2}+n}(\Delta_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}};{\underline{h}}-{\underline{h}}^{\prime}}f_{3})\right\|_{L^{2}(\mu)}>0,$

where

$\displaystyle{\mathcal{D}}_{{\underline{h}},{\underline{h}}^{\prime}}(n):=T_{2}^{n}\prod_{{\underline{\epsilon}}\in\\{0,1\\}^{s_{1}}}{\mathcal{C}}^{|{\underline{\epsilon}}|}{\mathcal{D}}_{s^{\prime},T_{2}}(\Delta_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}};{\underline{h}}^{{\underline{\epsilon}}}}\tilde{f}_{2}).$

Thus, ${\mathcal{D}}_{{\underline{h}},{\underline{h}}^{\prime}}$ is a product
of $2^{s_{1}}$ elements of ${\mathfrak{D}}_{s^{\prime}}$. Invoking Proposition
B.1, a version of Proposition 8.1 that allows us to ignore the dual term
${\mathcal{D}}_{{\underline{h}},{\underline{h}}^{\prime}}$, we deduce that
every average

$\displaystyle{\mathbb{E}}_{n\in[N]}\,T_{1}^{n}(\Delta_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}};{\underline{h}}-{\underline{h}}^{\prime}}f_{1})\cdot{\mathcal{D}}_{{\underline{h}},{\underline{h}}^{\prime}}(n^{2})\cdot
T_{3}^{n^{2}+n}(\Delta_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}};{\underline{h}}-{\underline{h}}^{\prime}}f_{3})$

is controlled by
$\lvert\\!|\\!|\Delta_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}};{\underline{h}}-{\underline{h}}^{\prime}}f_{3}|\\!|\\!\rvert_{s^{\prime},T_{3}}$
for some $s^{\prime}\in{\mathbb{N}}$ that depends only on the vectors
${\textbf{b}}_{1},\ldots,{\textbf{b}}_{s}$. We combine this insight with
Proposition A.1 and a pigeonholing argument like in the ping
step252525Specifically, we invoke Proposition A.1 for averages of the form
${\mathbb{E}}_{n\in[N]}\,T_{1}^{n}g_{1}\cdot\prod_{j=1}^{2^{s_{1}}}T_{2}^{n^{2}}g^{\prime}_{j}\cdot
T_{3}^{n^{2}+n}g_{3}$, where $g^{\prime}_{j}$ is
${\mathcal{Z}}_{s_{1}}(T_{2})$-measurable for each $j\in[2^{s_{1}}]$. One can
show that an average like this is qualitatively controlled by
$\lvert\\!|\\!|g_{3}|\\!|\\!\rvert_{s^{\prime},T_{3}}$ by approximating
functions $g^{\prime}_{1},\ldots,g^{\prime}_{2^{s_{1}}}$ by linear
combinations of dual functions using Proposition 3.2 and then applying
Proposition B.1. to conclude that

(91)
$\displaystyle\liminf_{H\to\infty}{\mathbb{E}}_{{\underline{h}},{\underline{h}}^{\prime}\in[H]^{s}}\lvert\\!|\\!|\Delta_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}};{\underline{h}}-{\underline{h}}^{\prime}}f_{3}|\\!|\\!\rvert_{s^{\prime},T_{3}}>0.$

Together with Lemma 5.2, the inductive formula (27), and Hölder inequality,
the inequality (91) gives us

$\lvert\\!|\\!|f_{3}|\\!|\\!\rvert_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}},{\mathbf{e}}_{3}^{\times
s^{\prime}}}>0,$

which is precisely what we claimed to prove. ∎

###  8.2. Introducing the formalism for longer families

The idea for handling longer families is to reduce an arbitrary average to an
average of smaller “type”. For instance, if
$p_{1},p_{2},p_{3}\in{\mathbb{Z}}[n]$ have the same degree, an arbitrary
average

$\displaystyle{\mathbb{E}}_{n\in[N]}\,T_{1}^{p_{1}(n)}f_{1}\cdot
T_{2}^{p_{2}(n)}f_{2}\cdot T_{3}^{p_{3}(n)}f_{3},$

has type $(1,1,1)$ (in the sense that each of $T_{1},T_{2},T_{3}$ occurs
exactly once) and can be reduced to an average of type $(2,1,0)$ (meaning that
$T_{1}$ occurs twice, $T_{2}$ occurs once, $T_{3}$ does not occur), or
$(1,2,0)$ ($T_{1},T_{2},T_{3}$ occur once, twice and zero times respectively).
We then show that an average of type $(2,1,0)$ reduces to an average of type
$(3,0,0)$ ($T_{1}$ occurs three times), which can be directly controlled using
Proposition B.1.

We now introduce a handy formalism for the induction scheme in the proof of
Theorem 2.8. Specifically, we conceptualise important properties of averages

(92)
$\displaystyle{\mathbb{E}}_{n\in[N]}\,\prod_{j\in[\ell]}T_{\eta_{j}}^{p_{j}(n)}f_{j}\cdot\prod_{j\in[L]}{\mathcal{D}}_{j}(q_{j}(n)).$

For an average (92), we let $\ell$ be its _length_ ,
$d:=\max\limits_{j\in[\ell]}\deg p_{j}$ be its _degree_ , and
$\eta:=(\eta_{1},\ldots,\eta_{\ell})$ be its _indexing tuple_. Furthermore, we
define

$\displaystyle{\mathfrak{L}}:=\\{j\in[\ell]\colon\ \deg p_{j}=d\\}$

to be the set of indices corresponding to polynomials of maximum degree. We
may denote ${\mathfrak{L}}={\mathfrak{L}}(p_{1},\ldots,p_{\ell})$ to signify
dependence on the specific polynomial family.

The induction scheme is guided by the principle that while proving Theorem 2.8
for a given average, we invoke the result for “simpler” averages. This
relative simplicity is formalised by the following notion of type. We let the
_type_ of (92) to be the tuple $w:=(w_{1},\ldots,w_{\ell})$. Each entry
$w_{t}$ is defined by

$\displaystyle w_{t}:=|\\{j\in{\mathfrak{L}}\colon\
\eta_{t}=j\\}|=|\\{j\in[\ell]\colon\ \eta_{t}=j,\ \deg p_{t}=d\\}|,$

and it represents the number of times the transformation $T_{t}$ appears in
the average with a polynomial iterate of maximum degree. For instance, the
average

$\displaystyle{\mathbb{E}}_{n\in[N]}\,T_{1}^{n^{3}}f_{1}\cdot
T_{1}^{n^{2}}f_{2}\cdot T_{2}^{n^{3}+n}f_{3}\cdot T_{1}^{n}f_{4}\cdot
T_{2}^{2n^{3}+n}f_{5}\cdot{\mathcal{D}}_{1}(n^{4})\cdot{\mathcal{D}}_{2}(n^{6})$

has length 5, degree 3, indexing tuple (1, 1, 2, 1, 2) and type $(1,2)$
because the transformation $T_{1}$ appears only once with the polynomial of
maximum degree (at the index $j=1$) while $T_{2}$ has an iterate of maximum
degree twice, at $j=3,5$. It does not matter that the polynomials in dual
terms have higher degrees since our notion of type completely ignores how many
dual terms we have and what polynomials they involve.

To organise the induction scheme, we need to define an ordering on types $w$
with a fixed length $|w|:=w_{1}+\cdots+w_{\ell}$. There are several good ways
of doing so; we choose the most efficient one. Let
$\operatorname{Supp}(w)=\\{t\in[\ell]:\ w_{t}>0\\}$. For distinct integers
$m,i\in[\ell]$ with $m\in\operatorname{Supp}(w)$, we define the type operation

$\displaystyle(\sigma_{mi}w)_{t}:=\begin{cases}w_{t},\;&t\neq m,i\\\
w_{m}-1,\;&t=m\\\ w_{i}+1,\;&t=i.\end{cases}.$

For instance, $\sigma_{12}(2,3,7)=(1,4,7)$. Letting
$w^{\prime}:=\sigma_{mi}w$, we set $w^{\prime}<w$ if $w_{m}\leq w_{i}$. In
particular, $(1,4,7)<(2,3,7)$ in the example above. We note that the tuple
$(2,3,7)$ of higher type has smaller variance than the tuple $(1,4,7)$ of
smaller type; this is a consequence of the fact that while passing from
$(2,3,7)$ to $(1,4,7)$, we decrease by 1 the smallest nonzero value 2. This
observation carries forward more generally: if the condition $w_{m}\leq w_{i}$
is satisfied, then an easy computation shows that $w^{\prime}$ has strictly
higher variance than $w$, or equivalently
${w^{\prime}_{1}}^{2}+\cdots+{w^{\prime}_{\ell}}^{2}>w_{1}^{2}+\cdots+w_{\ell}^{2}$.
Thanks to this fact, we can extend the ordering $<$ to all tuples
${\mathbb{N}}_{0}^{\ell}$ of fixed length $|w|$ by transitivity, and so for
two type tuples $w,w^{\prime}\in{\mathbb{N}}_{0}^{\ell}$, we let
$w^{\prime}<w$ if there exist types $w_{0},\ldots,w_{r}$ with $w_{0}=w$,
$w_{r}=w^{\prime}$, such that for every $l=0,\ldots,r-1$, we have
$w_{l+1}=\sigma_{mi}w_{l}$ for distinct $m,i\in{\mathfrak{L}}$ with
$w_{lm}\leq w_{li}$. For instance, this ordering induces the following chains
of types:

$\displaystyle(4,0,0)<(3,1,0)<(2,2,0)<(2,1,1)\quad\textrm{and}\quad(0,4,0)<(1,3,0)<(2,2,0)<(2,1,1),$

and we could use either sequence of steps in our argument.

The ordering $<$ on types organises the way in which we pass from averages of
indexing tuple $\eta$ to averages of indexing tuple $\eta^{\prime}$. After
applying the type operations finitely many types to a type $w^{\prime}$, we
arrive at a type $w$ with the property that $w_{t}=0$ for all but one value
$t\in{\mathfrak{L}}$. These types $w$ will serve as the basis for our
induction procedure because they allow us to directly invoke Proposition B.1,
therefore we call them _basic_. For instance, $(4,0,0)$ and $(0,4,0)$ in the
example above are basic types but $(3,1,0),(1,3,0),(2,2,0),(2,1,1)$ are
not262626As indicated before, we could define the ordering on types in a
different way. For instance, we could define $w^{\prime}:=\sigma_{mi}w$ if
$m<i$ (or if $m>i$) and extend it by transitivity to all tuples of fixed
length. The choice of the order is ultimately a matter of taste; ours is the
most efficient one in that it allows for the swiftest passage to tuples of
basic type..

###  8.3. The induction scheme

Theorem 2.8 follows from the result below upon setting $L:=0$ and letting
$\eta$ be the identity map.

######  Proposition 8.4.

Let $d,\ell,L\in{\mathbb{N}}$, $\eta\in[\ell]^{\ell}$ be an indexing tuple and
$p_{1},\ldots,p_{\ell},q_{1},\ldots,q_{L}\in{\mathbb{Z}}[n]$ be polynomials
such that $p_{1},\ldots,p_{\ell}$ are pairwise independent. Then there exists
$s\in{\mathbb{N}}$ such that for every system
$(X,{\mathcal{X}},\mu,T_{1},\ldots,T_{\ell})$, 1-bounded functions
$f_{1},\ldots,f_{\ell}\in L^{\infty}(\mu)$, and sequences of dual functions
${\mathcal{D}}_{1},\ldots,{\mathcal{D}}_{L}\in{\mathfrak{D}}_{d}$, we have

(93)
$\displaystyle\lim_{N\to\infty}\left\|{\mathbb{E}}_{n\in[N]}\,\prod_{j\in[\ell]}T_{\eta_{j}}^{p_{j}(n)}f_{j}\cdot\prod_{j\in[L]}{\mathcal{D}}_{j}(q_{j}(n))\right\|_{L^{2}(\mu)}=0$

whenever $\lvert\\!|\\!|f_{j}|\\!|\\!\rvert_{s,T_{\eta_{j}}}=0$.

In the seminorm smoothening argument, we use the following quantitative
version which follows by combining Propositions 8.4 and A.1.

######  Corollary 8.5.

Let $d,\ell,L\in{\mathbb{N}}$, $\eta\in[\ell]^{\ell}$ be an indexing tuple and
$p_{1},\ldots,p_{\ell},q_{1},\ldots,q_{L}\in{\mathbb{Z}}[n]$ be polynomials
such that $p_{1},\ldots,p_{\ell}$ are pairwise independent. Let
$s\in{\mathbb{N}}$ be as in Proposition 8.4 and
$(X,{\mathcal{X}},\mu,T_{1},\ldots,T_{\ell})$ be a system. Then for every
$\varepsilon>0$ there exists $\delta>0$ such that for all 1-bounded functions
$f_{1},\ldots,f_{\ell}\in L^{\infty}(\mu)$ and sequences of dual functions
${\mathcal{D}}_{1},\ldots,{\mathcal{D}}_{L}\in{\mathfrak{D}}_{d}$, we have

(94)
$\displaystyle\lim_{N\to\infty}\left\|{\mathbb{E}}_{n\in[N]}\,\prod_{j\in[\ell]}T_{\eta_{j}}^{p_{j}(n)}f_{j}\cdot\prod_{j\in[L]}{\mathcal{D}}_{j}(q_{j}(n))\right\|_{L^{2}(\mu)}<\varepsilon$

whenever $\lvert\\!|\\!|f_{j}|\\!|\\!\rvert_{s,T_{\eta_{j}}}<\delta$ for some
$j\in[\ell]$.

######  Remark.

Since ${\mathcal{D}}_{j}(q_{j}(n))$ has the form $T_{\pi_{j}}^{q_{j}(n)}g_{j}$
for some $\pi_{j}\in[\ell]$ and $g_{j}\in L^{\infty}(\mu)$, the previous limit
exists by [35]. The same comment applies for all limits involving dual
sequences that appear in the remaining of this section.

###### Proof.

We prove Corollary 8.5 for fixed $d,\ell,L,\eta$ by assuming Proposition 8.4
for the same parameters.

Fix $\pi\in[\ell]^{L}$. We first prove the following claim: for all
$f_{1},\ldots,f_{\ell},g_{1},\ldots,g_{L}\in L^{\infty}(\mu)$ with
${\mathcal{Z}}_{d}(T_{\pi_{j}})$-measurable $g_{j}$ for each $j\in[L]$, we
have

(95)
$\displaystyle\lim_{N\to\infty}\left\|{\mathbb{E}}_{n\in[N]}\,\prod_{j\in[\ell]}T_{\eta_{j}}^{p_{j}(n)}f_{j}\cdot\prod_{j\in[L]}T^{q_{j}(n)}_{\pi_{j}}g_{j}\right\|_{L^{2}(\mu)}=0$

whenever $\lvert\\!|\\!|f_{j}|\\!|\\!\rvert_{s,T_{\eta_{j}}}=0$ for some
$j\in[\ell]$.

Fix $f_{1},\ldots,f_{\ell},g_{1},\ldots,g_{L}$ and suppose that (95) fails.
Using Proposition 3.2 and the pigeonhole principle, we deduce that there exist
dual functions $g_{1}^{\prime},\ldots,g_{L}^{\prime}$ of
$T_{\pi_{1}},\ldots,T_{\pi_{L}}$ of degree $d$ such that

$\displaystyle\lim_{N\to\infty}\left\|{\mathbb{E}}_{n\in[N]}\,\prod_{j\in[\ell]}T_{\eta_{j}}^{p_{j}(n)}f_{j}\cdot\prod_{j\in[L]}T^{q_{j}(n)}_{\pi_{j}}g_{j}^{\prime}\right\|_{L^{2}(\mu)}>0.$

Setting ${\mathcal{D}}_{j}(n):=T_{\pi_{j}}^{n}g^{\prime}_{j}$ and using
Proposition 8.4, gives that
$\lvert\\!|\\!|f_{j}|\\!|\\!\rvert_{s,T_{\eta_{j}}}>0$ for all $j\in[\ell]$,
and so the claim follows.

Combining the claim above with Proposition A.1 for
${\mathcal{Y}}_{1}=\cdots={\mathcal{Y}}_{\ell}={\mathcal{X}}$ and
${\mathcal{Y}}_{\ell+1}=\cdots={\mathcal{Y}}_{\ell+L}={\mathcal{Z}}_{d}(T_{\pi_{j}})$,
we deduce the following: for every $\varepsilon>0$ there exists
$\delta_{\pi}>0$ such that for all 1-bounded functions
$f_{1},\ldots,f_{\ell},g_{1},\ldots,g_{L}\in L^{\infty}(\mu)$ with $g_{j}\in
L^{\infty}({\mathcal{Z}}_{d}(T_{\pi_{j}}))$, we have

$\displaystyle\lim_{N\to\infty}\left\|{\mathbb{E}}_{n\in[N]}\,\prod_{j\in[\ell]}T_{\eta_{j}}^{p_{j}(n)}f_{j}\cdot\prod_{j\in[L]}T^{q_{j}(n)}_{\pi_{j}}g_{j}\right\|_{L^{2}(\mu)}<\varepsilon$

whenever $\lvert\\!|\\!|f_{j}|\\!|\\!\rvert_{s,T_{\eta_{j}}}<\delta_{\pi}$ for
some $j\in[\ell]$.

Corollary 8.5 follows by taking
$\delta:=\min(\delta_{\pi}\colon\,\pi\in[\ell]^{L})$ and recalling that dual
functions of $T_{j}$ of order $d$ are ${\mathcal{Z}}_{d}(T_{j})$-measurable,
hence for every $j\in[L]$, the sequence ${\mathcal{D}}_{j}(q_{j}(n))$ has the
form ${\mathcal{D}}_{j}(q_{j}(n))=T_{\pi_{j}}^{q_{j}(n)}g_{j}$ for some
$\pi_{j}\in[\ell]$ and a 1-bounded ${\mathcal{Z}}_{d}(T_{j})$-measurable
function $g_{j}\in L^{\infty}(\mu)$. ∎

Proposition 8.4 will be deduced from the following result.

######  Proposition 8.6.

Let $d,\ell,L\in{\mathbb{N}}$, $\eta\in[\ell]^{\ell}$ be an indexing tuple and
$p_{1},\ldots,p_{\ell},q_{1},\ldots,q_{L}\in{\mathbb{Z}}[n]$ be polynomials
such that $p_{1},\ldots,p_{\ell}$ are pairwise independent. Let
$(X,{\mathcal{X}},\mu,T_{1},\ldots,T_{\ell})$ be a system, $w$ be the type of
the average (92), and $m\in[\ell]$ be such that
$w_{\eta_{m}}=\min\limits_{t\in\operatorname{Supp}(w)}w_{t}$. Then there exist
$s\in{\mathbb{N}}$, independent of the system, such that for all 1-bounded
functions $f_{1},\ldots,f_{\ell}\in L^{\infty}(\mu)$ and sequences of
functions ${\mathcal{D}}_{1},\ldots,{\mathcal{D}}_{L}\in{\mathfrak{D}}_{d}$,
we obtain (93) whenever
$\lvert\\!|\\!|f_{m}|\\!|\\!\rvert_{s,T_{\eta_{m}}}=0$.

For instance, if $w:=(1,2,1,0,0,3,0)$, then (93) follows whenever
$\lvert\\!|\\!|f_{m}|\\!|\\!\rvert_{s,T_{\eta_{m}}}=0$ for $m\in[7]$
satisfying $\eta_{m}=1$ or $\eta_{m}=3$.

Proposition 8.6 is a straightforward consequence of Proposition B.1, followed
by an iterated application of the smoothening result given below.

######  Proposition 8.7.

Let $d,\ell,L,s\in{\mathbb{N}}$, $\eta\in[\ell]^{\ell}$ be an indexing tuple,
and $p_{1},\ldots,p_{\ell}$, $q_{1},\ldots,q_{L}\in{\mathbb{Z}}[n]$ be
polynomials such that $p_{1},\ldots,p_{\ell}$ are pairwise independent. Let
$(X,{\mathcal{X}},\mu,T_{1},\ldots,T_{\ell})$ be a system, $w$ be the type of
the average (92), and $m\in[\ell]$ be such that
$w_{\eta_{m}}=\min\limits_{t\in\operatorname{Supp}(w)}w_{t}$. Then for each
vectors ${\textbf{b}}_{1},\ldots,{\textbf{b}}_{s+1}$ satisfying (121) there
exists $s^{\prime}\in{\mathbb{N}}$, independent of the system, with the
following property: for all 1-bounded functions $f_{1},\ldots,f_{\ell}\in
L^{\infty}(\mu)$ and sequences of functions
${\mathcal{D}}_{1},\ldots,{\mathcal{D}}_{L}\in{\mathfrak{D}}_{d}$, if
$\lvert\\!|\\!|f_{m}|\\!|\\!\rvert_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s+1}}}=0$
implies (93), then so does
$\lvert\\!|\\!|f_{m}|\\!|\\!\rvert_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}},{\mathbf{e}}_{\eta_{m}}^{\times
s^{\prime}}}=0$.

We prove Propositions 8.4, 8.6, and 8.7, as well as Corollary 8.5, by
induction on the length $\ell\in{\mathbb{N}}$, and for each fixed $\ell$ we
further induct on type using the ordering $<$. Specifically, for fixed length
$\ell$ and basic types $w$, the three aforementioned propositions follow
trivially from Proposition B.1. This also covers the base case $\ell=1$, since
then the average has the basic type $(1)$. For an average of length $\ell$ and
a non-basic type $w$, we prove Proposition 8.7 by assuming Corollary 8.5 for
length $\ell-1$ as well as for length $\ell$ and type $w^{\prime}<w$. We then
infer Proposition 8.4 for length $\ell$ and type $w$ by combining this result
for length $\ell-1$ with Proposition 8.6 for length $\ell$ and type $w$.
Finally, Corollary 8.5 for length $\ell$ and type $w$ follows directly from
Proposition 8.4 for $\ell$ and $w$ as proved before, and Proposition 8.6 for
length $\ell$ and type $w$ follows from Proposition 8.7 for $\ell$ and $w$ by
a straightforward argument sketched below the statement of Proposition 8.6.

We illustrate the aforementioned induction scheme by analysing how Proposition
8.4 is derived for the average

(96)
$\displaystyle\lim_{N\to\infty}{\mathbb{E}}_{n\in[N]}\,T_{1}^{n^{3}-n}f_{1}\cdot
T_{2}^{n^{3}}f_{2}\cdot T_{3}^{n^{3}+n}f_{3}$

for $1$-bounded functions $f_{1},f_{2},f_{3}\in L^{\infty}(\mu)$. This average
has type $(1,1,1)$ because each of the transformations $T_{1},T_{2},T_{3}$ has
a polynomial iterate of degree 3 exactly once. By Proposition B.1, this
average is controlled by the seminorm
$\lvert\\!|\\!|f_{3}|\\!|\\!\rvert_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s+1}}}$
for some vectors
${\textbf{b}}_{1},\ldots,{\textbf{b}}_{s+1}\in\\{{\mathbf{e}}_{3},{\mathbf{e}}_{3}-{\mathbf{e}}_{2},{\mathbf{e}}_{3}-{\mathbf{e}}_{1}\\}$.
The goal of Proposition 8.7 is to show that (96) is also controlled by the
seminorm
$\lvert\\!|\\!|f_{3}|\\!|\\!\rvert_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}},{\mathbf{e}}_{3}^{\times
s^{\prime}}}$ for some $s^{\prime}\in{\mathbb{N}}$. Applying this result
iteratively, we deduce that (96) is controlled by the seminorm
$\lvert\\!|\\!|f_{3}|\\!|\\!\rvert_{s^{\prime\prime},T_{3}}$ for some
$s^{\prime\prime}\in{\mathbb{N}}$, and this is the content of Proposition 8.6.
An analogous argument gives control of (96) by the seminorms
$\lvert\\!|\\!|f_{1}|\\!|\\!\rvert_{s^{\prime\prime},T_{1}}$ and
$\lvert\\!|\\!|f_{2}|\\!|\\!\rvert_{s^{\prime\prime},T_{2}}$ (for possibly
different values $s^{\prime\prime}$).

The proof of Proposition 8.7 for (96) proceeds in the same two steps as the
proof of Proposition 8.3. Suppose for the sake of concreteness that
${\textbf{b}}_{s+1}={\mathbf{e}}_{3}-{\mathbf{e}}_{2}$. In the ping step, we
show that the average is controlled by
$\lvert\\!|\\!|f_{2}|\\!|\\!\rvert_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}},{\mathbf{e}}_{2}^{\times
s_{1}}}$ for some $s_{1}\in{\mathbb{N}}$. To show this, we assume for the sake
of contradiction that

$\displaystyle\lim_{N\to\infty}\left\|{\mathbb{E}}_{n\in[N]}\,T_{1}^{n^{3}-n}f_{1}\cdot
T_{2}^{n^{3}}f_{2}\cdot T_{3}^{n^{3}+n}f_{3}\right\|_{L^{2}(\mu)}>0.$

We then replace $f_{3}$ by a suitably defined function $\tilde{f}_{3}$ using
Proposition 5.6. Subsequently, we apply Proposition 5.7 (i) like in the proof
of Proposition 8.3. Having done so, we arrive at

$\displaystyle\liminf_{H\to\infty}{\mathbb{E}}_{{\underline{h}},{\underline{h}}^{\prime}\in[H]^{s}}\lim_{N\to\infty}\left\|{\mathbb{E}}_{n\in[N]}\,T_{1}^{n^{3}-n}f_{1,{\underline{h}},{\underline{h}}^{\prime}}\cdot
T_{2}^{n^{3}}f_{2,{\underline{h}},{\underline{h}}^{\prime}}\cdot
T_{3}^{n^{3}+n}u_{{\underline{h}},{\underline{h}}^{\prime}}\right\|_{L^{2}(\mu)}>0$

for
$f_{j,{\underline{h}},{\underline{h}}^{\prime}}:=\Delta_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}};{\underline{h}}-{\underline{h}}^{\prime}}f_{j}$
and 1-bounded functions $u_{{\underline{h}},{\underline{h}}^{\prime}}$
invariant under $T_{3}T_{2}^{-1}$. The invariance property of
$u_{{\underline{h}},{\underline{h}}^{\prime}}$ gives

(97)
$\displaystyle\liminf_{H\to\infty}{\mathbb{E}}_{{\underline{h}},{\underline{h}}^{\prime}\in[H]^{s}}\lim_{N\to\infty}\left\|{\mathbb{E}}_{n\in[N]}\,T_{1}^{n^{3}-n}f_{1,{\underline{h}},{\underline{h}}^{\prime}}\cdot
T_{2}^{n^{3}}f_{2,{\underline{h}},{\underline{h}}^{\prime}}\cdot
T_{2}^{n^{3}+n}u_{{\underline{h}},{\underline{h}}^{\prime}}\right\|_{L^{2}(\mu)}>0.$

Hence, for a set $B$ of $({\underline{h}},{\underline{h}}^{\prime})$ of
positive lower density, the averages inside (97) are bounded away from 0 by
some $\varepsilon>0$. Each of these averages takes the form

(98)
$\displaystyle\lim_{N\to\infty}{\mathbb{E}}_{n\in[N]}\,T_{1}^{n^{3}-n}g_{1}\cdot
T_{2}^{n^{3}}g_{2}\cdot T_{2}^{n^{3}+n}g_{3}$

for some $1$-bounded $g_{1},g_{2},g_{3}\in L^{\infty}(\mu)$. They have type
$(1,2,0)$, which is smaller than the type $(1,1,1)$ of the original average
(96) because $(1,2,0)=\sigma_{32}(1,1,1)$. At this point, we could invoke
Proposition 8.4 inductively to assert that (98) is controlled by
$\lvert\\!|\\!|g_{2}|\\!|\\!\rvert_{s_{1},T_{2}}$, but we need something
stronger. We apply Corollary 8.5, the quantitative version of Proposition 8.4,
to conclude that for $({\underline{h}},{\underline{h}}^{\prime})\in B$, we
have the estimate
$\lvert\\!|\\!|f_{2,{\underline{h}},{\underline{h}}^{\prime}}|\\!|\\!\rvert_{s_{1},T_{2}}>\delta$
for some $\delta>0$ and $s_{1}\in{\mathbb{N}}$. Arguing in the same way as in
the proof of Proposition 8.3, we infer from this that (96) is controlled by
$\lvert\\!|\\!|f_{2}|\\!|\\!\rvert_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}},{\mathbf{e}}_{2}^{\times
s_{1}}}$.

In the pong step, we use the newly established control by
$\lvert\\!|\\!|f_{2}|\\!|\\!\rvert_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}},{\mathbf{e}}_{2}^{\times
s_{1}}}$ to replace $f_{2}$ in the original average by $\tilde{f}_{2}$ using
Proposition 5.6. An application of Proposition 5.7 gives

(99)
$\displaystyle\liminf_{H\to\infty}{\mathbb{E}}_{{\underline{h}},{\underline{h}}^{\prime}\in[H]^{s}}\lim_{N\to\infty}\left\|{\mathbb{E}}_{n\in[N]}\,T_{1}^{n^{3}-n}f_{1,{\underline{h}},{\underline{h}}^{\prime}}\cdot\prod_{j\in[2^{s}]}{\mathcal{D}}_{j,{\underline{h}},{\underline{h}}^{\prime}}(n^{3})\cdot
T_{3}^{n^{3}+n}f_{3,{\underline{h}},{\underline{h}}^{\prime}}\right\|_{L^{2}(\mu)}>0$

for some
${\mathcal{D}}_{1,{\underline{h}},{\underline{h}}^{\prime}},\ldots,{\mathcal{D}}_{2^{s},{\underline{h}},{\underline{h}}^{\prime}}\in{\mathfrak{D}}_{s_{1}}$.
Each of the averages inside (99) takes the form

(100)
$\displaystyle\lim_{N\to\infty}{\mathbb{E}}_{n\in[N]}\,T_{1}^{n^{3}-n}g_{1}\cdot\prod_{j\in[2^{s}]}{\mathcal{D}}_{j}(n^{3})\cdot
T_{3}^{n^{3}+n}g_{3}$

for some $1$-bounded $g_{1},g_{2},g_{3}\in L^{\infty}(\mu)$. Such averages
have length 2, and so we once again quote Corollary 8.5
inductively272727Specifically, we invoke it for averages of the form
${\mathbb{E}}_{n\in[N]}T_{1}^{n^{3}-n}g_{1}\cdot\prod_{j=1}^{L}T_{2}^{n^{3}}g^{\prime}_{j}\cdot
T_{3}^{n^{3}+n}g_{3}$, where $g^{\prime}_{j}$ is
${\mathcal{Z}}_{s_{1}}(T_{2})$-measurable for each $j\in[L]$. Corollary 8.5
for these averages follows from Proposition 8.4 for averages of length 2 of
the form (100). to conclude that for a positive lower density set of tuples
$({\underline{h}},{\underline{h}}^{\prime})$, we have
$\lvert\\!|\\!|f_{3,{\underline{h}},{\underline{h}}^{\prime}}|\\!|\\!\rvert_{s^{\prime},T_{3}}>\delta$
for some $s^{\prime}\in{\mathbb{N}}$ and $\delta>0$. Arguing as in the proof
of Proposition 8.3, we infer from this that
$\lvert\\!|\\!|f_{3}|\\!|\\!\rvert_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}},{\mathbf{e}}_{3}^{\times
s^{\prime}}}$ controls (98).

To sum up, in the proof of Proposition 8.7 for averages

$\displaystyle\lim_{N\to\infty}{\mathbb{E}}_{n\in[N]}\,T_{1}^{n^{3}-n}f_{1}\cdot
T_{2}^{n^{3}}f_{2}\cdot T_{3}^{n^{3}+n}f_{3},$

we invoke Corollary 8.5 for averages of the form

(101)
$\displaystyle\lim_{N\to\infty}{\mathbb{E}}_{n\in[N]}\,T_{1}^{n^{3}-n}f_{1}\cdot
T_{2}^{n^{3}}f_{2}\cdot T_{2}^{n^{3}+n}f_{3}$

in the ping step, and we do this for averages of the form

(102)
$\displaystyle\lim_{N\to\infty}{\mathbb{E}}_{n\in[N]}\,T_{1}^{n^{3}-n}f_{1}\cdot\prod_{j\in[2^{s}]}{\mathcal{D}}_{j}(n^{3})\cdot
T_{3}^{n^{3}+n}f_{3}$

in the pong step. The procedure remains the same for a more general average
(92) of type $w$. In the ping step of the proof of Proposition 8.7 for an
average (92), we invoke Corollary 8.5 for an average of length 3 and type
$\sigma_{\eta_{m}\eta_{i}}w$ for some
$\eta_{i}\in{\mathfrak{L}}\setminus\\{\eta_{m}\\}$, where $m\in{\mathfrak{L}}$
is an index such that
$w_{\eta_{m}}=\min\limits_{t\in\operatorname{Supp}(w)}w_{t}.$ This new average
has polynomial iterates $p^{\prime}_{1},\ldots,p^{\prime}_{\ell}$ which are
still pairwise independent. The fact that the pairwise independence of the
polynomials is preserved guarantees that we can apply Proposition B.1 without
running the risk that some of the vectors
${\textbf{b}}_{1},\ldots,{\textbf{b}}_{s+1}$ are 0. In the pong step, we use
Corollary 8.5 for averages of length $\ell-1$ with iterates
$p_{1},\ldots,p_{m-1},p_{m+1},\ldots,p_{\ell}$ .

###  8.4. Proofs

We now prove Proposition 8.7 for averages of length $\ell$ and type $w$
assuming Corollary 8.5 for averages of length $\ell-1$ as well as length
$\ell$ and type $w$.

###### Proof of Proposition 8.7.

We split the proof into two parts. The first part covers the case where the
vector $w$ is basic and can be dealt immediately using Proposition 8.1. This
can be thought of as the basis of the induction. The complementary non-basic
case will be dealt using the ping-pong strategy described above. This can be
thought of as the induction step.

Base case: $w$ is basic.

Suppose first that the vector $w$ is basic, i.e. $w_{t}=0$ for all but a
single value $t\in[\ell]$. That means that all the polynomials of maximum
degree are iterates of the same transformation $T_{t}$. Proposition 8.1
implies in this case that

$\displaystyle\lvert\\!|\\!|f_{m}|\\!|\\!\rvert_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s+1}}}=\lvert\\!|\\!|f_{m}|\\!|\\!\rvert_{{a_{1}}{\mathbf{e}}_{t},\ldots,{a_{s+1}}{\mathbf{e}}_{t}}$

for some $a_{1},\ldots,a_{s+1}\in{\mathbb{Z}}$ such that
${\textbf{b}}_{i}=a_{i}{\mathbf{e}}_{t}$ satisfy the condition (121). The
assumption that the polynomials $p_{1},\ldots,p_{\ell}$ are essentially
distinct implies that $a_{1},\ldots,a_{s+1}\neq 0$. Lemma 3.1 gives the
inequality

$\displaystyle\lvert\\!|\\!|f_{m}|\\!|\\!\rvert_{{a_{1}}{\mathbf{e}}_{t},\ldots,{a_{s+1}}{\mathbf{e}}_{t}}\ll_{a_{s+1}}\lvert\\!|\\!|f_{m}|\\!|\\!\rvert_{{a_{1}}{\mathbf{e}}_{t},\ldots,{a_{s}}{\mathbf{e}}_{t},{\mathbf{e}}_{t}}=\lvert\\!|\\!|f_{m}|\\!|\\!\rvert_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}},{\mathbf{e}}_{t}},$

which implies Proposition 8.7 in this case.

General case: $w$ is not basic.

We prove Proposition 8.7 for $\eta$ of type $w$ by assuming that Proposition
8.4 holds for $\ell-1$ as well as $\ell$ and all types $w^{\prime}<w$. For
simplicity of notation, we assume $m=\ell$.

By Proposition B.1 and the assumption that the polynomials
$p_{1},\ldots,p_{\ell}$ are essentially distinct, the vector
${\textbf{b}}_{s+1}$ is nonzero and equals
${\textbf{b}}_{s+1}=b_{\ell}{\mathbf{e}}_{\eta_{\ell}}-b_{i}{\mathbf{e}}_{\eta_{i}}$
for some $b_{\ell},b_{i}\in{\mathbb{Z}}$ and $i\in\\{0,\ldots,\ell-1\\}$. If
$i=0$, then $b_{\ell}\neq 0$ by the fact that ${\textbf{b}}_{s+1}$ is nonzero,
and the claim follows from the bound

(103)
$\displaystyle\lvert\\!|\\!|f_{\ell}|\\!|\\!\rvert_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}},b_{\ell}{\mathbf{e}}_{\eta_{\ell}}}\ll_{b_{\ell}}\lvert\\!|\\!|f_{\ell}|\\!|\\!\rvert_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}},{\mathbf{e}}_{\eta_{\ell}}}$

given by Lemma 3.1. If $\eta_{i}=\eta_{\ell}$, then the fact
${\textbf{b}}_{s+1}\neq\mathbf{0}$ implies that $b_{\ell}-b_{i}\neq 0$, and
the claim follows once again from (103). We therefore assume that
${\textbf{b}}_{s+1}=b_{\ell}{\mathbf{e}}_{\eta_{\ell}}-b_{i}{\mathbf{e}}_{\eta_{i}}$
for some $i\in{\mathfrak{L}}$ such that $\eta_{\ell}\neq\eta_{i}$, in which
case also $b_{\ell},b_{i}\neq 0$. The proof of Proposition 8.7 in this case
follows the same two-step strategy as the proof of Proposition 8.3. We first
obtain the control of (92) by
$\lvert\\!|\\!|f_{i}|\\!|\\!\rvert_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}},{\mathbf{e}}_{\eta_{i}}^{\times
s_{1}}}$ for some $s_{1}\in{\mathbb{N}}$. This is accomplished by using the
control by
$\lvert\\!|\\!|f_{\ell}|\\!|\\!\rvert_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s+1}}}$,
given by assumption, for an appropriately defined function $\tilde{f_{\ell}}$
in place of $f_{\ell}$. Subsequently, we repeat the procedure by applying the
newly established control by
$\lvert\\!|\\!|f_{i}|\\!|\\!\rvert_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}},{\mathbf{e}}_{\eta_{i}}^{\times
s_{1}}}$ for a function $\tilde{f}_{i}$ in place of $f_{i}$. This gives us the
claimed result.

Step 1 (ping): Obtaining control by a seminorm of $f_{i}$.

Suppose that

(104)
$\displaystyle\lim_{N\to\infty}\left\|{\mathbb{E}}_{n\in[N]}\,\prod_{j\in[\ell]}T_{\eta_{j}}^{p_{j}(n)}f_{j}\cdot\prod_{j\in[L]}{\mathcal{D}}_{j}(q_{j}(n))\right\|_{L^{2}(\mu)}>0.$

Then by a straightforward variant of Proposition 5.6 we have

$\displaystyle\lim_{N\to\infty}\left\|{\mathbb{E}}_{n\in[N]}\,\prod_{j\in[\ell-1]}T_{\eta_{j}}^{p_{j}(n)}f_{j}\cdot
T_{\eta_{\ell}}^{p_{\ell}(n)}\tilde{f}_{\ell}\cdot\prod_{j\in[L]}{\mathcal{D}}_{j}(q_{j}(n))\right\|_{L^{2}(\mu)}>0$

for some function

$\tilde{f}_{\ell}:=\lim_{k\to\infty}{\mathbb{E}}_{n\in[N_{k}]}\,T_{\eta_{\ell}}^{-p_{\ell}(n)}g_{k}\cdot\prod_{j\in[\ell-1]}T_{\eta_{\ell}}^{-p_{\ell}(n)}T_{\eta_{j}}^{p_{j}(n)}\overline{f}_{j}\cdot\prod_{j\in[L]}T_{\eta_{\ell}}^{-p_{\ell}(n)}\overline{{\mathcal{D}}}_{j}(q_{j}(n)),$

where the limit is a weak limit. Then our assumption gives

$\lvert\\!|\\!|\tilde{f}_{\ell}|\\!|\\!\rvert_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s+1}}}>0.$

By Proposition 5.7, we get the estimate

$\liminf_{H\to\infty}{\mathbb{E}}_{{\underline{h}},{\underline{h}}^{\prime}\in[H]^{s}}\\\
\lim_{N\to\infty}\left\|{\mathbb{E}}_{n\in[N]}\,\prod_{j\in[\ell-1]}T_{\eta_{j}}^{p_{j}(n)}(\Delta_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}};{\underline{h}}-{\underline{h}}^{\prime}}f_{j})\cdot
T_{\eta_{\ell}}^{p_{\ell}(n)}u_{{\underline{h}},{\underline{h}}^{\prime}}\cdot\prod_{j=1}^{L}{\mathcal{D}}^{\prime}_{j,{\underline{h}},{\underline{h}}^{\prime}}(q_{j}(n))\right\|_{L^{2}(\mu)}>0,$

where $u_{{\underline{h}},{\underline{h}}^{\prime}}$ are 1-bounded and
invariant under $T^{{\textbf{b}}_{s+1}}$, and for every $j\in[L]$,

$\displaystyle{\mathcal{D}}^{\prime}_{j,{\underline{h}},{\underline{h}}^{\prime}}(n):=\Delta_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}};{\underline{h}}-{\underline{h}}^{\prime}}{\mathcal{D}}_{j}(n)$

is a product of $2^{s}$ elements of ${\mathfrak{D}}_{d}$.

We recall that
$b_{\ell}{\mathbf{e}}_{\eta_{\ell}}-b_{i}{\mathbf{e}}_{\eta_{i}}$ for some
$i\in{\mathfrak{L}}$ such that $\eta_{\ell}\neq\eta_{i}$ and
$b_{\ell},b_{i}\neq 0$. It follows from the invariance property of
$u_{{\underline{h}},{\underline{h}}^{\prime}}$ that

(105)
$T_{\eta_{\ell}}^{b_{\ell}}u_{{\underline{h}},{\underline{h}}^{\prime}}=T_{\eta_{i}}^{b_{i}}u_{{\underline{h}},{\underline{h}}^{\prime}},$

where we use the identity
$T_{\eta_{\ell}}^{b_{\ell}}=T_{\eta_{i}}^{b_{i}}T^{{\textbf{b}}_{s+1}}$. Using
the triangle inequality we deduce the estimate

$\displaystyle\liminf_{H\to\infty}{\mathbb{E}}_{{\underline{h}},{\underline{h}}^{\prime}\in[H]^{s}}{\mathbb{E}}_{r\in[b_{\ell}]}$
$\displaystyle\lim_{N\to\infty}\left\|{\mathbb{E}}_{n\in[N]}\,\prod_{j\in[\ell-1]}T_{\eta_{j}}^{p_{j}(b_{\ell}n+r)}(\Delta_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}};{\underline{h}}-{\underline{h}}^{\prime}}f_{j})\cdot
T_{\eta_{\ell}}^{p_{\ell}(b_{\ell}n+r)}u_{{\underline{h}},{\underline{h}}^{\prime}}\cdot\prod_{j=1}^{L}{\mathcal{D}}^{\prime}_{j,{\underline{h}},{\underline{h}}^{\prime}}(q_{j}(n))\right\|_{L^{2}(\mu)}>0.$

After replacing $\liminf_{H\to\infty}$ with $\limsup_{H\to\infty}$, using
(105), and the pigeonhole principle, we get that for some
$r_{0}\in[b_{\ell}]$, we have

(106)
$\limsup_{H\to\infty}{\mathbb{E}}_{{\underline{h}},{\underline{h}}^{\prime}\in[H]^{s}}\lim_{N\to\infty}\left\|{\mathbb{E}}_{n\in[N]}\,\prod_{j\in[\ell]}T_{\eta^{\prime}_{j}}^{p^{\prime}_{j}(n)}f_{j,{\underline{h}},{\underline{h}}^{\prime}}\cdot\prod_{j=1}^{L}{\mathcal{D}}^{\prime}_{j,{\underline{h}},{\underline{h}}^{\prime}}(q_{j}(n))\right\|_{L^{2}(\mu)}>0,$

where

$\displaystyle p^{\prime}_{j}(n)$
$\displaystyle:=\begin{cases}p_{j}(b_{\ell}n+r_{0})-p_{j}(r_{0}),\;&j\in[\ell-1]\\\
\frac{b_{i}}{b_{\ell}}(p_{\ell}(b_{\ell}n+r_{0})-p_{\ell}(r_{0})),&j=\ell,\end{cases}$
$\displaystyle f_{j,{\underline{h}},{\underline{h}}^{\prime}}$
$\displaystyle:=\begin{cases}\Delta_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}};{\underline{h}}-{\underline{h}}^{\prime}}T_{\eta_{j}}^{p_{j}(r_{0})}f_{j},\;&j\in[\ell-1]\\\
T_{\eta_{\ell}}^{p_{\ell}(r_{0})}u_{{\underline{h}},{\underline{h}}^{\prime}},\;&j=\ell,\end{cases}$
$\displaystyle\eta^{\prime}_{j}$
$\displaystyle:=\begin{cases}\eta_{j},\;&j\in[\ell-1]\\\
\eta_{i},\;&j=\ell,\end{cases}$

and $q^{\prime}_{j}(n):=q_{j}(b_{\ell}n+r_{0})$.

Note that the polynomials $p^{\prime}_{1},\ldots,p^{\prime}_{\ell}$ are
pairwise independent. We make a small digression here regarding the necessity
of this assumption. It is crucial for us in order to be able to prove
Propositions 8.4 and 8.7 inductively that at each step of the induction
scheme, the polynomials $p_{1},\ldots,p_{\ell}$ are essentially distinct.
While essential distinctness of $p_{1},\ldots,p_{\ell}$ might not carry
forward to the essential distinctness of
$p^{\prime}_{1},\ldots,p^{\prime}_{\ell}$ (because essential distinctness is
not preserved under scaling), pairwise independence is preserved this way,
hence we make this assumption on the polynomials $p_{1},\ldots,p_{\ell}$ in
the first place. Moreover, it does not restrict generality since it is a
necessary assumption in the statement of Theorem 2.8.

We go back to the proof now. By assumption, $w_{\eta_{\ell}}$ minimises
$(w_{t})_{t\in\operatorname{Supp}(w)}$ and $\eta_{\ell}\neq\eta_{i}$.
Moreover, the structure of the set (121) implies that $w_{\eta_{i}}>0$, and so
$w_{\eta_{i}}\geq w_{\eta_{\ell}}$. Therefore, the type
$w^{\prime}=\sigma_{\eta_{\ell}\eta_{i}}w$ of the averages in (106) is
strictly smaller than $w$. We inductively apply Corollary 8.5 to each average
in (106). Together with the pairwise independence of the new polynomials
$p_{1}^{\prime},\ldots,p^{\prime}_{\ell}$, this allows us to conclude that
there exist $s_{1}\in{\mathbb{N}}$ (independent of the system or the
functions), $\varepsilon>0$, and a set $B\subset{\mathbb{N}}^{2s}$ of positive
upper density, such that for every
$({\underline{h}},{\underline{h}}^{\prime})\in B$ we have

$\lvert\\!|\\!|\Delta_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}};{\underline{h}}-{\underline{h}}^{\prime}}f_{i}|\\!|\\!\rvert_{s_{1},T_{\eta_{i}}}\geq\varepsilon.$

Hence,

$\displaystyle\limsup_{H\to\infty}{\mathbb{E}}_{{\underline{h}},{\underline{h}}^{\prime}\in[H]^{s}}\lvert\\!|\\!|\Delta_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}};{\underline{h}}-{\underline{h}}^{\prime}}f_{i}|\\!|\\!\rvert_{s_{1},T_{\eta_{i}}}>0.$

Then, using Lemma 5.2 and then (27) together with Hölder’s inequality, imply
that

$\lvert\\!|\\!|f_{i}|\\!|\\!\rvert_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}},{\mathbf{e}}_{\eta_{i}}^{\times
s_{1}}}>0,$

and so the seminorm
$\lvert\\!|\\!|f_{i}|\\!|\\!\rvert_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}},{\mathbf{e}}_{\eta_{i}}^{\times
s_{1}}}$ controls the average (92).

Step 2 (pong): Obtaining control by a seminorm of $f_{\ell}$.

To get the claim that
$\lvert\\!|\\!|f_{\ell}|\\!|\\!\rvert_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}},{\mathbf{e}}_{\eta_{\ell}}^{\times
s^{\prime}}}$ controls the average for some $s^{\prime}\in{\mathbb{N}}$, we
repeat the procedure once more with $f_{i}$ in place of $f_{\ell}$. From (104)
and a straightforward variant of Proposition 5.6 it follows that

$\displaystyle\lim_{N\to\infty}\left\|{\mathbb{E}}_{n\in[N]}\,\prod_{j\in[\ell],j\neq
i}T_{\eta_{j}}^{p_{j}(n)}f_{j}\cdot
T_{\eta_{i}}^{p_{i}(n)}\tilde{f_{i}}\cdot\prod_{j=1}^{L}{\mathcal{D}}_{j}(q_{j}(n))\right\|_{L^{2}(\mu)}>0$

for some function

$\tilde{f}_{i}:=\lim_{k\to\infty}{\mathbb{E}}_{n\in[N_{k}]}\,T_{\eta_{i}}^{-p_{i}(n)}g_{k}\cdot\prod_{j\in[\ell],j\neq
i}T_{\eta_{i}}^{-p_{i}(n)}T_{\eta_{j}}^{p_{j}(n)}\overline{f}_{j}\cdot\prod_{j=1}^{L}T_{\eta_{i}}^{-p_{i}(n)}\overline{{\mathcal{D}}}_{j}(q_{j}(n)),$

where the limit is a weak limit. Then the previous result gives

$\lvert\\!|\\!|\tilde{f}_{i}|\\!|\\!\rvert_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}},{\mathbf{e}}_{\eta_{i}}^{\times
s_{1}}}>0.$

Proposition 5.7 implies that

$\displaystyle\liminf_{H\to\infty}{\mathbb{E}}_{{\underline{h}},{\underline{h}}^{\prime}\in[H]^{s}}$
$\displaystyle\lim_{N\to\infty}\left\|{\mathbb{E}}_{n\in[N]}\,\prod_{j\in[\ell],j\neq
i}T_{{\eta_{j}}}^{p_{j}(n)}(\Delta_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}};{\underline{h}}-{\underline{h}}^{\prime}}f_{j})\cdot\prod_{j\in[L+1]}{\mathcal{D}}_{j,{\underline{h}},{\underline{h}}^{\prime}}(q_{j}(n))\right\|_{L^{2}(\mu)}>0,$

where

$\displaystyle{\mathcal{D}}_{j,{\underline{h}},{\underline{h}}^{\prime}}(n):=\begin{cases}\Delta_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}};{\underline{h}}-{\underline{h}}^{\prime}}{\mathcal{D}}_{j}(n),\;&j\in[L]\\\
T_{\eta_{i}}^{n}T^{-({\textbf{b}}_{1}h_{1}^{\prime}+\cdots+{\textbf{b}}_{s}h^{\prime}_{s})}\prod\limits_{{\underline{\epsilon}}\in\\{0,1\\}^{s}}{\mathcal{C}}^{|{\underline{\epsilon}}|}{\mathcal{D}}_{s_{1},T_{\eta_{i}}}(\Delta_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}};{\underline{h}}^{{\underline{\epsilon}}}}\tilde{f}_{i}),\;&j=L+1.\end{cases}$

Thus, the sequence of functions
${\mathcal{D}}_{j,{\underline{h}},{\underline{h}}^{\prime}}(n)$ is a product
of $2^{s}$ elements of ${\mathfrak{D}}_{d}$ if $j\in[L]$, and it is a product
of $2^{s}$ elements of ${\mathfrak{D}}_{s_{1}}$ for $j=L+1$. Consequently,
there exist $\varepsilon>0$ and a set $B^{\prime}\subset{\mathbb{N}}^{2s}$ of
positive lower density, such that for every
$({\underline{h}},{\underline{h}}^{\prime})\in B^{\prime}$, we have

(107)
$\displaystyle\lim_{N\to\infty}\left\|{\mathbb{E}}_{n\in[N]}\,\prod_{j\in[\ell],j\neq
i}T_{\eta_{j}}^{p_{j}(n)}(\Delta_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}};{\underline{h}}-{\underline{h}}^{\prime}}f_{j})\cdot\prod_{j\in[L+1]}{\mathcal{D}}_{j,{\underline{h}},{\underline{h}}^{\prime}}(q_{j}(n))\right\|_{L^{2}(\mu)}>\varepsilon.$

Each of the averages in (107) has length $\ell-1$. We therefore apply
Corollary 8.5 inductively to conclude that for every
$({\underline{h}},{\underline{h}}^{\prime})\in B^{\prime}$, we have

$\displaystyle\lvert\\!|\\!|\Delta_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}};{\underline{h}}-{\underline{h}}^{\prime}}f_{\ell}|\\!|\\!\rvert_{s^{\prime},T_{\eta_{\ell}}}>\delta$

for some $\delta>0$. Consequently, we deduce that

(108)
$\displaystyle\liminf_{H\to\infty}{\mathbb{E}}_{{\underline{h}},{\underline{h}}^{\prime}\in[H]^{s}}\lvert\\!|\\!|\Delta_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}};{\underline{h}}-{\underline{h}}^{\prime}}f_{\ell}|\\!|\\!\rvert_{s^{\prime},T_{\eta_{\ell}}}>0.$

Using Lemma 5.2 and then (27) together with Hölder’s inequality, we deduce
that
$\lvert\\!|\\!|f_{\ell}|\\!|\\!\rvert_{{{\textbf{b}}_{1}},\ldots,{{\textbf{b}}_{s}},{\mathbf{e}}_{\eta_{\ell}}^{\times
s^{\prime}}}>0$, as claimed. ∎

Finally, we prove Proposition 8.4 for averages of length $\ell$ and type $w$.

###### Proof of Proposition 8.4.

In the base case $\ell=1$, Proposition 8.4 follows directly from Proposition
B.1. We assume therefore that $\ell>1$. It follows from Proposition 8.6 that
there exist $m\in[\ell]$ and $s\in{\mathbb{N}}$ such that (93) holds whenever
$\lvert\\!|\\!|f_{m}|\\!|\\!\rvert_{s,T_{\eta_{m}}}=0$. Suppose now that

$\displaystyle\lim_{N\to\infty}\left\|{\mathbb{E}}_{n\in[N]}\,\prod_{j\in[\ell]}T_{\eta_{j}}^{p_{j}(n)}f_{j}\prod_{j\in[L]}{\mathcal{D}}_{j}(q_{j}(n))\right\|_{L^{2}(\mu)}>0.$

Applying the fact that $\lvert\\!|\\!|f_{m}|\\!|\\!\rvert_{s,T_{\eta_{m}}}$
controls this average, Proposition 3.2, and the pigeonhole principle, we
replace $f_{m}$ by a dual function of level $s$, so that we have

$\displaystyle\lim_{N\to\infty}\left\|{\mathbb{E}}_{n\in[N]}\,\prod_{j\in[\ell],j\neq
m}T_{\eta_{j}}^{p_{j}(n)}f_{j}\cdot{\mathcal{D}}(p_{m}(n))\cdot\prod_{j\in[L]}{\mathcal{D}}_{j}(q_{j}(n))\right\|_{L^{2}(\mu)}>0$

for some ${\mathcal{D}}\in{\mathfrak{D}}_{s}$. This average has length
$\ell-1$, and we apply Proposition 8.4 inductively to deduce that the
seminorms $\lvert\\!|\\!|f_{j}|\\!|\\!\rvert_{s_{j},T_{\eta_{j}}}$ control
(92) for some $s_{j}\in{\mathbb{N}}$ and all $j\in[\ell]$ independent of the
system or the functions. ∎

##  Appendix A Soft quantitative seminorm estimates

###  A.1. Main result

For the purposes of this section only, a system
$(X,{\mathcal{X}},\mu,T_{1},\ldots,T_{\ell})$ is a collection of (not
necessarily commuting) measure preserving transformations
$T_{1},\ldots,T_{\ell}$ acting on a probability space $(X,{\mathcal{X}},\mu)$.
We are interested in mean convergence properties of the averages

(109) ${\mathbb{E}}_{n\in[N]}\,T_{1}^{a_{1}(n)}f_{1}\cdots
T_{\ell}^{a_{\ell}(n)}f_{\ell}\,$

where $f_{1},\ldots,f_{\ell}\in L^{\infty}(\mu)$.

If $X$ is a normed space, we denote its closed unit ball with $B_{X}$.

######  Definition.

Let $(X,{\mathcal{X}},\mu,T_{1},\ldots,T_{\ell})$ be a system,
$a_{1},\ldots,a_{\ell}\colon{\mathbb{N}}\to{\mathbb{Z}}$ be sequences,
$\lvert\\!|\\!|\cdot|\\!|\\!\rvert$ be a seminorm on $L^{\infty}(\mu)$, and
$Y_{1},\ldots,Y_{\ell}$ be subspaces of $L^{\infty}(\mu)$.

  * •

We say that for some $m\in[\ell]$ the seminorm
$\lvert\\!|\\!|f_{m}|\\!|\\!\rvert$ controls the averages (109) for the
subspaces $Y_{1},\ldots,Y_{\ell}$ (and the given system), if
$\lvert\\!|\\!|f_{m}|\\!|\\!\rvert=0$ implies that the averages (109) converge
to $0$ in $L^{2}(\mu)$ for every $f_{1}\in Y_{1},\ldots,f_{\ell}\in Y_{\ell}$.

  * •

We say that the collection $a_{1},\ldots,a_{\ell}$ is good for mean
convergence for the subspaces $Y_{1},\ldots,Y_{\ell}$ (and the given system),
if the averages (109) converge in $L^{2}(\mu)$ for all $f_{1}\in
Y_{1},\ldots,f_{\ell}\in Y_{\ell}$.

The next result deals with averages that are controlled by the seminorms
$\lvert\\!|\\!|\cdot|\\!|\\!\rvert_{s}$ and is used in our seminorm
smoothening argument in Section 8.

######  Proposition A.1.

Let $(X,{\mathcal{X}},\mu,T_{1},\ldots,T_{\ell})$ be a system and
${\mathcal{Y}}_{1},\ldots,{\mathcal{Y}}_{\ell}\subseteq{\mathcal{X}}$ be
sub-$\sigma$-algebras. Suppose that the sequences
$a_{1},\ldots,a_{\ell}\colon{\mathbb{N}}\to{\mathbb{Z}}$ are good for mean
convergence for the subspaces
$L^{\infty}({\mathcal{Y}}_{1},\mu),\ldots,L^{\infty}({\mathcal{Y}}_{\ell},\mu)$.
Suppose moreover that for the same subspaces the seminorm
$\lvert\\!|\\!|f_{m}|\\!|\\!\rvert_{s}$ controls the averages (109) for some
$m\in[\ell]$ and $s\in{\mathbb{N}}$. Then for every $\varepsilon>0$ there
exists $\delta>0$ such that if $f_{1}\in
L^{\infty}({\mathcal{Y}}_{1},\mu),\ldots,f_{\ell}\in
L^{\infty}({\mathcal{Y}}_{\ell},\mu)$ are $1$-bounded and
$\lvert\\!|\\!|f_{m}|\\!|\\!\rvert_{s}\leq\delta$, then

$\lim_{N\to\infty}\left\|{\mathbb{E}}_{n\in[N]}\,T_{1}^{a_{1}(n)}f_{1}\cdots
T_{\ell}^{a_{\ell}(n)}f_{\ell}\right\|_{L^{2}(\mu)}\leq\varepsilon.$

######  Remarks.

$\bullet$ The important point is that the choice $\delta$ does not depend on
the $1$-bounded functions $f_{1},\ldots,f_{\ell}$ (but we do not claim
independence of the system or the subspaces).

$\bullet$ It is possible to prove variants of these results that do not have
mean convergence assumptions, but then the conclusion will be a subsequential
weak convergence property.

$\bullet$ Using straightforward modifications, similar variants can be proved
for other averaging schemes (averages along specific Følner sequences,
weighted, logarithmic, etc).

$\bullet$ In applications, the $\sigma$-algebras
${\mathcal{Y}}_{1},\ldots,{\mathcal{Y}}_{\ell}$ are factors of
$T_{1},\ldots,T_{\ell}$, but the result does not require this assumption.

###  A.2. A more abstract statement

We are going to deduce Propositions A.1 from a more abstract statement (see
Proposition A.2) that deals with continuous multilinear functionals on
reflexive Banach spaces.

######  Definition.

Let $(X_{j},\left\|\cdot\right\|_{X_{j}})$ be a normed space for every
$j\in[\ell]$.

  * •

On the product space $X_{1}\times\cdots\times X_{\ell}$, we always consider
the norm $\left\|\cdot\right\|_{X_{1}\times\cdots\times X_{\ell}}$ defined by

$\left\|(x_{1},\ldots,x_{\ell})\right\|_{X_{1}\times\cdots\times
X_{\ell}}:=\left\|x_{1}\right\|_{X_{1}}+\cdots+\left\|x_{\ell}\right\|_{X_{\ell}}.$

  * •

A mapping $A\colon X_{1}\times\cdots\times X_{\ell}\to{\mathbb{R}}$ is called
a multilinear functional, if it is linear with respect to each of its $\ell$
coordinates when the other $\ell-1$ coordinates are fixed.

It is easy to verify that a multilinear functional $A\colon
X_{1}\times\cdots\times X_{\ell}\to{\mathbb{R}}$ is continuous if and only if
there exists $C>0$ such that

(110) $|A(x_{1},\ldots,x_{\ell})|\leq
C\prod_{j\in[\ell]}\left\|x_{j}\right\|_{X_{j}}$

for all $(x_{1},\ldots,x_{\ell})\in X_{1}\times\cdots\times X_{\ell}$. The
forward implication is immediate since continuity at $0$ implies
$1$-boundedness on some ball centered at the origin, and then multilinearity
easily implies the stated estimate. One gets the reverse implication by using
a telescoping argument. We start by noting that

$|A(y_{1},\ldots,y_{\ell})-A(x_{1},\ldots,x_{\ell})|\leq\sum_{j=1}^{\ell}|A(x_{1},\ldots,x_{j-1},y_{j},\ldots,y_{\ell})-A(x_{1},\ldots,x_{j},y_{j+1},\ldots,y_{\ell})|.$

Using multilinearity we get that the right hand side is equal to

$\sum_{j=1}^{\ell}|A(x_{1},\ldots,x_{j-1},y_{j}-x_{j},\ldots,y_{\ell})|,$

and for $x_{j},y_{j}\in B_{X_{j}}$, $j\in[\ell],$ this can be bounded using
(110) by

$C\sum_{j=1}^{\ell}\left\|y_{j}-x_{j}\right\|_{X_{j}}.$

For future use, we record the following uniform continuity property of
continuous multilinear functionals that follows from the previous discussion:
for every $(x_{1},\ldots,x_{\ell-1})\in X_{1}\times\cdots\times X_{\ell-1}$,
we have

(111)
$\lim_{(y_{1},\ldots,y_{\ell-1})\to(x_{1},\ldots,x_{\ell-1})}\sup_{x_{\ell}\in
B_{X_{\ell}}}|A(y_{1},\ldots,y_{\ell-1},x_{\ell})-A(x_{1},\ldots,x_{\ell-1},x_{\ell})|=0$

where convergence is considered with the norm
$\left\|\cdot\right\|_{X_{1}\times\cdots\times X_{\ell-1}}$ on
$X_{1}\times\cdots\times X_{\ell-1}$.

######  Definition.

Let $\lvert\\!|\\!|\cdot|\\!|\\!\rvert_{1}$ be a seminorm on a linear space
$X$ and $\lvert\\!|\\!|\cdot|\\!|\\!\rvert_{2}$ be a seminorm on a subspace
$X^{\prime}$ of $X$. Let also $X_{0}\subseteq X^{\prime}$ be a subset. We say
that

  * •

$\lvert\\!|\\!|\cdot|\\!|\\!\rvert_{2}$ controls the convergence of
$\lvert\\!|\\!|\cdot|\\!|\\!\rvert_{1}$ on $X_{0}$ if whenever $x_{n},x\in
X_{0}$, $n\in{\mathbb{N}}$, and
$\lim_{n\to\infty}\lvert\\!|\\!|x_{n}-x|\\!|\\!\rvert_{2}=0$, we have
$\lim_{n\to\infty}\lvert\\!|\\!|x_{n}-x|\\!|\\!\rvert_{1}=0$.

  * •

$\lvert\\!|\\!|\cdot|\\!|\\!\rvert_{2}$ dominates
$\lvert\\!|\\!|\cdot|\\!|\\!\rvert_{1}$ on $X_{0}$ if for some $C>0$, we have
$\lvert\\!|\\!|x|\\!|\\!\rvert_{1}\leq C\lvert\\!|\\!|x|\\!|\\!\rvert_{2}$ for
every $x\in X_{0}$.

The next result enables us to infer quantitative information from purely
qualitative input. The subspaces $X^{\prime}_{j}$ are introduced for purely
technical reasons (see the first remark following the theorem), and on a first
reading one is advised to assume that the spaces
$X^{\prime}_{1}=\cdots=X^{\prime}_{\ell}=X_{1}=\cdots=X_{\ell}$ are all the
same.

######  Proposition A.2.

Let $(X_{j},\left\|\cdot\right\|_{X_{j}})$, $j\in[\ell]$, be separable,
reflexive, Banach spaces, and $A\colon X_{1}\times\cdots\times
X_{\ell}\to{\mathbb{R}}$ be a continuous multilinear functional. For
$j\in[\ell]$, let $X_{j}^{\prime}$ be $\left\|\cdot\right\|_{X_{j}}$-dense
subspaces of $X_{j}$ and $\left\|\cdot\right\|_{X_{j}^{\prime}}$ be a norm on
$X_{j}^{\prime}$ that dominates the norm $\left\|\cdot\right\|_{X_{j}}$ on
$X^{\prime}_{j}$ and such that $B_{X_{j}^{\prime}}$ is
$\left\|\cdot\right\|_{X_{j}}$-closed.282828Throughout this statement
$B_{X_{j}^{\prime}}$ is defined using the norm
$\left\|\cdot\right\|_{X_{j}^{\prime}}$. Let also $m\in[\ell]$ and
$\lvert\\!|\\!|\cdot|\\!|\\!\rvert$ be a seminorm on $X_{m}^{\prime}$, such
that the norm $\left\|\cdot\right\|_{X_{m}}$ controls the convergence of the
seminorm $\lvert\\!|\\!|\cdot|\\!|\\!\rvert$ on $B_{X_{m}^{\prime}}$. Then the
following two statements are equivalent:

  1. (i)

(Qualitative property) Whenever $x_{m}\in X_{m}^{\prime}$ satisfies
$\lvert\\!|\\!|x_{m}|\\!|\\!\rvert=0$, we have for all $x_{j}\in
X_{j}^{\prime}$, $j\in[\ell]$, $j\neq m$, that

$A(x_{1},\ldots,,x_{\ell})=0.$

  2. (ii)

(Quantitative property) For every $\varepsilon>0$ there exists $\delta>0$ such
that if some $x_{m}\in B_{X_{m}^{\prime}}$ satisfies
$\lvert\\!|\\!|x_{m}|\\!|\\!\rvert\leq\delta$, then

$\sup_{x_{j}\in B_{X_{j}},j\in[\ell],j\neq
m}|A(x_{1},\ldots,x_{\ell})|\leq\varepsilon.$

######  Remarks.

$\bullet$ In the proof of Proposition A.1, we will use this result for
$X_{j}:=L^{p}({\mathcal{Y}}_{j},\mu)$,
$X^{\prime}_{j}:=L^{\infty}({\mathcal{Y}}_{j},\mu)$,
$\left\|\cdot\right\|_{X_{j}}:=\left\|\cdot\right\|_{L^{p}(\mu)}$,
$\left\|\cdot\right\|_{X^{\prime}_{j}}:=\left\|\cdot\right\|_{L^{\infty}(\mu)}$,
$j\in[\ell]$,
$\lvert\\!|\\!|\cdot|\\!|\\!\rvert:=\lvert\\!|\\!|\cdot|\\!|\\!\rvert_{s}$,
for appropriate values of $p\in(1,+\infty)$ and $s\in{\mathbb{N}}$, the
functional

$A(f_{0},\ldots,f_{\ell}):=\lim_{N\to\infty}{\mathbb{E}}_{n\in[N]}\int
f_{0}\cdot T_{1}^{a_{1}(n)}f_{1}\cdots T_{\ell}^{a_{\ell}(n)}f_{\ell}\,d\mu$

and $\sigma$-algebras ${\mathcal{Y}}_{1},\ldots,{\mathcal{Y}}_{\ell}$ (which
in our applications in Section 8 are factors of $T_{1},\ldots,T_{\ell}$). In
our applications, it will be possible to show that there exists
$s\in{\mathbb{N}}$ such that if $f_{0}\in L^{\infty}({\mathcal{X}},\mu)$,
$f_{j}\in L^{\infty}({\mathcal{Y}}_{j},\mu)$ for every $j\in[\ell]$ and
additionally $\lvert\\!|\\!|f_{m}|\\!|\\!\rvert_{s}=0$, then
$A(f_{0},\ldots,f_{\ell})=0$. Establishing this implication for functions
$f_{j}\in L^{p}({\mathcal{Y}}_{j},\mu)$, $j\in[\ell]$, for some appropriate
$p\in{\mathbb{N}}$ is strictly harder and cannot be derived using an
approximation argument from $L^{\infty}(\mu)$ functions. This is the reason
why we introduce the auxiliary spaces
$X^{\prime}_{1},\ldots,X^{\prime}_{\ell}$ in the statement.

$\bullet$ The quantity $\delta$ is going to depend on the functional $A$,
spaces $X_{j},X_{j}^{\prime}$, index $\ell$, norms
$\left\|\cdot\right\|_{X_{j}},\left\|\cdot\right\|_{X^{\prime}_{j}}$, seminorm
$\lvert\\!|\\!|\cdot|\\!|\\!\rvert$, and the variable $\varepsilon$. What is
important for our applications is that it does not depend on elements
$x_{j}\in B_{X_{j}}$ for $j\in[\ell],j\neq m$ or $x_{m}\in B_{X^{\prime}_{m}}$
(as long as $\lvert\\!|\\!|x_{m}|\\!|\\!\rvert\leq\delta$).

The proof of Proposition A.2 uses two classical Banach space results: Mazur’s
lemma (see next subsection) is used to get the needed uniformity with respect
to the variable $x_{m}$, and then the Baire category theorem is used to get
uniformity with respect to the other variables.

###  A.3. Mazur’s lemma

If $A$ is a subset of a linear space we denote with $\text{co}(A)$ the set of
(finite) convex combinations of elements of $A$. The next is a standard fact
from Banach space theory.

######  Lemma A.3 (Mazur’s Lemma).

Let $X$ be a normed space and suppose that the sequence $(x_{n})$, of elements
of $X$, converges weakly to some $x\in X$. Then there exist
$y_{n}\in\text{co}(x_{1},x_{2},\ldots)$, $n\in{\mathbb{N}}$, such that the
sequence $(y_{n})$ converges strongly to $x$.

We are going to use the following consequence.

######  Corollary A.4.

Let $X$ be a reflexive Banach space and suppose that $(x_{n})$ is a bounded
sequence of elements of $X$. Then there exist
$y_{n}\in\text{co}(x_{n+1},x_{n+2},\ldots)$, $n\in{\mathbb{N}}$, and $x\in X$,
such that the sequence $(y_{n})$ converges strongly to $x$.

######  Remarks.

$\bullet$ We are going to use this for $X:=L^{p}(\mu)$ for appropriate values
of $p\in{\mathbb{N}}$.

$\bullet$ If $X$ is a Hilbert space, more is true and rather easy to prove. If
$x_{n}$ converges to $x_{0}$ weakly, then there exists a subsequence
$k_{n}\to\infty$ such that $\frac{x_{k_{1}}+\cdots+x_{k_{n}}}{n}\to x_{0}$
strongly. To see this we can assume that $x_{0}=0$ and
$\left\|x_{n}\right\|\leq 1$ for every $n\in{\mathbb{N}}$. Since $\langle
x_{n},x\rangle\to 0$ for every $x\in X$, we can choose inductively $k_{n}$
such that $|\langle x_{k_{n}},x_{k_{i}}\rangle|\leq\frac{1}{2^{n}}$,
$i=1,\ldots,n-1$ for every $n\in{\mathbb{N}}$. Then
$\left\|\frac{x_{k_{1}}+\cdots+x_{k_{n}}}{n}\right\|^{2}\leq\frac{3}{n}$ for
every $n\in{\mathbb{N}}$. A result of Banach-Saks claims that a similar result
is true when $X=L^{p}(\mu)$ is separable (which is the case since we work with
Lebesgue spaces) for any $p\in(1,+\infty)$ but this is much harder to prove
(see for example page 101 in [36]).

###### Proof.

Since $X$ is reflexive, its unit ball is weakly compact, hence the sequence
$(x_{n})$ has a subsequence $(x_{n_{k}})$ that converges weakly to some $x\in
X$. Since the sequence $x_{n_{1}},x_{n_{2}},\ldots$ converges weakly to $x$
the same holds for the sequence $x_{n_{k}},x_{n_{k+1}},\ldots$ for every
$k\in{\mathbb{N}}$. Hence, by the previous lemma, for every $k\in{\mathbb{N}}$
there exists $y_{k}\in\text{co}(x_{n_{k}},x_{n_{k+1}},\ldots)$ that satisfies
$\left\|x-y_{k}\right\|\leq 1/k$. Since $n_{k}\geq k$ the conclusion follows.
∎

###  A.4. Proof of Proposition A.2

The reverse implication is trivial, so we are going to prove the forward
implication. We assume property (i), and we are going to establish property
(ii). Without loss of generality, we can assume that $m=\ell$.

First Step. (Uniformity with respect to $x_{\ell}$) We claim that under the
stated assumptions the following holds: if $(x_{1},\ldots,x_{\ell-1})\in
X_{1}^{\prime}\times\cdots\times X_{\ell-1}^{\prime}$ are fixed, then there
exists $\delta>0$ (depending on $x_{1},\ldots,x_{\ell-1}$ but not on
$x_{\ell}$) such that if $x_{\ell}\in B_{X^{\prime}_{\ell}}$ satisfies
$\lvert\\!|\\!|x_{\ell}|\\!|\\!\rvert\leq\delta$, then
$|A(x_{1},\ldots,x_{\ell})|\leq\varepsilon$. By considering the element
$-x_{\ell}$ in place of $x_{\ell}$ and using that
$A(x_{1},\ldots,-x_{\ell})=-A(x_{1},\ldots,x_{\ell})$, it suffices to verify
that $A(x_{1},\ldots,x_{\ell})\leq\varepsilon$.

Arguing by contradiction, suppose that there exist $\varepsilon_{0}>0$ and
elements $x_{\ell,k}\in B_{X^{\prime}_{\ell}}$, $k\in{\mathbb{N}}$, such that

(112) $\lim_{k\to\infty}\lvert\\!|\\!|x_{\ell,k}|\\!|\\!\rvert=0$

and

(113) $A(x_{1},\ldots,x_{\ell-1},x_{\ell,k})\geq\varepsilon_{0},\quad
k\in{\mathbb{N}}.$

Since $\left\|\cdot\right\|_{X^{\prime}_{\ell}}$ dominates the norm
$\left\|\cdot\right\|_{X_{\ell}}$ on $X^{\prime}_{\ell}$, the sequence
$(x_{\ell,k})_{k\in{\mathbb{N}}}$ is
$\left\|\cdot\right\|_{X_{\ell}}$-bounded. Since $X_{\ell}$ is a reflexive
Banach space, by Corollary A.4, there exist $x_{\ell}\in X_{\ell}$, sequence
of integers $J_{k}\to\infty$, and $c_{1,k},\ldots,c_{J_{k},k}\in[0,1]$,
$k\in{\mathbb{N}}$, with sum $1$, such that

(114)
$\lim_{k\to\infty}\left\|\sum_{j=1}^{J_{k}}\,c_{j,k}\,x_{\ell,k+j}-x_{\ell}\right\|_{X_{\ell}}=0.$

Furthermore, since $x_{\ell,k}\in B_{X^{\prime}_{\ell}}$ for all
$k\in{\mathbb{N}}$, $c_{j,k}\geq 0$ and $\sum_{j=1}^{J_{k}}\,c_{j,k}=1$, we
have that $\sum_{j=1}^{J_{k}}\,c_{j,k}x_{\ell,k+j}\in B_{X^{\prime}_{\ell}}$
for all $k\in{\mathbb{N}}$, hence using our hypothesis that
$B_{X^{\prime}_{\ell}}$ is $\left\|\cdot\right\|_{X_{\ell}}$-closed and
identity (114), we get that $x_{\ell}\in B_{X^{\prime}_{\ell}}$. We claim that
this element $x_{\ell}$ contradicts our assumptions.

On the one hand, (113) combined with $c_{j,k}\geq 0$ and
$\sum_{j=1}^{J_{k}}\,c_{j,k}=1$ and the linearity of $A$ with respect to the
variable $x_{\ell}$, give

(115)
$A\big{(}x_{1},\ldots,x_{\ell-1},\sum_{j=1}^{J_{k}}\,c_{j,k}\,x_{\ell,k+j}\big{)}=\sum_{j=1}^{J_{k}}\,c_{j,k}\,A(x_{1},\ldots,x_{\ell-1},x_{\ell,k+j})\geq\varepsilon_{0},\quad
k\in{\mathbb{N}}.$

Using the $\left\|\cdot\right\|_{X_{\ell}}$-continuity of $A$ with respect to
the variable $x_{\ell}$ and equations (114) and (115), we deduce

$A(x_{1},\ldots,x_{\ell})\geq\varepsilon_{0}.$

On the other hand, we have

$\lvert\\!|\\!|x_{\ell}|\\!|\\!\rvert\leq\limsup_{k\to\infty}\lvert\\!|\\!|\sum_{j=1}^{J_{k}}\,c_{j,k}\,x_{\ell,k+j}|\\!|\\!\rvert\leq\limsup_{k\to\infty}\sum_{j=1}^{J_{k}}\,c_{j,k}\,\lvert\\!|\\!|x_{\ell,k+j}|\\!|\\!\rvert\leq\limsup_{k\to\infty}\sup_{j\in{\mathbb{N}}}\lvert\\!|\\!|x_{\ell,k+j}|\\!|\\!\rvert=0,$

where the first estimate follows from (114) and our assumption that the norm
$\left\|\cdot\right\|_{X_{\ell}}$ controls the convergence of the seminorm
$\lvert\\!|\\!|\cdot|\\!|\\!\rvert$ on $B_{X^{\prime}_{\ell}}$, the second
estimate holds since $c_{j,k}\geq 0$ and $\lvert\\!|\\!|\cdot|\\!|\\!\rvert$
is a seminorm, the third estimate holds since $c_{j,k}\geq 0$ and
$\sum_{j=1}^{J_{k}}\,c_{j,k}=1$, and the last identity because of (112).
Hence, $\lvert\\!|\\!|x_{\ell}|\\!|\\!\rvert=0$.

Combining the previous two facts, we have that
$\lvert\\!|\\!|x_{\ell}|\\!|\\!\rvert=0$ but $A(x_{1},\ldots,x_{\ell})\neq 0$.
This contradicts property (i) and concludes the proof of the claim.

Second Step. (Uniformity with respect to $x_{1},\ldots,x_{\ell-1}$, part I) We
plan to get some uniformity with respect to the variables
$x_{1},\ldots,x_{\ell-1}$ on some non-empty open subset of
$X_{1}\times\cdots\times X_{\ell-1}$ using the Baire category theorem.

For each $j\in[\ell-1]$, let $Y_{j}\subset X^{\prime}_{j}$ be a countable
dense set in $(X_{j},\left\|\cdot\right\|_{X_{j}})$; such a set can be found
since, by assumption, $X^{\prime}_{j}$ is $\left\|\cdot\right\|_{X_{j}}$-dense
in $X_{j}$ and $(X^{\prime}_{j},\left\|\cdot\right\|_{X_{j}})$ is separable
(since $(X_{j},\left\|\cdot\right\|_{X_{j}})$ is separable by assumption). Let
also $\varepsilon>0$. Using the first step we get that for every
$y=(y_{j})_{j\in[\ell-1]}\in Y_{1}\times\cdots\times Y_{\ell-1}$ there exists
$\delta_{y}=\delta_{y}(\varepsilon)$, such that if $x_{\ell}\in
B_{X^{\prime}_{\ell}}$ satisfies
$\lvert\\!|\\!|x_{\ell}|\\!|\\!\rvert\leq\delta_{y}$, then

$A(y_{1},\ldots,y_{\ell-1},x_{\ell})\leq\varepsilon/2.$

Let now $(x_{1},\ldots,x_{\ell-1})\in X_{1}\times\cdots\times X_{\ell-1}$ be
arbitrary. Since $Y_{1}\times\cdots\times Y_{\ell-1}$ is
$\left\|\cdot\right\|_{X_{1}\times\cdots\times X_{\ell-1}}$-dense in
$X_{1}\times\cdots\times X_{\ell-1}$ and $A$ satisfies (111) (since it is
multilinear and continuous), there exists $y=(y_{j})_{j\in[\ell-1]}\in
Y_{1}\times\cdots\times Y_{\ell-1}$ such that

$\sup_{x_{\ell}\in
B_{X^{\prime}_{\ell}}}|A(y_{1},\ldots,y_{\ell-1},x_{\ell})-A(x_{1},\ldots,x_{\ell-1},x_{\ell})|\leq\varepsilon/2.$

Combining the above, we get that for every $(x_{j})_{j\in[\ell-1]}\in
X_{1}\times\cdots\times X_{\ell-1}$ there exists $y=(y_{j})_{j\in[\ell-1]}\in
Y_{1}\times\cdots\times Y_{\ell-1}$ (depending on $(x_{j})_{j\in[\ell-1]}$ and
on $\varepsilon$) such that the following holds: for all $x_{\ell}\in
B_{X^{\prime}_{\ell}}$ with
$\lvert\\!|\\!|x_{\ell}|\\!|\\!\rvert\leq\delta_{y}$, we have

(116) $|A(x_{1},\ldots,x_{\ell-1},x_{\ell})|\leq\varepsilon.$

For $y\in Y_{1}\times\cdots\times Y_{\ell-1}$, we define the subset
$F_{y}=F_{y}(\varepsilon)$ of $X_{1}\times\cdots\times X_{\ell-1}$ by

(117) $F_{y}:=\\{(x_{j})_{j\in[\ell-1]}\in X_{1}\times\cdots\times
X_{\ell-1}\colon\eqref{E:gfegen}\text{ holds for all }x_{\ell}\in
B_{X^{\prime}_{\ell}}\text{ with
}\lvert\\!|\\!|x_{\ell}|\\!|\\!\rvert\leq\delta_{y}\\}.$

Since the multilinear functional $A$ is continuous, we deduce that for every
$y\in Y_{1}\times\cdots\times Y_{\ell-1}$, the subset $F_{y}$ of
$X_{1}\times\cdots\times X_{\ell-1}$ is
$\left\|\cdot\right\|_{X_{1}\times\cdots\times X_{\ell-1}}$-closed and by our
previous discussion, we have

$X_{1}\times\cdots\times X_{\ell-1}=\bigcup_{y\in Y_{1}\times\cdots\times
Y_{\ell-1}}F_{y}.$

Since the set $Y_{1}\times\cdots\times Y_{\ell-1}$ is countable and the space
$X_{1}\times\cdots\times X_{\ell-1}$ is
$\left\|\cdot\right\|_{X_{1}\times\cdots\times X_{\ell-1}}$-complete, the
Baire category theorem gives that some $F_{y}$ has non-empty interior with
respect to the norm $\left\|\cdot\right\|_{X_{1}\times\cdots\times
X_{\ell-1}}$. Hence, there exist $y_{0}=y_{0}(\varepsilon)\in
Y_{1}\times\cdots\times Y_{\ell-1}$,
$x^{\prime}_{1}=x^{\prime}_{1}(\varepsilon)\in
X_{1},\ldots,x^{\prime}_{\ell-1}=x^{\prime}_{\ell-1}(\varepsilon)\in
X_{\ell-1}$ and $r_{0}=r_{0}(\varepsilon)>0$ such that

(118) $\prod_{j\in[\ell-1]}D_{X_{j}}(x^{\prime}_{j},2r_{0})\subset F_{y_{0}}$

where $D_{X_{j}}(x_{0},r):=\\{x\in
X_{j}\colon\left\|x-x_{0}\right\|_{X_{j}}<r\\}$.

Third Step. (Uniformity with respect to $x_{1},\ldots,x_{\ell-1}$, part II) We
are going to use the multilinearity of the expression
$A(x_{1},\ldots,x_{\ell})$ in order to transfer the estimate (116) from the
open subset of $X_{1}\times\cdots\times X_{\ell-1}$ given in (118) to
$B_{X_{1}}\times\cdots\times B_{X_{\ell-1}}$.

Let $(x_{1},\ldots,x_{\ell-1})\in B_{X_{1}}\times\cdots\times B_{X_{\ell-1}}$.
Then by (118), we have

$(x^{\prime}_{j}+r_{0}x_{j})_{j\in[\ell-1]}\in\prod_{j\in[\ell-1]}D(x^{\prime}_{j},2r_{0})\subset
F_{y_{0}},$

hence by (117), we get that

(119)
$|A(x^{\prime}_{1}+r_{0}x_{1},\ldots,x^{\prime}_{\ell-1}+r_{0}x_{\ell-1},x_{\ell})|\leq\varepsilon$

holds for all $(x_{1},\ldots,x_{\ell-1})\in B_{X_{1}}\times\cdots\times
B_{X_{\ell-1}}$, and $x_{\ell}\in B_{X^{\prime}_{\ell}}$ with
$\lvert\\!|\\!|x_{\ell}|\\!|\\!\rvert\leq\delta_{y_{0}}$. For each
$j\in[\ell-1]$, we split
$x_{j}=\frac{1}{r_{0}}((x^{\prime}_{j}+r_{0}x_{j})-x^{\prime}_{j})$, and we
use multilinearity to decompose $A(x_{1},\ldots,x_{\ell})$ into $2^{\ell-1}$
terms, so that

(120)
$\displaystyle|A(x_{1},\ldots,x_{\ell})|\leq\frac{1}{r_{0}^{\ell-1}}\sum_{{\underline{\epsilon}}\in\\{0,1\\}^{\ell-1}}|A(x_{1}^{\prime}+\epsilon_{1}r_{0}x_{1},\ldots,x^{\prime}_{\ell-1}+\epsilon_{\ell-1}r_{0}x_{\ell-1},x_{\ell})|,$

where ${\underline{\epsilon}}=(\epsilon_{1},\ldots,\epsilon_{\ell-1})$.
Recalling that $x^{\prime}_{j},x^{\prime}_{j}+r_{0}x_{j}\in
D(x^{\prime}_{j},2r_{0})$, we use (119) and (120) to conclude that

$|A(x_{1},\ldots,x_{\ell})|\leq(2/r_{0})^{\ell-1}\varepsilon$

for all $(x_{1},\ldots,x_{\ell-1})\in B_{X_{1}}\times\cdots\times
B_{X_{\ell-1}}$ and $x_{\ell}\in B_{X^{\prime}_{\ell}}$ with
$\lvert\\!|\\!|x_{\ell}|\\!|\\!\rvert\leq\delta_{y_{0}}$. Hence, if
$\lvert\\!|\\!|x_{\ell}|\\!|\\!\rvert\leq(r_{0}/2)^{\ell-1}\delta_{y_{0}}$, we
have
$\lvert\\!|\\!|(2/r_{0})^{\ell-1}x_{\ell}|\\!|\\!\rvert\leq\delta_{y_{0}}$,
and the previous estimate implies

$|A(x_{1},\ldots,x_{\ell-1},(2/r_{0})^{\ell-1}x_{\ell})|\leq(2/r_{0})^{\ell-1}\varepsilon$

for all $(x_{1},\ldots,x_{\ell-1})\in B_{X_{1}}\times\cdots\times
B_{X_{\ell-1}}$. We deduce from this and the linearity of the expression
$A(x_{1},\ldots,x_{\ell})$ with respect to the variable $x_{\ell}$, that for
$\delta:=\delta_{y_{0}}(r_{0}/2)^{\ell-1}$, the inequality

$|A(x_{1},\ldots,x_{\ell})|\leq\varepsilon$

holds for all $(x_{1},\ldots,x_{\ell-1})\in B_{X_{1}}\times\cdots\times
B_{X_{\ell-1}}$ and $x_{\ell}\in B_{X^{\prime}_{\ell}}$ with
$\lvert\\!|\\!|x_{\ell}|\\!|\\!\rvert\leq\delta$. This completes the proof of
Proposition A.2.

###  A.5. Proof of the main result

We are going to use Proposition A.2 and some pretty standard maneuvers in
order to deduce Proposition A.1.

###### Proof of Proposition A.1.

We can assume that all functions are real valued and $m=\ell$.

We plan to apply Proposition A.2. For $j\in[\ell]$, we let

$\displaystyle
X_{j}:=L^{\ell+1}({\mathcal{Y}}_{j},\mu),\quad\left\|\cdot\right\|_{X_{j}}:=\left\|\cdot\right\|_{L^{\ell+1}(\mu)},$
$\displaystyle
X^{\prime}_{j}:=L^{\infty}({\mathcal{Y}}_{j},\mu),\quad\left\|\cdot\right\|_{X^{\prime}_{j}}:=\left\|\cdot\right\|_{L^{\infty}(\mu)},\quad\lvert\\!|\\!|\cdot|\\!|\\!\rvert:=\lvert\\!|\\!|\cdot|\\!|\\!\rvert_{s}.$

Fix $j\in[\ell]$ and recall the following facts:

  * •

$(X_{j},\left\|\cdot\right\|_{X_{j}})=(L^{\ell+1}({\mathcal{Y}}_{j},\mu),\left\|\cdot\right\|_{L^{\ell+1}(\mu)})$
is a separable, reflexive Banach space;

  * •

$\left\|\cdot\right\|_{L^{\ell+1}(\mu)}\leq\left\|\cdot\right\|_{L^{\infty}(\mu)}$,
hence $\left\|\cdot\right\|_{L^{\infty}(\mu)}$ dominates
$\left\|\cdot\right\|_{L^{\ell+1}(\mu)}$ on
$L^{\infty}({\mathcal{Y}}_{j},\mu)$;

  * •

the norm $\left\|\cdot\right\|_{L^{\ell+1}(\mu)}$ controls the convergence of
$\lvert\\!|\\!|\cdot|\\!|\\!\rvert_{s}$ on
$B_{L^{\infty}({\mathcal{Y}}_{j},\mu)}$292929It suffices to show that if
$g_{n}\in L^{\infty}(\mu)$, $n\in{\mathbb{N}}$, are $1$-bounded and
$\lim_{n\to\infty}\left\|g_{n}\right\|_{L^{\ell+1}(\mu)}=0$, then also
$\lim_{n\to\infty}\left\|g_{n}\right\|_{s}=0$. If $\ell+1\geq 2^{s}$ this
follows from
$\left\|g_{n}\right\|_{s}\leq\left\|g_{n}\right\|_{L^{2^{s}}(\mu)}\leq\left\|g_{n}\right\|_{L^{\ell+1}(\mu)}$.
If $\ell+1<2^{s}$, then for $1$-bounded functions $g_{n}$, we have
$\left\|g_{n}\right\|_{s}^{2^{s}}\leq\left\|g_{n}\right\|_{L^{1}(\mu)}\leq\left\|g_{n}\right\|_{L^{\ell+1}(\mu)}$.;

  * •

the space $L^{\infty}({\mathcal{Y}}_{j},\mu)$ is $L^{\ell+1}(\mu)$-dense in
$L^{\ell+1}({\mathcal{Y}}_{j},\mu)$, and
$B_{L^{\infty}({\mathcal{Y}}_{j},\mu)}$ is
$\left\|\cdot\right\|_{L^{\ell+1}(\mu)}$-closed.

We define the multilinear functional $A\colon X_{0}\times\cdots\times
X_{\ell}\to{\mathbb{R}}$, where $X_{0}:=L^{\ell+1}({\mathcal{X}},\mu)$ and
$X^{\prime}_{0}:=L^{\infty}({\mathcal{X}},\mu)$, by the formula

$A(f_{0},f_{1},\ldots,f_{\ell}):=\lim_{N\to\infty}{\mathbb{E}}_{n\in[N]}\int
f_{0}\cdot T_{1}^{a_{1}(n)}f_{1}\cdots T_{\ell}^{a_{\ell}(n)}f_{\ell}\,d\mu.$

Note that by our good convergence assumption, the previous limit exists
whenever $(f_{0},\ldots,f_{\ell})\in X^{\prime}_{0}\times\cdots\times
X^{\prime}_{\ell}$, i.e. functions $f_{0},\ldots,f_{\ell}\in L^{\infty}(\mu)$
measurable with regards to
${\mathcal{X}},{\mathcal{Y}}_{1},\ldots,{\mathcal{Y}}_{\ell}$ respectively.
Approximating elements of $X_{0}\times\cdots\times X_{\ell}$ by elements of
$X^{\prime}_{0}\times\cdots\times X^{\prime}_{\ell}$ in
$\left\|\cdot\right\|_{X_{0}\times\cdots\times X_{\ell}}$, we deduce that the
limit converges for all $(f_{0},\ldots,f_{\ell})\in X_{0}\times\cdots\times
X_{\ell}$.

Using Hölder’s inequality and the fact that the transformations
$T_{1},\ldots,T_{\ell}$ preserve the measure $\mu$, we get

$|A(f_{0},\ldots,f_{\ell})|\leq\prod_{j=0}^{\ell}\left\|f_{j}\right\|_{L^{\ell+1}(\mu)}$

for all $(f_{0},\ldots,f_{\ell})\in X_{0}\times\cdots\times X_{\ell}$. Hence,
$A$ is continuous. Note also that by assumption, the seminorm
$\lvert\\!|\\!|f_{\ell}|\\!|\\!\rvert_{s}$ controls the averages (109) for the
subspaces $X_{1},\ldots,X_{\ell}$, hence, using the Cauchy-Schwarz inequality
we get that if $f_{0},\ldots,f_{\ell}\in L^{\infty}(\mu)$ are measurable with
regards to ${\mathcal{X}},{\mathcal{Y}}_{1},\ldots,{\mathcal{Y}}_{\ell}$
respectively and $\lvert\\!|\\!|f_{\ell}|\\!|\\!\rvert_{s}=0$, then
$A(f_{0},\ldots,f_{\ell})=0$.

From the previous discussion we get that the assumptions and property (i) of
Proposition A.2 are satisfied and we deduce that property (ii) holds, namely:

For every $\varepsilon>0$ there exists $\delta>0$ such that if
$f_{0},\ldots,f_{\ell}\in L^{\infty}(\mu)$ are $1$-bounded and measurable with
regards to ${\mathcal{X}},{\mathcal{Y}}_{1},\ldots,{\mathcal{Y}}_{\ell}$
respectively, and additionally if