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# Joint ergodicity of Hardy field sequences

Konstantinos Tsinas  University of Crete, Department of mathematics and
applied mathematics, Voutes University Campus, Heraklion 71003, Greece
<EMAIL_ADDRESS>
###### Abstract.

We study mean convergence of multiple ergodic averages, where the iterates
arise from smooth functions of polynomial growth that belong to a Hardy field.
Our results include all logarithmico-exponential functions of polynomial
growth, such as the functions $t^{3/2},t\log t$ and $e^{\sqrt{\log t}}$. We
show that if all non-trivial linear combinations of the functions
$a_{1},...,a_{k}$ stay logarithmically away from rational polynomials, then
the $L^{2}$-limit of the ergodic averages
$\frac{1}{N}\sum_{n=1}^{N}f_{1}(T^{\lfloor a_{1}(n)\rfloor}x)\cdot...\cdot
f_{k}(T^{\lfloor a_{k}(n)\rfloor}x)$ exists and is equal to the product of the
integrals of the functions $f_{1},...,f_{k}$ in ergodic systems, which
establishes a conjecture of Frantzikinakis. Under some more general conditions
on the functions $a_{1},...,a_{k}$, we also find characteristic factors for
convergence of the above averages and deduce a convergence result for weak-
mixing systems.

###### Key words and phrases:

Ergodic averages, joint ergodicity, Hardy fields

###### 2020 Mathematics Subject Classification:

Primary: 37A44; Secondary: 28D05, 05D10, 11B30.

The author was supported by the Research Grant - ELIDEK HFRI-FM17-1684.

###### Contents

  1. 1 Introduction and main results
  2. 2 Background material
  3. 3 Characteristic factors for Hardy sequences
  4. 4 Bounds of polynomial averages
  5. 5 The sub-linear plus polynomial case
  6. 6 The general case of Proposition 3.1
  7. A Some properties of Hardy sequences

##  1\. Introduction and main results

The study of the multiple ergodic averages

(1) $\frac{1}{N}\sum_{n=1}^{N}f_{1}(T^{a_{1}(n)}x)\cdot...\cdot
f_{k}(T^{a_{k}(n)}x)$

for general sequences $a_{1}(n),...,a_{k}(n)$ of integers, where $T$ is an
invertible measure preserving map on a probability space
$(X,\mathcal{X},\mu)$, has been an active field of research in recent years.
Born from Furstenberg’s proof of Szemerédi’s theorem [17] with ergodic
theoretic tools, mean convergence of the averages (1) has been established for
a wide variety of sequences. In this article, our main result is that if the
sequences $a_{1},...,a_{k}$ arise from smooth functions of polynomial
growth111A function $f$ is said to have polynomial growth, if there exists a
positive integer $d$, such that the ratio $\frac{f(t)}{t^{d}}$ converges to 0,
as $t\to+\infty$. belonging to a Hardy field [18, 19] and satisfy certain
independence assumptions, then they are jointly ergodic, that is the
$L^{2}$-limit of the averages in (1) exists and is equal to the product of the
integrals of the functions $f_{1},...,f_{k}$, whenever the underlying system
$(X,\mathcal{X},\mu,T)$ is ergodic. Some typical examples of sequences that we
study are the polynomial sequences with real coefficients, the sequences
$\lfloor n^{3/2}\rfloor$, $\lfloor n\log n\rfloor$, $\lfloor n\log\log
n+\exp(\sqrt{\log(n^{2}+1))}\rfloor$, $\lfloor n^{\sqrt{2}}/\log^{2}n\rfloor$
and, in general, sequences arising from functions in the Hardy field
$\mathcal{LE}$ of logarithmico-exponential functions. Our main results also
establish a conjecture of Frantzikinakis, namely [13, Problem 23] (first
appearing in [11]), which is the content of Theorem 1.2. Furthermore, it gives
a partial answer to [13, Problem 22], which asks for general convergence of
averages of functions from a Hardy field and generalizes several known
results. In the case of weak mixing systems, we can relax our assumptions on
the functions $a_{1},...,a_{k}$ even further and establish a Furstenberg type
weak-mixing theorem (generalizing the results in [1]), which gives a positive
answer to [11, Problem 3].

###  1.1. Statement of the problem and main results

In order to state our theorems below, we shall work with Hardy fields
$\mathcal{H}$ that contain the Hardy field $\mathcal{LE}$ of logarithmico-
exponential functions and are closed under composition and compositional
inversion of functions. All the subsequent results in this section will be
stated under the above assumption. More background on Hardy fields will be
presented in Section 2, where we also present a Hardy field that satisfies the
above property. Before we begin, we present below a theorem concerning the
case of single ergodic averages. This is a consequence of Theorems 3.2 and 3.3
in [6]. More precisely, those two theorems handle the case that the function
$a$ below has at least linear growth, but the case that $a$ has sub-linear
growth rate follows using the same arguments and the equidistribution results
in [5]. The notions of a (measure-preserving) system and ergodicity are
defined in Section 2.

We denote by $C\mathbb{Z}[t]$ the collection of all real multiples of integer
polynomials on some variable $t$.

######  Theorem 1.

[6] Let $a\in\mathcal{H}$ be a function of polynomial growth that satisfies
the following condition:

(A) $\ \lim\limits_{t\to+\infty}\frac{|a(t)-p(t)|}{\log t}=+\infty\text{ for
any polynomial}\ p\in C\mathbb{Z}[t].$

Then, for any measure preserving system $(X,\mu,T)$ and function $f\in
L^{2}(\mu)$, the averages

$\frac{1}{N}\sum_{n=1}^{N}T^{\lfloor a(n)\rfloor}f$

converge in mean to the conditional expectation $\mathcal{E_{\mu}}(f|I(T))$,
where $I(T)$ is the invariant factor of the system $(X,\mu,T)$.

######  Remark.

The above condition is sufficient, but not necessary for convergence in the
single iterate case. However, it encompasses most typical functions in
$\mathcal{H}$ that are not rational polynomials.

We will give the following definition.

######  Definition 1.1.

We will call a function $a\in\mathcal{H}$ 1-good, if it satisfies (A).

The term 1-good indicates the relation of these functions with single ergodic
averages averages.

We show that a natural extension of the above condition implies norm
convergence in the case of multiple averages. If $a_{1},...,a_{k}$ are general
sequences or functions, we will denote by
$\mathcal{L}(a_{1},...,a_{k})\subseteq\mathcal{H}$ the set of non-trivial
linear combinations of the functions $a_{1},...,a_{k}$ (here $\mathcal{H}$ is
a vector space over $\mathbb{R}$). The following theorem is the main result of
this article:

######  Theorem 1.2.

Let $a_{1},...,a_{k}\in\mathcal{H}$ have polynomial growth and assume that
every function in $\mathcal{L}(a_{1},...,a_{k})$ is 1-good. Then, for any
ergodic measure preserving system $(X,\mu,T)$ and functions
$f_{1},...,f_{k}\in L^{\infty}(\mu)$, the averages

(2) $\frac{1}{N}\sum_{n=1}^{N}T^{\lfloor a_{1}(n)\rfloor}f_{1}\cdot...\cdot
T^{\lfloor a_{k}(n)\rfloor}f_{k}$

converge in mean to the product of the integrals $\int f_{1}\
d\mu\cdot...\cdot\int f_{k}\ d\mu.$

######  Remark.

It is a consequence of our proof that the condition on the linear combinations
of the functions $a_{1},...,a_{k}$ can be substituted by the following more
general assumption: for any real numbers $t_{1},...,t_{k}\in[0,1)$, not all of
them zero, we have

$\lim\limits_{N\to+\infty}\frac{1}{N}\sum_{n=1}^{N}e(t_{1}\lfloor
a_{1}(n)\rfloor+\cdots+t_{k}\lfloor a_{k}(n)\rfloor)=0.$

Actually, this is a necessary and sufficient condition in order to have
convergence to the product of the integrals in every ergodic system. This is a
consequence of Theorem 2 in the next section.

If we do not impose an ergodicity assumption on the system $(X,\mu,T)$, then
we can show that the averages in the above theorem converge to the product

$\mathcal{E}_{\mu}(f_{1}|\mathcal{I}_{T})\cdot...\cdot\mathcal{E}_{\mu}(f_{k}|\mathcal{I}_{T}),$

where $\mathcal{E}_{\mu}(f|\mathcal{I}_{T})$ is again the projection of $f$ to
the invariant factor of the system. This follows from a standard ergodic
decomposition argument, and thus, we will usually assume below that the system
$(X,\mu,T)$ is ergodic.

This theorem extends known results about ergodic averages of functions from a
Hardy field. In the case of real polynomials, Theorem 1.2 was established in
[23]. Theorem 1.2 was also proven in [11] when all functions $a_{1},...,a_{k}$
have different growth rates and satisfy $t^{N_{i}+\varepsilon}\ll
a_{i}(t)\prec t^{N_{i}+1}$ for non-negative integers $N_{i}$ and some
$\varepsilon>0$. In addition, Theorem 1.2 was established in [4] under a
variant of our condition. More precisely, an independence condition on the
functions $a_{1},...,a_{k}$ and on all of their derivatives was imposed. It
was proven, however, that if we use a weaker averaging scheme than Cesáro
averages, we can establish uniform convergence results for the corresponding
multiple ergodic averages222 In our setting, if we substitute the standard
Cesáro averages in (2) with uniform ones, then Theorem 1.2 is known to fail.
This is because of the fact that, if a function $f\in\mathcal{H}$ satisfies
$t^{k}\prec f(t)\prec t^{k+1}$ for some non-negative integer $k$, we can find
arbitrarily large intervals, such that $\lfloor f(n)\rfloor$ takes only odd
(or only even) values. Then, this assertion fails for the rotation by $1/2$ on
the torus $\mathbb{R}/\mathbb{Z}$. (cf. [3] for similar arguments and some
nice multiple recurrence and combinatorial results). Finally, Theorem 1.2 was
established recently for linear combinations of tempered functions from a
Hardy field and real polynomials in [14] (for functions $f$ belonging to
$\mathcal{H}$, the tempered condition is equivalent to the relation $t^{k}\log
t\prec f(t)\ll t^{k+1},$ for some non-negative integer $k$). Our result is
more general, since for example we can see that it covers even simple
collections of functions like $\\{t\log t,t^{2}\log t\\}$, for which
convergence has not been established in the literature.

A variant of Theorem 1.2 for commuting transformations was proven in [12]
(under more restrictive conditions). Our methods fail to extend Theorem 1.2 to
this case, the main reason being that we cannot establish seminorm estimates
for convergence of the averages (2). Indeed, even in the case when the
iterates are integer polynomials, characteristic factors have only been
described in some special cases like [7] or [20] and more recently in [9] and
[8], where joint ergodicity of commuting transformations along polynomials was
studied. Finally, we also remark that a similar problem regarding tempered
functions of different growth that do not necessarily belong to some Hardy
field was handled in [25].

###  1.2. Characteristic factors and the case of weak-mixing systems

If our only objective is to find characteristic factors for the averages in
(2), we can relax the conditions of Theorem 1.2 considerably. More precisely,
we have the following theorem which appeared as a conjecture in [11, Problem
3]. The notion of the Host-Kra factor of a system is defined in the following
section.

######  Theorem 1.3.

Assume that the functions $a_{1},...,a_{k}\in\mathcal{H}$ have polynomial
growth and satisfy

$\lim\limits_{t\to+\infty}\frac{|a_{i}(t)|}{\log t}=+\infty\ \ \ \text{ for
all }\ 1\leq i\leq k$

and

$\lim\limits_{t\to+\infty}\frac{|a_{i}(t)-a_{j}(t)|}{\log t}=+\infty\ \ \
\text{ for all }\ \ i\neq j.$

Then, there exists a positive integer $s$ such that, for any measure
preserving system $(X,\mu,T)$, we have

$\lim\limits_{N\to\infty}\big{\lVert}\frac{1}{N}\sum_{n=1}^{N}T^{\lfloor
a_{1}(n)\rfloor}f_{1}\cdot...\cdot T^{\lfloor
a_{k}(n)\rfloor}f_{k}-\frac{1}{N}\sum_{n=1}^{N}T^{\lfloor
a_{1}(n)\rfloor}\widehat{f}_{1}\cdot...\cdot T^{\lfloor
a_{k}(n)\rfloor}\widehat{f}_{k}\big{\rVert}_{L^{2}(\mu)}=0,$

where $\widehat{f}_{i}:=\mathcal{E}_{\mu}(f_{i}|Z_{s}(X))$ is the projection
of $f_{i}$ to the $s$-step Host-Kra factor of the system.

The conditions above are necessary (one can consider some weakly-mixing
systems that are not strongly-mixing to see this). Since for weak-mixing
systems, the Host-Kra factors of any order are trivial, we get the following
corollary, which extends the results in [1, Theorem 1.2] where the iterates
are polynomials taking integer values on the integers, as well as some of the
results in [2] involving tempered functions.

######  Corollary 1.4.

Assume that the functions $a_{1},...,a_{k}\in\mathcal{H}$ have polynomial
growth and satisfy

$\lim\limits_{t\to+\infty}\frac{|a_{i}(t)|}{\log t}=+\infty\ \ \ \text{ for
all }\ 1\leq i\leq k$

and

$\lim\limits_{t\to+\infty}\frac{|a_{i}(t)-a_{j}(t)|}{\log t}=+\infty\ \ \
\text{ for all }\ \ i\neq j.$

Then, for any weak-mixing system $(X,\mu,T)$, we have

$\lim\limits_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}T^{\lfloor
a_{1}(n)\rfloor}f_{1}\cdot...\cdot T^{\lfloor a_{k}(n)\rfloor}f_{k}=\int
f_{1}\ d\mu\cdot...\cdot\int f_{k}\ d\mu,$

where convergence takes place in $L^{2}(\mu)$.

######  Remark.

The proof of this restricted form of Theorem 1.3 still requires a large
portion of the arguments that are used in this article.

###  1.3. Combinatorial Applications

As a corollary of Theorem 1.2, we get the following multiple recurrence
result.

######  Corollary 1.5.

Let $a_{1},...,a_{k}$ be functions from a Hardy field $\mathcal{H}$ such that
every non-trivial linear combination of the functions is 1-good. Then, for any
measure preserving system $(X,\mu,T)$ and any set $A\subset X$ with
$\mu(A)>0$, we have

$\lim\limits_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\ \mu(A\cap T^{-\lfloor
a_{1}(n)\rfloor}A\cap\cdots\cap T^{-\lfloor
a_{k}(n)\rfloor}A)\geq\mu(A)^{k+1}.$

A similar result was established in [4] with $\limsup$ in place of the limit,
but under more general conditions on the functions $a_{1},...,a_{k}$.

Utilizing Furstenberg’s correspondence principle, we can deduce a
combinatorial result about large sets of integers. First of all, we give a
definition of the asymptotic density of a set.

Assume $\Lambda\subset\mathbb{N}$. Then, we define the upper density of the
set $\Lambda$ as the limit

${\bar{d}}(\Lambda):=\limsup\limits_{N\to\infty}\frac{|\Lambda\cap[1,N]|}{N}$

and the lower density $\underline{d}$ is defined similarly with $\liminf$
instead of $\limsup$. If those limits coincide, then we say that the set
$\Lambda$ has natural density $d$ equal to the limit.

######  Theorem  (Furstenberg’s correspondence principle).

For any set $\Lambda\subset\mathbb{N}$ with positive upper density, there
exists an invertible measure preserving system $(X,\mu,T)$ and a measurable
set $A\subset X$, such that $\overline{d}(\Lambda)=\mu(A)$ and for any
$r_{1},...,r_{k}\in\mathbb{Z}$, we have

$\bar{d}(\Lambda\cap(\Lambda-r_{1})\cap\cdots\cap(\Lambda-r_{k}))\geq\mu(A\cap
T^{-r_{1}}A\cap\cdots\cap T^{-r_{k}}A).$

######  Corollary 1.6.

Let $\Lambda\subset\mathbb{N}$ have positive upper density and let
$a_{1},...,a_{k}$ be functions from a Hardy field $\mathcal{H}$, such that
every non-trivial linear combination of these functions is 1-good. Then,

$\liminf\limits_{N\to\infty}\frac{1}{N}\sum_{n=1}^{N}\bar{d}(\Lambda\cap(\Lambda-\lfloor
a_{1}(n)\rfloor)\cap\cdots\cap(\Lambda-\lfloor
a_{k}(n)\rfloor))\geq(\bar{d}(\Lambda))^{k+1}.$

###  1.4. General overview of the proof and organization of the paper

Similarly to the work in [12, 11, 4], our approach is to show that the Host-
Kra factor introduced in [21] (see also [22] for a presentation of the general
theory) is characteristic for convergence of our averages. This is a technique
used extensively in the literature to reduce the problem of convergence in
general measure preserving systems to the case where the system is a rotation
in a nilpotent homogeneous space. However, we shall use a recent result of
Frantzikinakis [14] which roughly asserts that in order to prove Theorem 1.2,
we only need to prove that the Host-Kra factor is characteristic for the
averages in (2) plus some simple equidistribution results on the torus (which
are simple consequences of the equidistribution results in [5]). This bypasses
the usual hassle of proving convergence of the corresponding averages in
nilmanifolds. We remark that this technique can only be used when we expect
convergence of certain ergodic averages to the product of the integrals of the
involved functions, which is the case in this article.

Furthermore, there are also several differences between our methods and the
methods used in [4] and [12] to establish seminorm estimates, where a standard
PET induction argument was utilized to reduce to the case of functions with
sub-linear growth rate. This technique restricts the cases that can be
handled, because the van der Corput operation may eventually yield functions
that do not satisfy the condition (A) (a typical example in this case is the
pair of functions $(t\log t,t\log\log t)$, which "drop" to functions that have
growth rate smaller than $\log t$ after applications of the van der Corput
inequality). In order to overcome this, we use the fact that Hardy field
functions of polynomial growth behave "locally" as polynomials. This
observation was used in [11] to handle the special case of the family
$\\{\lfloor a(n)\rfloor,2\lfloor(a(n)\rfloor,...,k\lfloor a(n)\rfloor\\}$,
where there is only one Hardy field function. It was shown in this case that
the corresponding multiple ergodic averages over small intervals converge to
0, provided that $f_{1}$ is orthogonal to one particular Host-Kra factor of
the system. This technique does not extend to the more general case that we
wish to cover here. However, we can prove instead that these averages can be
bounded by finite ergodic averages, where the iterates are polynomials. We use
then a double averaging trick and the asserted asymptotic bounds to show that
the Host-Kra factor is, indeed, characteristic for convergence of the multiple
averages in our setting. The price to pay is that our argument has to be
somewhat finitary in nature and this makes the proof slightly more technical
and cumbersome. We also note here that our work concerns finding bounds for
ergodic averages involving families of variable polynomials. Some general
convergence results regarding multiple ergodic averages of variable
polynomials were recently established in [26].

A general difficulty in the proofs is that functions in $f\in\mathcal{H}$ that
satisfy $f(t)\ll t^{\delta}$ for all $\delta>0$ (such as the functions $(\log
t)^{c}$, where $c>1$) behave differently from functions that dominate some
fractional power. We describe this difference more clearly in the Appendix,
where we also provide several propositions and lemmas that will be used
extensively in Sections 3 through 6. In addition, we will revisit the ideas
discussed above in Section 3 and also present some examples that we believe
help illustrate the argument of the main proofs.

Our results do not cover the case of general convergence (not necessarily to
the product of the integrals) of the averages in (2). In this case, the 1-good
assumption on the functions can be relaxed further to include more functions,
like the polynomials with integer coefficients. In order to establish this, we
need to deal with the case of convergence in nilsystems, which will be done in
a subsequent article.

###  1.5. Some open problems

An interesting problem that arose when trying to prove our main result is
whether sequences of the form $\lfloor a(n)\rfloor^{\ell}$ are good for the
multiple ergodic theorem, where $\ell$ is a natural number. In the special
case $\ell=2$, we present the following problem:

######  Conjecture 1.

Let $c_{1},...,c_{k}$ be distinct positive non-integers. Do the averages

$\frac{1}{N}\sum_{n=1}^{N}T^{\lfloor n^{c_{1}}\rfloor^{2}}f_{1}\cdot...\cdot
T^{\lfloor n^{c_{k}}\rfloor^{2}}f_{k}$

converge in mean?

Conjecture 1 seems non-trivial even in the case where the fractional powers
$n^{c_{i}}$ are replaced by general (non-integer) real polynomials. In the
case $k=1$, we can use the spectral theorem and the equidistribution results
presented in [10] to give a positive answer. In particular, it was proven in
the same article that $\lfloor a(n)\rfloor^{k}$ is good for the ergodic
theorem when $a$ stays logarithmically far from real multiples of integer
polynomials (this is condition (A)). If we apply the van der Corput
inequality, the resulting sequences at each step become very complicated and
may oscillate substantially. As a consequence, the classical methods of
finding characteristic factors do not seem to yield a result in this case.

Furthermore, we expect that the above averages are jointly ergodic for totally
ergodic systems:

######  Conjecture 2.

Let $c_{1},...,c_{k}$ be distinct positive non-integers and let $(X,\mu,T)$ be
a totally ergodic system. Show that the averages

$\frac{1}{N}\sum_{n=1}^{N}T^{\lfloor n^{c_{1}}\rfloor^{2}}f_{1}\cdot...\cdot
T^{\lfloor n^{c_{k}}\rfloor^{2}}f_{k}$

converge to the product of the integrals $\int f_{1}\ d\mu\cdot...\cdot\int
f_{k}\ d\mu$ for any functions $f_{1},...,f_{k}\in L^{\infty}(\mu)$.

The above problem is interesting even for weak-mixing systems. In addition,
the problem of multiple recurrence is also open. In its simplest form, we have
the following open question.

######  Conjecture 3.

Show that any set of positive upper density contains patterns of the form

$\\{m,m+\lfloor n^{a}\rfloor^{2},m+\lfloor n^{b}\rfloor^{2},\
m,n\in\mathbb{N}\\},$

where $a,b>1$ are distinct non-integers.

In order to establish this, it may be possible to sidestep the more difficult
problem of proving convergence of the corresponding ergodic averages (see, for
example, the arguments in [16]). We do not concern ourselves with this here,
however.

###  1.6. Acknowledgements

I would like to thank my advisor Nikos Frantzikinakis for helpful discussions.
I thank the anonymous referee for useful remarks and corrections on the
previous version of this article. Finally, I would like to thank Dibyendu De
for pointing out corrections in several parts of the previous version.

### Notational conventions

We use $\mathbb{N}$ to denote the set of natural numbers, while
$\mathbb{Z}^{+}$ denotes the non-negative integers. For two sequences $a_{N}$
and $b_{N}$, we say that $b_{N}$ dominates $a_{N}$ and write $a_{N}\prec
b_{N}$ or $a_{N}=o(b_{N})$, if and only if the fraction
$\Big{|}\frac{a_{N}}{b_{N}}\Big{|}$ tends to 0 as $N\to\infty$ and we write
$a_{N}\sim b_{N}$, when this limit is is a finite non-zero real number. In the
latter case, we say that the sequences $a_{N}$ and $b_{N}$ have the same
growth rate. In addition, we write $a_{N}\ll b_{N}$ or $a_{N}=O(b_{N})$ if
there exists a constant $C$ such that, $|a_{N}|\leq C|b_{N}|$ for all
$N\in\mathbb{N}$. When we want to express dependence on some parameters
$c_{1},...,c_{k}$ in the above bounds, we will use the notation
$a_{N}=O_{c_{1},..,c_{k}}(b_{N})$ instead. We use similar asymptotic notation
when we compare growth rates of functions in some real variable $t$.
Furthermore, for a real valued function $f$ we will denote by $f^{(k)}$ the
$k$-th order derivative of $f$, assuming it is well defined.

We will sometimes use bold letters to distinguish between scalar and vector
valued quantities. For a positive integer $M$, we will use $[M]$ to denote the
set $\\{1,2,...,M\\}$. Given a sequence $a(n)$ and a real number $x\geq 1$, we
will use the averaging notation

$\underset{1\leq n\leq x}{\mathbb{E}}a(n):=\frac{1}{\lfloor
x\rfloor}\sum_{n=1}^{\lfloor x\rfloor}a(n).$

Consider a positive integer $s$. We will denote by $[[s]]$ the set
$\\{0,1\\}^{s}$ of ordered $s$-tuples of zeroes and ones, which contains
$2^{s}$ elements. For elements of the set $[[s]]$, we will use the notation
$\underline{\varepsilon}$ instead of bold letters. For convenience, we will
write $\underline{0},\underline{1}$ for the elements $(0,0,...0)$ and
$(1,1,...,1)$ of $[[s]]$ respectively. We will also define
$|\underline{\varepsilon}|$ to be the sum of elements of
$\underline{\varepsilon}$. For a finite set $Y$, we will similarly use the
notation $Y^{[[s]]}$ to denote the set $Y^{2^{s}}$. Each element ${\bf h}\in
Y^{[[s]]}$ can be represented as ${\bf
h}=(h_{\underline{\varepsilon}},\underline{\varepsilon}\in[[s]])$ where each
$h_{\underline{\varepsilon}}$ belongs to $Y$.

For complex numbers $z$, we define the operator $\mathcal{C}^{k}z$, where
$\mathcal{C}^{k}z:=z$, if $k$ is an even number and
$\mathcal{C}^{k}z:=\bar{z}$ otherwise. Finally, we use the notation
$e(t):=e^{2\pi it}$ for $t\in\mathbb{R}$.

##  2\. Background material

###  2.1. Preliminaries on Hardy fields

Let $\mathcal{B}$ denote the set of germs at infinity of real valued functions
defined on a half-line $[x,+\infty]$. Then, $(\mathcal{B},+,\cdot)$ is a ring.
A sub-field $\mathcal{H}$ of $\mathcal{B}$ that is closed under
differentiation is called a Hardy field. We will say that $a(n)$ is a Hardy
sequence, if for $n\in\mathbb{N}$ large enough we have $a(n)=f(n)$ for some
function $f\in\mathcal{H}$. We will make some small abuse of language and
sometimes also refer to sequences of the form $\lfloor f(n)\rfloor$ as Hardy
sequences.

An example of a Hardy field is the field $\mathcal{LE}$ of logarithmico-
exponential functions. These are the functions defined on some half line of
$\mathbb{R}$ by a finite combination of the operations $+,-,\cdot,\div,\
\exp$, $\log$ and composition of functions acting on a real variable $t$ and
real constants. The set $\mathcal{LE}$ contains functions such as the
polynomials $p(t)$, $t^{c}$ for all real $c>0$, $t\log t$, $t^{(\log t)^{2}}$
and $e^{\sqrt{t}}/t^{2}$.

The main advantage when working with functions in a Hardy field (instead of
just the $C^{\infty}$ functions) is that any two functions $f,g\in\mathcal{H}$
are comparable. That means that the limit

$\lim\limits_{t\to\infty}\frac{f(t)}{g(t)}$

exists and thus it makes sense to talk about and compare their growth rates.
In addition, since every function in our Hardy field has a multiplicative
inverse, we can easily infer that every function in $\mathcal{H}$ is
eventually monotone (and, therefore, has constant sign eventually).

It will be crucial in the proof to assume that $\mathcal{H}$ is closed under
composition and compositional inversion of functions, when defined. More
precisely, if $f,g\in\mathcal{H}$ are such that
$\lim\limits_{t\to+\infty}g(t)=+\infty$, then we have that $f\circ
g\in\mathcal{H}$ and $g^{-1}\in\mathcal{H}$. The Hardy field $\mathcal{LE}$
does not have this property. This can be achieved by working with the Hardy
field $\mathcal{P}$ of Pfaffian functions [24], which contains $\mathcal{LE}$
and satisfies the previously mentioned assumptions. This field can be defined
inductively as follows:
i) Let $\mathcal{P}_{1}$ be the set of the smooth functions satisfying the
differential equation $f^{\prime}=p(t,f)$ for some polynomial $p$ with integer
coefficients.
ii) Let $\mathcal{P}_{k}$ be the set of the smooth functions satisfying the
differential equation $f^{\prime}=p(t,f_{1},...,f_{k})$ for some polynomial
$p$ with integer coefficients and $f_{i}\in P_{i}$ for $1\leq i\leq k-1$. Then
$\mathcal{P}$ contains all germs at infinity of the set
$\cup_{i=1}^{\infty}\mathcal{P}_{i}$.

From now on, we will assume that $\mathcal{H}$ has all the above properties.
In the appendix, we have gathered some lemmas regarding growth rates of
functions in $\mathcal{H}$, which will play a crucial role in the
approximations in the following sections.

Finally, we give some definitions for functions whose growth rate is of
particular interest.

######  Definition 2.1.

We say that a function $f\in\mathcal{H}$ has sub-linear growth rate (or is
sub-linear), if $f(t)\prec t$. We say that a function $f\in\mathcal{H}$ has
sub-fractional growth rate (or is sub-fractional), if for all $\delta>0$, we
have $f(t)\ll t^{\delta}$.

Typical examples of sub-linear functions are $\sqrt{t}$, $e^{\sqrt{\log t}}$
and $\log^{3}(t)$. Among these, the functions $e^{\sqrt{\log t}}$ and
$\log^{3}(t)$ are also sub-fractional, while the first one is not sub-
fractional.

######  Definition 2.2.

We will call a function $f\in\mathcal{H}$ of polynomial growth strongly non-
polynomial, if there exists a non-negative integer $d$, such that

$t^{d}\prec f(t)\prec t^{d+1}.$

For example, the functions $t^{3/2}$ and $\log^{3}(t)$ are strongly non-
polynomial, while the function $t^{2}+\sqrt{t}$ is not.

###  2.2. Background in ergodic theory

####  2.2.1. Ergodicity and factors

A measure preserving system is a probability space $(X,\mathcal{X},\mu)$
equipped with an invertible measure preserving transformation $T$. We call a
system ergodic, if the only $T$-invariant functions in $L^{\infty}(\mu)$ are
the constant ones. The system $(X,\mathcal{X},\mu,T)$ is called weak-mixing,
if the product system $(X\times
X,\mathcal{X}\times\mathcal{X},\mu\times\mu,T\times T)$ is ergodic. We say the
system $(Y,\mathcal{Y},\nu,S)$ is a factor of $(X,\mathcal{X},\mu,T)$, if
there exist $X^{\prime}\subset X$, $Y^{\prime}\subset Y$ of full measure that
are invariant under $T$ and $S$ respectively and a map $p:X^{\prime}\to
Y^{\prime}$ such that $\nu=\mu\circ p^{-1}$ and $p\circ T(x)=S\circ p(x)$ for
all $x\in X^{\prime}$. If $p$ is a bijection, we say that the two systems are
isomorphic. A factor of the system $(X,\mathcal{X},\mu,T)$ corresponds to a
$T$-invariant sub-$\sigma$-algebra of $\mathcal{X}$ (in the above example this
$\sigma$-algebra is $p^{-1}(\mathcal{Y})$). From now on, we will omit the
$\sigma$-algebra $\mathcal{X}$ from the quadruple $(X,\mathcal{X},\mu,T)$.

####  2.2.2. Host-Kra seminorms and factors

Let $(X,\mu,T)$ be an invertible measure preserving system and let $f\in
L^{\infty}(\mu)$. We define the Host-Kra uniformity seminorms inductively as
follows:

$\lvert\\!|\\!|f|\\!|\\!\rvert_{0,T}:=\int f\ d\mu$

and, for $s\in\mathbb{Z}^{+}$,

(3)
$\lvert\\!|\\!|f|\\!|\\!\rvert_{s+1,T}^{2^{s+1}}:=\lim\limits_{H\to\infty}\underset{0\leq
h\leq H}{\mathbb{E}}\lvert\\!|\\!|\bar{f}\cdot
T^{h}f|\\!|\\!\rvert_{s,T}^{2^{s}}.$

The existence of the limits above was proven in [21] in the ergodic case (for
the non-ergodic case, see [22] for a proof) and it was also established that
the $\lvert\\!|\\!|\cdot|\\!|\\!\rvert_{s}$ are indeed seminorms (for $s\neq
0)$. More importantly, it was also shown in the same article that the
seminorms $\lvert\\!|\\!|f|\\!|\\!\rvert_{s,T}$ define a factor $Z_{s-1}(X)$
of $X$, which is characterized by the following property:

$f\perp L^{2}(Z_{s-1}(X))\iff\lvert\\!|\\!|f|\\!|\\!\rvert_{s,T}=0.$

It can be shown that the factors $Z_{s}(X)$ form an increasing sequence of
factors. This follows from the inequality
$\lvert\\!|\\!|f|\\!|\\!\rvert_{s,T}\leq\lvert\\!|\\!|f|\\!|\\!\rvert_{s+1,T}$,
for all non-negative integers $s$. For weak-mixing systems, it can be shown
that all the factors $Z_{s}(X)$ are trivial.

Furthermore, it is easy to prove that $\lvert\\!|\\!|\bar{f}\otimes
f|\\!|\\!\rvert_{s,T\times T}\leq\lvert\\!|\\!|f|\\!|\\!\rvert_{s+1,T}^{2}$,
where $\bar{f}\otimes f$ denotes the function $(x,y)\to\overline{f(x)}f(y)$ on
$(X\times X,\mu\times\mu,T\times T)$. Finally, when there is no danger of
confusion, we will omit the subscript $T$ in the seminorms and write simply
$\lvert\\!|\\!|f|\\!|\\!\rvert_{s}$.

####  2.2.3. Joint ergodicity of sequences

Let $a_{1}(n),...,a_{k}(n)$ be sequences of integers. Following the
terminology in [14], we call these sequences jointly ergodic, if for any
ergodic measure preserving system $(X,\mu,T)$ and functions
$f_{1},...,f_{k}\in L^{\infty}(\mu)$, we have

$\lim\limits_{N\to+\infty}\frac{1}{N}\sum_{n=1}^{N}T^{a_{1}(n)}f_{1}\cdot...\cdot
T^{a_{k}(n)}f_{k}=\int f_{1}\ d\mu\cdot....\cdot\int f_{k}\ d\mu,$

where convergence takes place in $L^{2}(\mu)$. We also give the following
definitions:

######  Definition 2.3.

We say that a collection of sequences $a_{1},...,a_{k}$ of integers:
i) is good for seminorm estimates, if for every ergodic system $(X,\mu,T)$
there exists an $s\in\mathbb{N}$, such that if $f_{1},...,f_{k}\in
L^{\infty}(\mu)$ and $\lvert\\!|\\!|f_{\ell}|\\!|\\!\rvert_{s}=0$ for some
$\ell\in\\{1,...,k\\}$, then333In [14], this property is called ”very good for
seminorm estimates”.

$\lim\limits_{N\to+\infty}\frac{1}{N}\sum_{n=1}^{N}T^{a_{1}(n)}f_{1}\cdot...\cdot
T^{a_{k}(n)}f_{k}=0$

in $L^{2}(\mu)$.
ii) is good for equidistribution, if for all $t_{1},...,t_{k}\in[0,1)$, not
all of them zero, we have

$\lim\limits_{N\to+\infty}\frac{1}{N}\sum_{n=1}^{N}e(t_{1}a_{1}(n)+\cdots+t_{k}a_{k}(n))=0.$

The main result in [14], which we are also going to use is the following:

######  Theorem 2.

[14, Theorem 1.1] Let $a_{1},...,a_{k}$ be a collection of sequences of
integers. Then, the following are equivalent:
i) The sequences $a_{1},...,a_{k}$ are jointly ergodic.
ii) The sequences $a_{1},...,a_{k}$ are good for seminorm estimates and good
for equidistribution.

###### Proof that Theorem 1.2 follows from Theorem 1.3.

Note that every 1-good function dominates the logarithmic function $\log t$.
Therefore, if the functions $a_{1},...,a_{k}$ are such that every non-trivial
linear combination of them is 1-good, then the hypotheses of Theorem 1.3 are
satisfied, which means that the sequences $\lfloor a_{1}(n)\rfloor,...,\lfloor
a_{k}(n)\rfloor$ are good for seminorm estimates. Therefore, due to Theorem 2
we only need to prove that they are good for equidistribution. This, however,
follows from the equidistribution results in [5] and has been established in
[14, Proposition 6.3]. ∎

##  3\. Characteristic factors for Hardy sequences

In this section, we present the main proposition that asserts that the Host-
Kra factors of a given system are characteristic for the convergence of the
averages (2). That means that if we substitute the functions $f_{i}$ by their
projections on $Z_{s}(X)$ for some suitable $s\in\mathbb{N}$, then the
limiting behavior of the average in (2) remains unchanged. We will also make
some small reductions to the original problem and prove some useful lemmas. We
also provide a brief overview of the proof and some examples that present the
main ideas, while avoiding most of the technicalities. The following
proposition will be proven in subsequent sections.

######  Proposition 3.1.

Assume that the functions $a_{1},a_{2},...,a_{k}\in\mathcal{H}$ have
polynomial growth and suppose that the following two conditions hold:

i) The functions $a_{1},...,a_{k}$ dominate the logarithmic function $\log t$.

ii) The pairwise differences $a_{i}-a_{j}$ dominate the logarithmic function
$\log t$ for any $i\neq j$.

Then, there exists a positive integer $s$ depending only on the functions
$a_{1},...,a_{k}$, such that for any measure preserving system $(X,\mu,T)$,
functions $f_{1}\in L^{\infty}(\mu)$ and $f_{2,N}...,f_{k,N}\in
L^{\infty}(\mu)$, all bounded by $1$, with $f_{1}\perp Z_{s}(X)$, the
expression

(4) $\sup_{|c_{n}|\leq 1}\lVert\underset{1\leq n\leq N}{\mathbb{E}}\ c_{n}\
T^{\lfloor a_{1}(n)\rfloor}f_{1}\cdot T^{\lfloor
a_{2}(n)\rfloor}f_{2,N}\cdot...\cdot T^{\lfloor
a_{k}(n)\rfloor}f_{k,N}\rVert_{L^{2}(\mu)}$

converges to 0, as $N\to+\infty$.

######  Remarks.

i) It is possible to establish Proposition 3.1 under the weaker assumption
that only the functions $a_{1},a_{1}-a_{2},...,a_{1}-a_{k}$ dominate the
logarithmic function, but this requires a few more details in the proof and is
not required for the proof of Theorem 1.3.
ii) It may be possible to establish that the number $s$ does not, in fact,
depend on the functions $a_{1},...,a_{k}$, but it can be bounded by a function
involving the number $k$ of functions and the highest degree444This means the
smallest integer $d$, for which $a_{i}(t)\ll t^{d}$ for all $1\leq i\leq k$.
$d$ of the involved functions. However, we do not concern ourselves here with
the optimal value of $s$. In particular, we will use polynomial expansions of
the functions $a_{1},...,a_{k}$ with degrees very large compared to the number
$d$, which means that any possible dependence on $d$ will be lost in the
proof.

It is obvious that Proposition 3.1 implies Theorem 1.3 (this follows from a
standard telescoping argument). Therefore, all of our remaining results follow
if we establish this proposition.

The reason that we work with sequences of functions and the bounded sequence
$c_{n}$ is because that will be helpful in some spots to absorb some of the
error terms that will appear in the iterates and also allows us to "transform"
the sequences in the iterates, so that we can reduce our problem to the case
that the first sequence $a_{1}$ has some specific properties depending on the
situation. As an example, we claim that we only need to consider the case when
the function $a_{1}(t)$ has maximal growth in the family
$\\{a_{1},...,a_{k}\\}$. Indeed, suppose that this is not the case. Then,
there exists a function $a_{i}$ for some $i\in\\{1,...,k\\}$ with $a_{1}\prec
a_{i}$. Without loss of generality, assume that the function $a_{k}$ has
maximal growth rate. It is sufficient to show that for any sequence of
functions $g_{N}$ with $||g_{N}||_{L^{\infty}(\mu)}\leq 1$, we have

$\lim\limits_{N\to\infty}\underset{1\leq n\leq N}{\mathbb{E}}c_{n,N}\int
g_{N}\ T^{\lfloor a_{1}(n)\rfloor}f_{1}\cdot...\cdot T^{\lfloor
a_{k}(n)\rfloor}f_{k,N}\ d\mu=0.$

Then, we can choose the function $g_{N}$ to be the conjugate of the average

$\underset{1\leq n\leq N}{\mathbb{E}}c_{n,N}\int T^{\lfloor
a_{1}(n)\rfloor}f_{1}\cdot...\cdot T^{\lfloor a_{k}(n)\rfloor}f_{k,N}\ d\mu$

to get our claim. Composing with $T^{-\lfloor a_{k}(n)\rfloor}$ and applying
the Cauchy-Schwarz inequality, it is sufficient to show that

$\lim\limits_{N\to+\infty}\sup_{|c_{n}|\leq 1}\lVert\underset{1\leq n\leq
N}{\mathbb{E}}\ c_{n}\ T^{-\lfloor a_{k}(n)\rfloor}g_{N}\cdot T^{\lfloor
a_{1}(n)\rfloor-\lfloor a_{k}(n)\rfloor}f_{1}\cdot...\cdot T^{\lfloor
a_{k-1}(n)\rfloor-\lfloor a_{k}(n)\rfloor}f_{k-1,N}\rVert_{L^{2}(\mu)}=0.$

We can write $\lfloor a_{i}(n)\rfloor-\lfloor a_{k}(n)\rfloor=\lfloor
a_{i}(n)-a_{k}(n)\rfloor+e_{i,n}$, where the errors $e_{i,n}$ take values in
$\\{0,\pm 1\\}$. Using Lemma 3.2 below, the errors can be absorbed by the
supremum outside the average and, therefore, the function that corresponds to
$f_{1}$ is equal to $a_{1}-a_{k}$, which now has maximal growth rate among the
new family of functions. It is also easy to check that the new family
satisfies the conditions of Proposition 3.1.

This notion of absorbing the errors that we described above can be made more
precise by the next lemma.

######  Lemma 3.2.

Assume that the integers $e_{i,n,N}$ take values in a finite set $S$. Then,
for any sequences $a_{i,N}$ of integers, complex numbers $c^{\prime}_{n,N}$
bounded in magnitude by 1 and any 1-bounded functions $f_{i,N}$, we have

$\displaystyle\lVert\underset{1\leq n\leq N}{\mathbb{E}}\ c^{\prime}_{n,N}\
T^{a_{1,N}(n)+e_{1,n,N}}f_{1,N}\cdot...\cdot
T^{a_{k,N}(n)+e_{k,n,N}}f_{k,N}\rVert_{L^{2}(\mu)}$ $\displaystyle\ll_{k,S}$
$\displaystyle\sup_{|c_{n,N}|\leq 1}\ \sup_{||f_{2}||_{\infty}\leq
1,...,||f_{k}||_{\infty}\leq 1}\ \lVert\underset{1\leq n\leq N}{\mathbb{E}}\
c_{n,N}\ T^{a_{1,N}(n)}f_{1,N}\cdot T^{a_{2,N}(n)}f_{2}\cdot...\cdot
T^{a_{k,N}(n)}f_{k}\rVert_{L^{2}(\mu)}$ .

As a consequence, there exist 1-bounded functions $f^{\prime}_{i,N}$, such
that the original expression is bounded by a constant multiple of the quantity

$\sup_{|c_{n,N}|\leq 1}\lVert\mathbb{E}_{1\leq n\leq N}\ c_{n,N}\
T^{a_{1,N}(n)}f_{1,N}\ T^{a_{2,N}(n)}f^{\prime}_{2,N}\cdot...\cdot
T^{a_{k,N}(n)}f^{\prime}_{k,N}\rVert_{L^{2}(\mu)}+o_{N}(1).$

###### Proof.

We partition the integers $n$ into a finite number of sets, in which all the
quantities $e_{i,n,N}$ are constant (as $n$ varies). There are at most
$|S|^{k}$ such sets. If $A_{1},...,A_{|S|^{k}}$ are these sets, then we have

$\displaystyle\lVert\underset{1\leq n\leq N}{\mathbb{E}}c^{\prime}_{n,N}\
T^{a_{1,N}(n)+e_{1,n,N}}f_{1,N}\cdot...\cdot
T^{a_{k,N}(n)+e_{k,n,N}}f_{k,N}\rVert_{L^{2}(\mu)}\leq$
$\displaystyle\sum_{i=1}^{|S|^{k}}\big{\lVert}\frac{1}{N}\sum_{n\in A_{i}}\
c^{\prime}_{n,N}\ T^{a_{1,N}(n)+e_{1,n,N}}f_{1,N}\cdot...\cdot
T^{a_{k,N}(n)+e_{k,n,N}}f_{k,N}\big{\rVert}_{L^{2}(\mu)}\leq$
$\displaystyle|S|^{k}\max_{1\leq
i\leq|S|^{k}}\big{\lVert}\frac{1}{N}\sum_{1\leq n\leq N}\
c^{\prime}_{n,N}\mathbbm{1}_{A_{i}}(n)\ T^{a_{1,N}(n)}f_{1,N}\cdot...\cdot
T^{a_{k,N}(n)+e_{k,n,N}-e_{1,n,N}}f_{k,N}\big{\rVert}_{L^{2}(\mu)}\leq$
$\displaystyle|S|^{k}\sup_{|c_{n,N}|\leq 1}\ \sup_{||f_{2}||_{\infty}\leq
1,...,||f_{k}||_{\infty}\leq 1}\lVert\underset{1\leq n\leq
N}{\mathbb{E}}c_{n,N}\ T^{a_{1,N}(n)}f_{1,N}\ T^{a_{2,N}(n)}f_{2}\cdot...\cdot
T^{a_{k,N}(n)}f_{k}\rVert_{L^{2}(\mu)},$

which is the required result. In the second to last relation, we composed with
$T^{-e_{1,n,N}}$, because $e_{1,n,N}$ is constant when $n$ is restricted to
the set $A_{i}$. ∎

######  Remark.

In the following sections, we will encounter situations where we have some
error terms in the iterates. The above lemma is not applied verbatim to all
cases below. However, the reasoning presented above (i.e. partitioning into
sets where the error sequences are constant) can be applied directly every
time to remove these error terms. In particular, we can also show (using the
same arguments) that a similar statement holds for double averages, that is,
if $I_{r}$ are a sequence of intervals with lengths going to infinity, $d$ is
a natural number and the error terms $e_{i,n,R}$ take values on a finite set
$S$ of integers, then

$\underset{1\leq r\leq R}{\mathbb{E}}\big{\lVert}\underset{n\in
I_{r}}{\mathbb{E}}c^{\prime}_{n,R}\
T^{a_{1,R}(n)+e_{1,n,R}}f_{1,R}\cdot...\cdot
T^{a_{k,R}(n)+e_{k,n,R}}f_{k,R}\big{\rVert}_{L^{2}(\mu)}^{d}\ll_{S,k,d}\\\
\sup_{||f_{2}||_{\infty}\leq 1,...,||f_{k}||_{\infty}\leq 1}\ \underset{1\leq
r\leq R}{\mathbb{E}}\ \sup_{|c_{n,R}|\leq 1}\ \big{\lVert}\underset{n\in
I_{r}}{\mathbb{E}}c_{n,R}\ T^{a_{1,R}(n)}f_{1,R}\cdot
T^{a_{2,R}(n)}f_{2}...\cdot T^{a_{k,R}(n)}f_{k}\big{\rVert}_{L^{2}(\mu)}^{d},$

where we also use the Hölder inequality (which gives dependence on the
exponent $d$ in the implicit constants). Therefore, instead of using the same
argument repeatedly, we will cite this lemma in such instances and add a
comment when a modified version is required.

###  3.1. Overview of the proof

Our main objective is to reduce our problem to the study of ergodic averages
of some variable polynomials. Therefore, we will first study asymptotic bounds
for certain polynomial families in Section 4, since they will be required for
the proof of Proposition 3.1. This will rely on the van der Corput inequality
and an induction argument on the complexity of the family. In Section 5, we
will establish bounds for Hardy sequences of a specific form, namely when the
involved functions are a sum of a sub-linear function and a polynomial. This
will also be required for the general case. In Section 6, we shall finish the
proof.

The main idea is that we can approximate the given Hardy functions by Taylor
polynomials (possibly constant) in suitable smaller intervals (with lengths
going to infinity). We shall reduce our problem to proving a statement of the
form

(5) $\lim\limits_{R\to+\infty}\underset{1\leq r\leq
R}{\mathbb{E}}\lVert\underset{n\in I_{r}}{\mathbb{E}}\ c_{n,R}\ T^{\lfloor
p_{1,r}(n)\rfloor}f_{1,r}\cdot...\cdot T^{\lfloor
p_{k,r}(n)\rfloor}f_{k,r}\rVert_{L^{2}(\mu)}^{2^{t}}=0,$

where the iterates are variable polynomials and $f_{1,r}$ has the form

$f_{1,r}=f_{1}\cdot T^{\lfloor b_{1}(r)\rfloor}h_{1}\cdot...\cdot T^{\lfloor
b_{\ell}(r)\rfloor}h_{\ell}$

for sub-linear functions $b_{1},...,b_{\ell}$ and $h_{1},...,h_{\ell}\in
L^{\infty}(\mu)$.

After this reduction, we bound the innermost average using the results from
Section 4. More precisely, we claim that the inner average can be bounded by a
quantity of the form

$\underset{{\bf m}\in[-M,M]^{t}}{\mathbb{E}}\Big{|}\int T^{\lfloor
q_{1}(r,{\bf m})\rfloor}g_{r,1}\cdot...\cdot T^{\lfloor q_{\ell}(r,{\bf
m})\rfloor}g_{r,\ell}\ d\mu\Big{|}$

plus some small error terms, where $M$ is a finite integer (independent from
the rest of our parameters) and all the functions $g_{r,i}$ are either
$\tilde{f}_{r}$ or $\overline{\tilde{f}_{r}}$. In addition, the functions
$q_{i}(r,{\bf m})$ in the iterates are such that, for (almost all) ${\bf
m}\in\mathbb{Z}^{l}$, they can be written as a sum of a sublinear function
plus a polynomial, which is the special case that we discussed above. Thus,
taking first the limit $R\to+\infty$ to use the bounds established in the
special case and then taking the limits $M\to+\infty$, we shall reach our
conclusion.

The fact that we can reduce our original problem to (5) is based on the
following elementary lemma.

######  Lemma 3.3.

Let $d$ be a positive integer and consider a two-parameter sequence
$\big{(}A_{R,n}\big{)}_{R,n\in\mathbb{N}}$ in a normed space such that $\lVert
A_{R,n}\rVert\leq 1$ for all possible choices of $R,n\in\mathbb{N}$. Let
$L(t)\in\mathcal{H}$ be an eventually positive function such that $1\prec
L(t)\prec t$ and assume that

$\limsup\limits_{R\to+\infty}\underset{1\leq r\leq R}{\mathbb{E}}\
\big{\lVert}\underset{r\leq n\leq
r+L(r)}{\mathbb{E}}A_{R,n}\big{\rVert}^{d}\leq C$

for some $C>0$. Then, we also have

$\limsup\limits_{R\to+\infty}\big{\lVert}\underset{1\leq n\leq
R}{\mathbb{E}}A_{R,n}\big{\rVert}\leq C^{1/d}.$

###### Proof.

Combining the power mean inequality and the triangle inequality, we can easily
deduce that

$\underset{1\leq r\leq R}{\mathbb{E}}\ \big{\lVert}\underset{r\leq n\leq
r+L(r)}{\mathbb{E}}A_{R,n}\big{\rVert}^{d}\geq\big{\lVert}\underset{1\leq
r\leq R}{\mathbb{E}}\big{(}\underset{r\leq n\leq
r+L(r)}{\mathbb{E}}A_{R,n}\big{)}\big{\rVert}^{d}.$

Therefore, our result will follow if we show that

$\big{\lVert}\underset{1\leq r\leq R}{\mathbb{E}}\big{(}\underset{r\leq n\leq
r+L(r)}{\mathbb{E}}A_{R,n}\big{)}-\underset{1\leq n\leq
R}{\mathbb{E}}A_{R,n}\big{\rVert}=o_{R}(1).$

Let $u$ be the compositional inverse of the function $t+L(t)$. Our assumptions
on the Hardy field $\mathcal{H}$ imply that $u\in\mathcal{H}$. In addition, it
is easy to check that $\lim\limits_{t\to+\infty}u(t)/t=1$. Now, we have

$\underset{1\leq r\leq R}{\mathbb{E}}\big{(}\ \underset{r\leq n\leq
r+L(r)}{\mathbb{E}}A_{R,n}\big{)}=\frac{1}{R}\big{(}\sum_{n=1}^{R}p_{R}(n)A_{R,n}+\sum_{n=R+1}^{R+L(R)}p_{R}(n)A_{R,n}\big{)}$

for some real numbers $p_{R}(n)$. Assuming that $n$ (and thus $R$) is
sufficiently large (so that $u(n)$ is positive) we can calculate $p_{R}(n)$ to
be equal to

$p_{R}(n)=\frac{1}{L(\lfloor
u(n)\rfloor)+1}+\cdots+\frac{1}{L(n)+1}+o_{n}(1),$

since the number $A_{R,n}$ appears on the average $\underset{r\leq n\leq
r+L(r)}{\mathbb{E}}$ if and only if $u(n)\leq r\leq n$. Note that $p_{R}(n)$
is actually independent of $R$ (for $n$ large enough) and therefore, we will
denote it simply as $p(n)$ from now on. We claim that

(6) $\lim_{n\to+\infty}p(n)=1.$

Let us first see how this finishes the proof. Since for $n$ large enough we
must have $p(n)\leq 2$, we can easily deduce that

$\frac{1}{R}\sum_{n=R+1}^{R+L(R)}p(n)A_{R,n}=o_{R}(1).$

Here, we used the fact that $L(t)\prec t$. In addition, we have

$\big{\lVert}\frac{1}{R}\sum_{n=1}^{R}p(n)A_{R,n}-\frac{1}{R}\sum_{n=1}^{R}A_{R,n}\big{\rVert}\leq\frac{1}{R}\sum_{n=1}^{R}|p(n)-1|,$

which is also $o_{R}(1)$. Combining the above we reach the desired conclusion.

In order to establish (6), we observe that $L(t)$ is eventually strictly
increasing, and therefore, we can easily get

$\int_{\lfloor u(n)\rfloor}^{n+1}\frac{1}{L(t)+1}\ dt\leq
p(n)\leq\int_{\lfloor u(n)\rfloor-1}^{n}\frac{1}{L(t)+1}\ dt.$

Thus, it suffices to show that the integrals on both sides of the above
inequality converge to 1. It is straightforward to check that each of these
integrals is $o_{n}(1)$ close to the integral

$I_{n}=\int_{u(n)}^{n}\frac{1}{L(t)+1}\ dt.$

Therefore, we only need to prove that $I_{n}\to 1$. Using the mean value
theorem, we can find a real number $h_{n}\in[u(n),n]$ such that,

$I_{n}=\frac{n-u(n)}{L(h_{n})+1}=\frac{L(u(n))}{L(h_{n})+1}.$

The last equality follows easily from the definition of $u$. Since $L$ is
eventually strictly increasing, we conclude that $I_{n}$ is smaller than
$L(u(n))/(L(u(n))+1)\leq 1$. In addition, we also have

$I_{n}\geq\frac{L(u(n))}{L(n)+1}.$

The result follows if we show (note that the function $u^{-1}$ is onto in a
half line of $\mathbb{R}$)

$\lim\limits_{t\to+\infty}\frac{L(t)}{L(u^{-1}(t))+1}=1.$

However,

$\frac{L(t)}{L(u^{-1}(t))+1}=\frac{L(t)}{L(t+L(t))+1}=\frac{L(t)}{L(t+L(t))}+o_{t}(1).$

Using the mean value theorem, we can write

$L(t+L(t))=L(t)+L(t)L^{\prime}(x_{t})\ ,$

where $x_{t}\in[t,t+L(t)]$. Thus,

$\frac{L(t+L(t))}{L(t)}=1+L^{\prime}(x_{t})=1+o_{t}(1)\ ,$

since $L^{\prime}(t)\ll L(t)/t\prec 1$. The result follows. ∎

###  3.2. Two examples

a) Whenever we use $\ll$ without indices in this example, we imply that the
constants are absolute. Assume that $a(t)=t\log t+\log^{3}t$, $b(t)=t\log t$
and $c(t)=\sqrt{t}$. We want to show that there exists $s\in\mathbb{N}$, such
that, if $\lvert\\!|\\!|f|\\!|\\!\rvert_{s}=0$, then

$\underset{1\leq n\leq N}{\mathbb{E}}T^{\lfloor n\log
n+\log^{3}n\rfloor}f\cdot T^{\lfloor n\log n\rfloor}g_{1}\cdot
T^{\lfloor\sqrt{n}\rfloor}g_{2}$

converges to 0 in $L^{2}$ as $N\to+\infty$. Here, $g_{1}$ and $g_{2}$ are
arbitrary 1-bounded functions in $L^{\infty}(\mu)$. In view of Lemma 3.3, it
suffices to show that

(7) $\underset{1\leq r\leq R}{\mathbb{E}}\big{\lVert}\underset{r\leq n\leq
r+L(r)}{\mathbb{E}}T^{\lfloor n\log n+\log^{3}n\rfloor}f\cdot T^{\lfloor n\log
n\rfloor}g_{1}\cdot
T^{\lfloor\sqrt{n}\rfloor}g_{2}\big{\rVert}_{L^{2}(\mu)}^{2^{d}}=\underset{1\leq
r\leq R}{\mathbb{E}}A_{r}$

converges to 0 as $R\to+\infty$, for some sub-linear function
$L(t)\in\mathcal{H}$ and an integer $d$, both of which we will choose later.

###  Step 1: Reduction to averages of variable polynomials.

We observe that

$A_{r}=\big{\lVert}\underset{0\leq h\leq L(r)}{\mathbb{E}}\
T^{\lfloor(r+h)\log(r+h)+\log^{3}(r+h)\rfloor}f\cdot
T^{\lfloor(r+h)\log(r+h)\rfloor}g_{1}\cdot
T^{\lfloor\sqrt{r+h}\rfloor}g_{2}\big{\rVert}_{L^{2}(\mu)}^{2^{d}}.$

Now, we can use the Taylor expansion to write

$(r+h)\log(r+h)=-\frac{h^{3}}{6x_{h}^{2}}+\frac{h^{2}}{2r}+h(\log r+1)+r\log
r,\ \text{ for some}\ \ \ x_{h}\in[r,r+h],$

and

$\sqrt{r+h}=-\frac{h^{2}}{8(x^{\prime}_{h})^{3/2}}+\frac{h}{2\sqrt{r}}+\sqrt{r},\
\text{ for some}\ \ \ x^{\prime}_{h}\in[r,r+h],$

for every $0\leq h\leq L(r)$. Since

$\Big{|}\frac{h^{3}}{6x_{h}^{2}}\Big{|}\leq\frac{L(r)^{3}}{r^{2}}$

and

$\Big{|}\frac{h^{2}}{8(x^{\prime}_{h})^{3/2}}\Big{|}\leq\frac{L^{2}(r)}{8r^{3/2}},$

we conclude that these two last terms are both $o_{r}(1)$, provided that we
choose the function $L(t)$ to satisfy $L(t)\prec t^{2/3}$. We also choose
$L(t)\succ t^{1/2}$, so that both the 2-degree term in the expansion of
$(r+h)\log(r+h)$ and the 1-degree term in the expansion of $\sqrt{r+h}$ are
not bounded (for $h$ taking values in the range $[0,L(r)]$). In addition,
under the above assumptions, we can also show that

$\max_{0\leq h\leq L(r)}|\log^{3}(r+h)-\log^{3}(r)|=o_{r}(1)$

using the mean-value theorem. Therefore, we have555In this example, we split
and combine the integer parts freely, which is not true in general. In our
main proof, we explain this argument using Lemma 3.2.

(8) $A_{r}\simeq\big{\lVert}\underset{0\leq h\leq L(r)}{\mathbb{E}}\
T^{\lfloor\frac{h^{2}}{2r}+h(\log r+1)+r\log r+\log^{3}r\rfloor}f\cdot
T^{\lfloor\frac{h^{2}}{2r}+h(\log r+1)+r\log r\rfloor}g_{1}\cdot
T^{\lfloor\frac{h}{2\sqrt{r}}+\sqrt{r}\rfloor}g_{2}\big{\rVert}_{L^{2}(\mu)}^{2^{d}}=\\\
\big{\lVert}\underset{0\leq h\leq L(r)}{\mathbb{E}}\
T^{\lfloor\frac{h^{2}}{2r}+h(\log r+1)+r\log r\rfloor}(g_{1}\cdot
T^{\lfloor\log^{3}r\rfloor}f)\cdot
T^{\lfloor\frac{h}{2\sqrt{r}}+\sqrt{r}\rfloor}g_{2}\big{\rVert}_{L^{2}(\mu)}^{2^{d}},$

which is an average where the iterates are polynomials in $h$. The fact that
the $o_{r}(1)$ terms can be discarded follows from Lemma 3.2 and will be
explained more thoroughly in the formal proof. Note that the iterates have now
become polynomials in the variable $h$.

######  Remark.

In the proof of Proposition 3.1 in Section 6, we will choose the function
$L(t)$ in order to have a common polynomial expansion as above. Although in
this example this is easily done by hand, this will be accomplished in the
general case using some lemmas and propositions that are proven in the
appendix.

We will use the van der Corput inequality (Lemma 4.4):

$|\underset{1\leq n\leq N}{\mathbb{E}}\
a_{n}|^{2^{d}}\ll_{d}\frac{1}{M}+\underset{|m|\leq
M}{\mathbb{E}}|\underset{1\leq n\leq N}{\mathbb{E}}\langle
a_{n+m},a_{m}\rangle|^{2^{d-1}}+o_{N}(1)\ ,$

which holds as long as $M=o(N)$.

We will deal with a simpler case here, since (8) requires many applications of
the van der Corput inequality and the estimates are quite complicated. We
shall find a bound for the average

$\underset{1\leq r\leq R}{\mathbb{E}}\big{\lVert}\underset{0\leq h\leq
L(r)}{\mathbb{E}}T^{\lfloor\frac{h^{2}}{2r}\rfloor}f_{r}\big{\rVert}^{4}=\underset{1\leq
r\leq R}{\mathbb{E}}A_{r}^{4}\ ,$

where $f_{r}=g_{1}\cdot T^{\lfloor\log^{3}(r)\rfloor}f$.

### Step 2: A change of variables trick and bounds for the polynomial averages

First of all, we can write $h=k\lfloor\sqrt{2r}\rfloor+s$, where the integers
$k,s$ satisfy $0\leq k\leq L(r)/\lfloor\sqrt{2r}\rfloor$ and $0\leq
s\leq\lfloor\sqrt{2r}\rfloor-1$. Then, we have

$\frac{h^{2}}{2r}=\frac{k^{2}\lfloor\sqrt{2r}\rfloor^{2}}{2r}+\frac{2k\lfloor\sqrt{r}\rfloor
s}{2r}+\frac{s^{2}}{2r}.$

Note that

$\Big{|}\frac{k^{2}\lfloor\sqrt{2r}\rfloor^{2}}{2r}-k^{2}\Big{|}\leq
2k^{2}\frac{\\{\sqrt{2r}\\}}{\sqrt{2r}}\leq
2\frac{L^{2}(r)}{\lfloor\sqrt{2r}\rfloor^{2}\sqrt{2r}}.$

If we choose $L(t)$ to satisfy the additional hypothesis $L(t)\prec t^{3/4}$,
then we get that the above quantity is $o_{r}(1)$. In this example, we can
take $L(t)=t^{3/5}$ as our sub-linear function (observe that all of the
restrictions we imposed above are satisfied). Therefore, we can use the power
mean inequality to deduce that

(9) $A_{r}^{4}\leq\underset{0\leq s\leq\lfloor\sqrt{2r}\rfloor-1}{\mathbb{E}}\
\big{\lVert}\underset{1\leq
k\leq\frac{L(r)}{\lfloor\sqrt{2r}\rfloor}}{\mathbb{E}}T^{\lfloor
k^{2}+p_{s,r}(k)\rfloor}f_{r}\big{\rVert}^{4}$

for some linear polynomials $p_{s,r}(k)$. Denote by $A_{s,r}$ the innermost
average in the above relation.

We fix a positive integer parameter $M$. Applying the van der Corput
inequality twice, we deduce that

$A_{s,r}^{4}\ll\frac{1}{M}+\underset{|m_{1}|,|m_{2}|\leq
M}{\mathbb{E}}\Big{|}\int\bar{f_{r}}\cdot T^{2m_{1}m_{2}}f_{r}\
d\mu\Big{|}+o_{r}(1),$

where the implied constant is absolute (and, in particular, independent of
$M$). We omitted the routine computations here (the general case is more
complicated than this and is handled in Section 4). This bound holds
regardless of the choice of the polynomial $p_{s,r}(k)$. Using this bound in
(9) we deduce that

$A_{r}^{4}\ll\frac{1}{M}+\underset{|m_{1}|,|m_{2}|\leq
M}{\mathbb{E}}\Big{|}\int\overline{(g_{1}\cdot
T^{\lfloor\log^{3}r\rfloor}f)}\cdot T^{2m_{1}m_{2}}(g_{1}\cdot
T^{\lfloor\log^{3}r\rfloor}f)\ d\mu\Big{|}+o_{r}(1).$

Therefore, the quantity in (7) is $\ll$

(10) $\frac{1}{M}+\underset{1\leq r\leq R}{\mathbb{E}}\
\underset{|m_{1}|,|m_{2}|\leq M}{\mathbb{E}}\Big{|}\int\overline{(g_{1}\cdot
T^{\lfloor\log^{3}r\rfloor}f)}\cdot T^{2m_{1}m_{2}}(g_{1}\cdot
T^{\lfloor\log^{3}r\rfloor}f)\ d\mu\Big{|}+o_{R}(1)=\\\
\frac{1}{M}+\underset{|m_{1}|,|m_{2}|\leq M}{\mathbb{E}}\ \underset{1\leq
r\leq R}{\mathbb{E}}\Big{|}\int(\bar{g}_{1}\cdot T^{2m_{1}m_{2}}g_{1})\cdot
T^{\lfloor\log^{3}r\rfloor}(\bar{f}\cdot T^{2m_{1}m_{2}}f)\
d\mu\Big{|}+o_{R}(1).$

######  Remark.

In the proof of the general case, instead of the sub-linear function
$\lfloor\log^{3}(r)\rfloor$ in the iterates in (10), we may also have
functions of the form $\lfloor u(r)\rfloor^{k}$, where $u\in\mathcal{H}$ is a
sub-linear function and $k\in\mathbb{Z}^{+}$ (like
$\lfloor\sqrt{r}\rfloor^{3}$ and $\lfloor r^{2/3}\rfloor^{5}$). For instance,
assume we want to study the limit of the averages

$\underset{1\leq n\leq N}{\mathbb{E}}T^{\lfloor\sqrt{n}+n^{3}\rfloor}f\cdot
T^{\lfloor\sqrt{n}\rfloor}g.$

Using Lemma 3.3, it suffices to show that

$\underset{1\leq r\leq R}{\mathbb{E}}\big{\lVert}\underset{0\leq h\leq
L(r)}{\mathbb{E}}T^{\lfloor\sqrt{r+h}+(r+h)^{3}\rfloor}f\cdot
T^{\lfloor\sqrt{r+h}\rfloor}g\big{\rVert}_{L^{2}(\mu)}^{2^{d}}$

for some $d\in\mathbb{N}$ and some sub-linear function $L(t)\in\mathcal{H}$.
If we choose $L(t)$ appropriately, then we can write

$\sqrt{r+h}=\sqrt{r}+\frac{h}{2\sqrt{r}}+o_{r}(1)$

for $0\leq h\leq L(r)$. Now, using the change of variables $h=k\lfloor
2\sqrt{r}\rfloor+s$, we observe that the leading coefficient of the polynomial
$(r+h)^{3}$ in the iterates becomes $\lfloor 2\sqrt{r}\rfloor^{3}$. If we
proceed similarly as in step 2 above using repeated applications of the van
der Corput inequality, we will arrive at a similar bound as the one in (10),
but now the term $\lfloor 2\sqrt{r}\rfloor^{3}$ will appear in the iterates.

In order to combat this situation, we need another intermediate step in our
proof (this is Step 7 in Section 6). We shall use a lemma that allows us to
replace the sub-linear function $2\sqrt{r}$ by the identity function $a(r)=r$.
As an example, suppose we want to bound the limit of the averages

$\underset{1\leq r\leq R}{\mathbb{E}}T^{\lfloor\sqrt{r}\rfloor}f\cdot
T^{\lfloor\sqrt{r}\rfloor^{3}+\lfloor r^{2/5}\rfloor}g$

as $R\to+\infty$. We rewrite this expression as a function of $\sqrt{r}$

$\underset{1\leq r\leq R}{\mathbb{E}}T^{\lfloor\sqrt{r}\rfloor}f\cdot
T^{\lfloor\sqrt{r}\rfloor^{3}+\lfloor({\sqrt{r}})^{4/5}\rfloor}g.$

Then, we can prove that

$\limsup\limits_{R\to+\infty}\big{\lVert}\underset{1\leq r\leq
R}{\mathbb{E}}T^{\lfloor\sqrt{r}\rfloor}f\cdot
T^{\lfloor\sqrt{r}\rfloor^{3}+\lfloor({\sqrt{r}})^{4/5}\rfloor}g\big{\rVert}_{L^{2}(\mu)}\leq
C\limsup\limits_{R\to+\infty}\big{\lVert}\underset{1\leq r\leq
R}{\mathbb{E}}T^{r}f\cdot T^{r^{3}+\lfloor
r^{4/5}\rfloor}g\big{\rVert}_{L^{2}(\mu)}$

for some positive real number $C$. Now the functions in the iterates are sub-
linear functions and polynomials, which we are now able to handle (this is the
content of Section 5).

### Step 3: Dealing with the sub-linear function.

In this step we show that the quantity in (10) goes to 0, if we take
$R\to+\infty$ and then $M\to+\infty$. While steps 1 and 2 of this example
correspond to parts of the proof in Sections 4 and 6, this step corresponds to
the proofs in Section 5.

We observe that the function $\log^{3}(r)$ in the iterates is a sub-linear
function. We will show that

(11) $\lim\limits_{R\to+\infty}\underset{1\leq r\leq
R}{\mathbb{E}}\Big{|}\int(\bar{g}\cdot T^{2m_{1}m_{2}}g)\cdot
T^{\lfloor\log^{3}r\rfloor}(\bar{f}\cdot
T^{2m_{1}m_{2}}f)d\mu\Big{|}\ll\lvert\\!|\\!|\bar{f}\cdot
T^{2m_{1}m_{2}}f|\\!|\\!\rvert_{3}.$

In addition, the implicit constants do not depend on $m_{1},m_{2}$. Assuming
that (11) holds, we take the limit as $M\to+\infty$ (this can be done because
all implied asymptotic constants do not depend on $m_{1},m_{2}$) and we need
to show that

$\lim\limits_{M\to+\infty}\underset{|m_{1}|,|m_{2}|\leq
M}{\mathbb{E}}\lvert\\!|\\!|\bar{f}\cdot
T^{2m_{1}m_{2}}f|\\!|\\!\rvert_{3}=0.$

Applying the Hölder inequality, we are left with showing that

$\lim\limits_{M\to+\infty}\underset{|m_{1}|,|m_{2}|\leq
M}{\mathbb{E}}\lvert\\!|\\!|\bar{f}\cdot
T^{2m_{1}m_{2}}f|\\!|\\!\rvert_{3}^{8}=0.$

Using the definition of the Host-Kra seminorms, this relation reduces to an
ergodic average with polynomial iterates, which is well known to converge to
$0$ under our hypothesis on the function $f$ (namely, that
$\lvert\\!|\\!|f_{1}|\\!|\\!\rvert_{s}=0$ for some suitable $s\in\mathbb{N}$).

We now establish (11). It suffices to show that

$\lim\limits_{R\to+\infty}\underset{1\leq r\leq R}{\mathbb{E}}\Big{|}\int
g\cdot
T^{\lfloor\log^{3}r\rfloor}fd\mu\Big{|}\ll\lvert\\!|\\!|f|\\!|\\!\rvert_{3,T}$

for any 1-bounded functions $f\text{ and }g$, where the implied constant is
absolute. We square the above expression and apply the Cauchy-Schwarz
inequality to bound it by

$\underset{1\leq r\leq R}{\mathbb{E}}\int{G}\cdot
S^{\lfloor\log^{3}(r)\rfloor}F\ d(\mu\times\mu),$

where $F:=\overline{f}\otimes f$, $G:=\overline{g}\otimes g$ and $S:=T\times
T$. Then, (11) follows if we show

$\big{\lVert}\underset{1\leq r\leq
R}{\mathbb{E}}S^{\lfloor\log^{3}(r)\rfloor}F\big{\rVert}_{L^{2}(\mu\times\mu)}\ll\lvert\\!|\\!|f|\\!|\\!\rvert_{3,T}^{2}.$

We use Lemma 3.3 once more: it suffices to show that

$\limsup\limits_{r\to+\infty}\big{\lVert}\underset{r\leq n\leq
r+L(r)}{\mathbb{E}}S^{\lfloor\log^{3}(n)\rfloor}F\big{\rVert}_{L^{2}(\mu\times\mu)}\ll\lvert\\!|\\!|f|\\!|\\!\rvert_{3,T}^{2}\
,$

where $L(t)\in\mathcal{H}$ is sub-linear. Using the Taylor expansion, we can
write

$\log^{3}(r+h)=\log^{3}(r)+\frac{3\log^{2}r}{r}h-\frac{6\log
x_{h}-3\log^{2}x_{h}}{2x_{h}^{2}}h^{2},$

where $0\leq h\leq L(r)$ and $x_{h}\in[r,r+h]$. If we choose the function
$L(t)$ so that

$\frac{t}{\log^{2}t}\prec L(t)\prec\frac{t}{\log t},$

we can then deduce that the last term in the above expansion is $o_{r}(1)$.
Our problem reduces to

$\limsup\limits_{r\to+\infty}\big{\lVert}\underset{0\leq h\leq
L(r)}{\mathbb{E}}S^{\lfloor\log^{3}(r)+\frac{3\log^{2}r}{r}h\rfloor}F\big{\rVert}_{L^{2}(\mu\times\mu)}\ll\lvert\\!|\\!|f|\\!|\\!\rvert_{3,T}^{2}.$

We have again reduced our problem to finding a bound for an ergodic average
with (variable) polynomials. In order to finish the proof, we work similarly
as in the previous steps, using the change of variables trick and one
application of the van der Corput inequality (we also need to use the
inequality $\lvert\\!|\\!|F|\\!|\\!\rvert_{2,T\times
T}\leq\lvert\\!|\\!|f|\\!|\\!\rvert_{3,T}^{2}$).

b) In this second example we describe the strategy that will be used in the
special case that we discussed above, that is when our functions are sums of
sublinear functions and polynomials. This case is covered in full generality
in Section 5. We consider the triplet of functions in $\mathcal{H}$
$(t+\log^{3}t,t,\log^{2}t)$ and we shall show that there exists
$s\in\mathbb{N}$ so that, if $\lvert\\!|\\!|f|\\!|\\!\rvert_{s}=0$, then

$\underset{1\leq n\leq N}{\mathbb{E}}T^{\lfloor n+\log^{3}n\rfloor}f\cdot
T^{{n}}g_{1}\cdot T^{\lfloor\log^{2}n\rfloor}g_{2}$

converge to $0$ in mean ($g_{1},g_{2}$ are again arbitrary 1-bounded
functions).

### Step 1: Reducing to the case when all iterates have sub-linear growth.

We start by using Lemma 3.3 to reduce our problem to

(12) $\limsup\limits_{R\to+\infty}\underset{1\leq r\leq
R}{\mathbb{E}}\big{\lVert}\underset{r\leq n\leq r+L(r)}{\mathbb{E}}T^{\lfloor
n+\log^{3}n\rfloor}f\cdot T^{{n}}g_{1}\cdot
T^{\lfloor\log^{2}n\rfloor}g_{2}\big{\rVert}_{L^{2}(\mu)}^{2}=0$

for some sub-linear function $L(t)\in\mathcal{H}$. In this example, we will
choose the function $L(t)$, so that

$\max_{r\leq n\leq r+L(r)}|\log^{3}(n)-\log^{3}(r)|=o_{r}(1)\ \ \ \ \text{ and
}\ \ \ \max_{r\leq n\leq r+L(r)}|\log^{2}(n)-\log^{2}(r)|=o_{r}(1).$

For instance, the function $L(t)=\sqrt{t}$ can easily be checked to satisfy
the above. Therefore, if $r$ is very large, we can write

$\big{\lVert}\underset{r\leq n\leq r+L(r)}{\mathbb{E}}T^{\lfloor
n+\log^{3}n\rfloor}f\cdot T^{{n}}g_{1}\cdot
T^{\lfloor\log^{2}n\rfloor}g_{2}\big{\rVert}_{L^{2}(\mu)}=\\\
\big{\lVert}\underset{r\leq n\leq
r+L(r)}{\mathbb{E}}T^{n+\lfloor\log^{3}r\rfloor+e_{1,n}}f\cdot\
T^{n}g_{1}\cdot\
T^{\lfloor\log^{2}r\rfloor+e_{2,n}}g_{2}\big{\rVert}_{L^{2}(\mu)},$

where $e_{1,n},e_{2,n}\in\\{0,\pm 1\\}$. We assume here that all the error
terms are zero (in the main proof, we will invoke Lemma 3.2 to remove the
error terms). Therefore, we want to show that

$\limsup\limits_{R\to+\infty}\underset{1\leq r\leq
R}{\mathbb{E}}\big{\lVert}\underset{r\leq n\leq
r+L(r)}{\mathbb{E}}T^{n+\lfloor\log^{3}r\rfloor}f\cdot\ T^{n}g_{1}\cdot\
T^{\lfloor\log^{2}r\rfloor}g_{2}\big{\rVert}_{L^{2}(\mu)}^{2}=0.$

Since $\lVert g_{2}\rVert_{\infty}\leq 1$, we reduce our problem to

$\limsup\limits_{R\to+\infty}\underset{1\leq r\leq
R}{\mathbb{E}}\big{\lVert}\underset{r\leq n\leq
r+L(r)}{\mathbb{E}}T^{n}(g_{1}\cdot
T^{\lfloor\log^{3}r\rfloor}f)\big{\rVert}_{L^{2}(\mu)}^{2}=0.$

Note that the inner average is a polynomial average in the variable $n$. We
fix a positive integer $M$ and use the van der Corput inequality to deduce
that

$\big{\lVert}\underset{r\leq n\leq r+L(r)}{\mathbb{E}}T^{n}(g_{1}\cdot
T^{\lfloor\log^{3}r\rfloor}f)\big{\rVert}_{L^{2}(\mu)}^{2}\ll\frac{1}{M}+\underset{|m|\leq
M}{\mathbb{E}}\Big{|}\int\overline{(g_{1}\cdot
T^{\lfloor\log^{3}(r)\rfloor})}\cdot T^{m}(g_{1}\cdot
T^{\lfloor\log^{3}(r)\rfloor})\ d\mu\Big{|}+o_{r}(1)\ ,$

where the implied constant is absolute. Thus, we want to show that

$\frac{1}{M}+\underset{|m|\leq M}{\mathbb{E}}\ \underset{1\leq r\leq
R}{\mathbb{E}}\Big{|}\int(\overline{g_{1}}\cdot T^{m}g_{1})\cdot
T^{\lfloor\log^{3}(r)\rfloor}(\overline{f}\cdot T^{m}f)\ d\mu\Big{|}+o_{R}(1)$

goes to $0$, as $R\to+\infty$ and then as $M\to+\infty$.

### Step 2: Dealing with the sub-linear functions.

Our problem follows by taking the limit as $R\to+\infty$ and then using the
bound

(13) $\limsup\limits_{R\to+\infty}\underset{1\leq r\leq
R}{\mathbb{E}}\Big{|}\int(\overline{g_{1}}\cdot T^{m}g_{1})\cdot
T^{\lfloor\log^{3}(r)\rfloor}(\overline{f}\cdot T^{m}f)\
d\mu\Big{|}\ll\lvert\\!|\\!|\overline{f}\cdot T^{m}f|\\!|\\!\rvert_{3,T}.$

This was established in the previous example. Using this relation and taking
the limit $M\to+\infty$ (note that our asymptotic constants do not depend on
$M$), we reach the conclusion.

Since (13) follows from the previous example, we will describe our arguments
for a more representative case. We shall prove that

(14) $\limsup\limits_{N\to+\infty}\big{\lVert}\underset{1\leq n\leq
N}{\mathbb{E}}T^{\lfloor\log^{3}n+\log^{2}n\rfloor}f\cdot
T^{\lfloor\log^{3}n\rfloor}g_{1}\cdot
T^{\lfloor\log^{2}n\rfloor}g_{2}\big{\rVert}_{L^{2}(\mu)}\ll\lvert\\!|\\!|f|\\!|\\!\rvert_{4},$

where the implied constant is absolute. Using Lemma 3.3, it suffices to show
that

$\limsup\limits_{R\to+\infty}\underset{1\leq r\leq
R}{\mathbb{E}}\big{\lVert}\underset{r\leq n\leq
r+L(r)}{\mathbb{E}}T^{\lfloor\log^{3}n+\log^{2}n\rfloor}f\cdot
T^{\lfloor\log^{3}n\rfloor}g_{1}\cdot
T^{\lfloor\log^{2}n\rfloor}g_{2}\big{\rVert}_{L^{2}(\mu)}^{2}\ll\lvert\\!|\\!|f|\\!|\\!\rvert_{4}^{2}$

for some sub-linear function $L(t)\in\mathcal{H}$. We choose $L(t)=t(\log
t)^{-3/2}$. Using similar approximations as in the first example, we can show
that for any $0\leq h\leq L(r)$

$\log^{3}(r+h)=\log^{3}r+h\frac{3\log^{2}r}{r}+o_{r}(1),$

while

$\log^{2}(r+h)=\log^{2}r+o_{r}(1)$

for all $0\leq h\leq L(r)$. Disregarding the error terms $o_{r}(1)$ in this
example, it suffices to show that

$\limsup\limits_{R\to+\infty}\underset{1\leq r\leq
R}{\mathbb{E}}\big{\lVert}\underset{0\leq h\leq
L(r)}{\mathbb{E}}T^{\lfloor\log^{3}r+h\frac{3\log^{2}r}{r}\rfloor}\big{(}T^{\lfloor\log^{2}r\rfloor}f\cdot
g_{1}\big{)}\cdot
T^{\lfloor\log^{2}r\rfloor}g_{2}\big{\rVert}_{L^{2}(\mu)}^{2}\ll\lvert\\!|\\!|f|\\!|\\!\rvert_{4}^{2}.$

Since $g_{2}$ is bounded by 1, the above bound follows from

$\limsup\limits_{N\to+\infty}\underset{1\leq r\leq
R}{\mathbb{E}}\big{\lVert}\underset{0\leq h\leq
L(r)}{\mathbb{E}}T^{\lfloor\log^{3}r+h\frac{3\log^{2}r}{r}\rfloor}\big{(}T^{\lfloor\log^{2}r\rfloor}f\cdot
g_{1}\big{)}\big{\rVert}_{L^{2}(\mu)}\ll\lvert\\!|\\!|f|\\!|\\!\rvert_{4}^{2}.$

This is an average where the iterates are variable polynomials. Working
similarly to the previous example, we can show that

$\big{\lVert}\underset{0\leq h\leq
L(r)}{\mathbb{E}}T^{\lfloor\log^{3}r+h\frac{3\log^{2}r}{r}\rfloor}\big{(}T^{\lfloor\log^{2}r\rfloor}f\cdot
g_{1}\big{)}\big{\rVert}_{L^{2}(\mu)}^{2}\ll\\\ \frac{1}{M}+\underset{|m|\leq
M}{\mathbb{E}}\Big{|}\int\overline{\big{(}T^{\lfloor\log^{2}r\rfloor}f\cdot
g_{1}\big{)}}\cdot T^{m}\big{(}T^{\lfloor\log^{2}r\rfloor}f\cdot g_{1}\big{)}\
d\mu\Big{|}+o_{r}(1).$

Thus, it suffices to show that

$\limsup\limits_{M\to+{\infty}}\underset{|m|\leq
M}{\mathbb{E}}\limsup\limits_{R\to+\infty}\underset{1\leq r\leq
R}{\mathbb{E}}\Big{|}\int(\bar{g_{1}}\cdot T^{m}g_{1})\cdot
T^{\lfloor\log^{2}r\rfloor}(\bar{f}\cdot T^{m}f)\
d\mu\Big{|}\ll\lvert\\!|\\!|f|\\!|\\!\rvert_{4}^{2}.$

Note that we started with three sub-linear functions in the iterates and now
we have an average with only one sub-linear function (our argument in the
general case is based on this induction scheme). The result follows by working
similarly to step 3 in the previous example.

##  4\. Bounds of polynomial averages

Our main goal in this section is to establish Proposition 4.5 below. Before
stating that proposition, we will first give some definitions.

###  4.1. Families of variable polynomials

Assume we are given a family $P_{N}=\\{p_{1,N},...,p_{k,N}\\}$ of essentially
distinct (i.e. their pairwise differences are non-constant polynomials)
variable polynomials, such that the degrees of the polynomials in $P_{N}$ and
of their pairwise differences are independent of $N$ (for $N$ large enough).
Then, we can assign to $p_{1,N}$ its own vector $(v_{1,N},...,v_{k,N})$, where
$v_{1,N}$ is the leading coefficient of $p_{1,N}$ and $v_{j,N}$ is the leading
coefficient of $p_{1,N}-p_{j,N}$ for $j\neq 1$. We symbolize this by
$\mathcal{S}(p_{1,N})$ and call this the leading vector of the family $P_{N}$
corresponding to $p_{1,N}$. We similarly define $\mathcal{S}(p_{i,N})$ for
every $i\in\\{1,...,k\\}$ and call it the leading vector corresponding to
$p_{i,N}$. Let us remark that the leading vector has no elements equal to 0,
because we have assumed that the polynomials are essentially distinct.
Finally, we call $P_{N}$ ordered, if the degrees of the polynomials $p_{i,N}$
are non-increasing. In this case, the polynomial $p_{1,N}$ has maximal degree
and we call it the leading polynomial. The leading vector of an ordered
polynomial family is defined as the leading vector corresponding to its
leading polynomial.

###  4.2. Types of polynomial families

We define the type $(d,w_{d},...,w_{1})$ of the polynomial family, where $d$
is the largest degree appearing in the polynomials of $P_{N}$ and $w_{i}$ is
the number of distinct leading coefficients of the members of $P_{N}$ with
degree exactly $i$ among all polynomials in the family. Note that for families
of variable polynomials, the value of this vector may depend on the variable
$N$. We order the types by the value of $d$ and then order types of same
degree lexicographically. We observe that a decreasing sequence of types must
eventually be constant. The type of a family is a classical quantity used in
the literature when an induction scheme on polynomial families is required.

###  4.3. Good sequences and nice polynomial families

Now, we define the notion of a nice polynomial family. Namely, we will deal
with polynomials whose coefficients are well-behaved sequences. Our arguments
fail to work in the general case where the coefficients can be arbitrary
sequences.

######  Definition 4.1.

a) A sequence $(a_{n})_{n\in\mathbb{N}}$ of real numbers is called "good", if
there exists a function $f\in\mathcal{H}$ with
$\lim\limits_{t\to+\infty}f(t)\neq 0$ such that

$\lim\limits_{n\to+\infty}\frac{a_{n}}{f(n)}=1.$

b) Let $P_{N}=\\{p_{1,N},...,p_{k,N}\\}$ be a collection of polynomials. The
family $P_{N}$ is called nice, if all the degrees of the polynomials $p_{i,N}$
and $p_{i,N}-p_{j,N}$ are independent of $N$ for $N$ large enough and their
leading coefficients are good sequences, for all admissible values of the
$i,j$.

Note that any good sequence has a limit (possibly infinite). An example of a
good sequence that is not a Hardy sequence is the sequence $\frac{\lfloor
N^{2/3}\rfloor}{\sqrt{N}}$, which is asymptotically close to $N^{1/6}$. In
particular, all sequences of the form $\lfloor f(n)\rfloor$, where the
function $f\in\mathcal{H}$ does not converge to $0$ (as $t\to+\infty$), are
good sequences, while, for example, $\lfloor\frac{1}{\log n}\rfloor$ is not a
good sequence.

######  Lemma 4.2.

The type of a nice polynomial family is well-defined (independent of $N$) for
$N$ large enough.

###### Proof.

This is fairly straightforward. Indeed, assume that the polynomials $p_{i,N}$
and $p_{j,N}$ of the given family have the same degree $s$. Let
$a_{i}(N),a_{j}(N),a_{ij}(N)$ be the leading coefficients of $p_{i,N},p_{j,N}$
and $p_{i,N}-p_{j,N}$, which are all good sequences. The degree of the
polynomial $p_{i,N}-p_{j,N}$ does not depend on $N$ for $N$ sufficiently
large. Then, we have either one of the following:
i) If the polynomial $p_{i,N}-p_{j,N}$ has degree equal to $s$, then for $N$
large enough, $a_{ij}(N)=a_{i}(N)-a_{j}(N)\neq 0$ and therefore the
polynomials $p_{i,N},p_{j,N}$ have distinct leading coefficients eventually.
ii) If the polynomial $p_{i,N}-p_{j,N}$ has degree smaller than $s$, then that
means that, for $N$ large enough, we have $a_{i}(N)-a_{j}(N)=0$ and the
polynomials $p_{i,N},p_{j,N}$ have equal leading coefficients eventually.
The claim easily follows. ∎

###  4.4. The van der Corput inequality

We shall rely heavily on the following variant of the van der Corput
inequality in our proofs.

######  Lemma 4.3.

For a sequence $u_{n}$ in a Hilbert space with $\lVert u_{n}\rVert\leq 1$ and
a quantity $M=o(N)$, we have

$\big{\lVert}\frac{1}{N}\sum_{n=0}^{N-1}u_{n}\big{\rVert}^{2^{d}}\ll_{d}\frac{1}{M}+\underset{-M\leq
m\leq M}{\mathbb{E}}\ \Big{|}\underset{0\leq n\leq N-1}{\mathbb{E}}\langle
u_{n+m},u_{n}\rangle\Big{|}^{2^{d-1}}+o_{N}(1).$

###### Proof.

This follows from the basic van der Corput inequality

$\big{\lVert}\frac{1}{N}\sum_{n=0}^{N-1}u_{n}\big{\rVert}\ll\frac{1}{M^{1/2}}+\big{(}\underset{-M\leq
m\leq M}{\mathbb{E}}\Big{|}\underset{0\leq n\leq N-1}{\mathbb{E}}\langle
u_{n+m},u_{n}\rangle\Big{|}\big{)}^{1/2}+\frac{M^{1/2}}{N^{1/2}}$

by successively squaring and applying the Cauchy-Schwarz inequality. ∎

We will use this inequality to derive asymptotic bounds for multiple ergodic
averages involving polynomials. The above inequality holds, in particular,
when $M$ is a fixed positive integer. We state here the equivalent result for
variable sequences, since this is more consistent with the notation used in
the proof below.

######  Lemma 4.4.

For sequences $(u_{n,N})_{n,N\in\mathbb{N}}$ in a Hilbert space with $\lVert
u_{n,N}\rVert\leq 1$ and a quantity $M=o(N)$, we have

$\big{\lVert}\frac{1}{N}\sum_{n=0}^{N-1}u_{n,N}\big{\rVert}^{2^{d}}\ll_{d}\frac{1}{M}+\underset{|m|\leq
M}{\mathbb{E}}\ \Big{|}\underset{0\leq n\leq N-1}{\mathbb{E}}\langle
u_{n+m,N},u_{n,N}\rangle\Big{|}^{2^{d-1}}+o_{N}(1).$

###  4.5. Bounds of polynomial averages

The remainder of the section will be dedicated to establishing the following
proposition:

######  Proposition 4.5.

Let $k,d$ be positive integers and let $M$ be a positive integer parameter.
Suppose ${\bf W}=(d,w_{d},...,w_{1})$ is a (d+1)-tuple of positive integers
that is also a type for some polynomial family. Then, there exist positive
integers $t=t(d,k,{\bf W})$, $s=s(d,k,{\bf W})$, a finite set $Y=Y(d,k,{\bf
W})$ of integers and integer polynomials in $t$ variables
$p_{\underline{\varepsilon},j},\text{ with }\underline{\varepsilon}\in[[s]]$
and $1\leq j\leq\ k$, that are at most linear in each variable666This means
that when regarded as polynomials only in one variable, then they are linear.
Examples are $p_{1}(m_{1},m_{2})=m_{1}-2m_{2}$ and
$p_{2}(m_{1},m_{2},m_{3})=m_{1}m_{2}-3m_{3}$., such that for any ordered nice
family of non-constant, essentially distinct polynomials

$P_{N}=\\{p_{1,N},...,p_{k,N}\\}$

of degree $d$ and type ${\bf W}$ with leading vector
$\mathcal{S}(P_{N})=\\{u_{1,N},...,u_{k,N}\\}$, any increasing sequence
$L_{N}\to\infty$, any measure preserving system $(X,\mu,T)$ and sequences of
1-bounded functions $f_{1,N},...,f_{k,N}$, we have

(15) $\sup_{|c_{n,N}|\leq 1}\big{\lVert}\underset{0\leq n\leq
L_{N}}{\mathbb{E}}\ c_{n,N}\prod_{i=1}^{k}T^{\lfloor
p_{i,N}(n)\rfloor}f_{i,N}\big{\rVert}_{L^{2}(\mu)}^{2^{t}}\ll_{d,k,{\bf W}}\\\
\frac{1}{M}+\sum_{{\bf h}\in Y^{[[s]]}}\underset{{\bf
m}\in[-M,M]^{t}}{\mathbb{E}}\Big{|}\int\prod_{\underline{\varepsilon}\in[[s]]}T^{\lfloor
A_{\underline{\varepsilon},N}({\bf
m})\rfloor+h_{\underline{\varepsilon}}}(\mathcal{C}^{|\underline{\varepsilon}|}f_{1,N})\
d\mu\Big{|}+o_{N}(1),$

where

$A_{\underline{\varepsilon},N}({\bf m})=\sum_{1\leq j\leq k}\
p_{\underline{\varepsilon},j}({\bf m})u_{j,N}$

are real polynomials in ${\bf m}$. In addition, we have the following:
i) For $\underline{\varepsilon}\neq\underline{0}$, we have that the polynomial
$A_{\underline{\varepsilon},N}({\bf m})$ is non-constant.
ii) The polynomials $A_{\underline{\varepsilon},N}({\bf m}),\
\underline{\varepsilon}\in[[s]]$ are pairwise essentially distinct.
iii) We have the relation

$A_{\underline{\varepsilon},N}({\bf m})+A_{\underline{\varepsilon}^{c},N}({\bf
m})=A_{\underline{1},N}({\bf m})$

for any $\underline{\varepsilon}\in[[s]]$. More generally, if
$\underline{\varepsilon},\underline{\varepsilon}^{\prime}\in[[s]]$ are such
that $\underline{\varepsilon}+\underline{\varepsilon}^{\prime}\in[[s]]$
777This means that the $i$-th entries of
$\underline{\varepsilon},\underline{\varepsilon}^{\prime}$ cannot
simultaneously be 1, for any admissible value of $i$., then

$A_{\underline{\varepsilon},N}({\bf
m})+A_{\underline{\varepsilon}^{\prime},N}({\bf
m})=A_{\underline{\varepsilon}+\underline{\varepsilon}^{\prime},N}({\bf m})$

iv) For any $\underline{\varepsilon}\in[[s]]$, we have that if

$c_{1}p_{\underline{\varepsilon},1}({\bf
m})+...+c_{k}p_{\underline{\varepsilon},k}({\bf m})$

is the zero polynomial for some $c_{1},...,c_{k}\in\mathbb{R}$, then we have
$c_{i}=0$ or $p_{\underline{\varepsilon},i}({\bf m})$ is the zero polynomial,
for every $1\leq i\leq k$.

######  Comment.

The $\sum\limits_{{\bf h}\in Y^{[[s]]}}$ means that we take the sum for all
choices of ${\bf h}=(h_{\underline{\varepsilon}},\
\underline{\varepsilon}\in[[s]])$ where $h_{\underline{\varepsilon}}\in Y$. In
addition, we will make a small abuse of notation and write $\underset{{\bf
m}\in[-M,M]^{t}}{\mathbb{E}}$ to denote the average over all ${\bf
m}\in\mathbb{Z}^{t}\cap[-M,M]^{t}$.

######  Remarks.

i) The polynomials $p_{\underline{\varepsilon},j}$ are independent of the
leading vector $\\{u_{1,N},...,u_{k,N}\\}$ and are, more importantly,
independent of the variable $N$.
ii) The existence of the errors $h_{\underline{\varepsilon}}$ is merely
technical and arises from the floor function in the last expression inside the
integral, since we cannot use Lemma 3.2 to remove the error terms in this
case. This will be more easily understood in the proof of the case of linear
polynomials that follows.
iii) The quantity $o_{N}(1)$ depends of course on the values of $d$ and $k$.
It also depends on the value of the fixed number $M$. However, this dependence
plays no role in arguments of the following sections (where we will usually
take limits first as $N\to+\infty$ and, then, as $M\to+\infty$). For ease of
notation, we will omit all other subscripts for the term $o_{N}(1)$.
iv) The final condition $iv)$ above implies that, for a fixed
$\underline{\varepsilon}\in[[s]]$, if we exclude all the constant polynomials
among the $p_{\underline{\varepsilon},j}$, the remaining polynomials are
linearly independent.

Ignoring the technical parts of the statement, the above proposition asserts
that when working with multiple averages on some polynomials that vary with
$N$, we can instead bound them by the averages of a polynomial correlation
sequence of only the function $f_{1,N}$. Even though the new polynomials
$A_{\underline{\varepsilon},N}$ have several variables, they only depend on
the sequences $u_{1,N},...,u_{k,N}$ and, assuming we they have good limiting
behavior, we can take the limits first as $N\to+\infty$ and then as
$M\to+\infty$ to get some nice bounds for the original averages. For instance,
in the case where we have a fixed function $f_{1,N}=f_{1}$ and the sequences
$u_{i,N}$ converge to non-zero real numbers, the above statement can be used
to prove that the $\limsup$ of the ergodic averages in the left-hand side of
(18) can be bounded by a power of $\lvert\\!|\\!|f_{1}|\\!|\\!\rvert_{s}$ for
some suitable positive integer $s$. This last assertion follows from minor
modifications to the argument present in [27] (to cover the case of real
polynomials instead of just integer polynomials).

###### Proof in the linear case.

Firstly, we shall establish Proposition 4.5 in the case where all the
polynomials have degree 1. Thus, assume that $p_{i,N}(t)=a_{i,N}t+b_{i,N}$
where $a_{i,N},b_{i,N}\in\mathbb{R}$ so that the variables $a_{i,N}$ are
(eventually) non-zero. The assumption that our polynomials are essentially
distinct implies that the numbers $a_{i,N}$ and $a_{j,N}$ are distinct. The
leading vector of $P_{N}$ is the set

$\\{a_{1,N},a_{1,N}-a_{2,N},...,a_{1,N}-a_{k,N}\\}$

and these are good sequences.

We induct on $k$. For $k=1$, we apply the van der Corput inequality to get

$\big{\lVert}\underset{0\leq n\leq L_{N}}{\mathbb{E}}\ c_{n,N}T^{\lfloor
a_{1,N}n+b_{1,N}\rfloor}f_{1,N}\big{\rVert}_{L^{2}(\mu)}^{2}\ll\\\
\frac{1}{M}+\underset{|m|\leq M}{\mathbb{E}}\Big{|}\underset{0\leq n\leq
L_{N}}{\mathbb{E}}\overline{c_{n,N}}c_{n+m,N}\int\overline{f_{1,N}}\cdot
T^{\lfloor a_{1,N}n+b_{1,N}+ma_{1,N}\rfloor-\lfloor
a_{1,N}n+b_{1,N}\rfloor}f_{1,N}\ d\mu\Big{|}+o_{N}(1).$

We rewrite the last quantity as

$\frac{1}{M}+\underset{|m|\leq M}{\mathbb{E}}\Big{|}\underset{0\leq n\leq
L_{N}}{\mathbb{E}}\overline{c_{n,N}}c_{n+m,N}\int\overline{f_{1,N}}\cdot
T^{\lfloor ma_{1,N}\rfloor+e_{n,m,N}}f_{1,N}\ d\mu\Big{|}+o_{N}(1),$

where $e_{n,m,N}\in\\{0,\pm 1\\}$ (the implied constant is independent of all
variables in the above relation). Let
$A_{z,m,N}=\\{n\in\mathbb{Z}^{+}{:}\;0\leq n\leq L_{N}\text{ and
}e_{n,m,N}=z\\}$ for $z\in\\{0,\pm 1\\}=Y$. Then, the innermost average can be
rewritten as

$\Big{|}\frac{1}{L_{N}}\sum_{z\in Y}\sum_{n\in
A_{z,N,m}}\overline{c_{n,N}}c_{n+m,N}\int\overline{f_{1,N}}\cdot T^{\lfloor
ma_{1,N}\rfloor+z}f_{1,N}\ d\mu|\leq\sum_{z\in
Y}\Big{|}\int\overline{f_{1,N}}\cdot T^{\lfloor ma_{1,N}\rfloor+z}f_{1,N}\
d\mu\Big{|},$

which, combined with the above, gives the desired result (for constants $t=1$
and $s=1$, polynomials $p_{1}(m)=ma_{1,N}$ and $p_{0}(m)=0$ and set
$Y=\\{0,\pm 1\\}$).

Now assume that we have proven the result for $k-1$ ($k\geq 2$), with the
constants of the proposition given by $t=k-1$ and $s=k-1$. Then, we use the
van der Corput inequality to get

$\big{\lVert}\underset{0\leq n\leq L_{N}}{\mathbb{E}}\
c_{n,N}\prod_{i=1}^{k}T^{\lfloor
a_{i,N}n+b_{i,N}\rfloor}f_{i,N}\big{\rVert}_{L^{2}(\mu)}^{2^{k}}\ll_{k}\frac{1}{M}+o_{N}(1)+\\\
\underset{|m|\leq M}{\mathbb{E}}\Big{|}\underset{0\leq n\leq
L_{N}}{\mathbb{E}}\overline{c_{n,N}}c_{n+m,N}\int\prod_{i=1}^{k}T^{\lfloor
a_{i,N}n+b_{i,N}\rfloor+\lfloor ma_{i,N}\rfloor+e_{i,m,n,N}}f_{1,N}\
T^{\lfloor a_{i,N}n+b_{i,N}\rfloor}\overline{f_{i,N}}\ d\mu\Big{|}^{2^{k-1}},$

which is smaller than

(16) $\underset{|m|\leq M}{\mathbb{E}}\Big{|}\underset{0\leq n\leq
L_{N}}{\mathbb{E}}\ \overline{c_{n,N}}c_{n+m,N}\ \int\prod_{i=1}^{k}T^{\lfloor
a_{i,N}n+b_{i,N}\rfloor-\lfloor a_{k,N}n+b_{k,N}\rfloor+\lfloor
ma_{i,N}\rfloor+e_{i,m,n,N}}f_{1,N}\cdot\\\ T^{\lfloor
a_{i,N}n+b_{i,N}\rfloor-\lfloor a_{k,N}n+b_{k,N}\rfloor}\overline{f_{i,N}}\
d\mu\Big{|}^{2^{k-1}}+1/M+o_{N}(1),$

where we again have $e_{i,m,,n,N}\in\\{0,\pm 1\\}$. In the last step, we
composed with $T^{-\lfloor a_{k,N}n+b_{k,N}\rfloor}$ inside the integral.

We have

$\lfloor a_{i,N}n+b_{i,N}\rfloor-\lfloor
a_{k,N}n+b_{k,N}\rfloor=\lfloor{(a_{i,N}-a_{k,N})n+b_{i,N}-b_{k,N}}\rfloor+e^{\prime}_{i,n,N}$

where $e^{\prime}_{i,n,N}\in\\{0,\pm 1\\}$. Therefore, we can rewrite the last
expression in (16) as

$\frac{1}{M}+\underset{|m|\leq M}{\mathbb{E}}\ \Big{|}\underset{0\leq n\leq
L_{N}}{\mathbb{E}}\ \overline{c_{n,N}}c_{n+m,N}\\\ \prod_{i=1}^{k}\int
T^{\lfloor(a_{i,N}-a_{k,N})n+b_{i,N}-b_{k,N}\rfloor+e^{\prime}_{i,n,N}}\big{(}\overline{f_{i,N}}\cdot
T^{\lfloor ma_{i,N}\rfloor+e_{i,m,n,N}}f_{i,N}\big{)}\
d\mu\Big{|}^{2^{k-1}}+o_{N}(1).$

Then, using the Cauchy-Schwarz inequality and the argument in Lemma 3.2, we
can bound the innermost average in the above expression by $O_{k}(1)$ times
the quantity

$A_{m,N}=\sup_{|c_{n,N}|\leq 1}\big{\lVert}\underset{0\leq n\leq
L_{N}}{\mathbb{E}}\ c_{n,N}\prod_{i=1}^{k-1}\
T^{\lfloor(a_{i,N}-a_{k,N})n+(b_{i,N}-b_{k,N})\rfloor}(\overline{f_{i,N}}\cdot
T^{\lfloor
ma_{i,N}\rfloor+e_{i,m,n,N}}f_{i,N})\big{\rVert}_{L^{2}(\mu)}^{2^{k-1}}.$

Now, we use the argument of Lemma 3.2 again to deduce that $A_{m,N}$ is
bounded by $O_{k}(1)$ times

$\sum_{\underset{1\leq i\leq k-1}{z_{i}\in\\{0,\pm 1\\}}}\sup_{|c_{n,N}|\leq
1}\big{\lVert}\underset{0\leq n\leq L_{N}}{\mathbb{E}}\
c_{n,N}\prod_{i=1}^{k-1}\
T^{\lfloor(a_{i,N}-a_{k,N})n+(b_{i,N}-b_{k,N})\rfloor}(\overline{f_{i,N}}\cdot
T^{\lfloor ma_{i,N}\rfloor+z_{i}}f_{i,N})\big{\rVert}_{L^{2}(\mu)}^{2^{k-1}}.$

We fix some ${\bf z}=(z_{1},...,z_{k-1})\in\\{0,\pm 1\\}^{k-1}$. If we take
the polynomial that corresponds to $\overline{f_{1,N}}\cdot T^{\lfloor
ma_{1,N}\rfloor+z_{1}}f_{1,N}$ to be the new leading polynomial, then the new
leading vector is the set

$\\{a_{1,N}-a_{k,N},a_{1,N}-a_{2,N},...,a_{1,N}-a_{k-1,N}\\}.$

By the induction hypothesis, there exists a finite set $Y_{k-1}$, for which

$\sup_{|c_{n,N}|\leq 1}\big{\lVert}\underset{0\leq n\leq L_{N}}{\mathbb{E}}\
c_{n,N}\prod_{i=1}^{k-1}\
T^{\lfloor(a_{i,N}-a_{k,N})n+(b_{i,N}-b_{k,N})\rfloor}(\overline{f_{i,N}}\cdot
T^{\lfloor
ma_{i,N}\rfloor+z_{i}}f_{i,N})\big{\rVert}_{L^{2}(\mu)}^{2^{k-1}}\ll_{k}\\\
\frac{1}{M}+\sum_{{\bf h}\in[[Y_{k-1}]]}\underset{|m_{1}|,...,|m_{k-1}|\leq
M}{\mathbb{E}}\Big{|}\int\prod_{\underline{\varepsilon}\in[[k-1]]}T^{\lfloor\sum_{1\leq
j\leq k-1}\
p_{\underline{\varepsilon},j}(m_{1},...,m_{k-1})(a_{1,N}-a_{j,N})\rfloor+h_{\underline{\varepsilon}}}\\\
\mathcal{C}^{|\underline{\varepsilon}|}(\overline{f_{1,N}}\cdot T^{\lfloor
ma_{1,N}\rfloor+z_{1}}f_{1,N})\ d\mu\Big{|}+o_{N}(1).$

Using the identification $[[k]]=[[k-1]]\times\\{0,1\\}$, we can write an
$\underline{\varepsilon}\in[[k]]$ as
$\underline{\varepsilon}=(\underline{\varepsilon_{1}},\varepsilon_{2})$ where
$\underline{\varepsilon_{1}}\in[[k-1]]$ and $\varepsilon_{2}\in\\{0,1\\}$. We
also write ${\bf m}=(m,m_{1},...,m_{k-1})$. Combining the integer parts, we
rewrite the last integral as

$\int\prod_{\varepsilon\in[[k]]]}T^{\lfloor\sum_{1\leq j\leq k-1}\
p^{\prime}_{\underline{\varepsilon},j}(m_{1},...,m_{k-1})(a_{1,N}-a_{j,N})+p^{\prime}_{\underline{\varepsilon},k}(m)a_{1,N}\rfloor+h^{\prime}_{\underline{\varepsilon},{\bf
m}}}\ \mathcal{C}^{|\underline{\varepsilon}|}f_{1,N}\ d\mu\ ,$

where

  1. (1)

$p^{\prime}_{\underline{\varepsilon},j}$ is the polynomial
$p_{\underline{\varepsilon}_{1},j}$ for $1\leq j\leq k-1$,

  2. (2)

the polynomial $p^{\prime}_{\underline{\varepsilon},k}$ is equal to $m$ when
$\underline{\varepsilon_{2}}=0$ and is zero otherwise and

  3. (3)

$h^{\prime}_{\underline{\varepsilon},{\bf
m}}=h_{\underline{\varepsilon}_{1}}+h_{2,\underline{\varepsilon},{\bf m}}$,
where 888 In particular, $h_{2,\underline{\varepsilon},{\bf m}}$ is the sum of
$z_{1}$ plus the error term appearing by combining $\lfloor ma_{1,N}\rfloor$
with the other integer part, whenever they both appear. Otherwise, it is zero.
Thus, it takes values on a finite set of integers.
$h_{2,\underline{\varepsilon},{\bf m}}\in\\{0,\pm 1,\pm 2\\}$. More
importantly, $h^{\prime}_{\underline{\varepsilon},m}$ takes values in a finite
set $Y_{k}$.

We observe that

$\Big{|}\int\prod_{\varepsilon\in[[k]]]}T^{\lfloor\sum_{1\leq j\leq k-1}\
p^{\prime}_{\underline{\varepsilon},j}(m_{1},...,m_{k-1})(a_{1,N}-a_{j,N})+p^{\prime}_{\underline{\varepsilon},k}(m)a_{1,N}\rfloor+h^{\prime}_{\underline{\varepsilon},{\bf
m}}}\ \mathcal{C}^{|\underline{\varepsilon}|}f_{1,N}\ d\mu\Big{|}\leq\\\
\sum_{{\bf
h}\in[[Y_{k}]]}\Big{|}\int\prod_{\varepsilon\in[[k]]]}T^{\lfloor\sum_{1\leq
j\leq k-1}\
p^{\prime}_{\underline{\varepsilon},j}(m_{1},...,m_{k-1})(a_{1,N}-a_{j,N})+p^{\prime}_{\underline{\varepsilon},k}(m)a_{1,N}\rfloor+h_{\underline{\varepsilon}}}\
\mathcal{C}^{|\underline{\varepsilon}|}f_{1,N}\ d\mu\Big{|}.$

Averaging over $m,m_{1},...,m_{k-1}$ and summing over ${\bf z}\in\\{0,\pm
1\\}^{k-1}$, we have that for the finite set $Y_{k}$ above, the original
expression is bounded by $O_{k}(1)$ times

$\frac{1}{M}+\sum_{{\bf h}\in[[Y_{k}]]}\ \underset{{\bf
m}\in[-M,M]^{k}}{\mathbb{E}}\Big{|}\int\prod_{\underline{\varepsilon}\in[[k]]}T^{\lfloor\sum_{1\leq
j\leq k}\ p^{\prime}_{\underline{\varepsilon},j}({\bf
m})u_{j,N}\rfloor+h_{\underline{\varepsilon}}}\
(\mathcal{C}^{|\underline{\varepsilon}|}f_{1,N})\ d\mu\Big{|}+o_{N}(1),$

where $u_{1,N}=a_{1,N}$ and $u_{j,N}=a_{1,N}-a_{j,N}$. The conclusion follows.
∎

######  Remark.

It follows from the above proof that the polynomials
$A_{\underline{\varepsilon},N}$ in the statement of Proposition 4.5 have the
following form:

$A_{\underline{\varepsilon},N}(m_{1},...,m_{k})=\underline{\varepsilon}\cdot(m_{1}u_{1,N},...,m_{k}u_{k,N})$

where "$\cdot$" denotes here the standard inner product on $\mathbb{R}^{k}$.
Thus, it is straightforward to check that the polynomials
$A_{\underline{\varepsilon},N}$ satisfy the conditions $i),ii)$, $iii)$ and
$iv)$ of Proposition 4.5. Note that all these polynomials have degree 1. This
will not be the case when working with polynomials of higher degree, where we
may have higher degree terms (like products of the form $m_{1}m_{2}$), but
they will be linear in each variable separately.

###  4.6. The PET induction.

For a polynomial $p_{N}$, a family $P_{N}$ and $h\in\mathbb{N}$, we define the
van der Corput operation (or vdC operation), where we form the family

$\\{p_{1,N}(t+h)-p_{N}(t),...,p_{k,N}(t+h)-p_{N}(t),\
p_{1,N}(t)-p_{N}(t),...,p_{k,N}(t)-p_{N}(t)\\}$

and remove polynomials of degree 0. We denote this new family by
$(p_{N},h)^{*}P_{N}$. At first glance, it is not obvious that this operation
is well defined, because the constant polynomials that we discard may be
different for different values of $N$. We will see that this is not the case
for nice polynomial families below. We will use the vdC operation successively
to reduce the "complexity" of a polynomial family. Our main observation is
that the leading vector of a polynomial family is well behaved under the vdC
operation.

Consider a family of variable polynomials $P_{N}=\\{p_{1,N},...,p_{k,N}\\}$
and let the leading vector of $P_{N}$ corresponding to $p_{1,N}$ be

$\mathcal{S}(P_{N})=\\{u_{1,N},...,u_{k,N}\\}.$

Fix any $1\leq i_{0}\leq k$, as well as the polynomial $p_{i_{0},N}$, which we
symbolize as $p_{N}$ from now on for convenience. Consider the new polynomial
family $P^{\prime}_{N,h}=(p_{N},h)^{\star}P_{N}$ that arises from the van der
Corput operation. Here, $h$ ranges over the non-zero integers.

######  Lemma 4.6.

Assume that the family $P_{N}$ of degree $d$ is nice and let
$(u_{1,N},...,u_{k,N})$ be its leading vector corresponding to $p_{1,N}$. For
every choice of polynomial $p_{N}$ above and the value of
$h\in\mathbb{Z}^{*}$, we have that each element of the leading vector of
$P^{\prime}_{N,h}$ corresponding to the new polynomial $p_{1,N}(t+h)-p_{N}(t)$
has one the following forms:

  * •

They are equal to one of the $u_{i,N}$ for some $2\leq i\leq k$.

  * •

They have the form $du_{1,N}h$.

  * •

They are the sum $du_{1,N}h+u_{i,N}$ for some $u_{i,N}$ with $i\neq 1$.

###### Proof.

Without loss of generality, we will assume that we have taken $p_{N}=p_{k,N}$
(the case $p_{N}=p_{1,N}$ is very similar). We want to study the leading
vector corresponding to the polynomial $p_{1,N}(t+h)-p_{k,N}(t)$. Therefore,
it is sufficient to find the leading coefficients of the polynomials

$\displaystyle\big{(}p_{1,N}(t+h)-p_{k,N}(t)\big{)}$
$\displaystyle-\big{(}p_{1,N}(t)-p_{k,N}(t)\big{)}$
$\displaystyle\big{(}p_{1,N}(t+h)-p_{k,N}(t)\big{)}$
$\displaystyle-\big{(}p_{i,N}(t+h)-p_{k,N}(t)\big{)}$
$\displaystyle\big{(}p_{1,N}(t+h)-p_{k,N}(t)\big{)}$
$\displaystyle-\big{(}p_{i,N}(t)-p_{k,N}(t)\big{)}$

for $2\leq i\leq k$. The leading coefficient of the first polynomial is always
$dhu_{1,N}$ and that satisfies our required property. The leading coefficient
of the second polynomial is always equal to the leading coefficient of
$p_{1,N}(t+h)-p_{i,N}(t+h)$ and this is always equal to the leading
coefficient of $p_{1,N}(t)-p_{i,N}(t)$ which belongs to the leading vector.
Finally, the leading coefficient of the third polynomial is equal to the
leading coefficient of $p_{1,N}(t+h)-p_{i,N}(t)$. Note that this polynomial
can be rewritten as

$(p_{1,N}(t+h)-p_{1,N}(t))+(p_{1,N}(t)-p_{i,N}(t)).$

The leading coefficient of the first polynomial is equal to $dhu_{1,N}$ as we
established above, while the second difference has leading coefficient
$u_{i,N}$ (by definition). Therefore, the leading coefficient of their sum is
either $dhu_{1,N},u_{i,N}$ or their sum $dhu_{1,N}+u_{i,N}$, which concludes
the proof. ∎

Observe that the particular form each element of the new leading vector has
does not depend on the value of $N$ (i.e. it cannot have the first form for
one value of $N$ and then the second form for some other value of $N$). This
follows from the fact that the type of the original family is independent of
$N$, if $N$ is large enough. We will now use this lemma to study how the van
der Corput operation affects the type of the original family.

######  Corollary 4.7.

Let $P_{N},p_{N}$ be as above and let $d$ be the degree of the family $P_{N}$.
Then, there exists a set of integers $Y$ with at most $O_{k,d}(1)$ elements
such that, for every $h\notin Y$, the polynomial family
$P^{\prime}_{N,h}=(p_{N},h)^{\star}P_{N}$ that arises from the van der Corput
operation is nice and its type is independent999 The type depends only on
which polynomial of the initial family we choose to be the polynomial $p_{N}$,
as well as the type of the original family. of the value of $h$.

###### Proof.

We denote by $u_{ii,N}$ the leading coefficient of $p_{i,N}$, while $u_{ij,N}$
denotes the leading coefficient of $p_{i,N}-p_{j,N}$ for $i\neq j$. These are
all good sequences by the definition of a nice family. Using Lemma 4.6, we can
prove that the leading coefficients of all the polynomials in
$P^{\prime}_{N,h}$ and of their differences can take one of the following
forms:
i) they are equal to some $u_{ij,N}$ with $i\neq j$,
ii) they have the form $ru_{ii,N}h$ for some $1\leq r\leq d$ or
iii) they have the form $ru_{ii,N}h+u_{ij,N}$ for some $1\leq r\leq d$.

We prove that these sequences are good for all except $O_{d,k}(1)$ values of
$h$. For all values of $1\leq i,j\leq k$ ($i\neq j$) and $1\leq r\leq d$, we
consider the set $A(i,j,r)$ of all possible sequences of the above three forms
(not all of them appear as leading coefficients, but this does not affect our
argument), where $h$ is some fixed non-zero integer. There are only finitely
many such sets. Note that for $h\neq 0$, the sequences of the first two forms
are always good. Now consider a sequence of the form $ru_{ii,N}h+u_{ij,N}$.
There exist functions $f_{1},f_{2}\in\mathcal{H}$, not converging to 0, such
that $|u_{ii,N}/f_{1}(N)|=1+o_{N}(1)$ and $|u_{ij,N}/f_{2}(N)|=1+o_{N}(1)$.
The function $rhf_{1}(t)+f_{2}(t)$ is obviously an element of $\mathcal{H}$.
In addition, for our fixed $r$, the relation

$\lim\limits_{t\to+\infty}rhf_{1}(t)+f_{2}(t)=0$

can hold only for at most one possible value of $h\in\mathbb{Z}$, which we
call a "bad value". Then, if $h$ is not a bad value, we have

$\Big{|}\frac{ru_{ii,N}h+u_{ij,N}}{rhf_{1}(N)+f_{2}(N)}\Big{|}=1+o_{N}(1).$

Indeed, this follows easily because the functions $f_{1}$ and $f_{2}$ are
comparable, which also means that all the involved sequences are comparable.
Thus, dividing the numerator and denominator of the above fraction by either
$f_{1}(N)$ or $f_{2}(N)$, we easily get the result. In conclusion, the
sequence $ru_{ii,N}h+u_{ij,N}$ is a good sequence for all non-bad values of
$h$.

Now, if we take all possible values of the $i,j,r$, we conclude that there are
at most $O_{d,k}(1)$ bad values of $h$.

We have shown that for every non-bad value of $h$, the family
$P^{\prime}_{N,h}$ is a nice polynomial family and, therefore, has a fixed
type (independent of $N$). We show that its type does not depend on $h$.
Therefore, consider two polynomials $q_{1},q_{2}$ of $P^{\prime}_{N,h}$ of the
same degree. We consider some possible cases:
a) If $q_{1}$ and $q_{2}$ have the form $p_{i,N}(t)-p_{N}(t)$, then whether or
not their leading coefficients are equal depends only on the type of the
original family and the choice of $p_{N}$ (and not on $h$).
b) If $q_{1}$ has the form $p_{i,N}(t+h)-p_{N}(t)$, while $q_{2}$ has the form
$p_{j,N}(t)-p_{N}(t)$, then their leading coefficients can be equal in only
two possible cases: if the polynomial $p_{N}$ has degree strictly larger than
the degree of both $p_{i,N}$ and $p_{j,N}$ (this depends only on the choice of
$p_{N}$, not on $h$), or if the polynomials $p_{i,N}(t+h)$ and $p_{j,N}(t)$
have the same degree (bigger than or equal to the degree of $p_{N}$) and equal
leading coefficients. In the second case, we must have that $p_{i,N}(t)$ and
$p_{j,N}(t)$ have equal leading coefficients, which depends only on the type
of the original family and not on $h$.
c) If $q_{1}$ and $q_{2}$ both have the form $p_{i,N}(t+h)-p_{N}(t)$, then the
result follows similarly as in the case a).

The fact that the degrees of the polynomials of the new family and of their
differences do not depend on $N$ and $h$ can also be established easily using
the preceding arguments. We omit the details. ∎

######  Proposition 4.8.

If $P_{N}=\\{p_{1,N},...,p_{k,N}\\}$ is an ordered polynomial family, then
there exists a polynomial $p_{N}\in P_{N}$, such that for all, except at most
one value of $h\in\mathbb{Z}$, the polynomial family
$P^{\prime}_{N,h}=(p_{N},h)^{*}P_{N}$ has type strictly smaller than the type
of $P_{N}$ and its leading polynomial is the polynomial
$p_{1,N}(t+h)-p_{N}(t)$.

###### Proof.

We describe the operation that reduces the type. At each step, we choose a
polynomial $p_{N}\in P_{N}$ that has minimal degree in the family. For an
$h\in\mathbb{Z}$, apply the van der Corput operation. This forms a polynomial
family

(17) $P^{\prime}_{N}=\\{p_{1,N}(t+h)-p_{N}(t),...,p_{k,N}(t+h)-p_{N}(t),\
p_{1,N}(t)-p_{N}(t),...,p_{k,N}(t)-p_{N}(t)\\}$

and choose $p_{1,N}(t+h)-p_{N}(t)$ to be the new leading polynomial. We
distinguish between some cases:

a) Assume that the polynomials $p_{1,N}$ and $p_{k,N}$ have distinct degrees.
Then, choose $p_{N}=p_{k,N}$, which by the "ordered" assumption has minimal
degree. We notice that the polynomial $p_{1,N}(t+h)-p_{N}(t)$ has maximal
degree in the polynomial family. We check that the type of the polynomial
family is reduced. Indeed, if the degree of $p_{k,N}(t)$ is $d^{\prime}$, then
the number $w_{d}^{\prime}$ is reduced, while all the numbers $w_{i}$ are left
unchanged for $i>d^{\prime}.$

b) Suppose the polynomials $p_{1,N}$ and $p_{k,N}$ have the same degree and
not all leading coefficients in the family $P_{N}$ are equal. In particular,
we may assume, without loss of generality, that this holds for the polynomials
$p_{1,N}$ and $p_{k,N}$. Again, choose $p_{N}=p_{k,N}$. Then, the polynomial
$p_{1,N}(t+h)-p_{N}(t)$ has maximal degree in the new polynomial family. In
addition, the number $w_{d}$ is reduced, which means that the new family has
smaller type than the original.

c) Finally, assume that all polynomials have the same degree and the same
leading coefficient. We choose again $p_{N}=p_{k,N}$. The polynomial
$p_{1,N}(t+h)-p_{N}(t)$ has maximal degree equal to $d-1$ in $P^{\prime}_{N}$,
except possibly for one value of $h\in\mathbb{Z}$ (to see this, we can work
similarly as in the proof of Corollary 4.7). Also, the family $P^{\prime}_{N}$
has smaller type than $P_{N}$, since it has degree at most $d-1$. ∎

While for a given type ${\bf W}$ there are infinite types smaller than ${\bf
W}$, it is straightforward to see that a decreasing sequence of types is
eventually constant. Therefore, the type-reducing operation that we did above
will eventually terminate to a type of degree 1, namely we will reduce our
problem to the linear case, which we have already established. To summarize
all of the above, we have the following:

######  Corollary 4.9.

Let $P_{N}$ be a nice polynomial family of degree $d$, with $k$ polynomials
and with type ${\bf W}$. Then, there exists a $p_{N}\in P_{N}$, such that the
family $P^{\prime}_{N}=(p_{N},h)^{*}P_{N}$ is nice and has (fixed) type
smaller than ${\bf W}$ for all, except at most $O_{d,k}(1)$ values of $h$.

######  Definition 4.10.

We will call a van der Corput operation $(p_{N},h)^{*}P_{N}$ non-degenerate,
if the polynomial $p_{N}\in P_{N}$ is such, that the conditions of Corollary
4.9 hold.

Namely, the polynomial $p_{N}$ must be chosen, so that the resulting family
has type independent of $N,h$, provided that $N$ is sufficiently large and $h$
takes values outside a set of at most $O_{d,k}(1)$ elements (here, this
notation refers to the same asymptotic constant appearing in the statement of
Corollary 4.9). In view of the above corollary, we deduce that there always
exists a non-degenerate van der Corput operation. We will denote a non-
degenerate van der Corput operation simply by $(p_{N})^{*}P_{N}$ to indicate
the independence on the parameter $h$.

We are now ready to finish the proof of Proposition 4.5:

###### Proof of the higher degree case.

First of all, we shall explain how we will choose the parameters $t,s$. These
depend crucially on how the van der Corput operations are used (and there are
possibly many ways in which the successive van der Corput can be carried out),
which may lead to ambiguity.

Let ${\bf W}=(d,w_{d},...,w_{1})$ be the type of the given polynomial family.
We say that a triplet $(d^{\prime},k^{\prime},{\bf W}^{\prime})$ can be
reached by the triplet $(d,k,{\bf W})$ if there exists a sequence of non-
degenerate van der Corput operations that produces the families

$P_{1,N}=(q_{1,N})^{*}P_{N},...,P_{\ell,N}=(q_{\ell,N})^{*}P_{\ell-1,N},$

where the family $P_{\ell,N}$ consists of $k^{\prime}$ polynomials, has degree
$d^{\prime}$ and type ${\bf W}^{\prime}$.

Observe that the triplets that can be reached by the original triplet
$(d,k,{\bf W})$ are finitely many in number, since there are only finitely
many choices (depending on $d,k,{\bf W}$) for each polynomial $q_{i,N}$ at
each step. In particular, they all have degrees at most $d$, types strictly
smaller than ${\bf W}$ and the number $k^{\prime}$ can be bounded by a
function of $(d,k,{\bf W})$, since each van der Corput operation at most
doubles the number of polynomials in a family and this operation can occur
finitely many times as well. We also remark that we have already established
our claim for all polynomial families of degree $d=1$ (this will work as the
base case of our induction).

Let $S_{d,k,{\bf W}}$ be the set of triplets that can be possibly reached by
$(d,k,{\bf W})$, which is a finite set. We will use induction by considering
that our claim holds for all triplets in $S_{d,k,{\bf W}}$ and we will show
that the claim holds for families corresponding to our original family $P_{N}$
that corresponds to the triplet $(d,k,{\bf W})$.

Fix such a triplet $(d^{\prime},k^{\prime},{\bf W}^{\prime})$ and define
$t(d^{\prime},k^{\prime},{\bf W}^{\prime}),s(d^{\prime},k^{\prime},{\bf
W}^{\prime})$ to be the numbers appearing in the statement of Proposition 4.5.
Namely, if the nice ordered family

$Q_{N}=\\{q_{1,N},...,q_{k^{\prime},N}\\}$

has degree $d^{\prime}$ and type ${\bf W}^{\prime}$, then

(18) $\sup_{|c_{n,N}|\leq 1}\big{\lVert}\underset{0\leq n\leq
L_{N}}{\mathbb{E}}\ c_{n,N}\prod_{i=1}^{k^{\prime}}T^{\lfloor
q_{i,N}(n)\rfloor}f_{i,N}\big{\rVert}_{L^{2}(\mu)}^{2^{t(d^{\prime},k^{\prime},{\bf
W}^{\prime})}}\ll_{d^{\prime},k^{\prime},{\bf W}^{\prime}}\\\
\frac{1}{M}+\sum_{{\bf h}\in Y^{[[s(d^{\prime},k^{\prime},{\bf
W}^{\prime})]]}}\underset{{\bf m}\in[-M,M]^{t(d^{\prime},k^{\prime},{\bf
W}^{\prime})}}{\mathbb{E}}\Big{|}\int\prod_{\underline{\varepsilon}\in[[s(d^{\prime},k^{\prime},{\bf
W}^{\prime})]]}T^{\lfloor A_{\underline{\varepsilon},N}({\bf
m})\rfloor+h_{\underline{\varepsilon}}}(\mathcal{C}^{|\underline{\varepsilon}|}f_{1,N})\
d\mu\Big{|}+o_{N}(1),$

where we are being vague on the dependence of the polynomials
$A_{\underline{\varepsilon},N}$ on the parameters $(d^{\prime},k^{\prime},{\bf
W}^{\prime})$ and the family $Q_{N}$ in this relation, since this will not
concern us temporarily.

The number $t(d^{\prime},k^{\prime},{\bf W}^{\prime})$ is the number of times
we apply the van der Corput inequality in order to bound the left-hand side by
the quantity on the right-hand side. Now, we define

$t_{0}=\max\limits_{(d^{\prime},k^{\prime},{\bf W}^{\prime})\in S_{d,k,{\bf
W}}}t(d^{\prime},k^{\prime},{\bf W}^{\prime}),$

which, of course, is a parameter that depends only on $(d,k,{\bf W})$. Assume
that the number $t_{0}$ corresponds to a family $Q_{N}$. Then, it is obvious
that $Q_{N}$ can be reached by the original family $P_{N}$ in only one step.
Indeed, if there was a another family in the sequence of van der Corput
operations starting from $P_{N}$ to $Q_{N}$, then this family would have a
strictly larger parameter $t(\cdot)$ associated to it than $t_{0}$.

Assume that the family $Q_{N}$ has the the triplet
$(d^{\prime},k^{\prime},{\bf W}^{\prime})$ associated to it. Although the
parameter $t_{0}$ is well defined (and depends only on $(d,k,{\bf W})$), the
parameter $s(d^{\prime},k^{\prime},{\bf W}^{\prime})$ may not be, because
there may be another family $Q^{\prime}_{N}$ which has the same value $t_{0}$
for the first parameter, but different for the second. In this case, we simply
take $Q_{N}$ to be the one for which the parameter
$s(d^{\prime},k^{\prime},{\bf W})$ is also maximized (denote this simply by
$s$ from this point onward). Obviously, we have that $s$ depends only on
$(d,k,{\bf W})$.

For the family $Q_{N}$ constructed above, we can write
$Q_{N}=(p_{N})^{*}P_{N}$ for some $p_{N}\in P_{N}$. Without loss of
generality, assume that $p_{N}=p_{k,N}$ (the case where $p_{N}=p_{1,N}$ is
similar).

We apply the van der Corput inequality to get

(19) $\big{\lVert}\underset{0\leq h\leq L_{N}}{\mathbb{E}}\
c_{n,N}\prod_{i=1}^{k}T^{\lfloor
p_{i,N}(n)\rfloor}f_{i,N}\big{\rVert}_{L^{2}(\mu)}^{2^{t_{0}+1}}\ll_{t_{0}}\\\
\frac{1}{M}+\underset{|m|\leq M}{\mathbb{E}}\Big{|}\underset{0\leq n\leq
L_{N}}{\mathbb{E}}\ c_{n+m,N}\overline{c_{n,N}}\int\prod_{i=1}^{k}\ T^{\lfloor
p_{i,N}(n+m)\rfloor}f_{i,N}\cdot T^{\lfloor
p_{i,N}(n)\rfloor}\overline{f_{i,N}}\ d\mu\Big{|}^{2^{t_{0}}}+o_{N}(1).$

We compose with $T^{-\lfloor p_{N}(n)\rfloor}$ in the above integral, so that

$\displaystyle\underset{0\leq n\leq L_{N}}{\mathbb{E}}\
c_{n+m,N}\overline{c_{n,N}}\int\prod_{i=1}^{k}\ T^{\lfloor
p_{i,N}(n+m)\rfloor}f_{i,N}\cdot T^{\lfloor
p_{i,N}(n)\rfloor}\overline{f_{i,N}}\ d\mu=$ $\displaystyle\underset{0\leq
n\leq L_{N}}{\mathbb{E}}\ c_{n+m,N}\overline{c_{n,N}}\int\prod_{i=1}^{k}\
T^{\lfloor p_{i,N}(n+m)\rfloor-\lfloor p_{N}(n)\rfloor}f_{i,N}\cdot T^{\lfloor
p_{i,N}(n)\rfloor-\lfloor p_{N}(n)\rfloor{}}\overline{f_{i,N}}\ d\mu=$
$\displaystyle\underset{0\leq n\leq
L_{N}}{\mathbb{E}}c_{n+m,N}\overline{c_{n,N}}\int\prod_{i=1}^{k}\ T^{\lfloor
p_{i,N}(n+m)-p_{N}(n)\rfloor+e_{1,n,i,m,N}}f_{i,N}\cdot T^{\lfloor
p_{i,N}(n)-p_{N}(n)\rfloor+e_{2,n,i,N}}\overline{f_{i,N}}\ d\mu,$

where the numbers $e_{1,n,i,m,N}\text{ and }e_{2,n,i,N}$ take values in the
set $\\{0,\pm 1\\}$. We use the Cauchy-Schwarz inequality and then use Lemma
3.2 to bound the absolute value of the last quantity by a constant (depending
only on $k$) multiple of the expression

$\sup_{|c_{n,N}|\leq 1}\big{\lVert}\underset{0\leq n\leq
L_{N}}{\mathbb{E}}c_{n,N}\big{(}\prod_{i=1}^{k-1}T^{\lfloor
p_{i,N}(n+m)-p_{N}(n)\rfloor}f^{\prime}_{i,N}\cdot T^{\lfloor
p_{i,N}(n)-p_{N}(n)\rfloor}\overline{f^{\prime}_{i,N}}\big{)}\\\ T^{\lfloor
p_{k,N}(n+m)-p_{k,N}(n)\rfloor}f^{\prime}_{k,N}\big{\rVert}_{L^{2}{(\mu)}}+o_{N}(1)$

for some 1-bounded functions
$f^{\prime}_{1,N}=f_{1,N},f^{\prime}_{2,N},...,f^{\prime}_{k,N}$. Recall that
we chose $p_{N}=p_{k,N}$. The family of polynomials

$P^{\prime}_{N,m}=\\{p_{1,N}(t+m)-p_{k,N}(t),...,p_{k,N}(t+m)-p_{k,N}(t),\
p_{1,N}(t)-p_{k,N}(t),...,p_{k-1,N}(t)-p_{k,N}(t)\\}$

is nice and has (fixed) type ${\bf W}^{\prime}<{\bf W}$ independent of $m$ for
all, except at most $O_{d,k}(1)$ values of $m\in\mathbb{N}$ (it has the same
triplet $(d^{\prime},k^{\prime},{\bf W}^{\prime})$ of parameters as the family
$Q_{N}$ above). Let $Q$ be this finite set of "bad" values of $m$ and let

$\mathcal{S}(P^{\prime}_{N,m})=\\{u^{\prime}_{1,m,N},...,u^{\prime}_{k^{\prime},m,N}\\}$

be the leading vector of $P^{\prime}_{N,m}$, where $k^{\prime}\leq 2k-1$. For
all $m\notin Q$, we use the induction hypothesis to deduce that

(20) $\sup_{|c_{n,N}|\leq 1}\big{\lVert}\underset{0\leq n\leq
L_{N}}{\mathbb{E}}c_{n,N}\big{(}\prod_{i=1}^{k-1}T^{\lfloor
p_{i,N}(n+m)-p_{N}(n)\rfloor}f^{\prime}_{i,N}\cdot T^{\lfloor
p_{i,N}(n)-p_{N}(n)\rfloor}\overline{f^{\prime}_{i,N}}\big{)}\cdot\\\ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ T^{\lfloor
p_{k,N}(t+m)-p_{k,N}(t)\rfloor}f^{\prime}_{k,N}\big{\rVert}_{L^{2}{(\mu)}}^{2^{t_{0}}}\ll_{k,d,{\bf
W}^{\prime}}\\\ \frac{1}{M}+\ \sum_{{\bf
h}\in[[Y_{0}]]}\underset{(m_{1},...,m_{t})\in[-M,M]^{t}}{\mathbb{E}}\Big{|}\int\prod_{\underline{\varepsilon}\in[[s]]}T^{\lfloor\sum_{1\leq
j\leq k^{\prime}}\
p_{\underline{\varepsilon},j}(m_{1},...,m_{t})u^{\prime}_{j,m,N}\rfloor+h_{\underline{\varepsilon}}}(\mathcal{C}^{|\underline{\varepsilon}|}f_{1,N})\
d\mu\Big{|}+o_{N}(1)$

for a finite set $Y_{0}$ that depends only on $d^{\prime},{\bf
W}^{\prime}\text{ and }k^{\prime}$ (i.e. $d,{\bf W}$ and $k$). We will now set
the parameter $t=t(d,k,{\bf W})$ to be simply $t_{0}+1$.

We also observe that our induction imposes that the polynomials

$A_{\underline{\varepsilon},N,m}(m_{1},...,m_{t})=\sum_{1\leq j\leq
k^{\prime}}\ p_{\underline{\varepsilon},j}(m_{1},...,m_{t})u^{\prime}_{j,m,N}$

are non-constant and pairwise essentially distinct for any (non-zero) values
of the leading vector
$\\{u^{\prime}_{1,m,N},...,u^{\prime}_{k^{\prime},m,N}\\}$ and that all the
polynomials $p_{\underline{\varepsilon},j}$ are at most linear in each
variable. In addition, we claim that

(21)
$A_{\underline{\varepsilon},N,m}(m_{1},...,m_{t})+A_{\underline{\varepsilon^{\prime}},N,m}(m_{1},...,m_{t})=A_{\underline{\varepsilon}+\underline{\varepsilon}^{\prime},N}(m_{1},...,m_{t})\
\text{whenever }\
\underline{\varepsilon}+\underline{\varepsilon}^{\prime}\in[[s]].$

(we have seen that all of the above are true in the linear case). These are
the properties i)-iii) in Proposition 4.5.

All the $u^{\prime}_{j,m,N}$ have the form described by Lemma 4.6. Therefore,
we can write

(22)
$p_{\underline{\varepsilon},j}(m_{1},...,m_{t})u^{\prime}_{j,m,N}=p^{\prime}_{1,\underline{\varepsilon},\ell}(m,m_{1},...,m_{t})u_{\ell,N}+p^{\prime}_{2,\underline{\varepsilon},\ell^{\prime}}(m_{1},...,m_{t})u_{\ell^{\prime},N}.$

In order to describe the form of the new polynomials
$p^{\prime}_{1,\underline{\varepsilon},\ell},p^{\prime}_{2,\underline{\varepsilon},\ell^{\prime}}$,
we split into cases depending on the form of $u^{\prime}_{j,m,N}$ (cf. Lemma
4.6):
a) If $u^{\prime}_{j,m,N}$ is equal to some $u_{\ell,N}$ for $1\leq\ell\leq
k$, then we have $p^{\prime}_{2,\underline{\varepsilon},\ell^{\prime}}=0$ and

$p^{\prime}_{1,\underline{\varepsilon},\ell}(m,m_{1},...,m_{t})=p_{\underline{\varepsilon},j}(m_{1},...,m_{t})$

(thus $p^{\prime}_{1,\underline{\varepsilon},\ell}(m_{1},...,m_{t})$ is
constant as a polynomial in $m$).
b) If $u^{\prime}_{j,m,N}$ is equal to $dmu_{1,N}$ ($\ell=1$), then we have
again $p^{\prime}_{2,\underline{\varepsilon},\ell^{\prime}}=0$ and

$p^{\prime}_{1,\underline{\varepsilon},1}(m,m_{1},...,m_{t})=dmp_{\underline{\varepsilon},j}(m_{1},...,m_{t}).$

c) In the final case that $u^{\prime}_{j,m,N}=dmu_{1,N}+u_{\ell^{\prime},N}$
for some $\ell^{\prime}\neq 1$, then we have
$p^{\prime}_{2,\underline{\varepsilon},\ell^{\prime}}=p_{\underline{\varepsilon},j}(m_{1},...,m_{t})$
and

$p^{\prime}_{1,\underline{\varepsilon},1}(m,m_{1},...,m_{t})=dmp_{\underline{\varepsilon},j}(m_{1},...,m_{t}).$

Therefore, the new polynomials $p_{1,\underline{\varepsilon},\ell}$ and
$p_{2,\underline{\varepsilon},\ell^{\prime}}$ are at most linear in each of
the variables $m_{1},...,m_{t}$, as well as the new variable $m$. By grouping
the terms corresponding to the same $u_{\ell,N}$, we can rewrite

$\sum_{1\leq r\leq k^{\prime}}\
p_{\underline{\varepsilon},r}(m_{1},...,m_{t})u^{\prime}_{r,m,N}=\sum_{1\leq
r\leq k}q_{\underline{\varepsilon},r}(m,m_{1},....,m_{t})u_{r,N}$

for some new polynomials $q_{\underline{\varepsilon},r}$.

######  Claim 1.

The new polynomials $\sum_{1\leq r\leq
k}q_{\underline{\varepsilon},r}(m,m_{1},....,m_{t})u_{r,N}$ satisfy conditions
i), ii), iii) and iv) of Proposition 4.5, for any (non-zero) values of the
$u_{r,N}$.

###### Proof of the Claim.

The fact that they are non-constant is trivial, since otherwise one of the
polynomials

$\sum_{1\leq r\leq k^{\prime}}\
p_{\underline{\varepsilon},r}(m_{1},...,m_{t})u^{\prime}_{r,m,N}$

would be constant, which is at odds with the induction hypothesis. Assume that
condition ii) fails for two
$\underline{\varepsilon_{1}},\underline{\varepsilon_{2}}\in[[s]]$. Regarding
these two polynomials as polynomials only in $(m_{1},...,m_{t})$, (22) would
give that the polynomials

$\sum_{1\leq r\leq
k^{\prime}}p_{\underline{\varepsilon_{1}},r}(m_{1},...,m_{t})u^{\prime}_{r,m,N}\
\text{ and }\ \sum_{1\leq r\leq
k^{\prime}}p_{\underline{\varepsilon_{2}},r}(m_{1},...,m_{t})u^{\prime}_{r,m,N},$

are not essentially distinct, which is false by the induction hypothesis.
Therefore, we have established both i) and ii).

Now, we want to prove an analogue of (21) for our new polynomials. But this
follows trivially by (22) (the new polynomials are just a rewritten form of
the $A_{\underline{\varepsilon},N}$). This establishes that the new
polynomials satisfy condition iii) in the statement of Proposition 4.5.

Finally, we are going to prove that the new polynomials
$q_{\underline{\varepsilon},j}$ satisfy condition iv) of Proposition 4.5. Fix
an $\underline{\varepsilon}\in[[s]]$. We will assume that all
$q_{\underline{\varepsilon},j}$ are non-zero and we will show that they are
linearly independent (if there are identically zero polynomials among the
$q_{\underline{\varepsilon},j}$, we proceed similarly by ignoring these
polynomials). It suffices to show that if $a_{1},...,a_{k}$ are real numbers,
such that

$a_{1}q_{\underline{\varepsilon},1}(m,m_{1},...,m_{t})+...+a_{k}q_{\underline{\varepsilon},k}(m,m_{1},...,m_{t})$

is the zero polynomial, then all the numbers $a_{i}$ are zero. Recalling the
form of the $q_{\underline{\varepsilon},r}$, this becomes a linear combination
of the form

(23) $a_{1}P_{1,\underline{\varepsilon}}(m,m_{1},...,m_{t})+\sum_{i\in
I_{1}}b_{i}p_{\underline{\varepsilon},i}(m_{1},...,m_{t})$

for some $I_{1}\subset\\{1,2,...,k^{\prime}\\}$ and
$b_{i}\in\\{a_{2},...,a_{k}\\}$101010Observe that each one of the numbers
$a_{2},..,a_{k}$ appears in the set $\\{b_{i},i\in I_{1}\\}$ (maybe with
multiplicity), because we have assumed that each polynomial
$q_{\underline{\varepsilon},\ell},(\ell>1)$ is not the trivial polynomial
(otherwise, we ignore it).. In addition, the polynomial
$P_{1,\underline{\varepsilon}}$ has the form

$dm\sum_{i\in I_{2}}p_{\underline{\varepsilon},i}+\sum_{i\in
I_{3}}p_{\underline{\varepsilon},i}$

for some $I_{2},I_{3}\subset\\{1,2,...,k^{\prime}\\}$ with $I_{1}\cap
I_{2}=\emptyset$ and $I_{1}\cap I_{3}=\emptyset$. We argue by contradiction.
For $m=0$, the polynomial in (23) must be identically zero and this easily
yields that all the $b_{i}$ must be zero and that $a_{1}\sum_{i\in
I_{3}}p_{\underline{\varepsilon},i}$ is also the zero polynomial. The first
relation implies that $a_{2}=...=a_{k}=0$ by the induction hypothesis, while
the second implies that either $a_{1}=0$ (in which case we are done), or
$I_{3}=\emptyset$ (since the $p_{\underline{\varepsilon},i}$ are linearly
independent by the induction hypothesis). If $I_{3}=\emptyset$, then (23)
implies that the polynomial

$a_{1}dm\sum_{i\in I_{2}}p_{\underline{\varepsilon},i}$

is the zero polynomial. This implies that $a_{1}=0$ or $I_{2}=\emptyset.$
However, we cannot have $I_{2}=I_{3}=\emptyset$, because that would imply that
the polynomial $q_{\underline{\varepsilon},1}$ is identically zero, which is
absurd (since we assumed that we have already discarded the zero polynomials
among the $q_{\underline{\varepsilon},i}$). Our claim follows. ∎

Combining all of the above we rewrite (20) as

$\sup_{|c_{n,N}|\leq 1}\big{\lVert}\underset{0\leq n\leq
L_{N}}{\mathbb{E}}c_{n,N}\big{(}\prod_{i=1}^{k-1}T^{\lfloor
p_{i,N}(n+m)-p_{N}(n)\rfloor}f_{i,N}\cdot T^{\lfloor
p_{i,N}(n)-p_{N}(n)\rfloor}\overline{f_{i,N}}\big{)}\cdot\\\ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ T^{\lfloor
p_{k,N}(t+m)-p_{k,N}(t)\rfloor}f_{k,N}\big{\rVert}_{L^{2}{(\mu)}}^{2^{t_{0}+1}}\ll_{d,k,{\bf
W}}\\\ \frac{1}{M}+\sum_{{\bf h}\in[[Y_{0}]]}\underset{|m_{1}|,...,|m_{t}|\leq
M}{\mathbb{E}}\
\Big{|}\int\prod_{\underline{\varepsilon}\in[[s]]}T^{\lfloor\sum_{1\leq r\leq
k}q_{\underline{\varepsilon},r}(m,m_{1},....,m_{t})u_{r,N}\rfloor+h_{\underline{\varepsilon}}}(\mathcal{C}^{|\underline{\varepsilon}|}f_{1,N})d\mu\Big{|}+o_{N}(1).$

We use the above bounds for all $-M\leq m\leq M$ in (19). The possible error
coming from the bad values of the set $Q$ can be absorbed by an
$O_{d,k}({1}/{M})$ term. Finally, we get

$\big{\lVert}\underset{0\leq h\leq
L_{N}}{\mathbb{E}}c_{n,N}\prod_{i=1}^{k}T^{\lfloor
p_{i,N}(n)\rfloor}f_{i,N}\big{\rVert}_{L^{2}(\mu)}^{2^{t}}\ll_{d,k,{\bf W}}\\\
\frac{1}{M}+\ \sum_{{\bf h}\in[[Y_{0}]]}\underset{|m|,|m_{1}|,...,|m_{t}|\leq
M}{\mathbb{E}}\Big{|}\int\prod_{\underline{\varepsilon}\in[[s]]}T^{\lfloor\sum_{1\leq
r\leq
k}q_{\underline{\varepsilon},r}(m,m_{1},....,m_{t})u_{r,N}\rfloor+h_{\underline{\varepsilon}}}(\mathcal{C}^{|\underline{\varepsilon}|}f_{1,N})d\mu\Big{|}+o_{N}(1),$

which is what we wanted to show. ∎

##  5\. The sub-linear plus polynomial case

In this section, we establish a particular case of Proposition 3.1, which we
shall also use in the general case in the next section. Let $\mathcal{S}$
denote the subset of $\mathcal{H}$ that contains the functions with sub-linear
growth rate and $\mathcal{P}\subseteq\mathcal{H}$ denotes the collection of
polynomials with real coefficients. Then, we let $\mathcal{S+P}$ denote the
collection of functions that can be written as a sum of a function in
$\mathcal{S}$ and a function in $\mathcal{P}$ (or equivalently, linear
combinations of functions in $\mathcal{S}$ and $\mathcal{P}$).

Let $a_{1},...,a_{k}$ be a collection of functions in $\mathcal{S+P}$. Then,
we can write $a_{i}=u_{i}+p_{i}$, where $u_{i}\in\mathcal{S}$ and $p_{i}$ is a
polynomial. We will also define the degree and type of the collection
$a_{1},a_{2},...,a_{k}$ using a similar notion to the degree and type of a
polynomial family defined in the previous section. More precisely, since we do
not impose that the polynomials $p_{1},...,p_{k}$ are essentially distinct, we
choose a maximal subset of the polynomials $p_{i}$ consisting of non-constant
and essentially distinct polynomials and we define the degree and type of the
collection $a_{1},...,a_{k}$ to be the degree and type of this new subfamily
of polynomials, respectively. Similarly, we define the leading vector of
$a_{1},...,a_{k}$ as the leading vector of the maximal subfamily that we
defined above. We can always choose this maximal subset to contain the
polynomial $p_{1}$. We define the cardinality of this new maximal subset to be
the size of the collection $a_{1},...,a_{k}$.

######  Proposition 5.1.

Let $M$ be a positive integer and let $a_{1},...,a_{k}$ be a collection of
functions in $\mathcal{S+P}$ with degree $d$, type ${\bf W}$ and size
$k^{\prime}\leq k$. Let $(c_{1},...,c_{k^{\prime}})$ be the leading vector of
the family $\\{a_{1},...,a_{k}\\}$. In addition, assume that
$a_{1}(t)\succ\log t$ and $a_{1}(t)-a_{j}(t)\succ\log t$ for $j\neq 1$. Then,
there exist positive integer $s,t$, a finite set $Y$ of integers and real
polynomials $p_{\underline{\varepsilon},j}$ in $t$ variables, where
$\underline{\varepsilon}\in[[s]]$ and $1\leq j\leq k$, all depending only on
$d,k^{\prime},{\bf W}$, such that, for any measure preserving system
$(X,\mu,T)$ and function $f_{1}\in L^{\infty}(\mu)$ bounded by 1, we have

(24) $\sup_{\lVert f_{2}\rVert_{\infty},...,\lVert f_{k}\rVert_{\infty}\leq
1}\sup_{|c_{n}|\leq 1}\big{\lVert}\underset{1\leq n\leq N}{\mathbb{E}}c_{n}\
T^{\lfloor a_{1}(n)\rfloor}f_{1}\cdot...\cdot T^{\lfloor
a_{k}(n)\rfloor}f_{k}\big{\rVert}_{L^{2}(\mu)}^{2^{t}}\ll_{d,k,k^{\prime},{\bf
W}}\\\ \frac{1}{M}+\sum_{{\bf h}\in
Y^{[[s]]}}\underset{m\in[-M,M]^{t}}{\mathbb{E}}\lvert\\!|\\!|\prod_{\underline{\varepsilon}\in[[s]]}T^{\lfloor
A_{\underline{\varepsilon}}({\bf
m})\rfloor+h_{\underline{\varepsilon}}}f_{1}|\\!|\\!\rvert_{2k+1}+o_{N}(1)$

where

$A_{\underline{\varepsilon}}({\bf
m})=\sum_{j=1}^{k^{\prime}}p_{\underline{\varepsilon},j}({\bf m})c_{j}.$

are pairwise essentially distinct polynomials.

Observe that the iterates inside the seminorm in (24) are real polynomials in
several variables. We can take $M\to+\infty$ and expand these seminorms to
arrive at an iterated limit of polynomial averages. It is possible to bound
these averages by a suitable seminorm of the function $f_{1}$ using the
results in [27] and get a simpler bound in (24). This necessitates that we
substitute the $O_{d,k,k^{\prime},{\bf W}}(1)$ implicit constant by an
$O_{a_{1},...,a_{k}}(1)$ constant and this is insufficient for our purposes in
the next section, where we will have to apply Proposition 5.1 for several
collections of functions simultaneously. However, in view of the above
discussion, we deduce the following:

######  Corollary 5.2.

Let $a_{1},...,a_{k}$ be a collection of functions in
$\mathcal{S}+\mathcal{P}$ such that $a_{1}(t)\succ\log t$ and
$a_{1}(t)-a_{j}(t)\succ\log t$ for $j\neq 1$. Then, there exists a positive
integer $s$ such that, for any measure preserving system $(X,\mu,T)$ and
1-bounded function $f_{1}\perp Z_{s}(X)$, we have

$\lim\limits_{N\to+\infty}\ \sup_{\lVert f_{2}\rVert_{\infty},...,\lVert
f_{k}\rVert_{\infty}\leq 1}\ \sup_{|c_{n}|\leq 1}\big{\lVert}\underset{1\leq
n\leq N}{\mathbb{E}}c_{n}\ T^{\lfloor a_{1}(n)\rfloor}f_{1}\cdot...\cdot
T^{\lfloor a_{k}(n)\rfloor}f_{k}\big{\rVert}_{L^{2}(\mu)}=0.$

We analyze the conditions imposed on the functions $a_{1},...,a_{k}$ more
closely: write each function $a_{i}$ in the form $a_{i}(t)=u_{i}(t)+p_{i}(t)$,
where $u_{i}\in\mathcal{S}$ and $p_{i}\in\mathcal{P}$. The condition
$a_{1}(t)\succ\log t$ implies that either $u_{1}(t)\succ\log t$ or $p_{1}(t)$
is a non-constant polynomial. Similarly, the second condition implies that
either $u_{1}(t)-u_{i}(t)\succ\log t$ or $p_{1}(t)-p_{j}(t)$ is a non-constant
polynomial.

Furthermore, we can make one more reduction. Writing again
$a_{i}(t)=u_{i}(t)+p_{i}(t)$ as above and using the same argument as in
Section 3 (see the discussion following the statement of Proposition 3.1), we
may assume that the function $u_{1}$ has the largest growth rate among the
functions $u_{i}$.

In order to establish the main result of this section, we will also use the
following proposition, which is special case of Proposition 5.1.

######  Proposition 5.3.

Let $a_{1},...,a_{k}$ be sub-linear functions in $\mathcal{H}$ and assume that
all the functions $a_{1},a_{1}-a_{2},...,a_{1}-a_{k}$ dominate $\log t$. Then,
for any measure preserving system $(X,\mu,T)$ and function $f_{1}\in
L^{\infty}(\mu)$ bounded by 1, we have

(25) $\limsup\limits_{N\to+\infty}\sup_{\lVert f_{2}\rVert_{\infty},...,\lVert
f_{k}\rVert_{\infty}\leq 1}\sup_{|c_{n}|\leq 1}\ \big{\lVert}\underset{1\leq
n\leq N}{\mathbb{E}}c_{n}\ T^{\lfloor a_{1}(n)\rfloor}f_{1}\cdot...\cdot
T^{\lfloor a_{k}(n)\rfloor}f_{k}\big{\rVert}_{L^{2}(\mu)}\ll_{k}\
\lvert\\!|\\!|f_{1}|\\!|\\!\rvert_{2k}.$

######  Remark.

The proof that Proposition 5.3 implies Proposition 5.1 corresponds to Step 1
in example b) of Section 3, while the proof of Proposition 5.3 corresponds to
step 2 of the same example.

###### Proof that Proposition 5.3 implies Proposition 5.1.

First of all, we write each $a_{i}(t)$ in the form $u_{i}(t)+p_{i}(t)$ as we
discussed above. Our main tool will be to use Lemma 3.3 in order to reduce our
problem to studying averages on small intervals, where the sublinear functions
$u_{i}$ will have a constant integer part.

Suppose that not all of the polynomials $p_{1}(t),...,p_{k}(t)$ are constant,
since the other case follows from Proposition 5.3 (that means the family has
degree $\geq 1$). We can assume, without loss of generality, that $p_{i}(0)=0$
for all $i$ (the constant terms can be absorbed by the functions $u_{i}$).
Therefore, let $L(t)\in\mathcal{H}$ be a sub-linear function to be chosen
later. In addition, we choose functions $f_{2,N},...,f_{k,N}$ so that the
average in the left-hand side of (24) is $1/N$ close to the supremum. We want
to bound

$\underset{1\leq r\leq R}{\mathbb{E}}\ \sup_{|c_{n,r}|\leq 1}\
\big{\lVert}\underset{r\leq n\leq r+L(r)}{\mathbb{E}}\ c_{n,r}\ T^{\lfloor
u_{1}(n)+p_{1}(n)\rfloor}f_{1}\cdot...\cdot T^{\lfloor
u_{k}(n)+p_{k}(n)\rfloor}f_{k,R}\big{\rVert}_{L^{2}(\mu)}^{2^{t}}$

for some integer parameter $t$, which we will choose later to depend only on
the quantities $d,k^{\prime},{\bf W}$ (thus, when applying Lemma 3.2 below to
remove the error terms in the iterates, we will always have that the implicit
constant depends only on $d,k^{\prime},{\bf W}$).

Recall that we have reduced our problem to the case that the function $u_{1}$
has the largest growth rate among the functions $u_{i}$. Now, we want to
choose the sub-linear function $L(t)\in\mathcal{H}$ so that the functions
$u_{i}(n)$ restricted to the interval $[r,r+L(r)]$ become very close to the
value $u_{i}(r)$. To achieve this, it suffices to take $L(t)\in\mathcal{H}$
such that

$1\prec L(t)\prec(u_{1}^{\prime}(t))^{-1}.$

To see that such a function exists, we only need to show that
$(u_{1}^{\prime}(t))^{-1}\succ 1$ which follows easily from the fact that
$u_{1}(t)\prec t$. Observe that for every $i\in\\{1,2,...,k\\}$ we must have
$L(t)\prec(u_{i}^{\prime}(t))^{-1}$, since $u_{1}$ has maximal growth among
the functions $u_{i}$. For every $n\in[r,r+L(r)]$, we observe that

$|u_{i}(n)-u_{i}(r)|\leq(n-r)\max_{x\in[r,r+L(r)]}|u_{i}^{\prime}(x)|.$

Since $|u_{i}^{\prime}(t)|\searrow 0$, we have that for $r$ large enough

$|u_{i}(n)-u_{i}(r)|\leq L(r)u_{i}^{\prime}(r)=o_{r}(1),\ \ \ \ \ \
n\in[r,r+L(r)].$

Therefore, for $r$ sufficiently large we have

$\lfloor u_{i}(n)+p_{i}(n)\rfloor=\lfloor u_{i}(r)\rfloor+\lfloor
p_{i}(n)\rfloor+e_{i,n},\ \ \ \ \ \ \ n\in[r,r+L(r)],$

where $e_{i,n}\in\\{0,\pm 1,\pm 2\\}$. Therefore, our original problem reduces