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# Pointwise convergence in nilmanifolds along smooth functions of polynomial
growth

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

We study the equidistribution of orbits of the form $b_{1}^{a_{1}(n)}\cdots
b_{k}^{a_{k}(n)}\Gamma$ in a nilmanifold $X$, where the sequences $a_{i}(n)$
arise from smooth functions of polynomial growth belonging to a Hardy field.
We show that under certain assumptions on the growth rates of the functions
$a_{1},...,a_{k}$, these orbits are uniformly distributed on some
subnilmanifold of the space $X$. As an application of these results and in
combination with the Host-Kra structure theorem for measure preserving
systems, as well as some recent seminorm estimates of the author for ergodic
averages concerning Hardy field functions, we deduce a norm convergence result
for multiple ergodic averages. Our method mainly relies on an equidistribution
result of Green-Tao on finite polynomial orbits of a nilmanifold.

###### Key words and phrases:

Ergodic averages, Equidistribution, Nilmanifolds, Hardy fields

###### 2020 Mathematics Subject Classification:

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

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

##  1\. Introduction and main results

###  1.1. Motivation

In recent years, there has been an active interest in determining the limiting
behavior of the multiple ergodic averages

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

for various sequences $a_{1}(n),...,a_{k}(n)$ of integers, where $T$ is an
invertible measure preserving transformation acting on a probability space
$(X,\mathcal{X},\mu)$ and $f_{1},..,f_{k}$ are functions in $L^{\infty}(\mu)$.
Through the breakthrough work of Furstenberg [6], which provided a new proof
of Szemerédi’s theorem using tools from ergodic theory, it has been understood
that the study of the averages in (1) has important applications in number
theory and combinatorics. In particular, we now have significant
generalizations of Szemerédi’s theorem, some of which have not been proven
with tools other than the study of ergodic averages.

An important tool in establishing convergence of the averages in (1) is the
structure theorem of Host-Kra [10], which in many cases reduces the above
problem to studying rotations on certain spaces with algebraic structure,
which are called nilmanifolds (we refer to the reader to [11] for a full
presentation of the theory). The study of nilmanifolds is important due to
aforementioned ties to ergodic theory, as well as the numerous applications in
combinatorics and number theory.

In this article, our main problem is to study the distribution of orbits in a
nilmanifold along sequences that arise from smooth functions and have
polynomial growth 111A sequence $a(n)$ has polynomial growth, if there exists
a positive integer $k$ such that
$\lim\limits_{n\to+\infty}\frac{a(n)}{n^{k}}=0.$ . We assume that our
functions are elements of a Hardy field (for the definition of a Hardy field,
we refer the reader to Section 2). In particular, we extend the uniform
distribution results in [3] and some of the results in [16], while also making
progress towards a conjecture of Frantzikinakis [3].

###  1.2. Main results

In order to state our results, we will assume that we have fixed a Hardy field
$\mathcal{H}$ that contains the Hardy field $\mathcal{LE}$ of logarithmico-
exponential functions. For the definition and properties of nilmanifolds and
Hardy fields, see Section 2.

The base case when our nilmanifold is a finite dimensional torus is known and
follows from equidistribution results of Boshernitzan. In particular, we
restate here Theorem 1.3 from [2]:

######  Theorem A.

Let the function $a\in\mathcal{H}$ have polynomial growth. Then, the sequence
$a(n)$ is uniformly distributed $\mod 1$ if and only if

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

Using Weyl’s equidistribution theorem and the previous result, we can easily
show that if the functions $a_{1},...,a_{k}$ have polynomial growth and each
non-trivial linear combination of them stays logarithmically away from real
multiples of integer polynomials, then the sequence
$(c_{1}a_{1}(n),...,c_{k}a_{k}(n))$ is uniformly distributed on
$\mathbb{T}^{k}$ for all non-zero real numbers $c_{1},...,c_{k}$. Our main
result is that a slightly weaker condition than the above is sufficient for
equidistribution in all nilsystems. This is the content of the next result. We
will use the notation $\mathcal{L}(a_{1},...,a_{k})$ to denote the collection
of all non-trivial linear combinations of the functions
$a_{1}(t),...,a_{k}(t)$ belonging to $\mathcal{H}$. The nilmanifolds
$\overline{(b^{\mathbb{R}}x)}$ are defined in Section 2.

######  Theorem 1.1.

Let $a_{1},...,a_{k}\in\mathcal{H}$ have polynomial growth and assume that
every function in $a\in\mathcal{L}(a_{1},...,a_{k})$ satisfies

(2) $\lim\limits_{t\to+\infty}\frac{|a(t)-p(t)|}{t^{\varepsilon}}=+\infty\
\text{ for any polynomial }\ p(t)\in\mathbb{Z}[t]$

for some $\varepsilon>0$. Then, we have the following:
(i) For any collection of nilmanifolds $X_{i}=G_{i}/\Gamma_{i}$ and elements
$b_{i}\in G_{i}$ and $x_{i}\in X_{i}$, the sequence

$\big{(}b_{1}^{\lfloor a_{1}(n)\rfloor}x_{1},...,b_{k}^{\lfloor
a_{k}(n)\rfloor}x_{k}\big{)}$

is equidistributed on the nilmanifold
$\overline{(b_{1}^{\mathbb{N}}x_{1})}\times\dots\times\overline{(b_{k}^{\mathbb{N}}x_{k})}$.
(ii) For any collection of nilmanifolds $X_{i}=G_{i}/\Gamma_{i}$ such that the
groups $G_{i}$ are connected, simply connected and elements $b_{i}\in G_{i}$
and $x_{i}\in X_{i}$, the sequence

$\big{(}b_{1}^{a_{1}(n)}x_{1},...,b_{k}^{a_{k}(n)}x_{k}\big{)}$

is equidistributed on the nilmanifold
$\overline{(b_{1}^{\mathbb{R}}x_{1})}\times\dots\times\overline{(b_{k}^{\mathbb{R}}x_{k})}$.

######  Remarks.

i) The connectedness assumptions imposed on the second part of the previous
theorem ensure that all elements of the form $b^{s}$ where $b\in G$ and
$s\in\mathbb{R}$ are well defined (see also Section 2 for the definition of
the element $b^{s}$).
ii) In regards to part (b) of the above theorem, we show the slightly more
general statement that the sequence $b_{1}^{a_{1}(n)}\cdots
b_{k}^{a_{k}(n)}\Gamma$ is equidistributed on the nilmanifold
$\overline{b_{1}^{\mathbb{R}}\cdots b_{k}^{\mathbb{R}}\Gamma}$.

Observe that, in contrast to Theorem A, we have the term $t^{\varepsilon}$ in
the denominator, which is just out of reach of the conjectured optimal term
$\log t$. As an example, using Theorem 1.1, we can prove that (we use the same
notation as above) for any elements $b_{1},b_{2}\in G$, the sequence
$(b_{1}^{n\log n},b_{2}^{n^{3/2}})$ is equidistributed on the nilmanifold
$(\overline{b_{1}^{\mathbb{R}}\Gamma},\overline{b_{2}^{\mathbb{R}}\Gamma})$.

We can relax the above linear independence condition and prove the following:

######  Theorem 1.2.

Let $a_{1},...,a_{k}\in\mathcal{H}$ have polynomial growth and assume that
every function in $a\in\mathcal{L}(a_{1},...,a_{k})$ satisfies either

(3) $\lim\limits_{t\to+\infty}\frac{|a(t)-p(t)|}{t^{\varepsilon}}=+\infty\
\text{ for any polynomial }\ p(t)\in\mathbb{Z}[t]$

for some $\varepsilon>0$, or

(4) $\text{the limit }\ \lim\limits_{t\to+\infty}a(t)\ \text{is a real
number}.$

Then, we have the following:
(i) For any collection of nilmanifolds $X_{i}=G_{i}/\Gamma_{i}$, elements
$b_{i}\in G_{i}$, $x_{i}\in X_{i}$ and continuous functions $f_{1},...,f_{k}$
with complex values, the averages

$\frac{1}{N}\sum_{i=1}^{N}f_{1}(b_{1}^{\lfloor a_{1}(n)\rfloor}x_{1})\dots
f_{k}(b_{k}^{\lfloor a_{k}(n)\rfloor}x_{k})$

converge.
(ii) For any collection of nilmanifolds $X_{i}=G_{i}/\Gamma_{i}$ such that the
groups $G_{i}$ are connected, simply connected and elements $b_{i}\in G_{i}$,
$x_{i}\in X_{i}$ and continuous functions $f_{1},...,f_{k}$ with complex
values, the averages

$\frac{1}{N}\sum_{i=1}^{N}f_{1}(b_{1}^{a_{1}(n)}x_{1})\dots
f_{k}(b_{k}^{a_{k}(n)}x_{k})$

converge.

The main difference between Theorem 1.1 and 1.2 is that in the second case we
allow for linear dependencies between the functions $a_{1}(t),...,a_{k}(t)$
(for example, we may have the functions $(t\log t,t^{3/2},t^{3/2}+t\log t)$).

Theorems 1.1 and 1.2 extend the equidistribution results in [3], where the
functions $a_{1},...,a_{k}$ were assumed to have distinct growth rates and
satisfy the growth conditions $t^{k_{i}}\log t\prec a_{i}(t)\prec
t^{k_{i}+1}$, where $k_{i}$ are positive integers. In [16], a similar theorem
was proven under a linear independence condition on the functions
$a_{1},...,a_{k}$ and all of their derivatives. As an example, the previous
results in the literature do not cover the pair of functions $(t\log
t,t^{2}\sqrt{\log t})$. More generally, our theorem concerns all families of
functions of the form $\sum_{i=1}^{k}c_{i}t^{a_{i}}(\log t)^{b_{i}}$, where
$a_{i}>0$ and $b_{i},c_{i}\in\mathbb{R}$, assuming, of course, that the linear
combinations of the involved functions satisfy either (3) or (4). However,
there are cases covered in the results of [16] that do not follow from the
methods of this article. These examples concern functions that grow slower
than fractional powers $t^{c}$, such as $\log^{A}t$ for $A>0$ or
$\exp(\sqrt{\log t})$. An example that is not covered by Theorem 1.2 is the
pair of functions $(\log^{2}t,t^{3/2})$ (although this one follows from the
results in [16]).

Using the above theorem and the results in [17] on characteristic factors, we
get a mean convergence result for multiple ergodic averages. Since the
seminorm estimates for such averages were proven in [17] under the assumption
that the Hardy field $\mathcal{H}$ was closed under composition and
compositional inversion of functions, we assume in the following theorem that
$\mathcal{H}$ has this property. An example of a Hardy field that contains the
field $\mathcal{LE}$ and has this property can be found in [17, Section 2]

######  Theorem 1.3.

Let $\mathcal{H}$ be a Hardy field as above and assume that the functions
$a_{1},...,a_{k}\in\mathcal{H}$ are as in Theorem 1.2. Then, for any measure
preserving system $(X,\mu,T)$ and any functions $f_{1},...,f_{k}\in
L^{\infty}(\mu)$, we have that the averages

(5) $\frac{1}{N}\sum_{n=1}^{N}T^{\lfloor a_{1}(n)\rfloor}f_{1}\dots T^{\lfloor
a_{k}(n)\rfloor}f_{k}$

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

It follows from the results in [17] that, if the functions $a_{1},...,a_{k}$
are as in Theorem 1.1 (actually, this condition can be weakened), then for any
ergodic measure preserving system $(X,\mu,T)$ and bounded functions
$f_{1},...,f_{k}$, the averages

$\frac{1}{N}\sum_{i=1}^{N}f_{1}(T^{\lfloor a_{1}(n)\rfloor}x)\dots
f_{k}(T^{\lfloor a_{k}(n)\rfloor}x)$

converge in the $L^{2}$-sense to the product of the integrals $\int
f_{1}d\mu\cdots\int f_{k}d\mu$. The methods used in that article cannot work
when there are linear dependencies between the functions $a_{1},...,a_{k}$
(since they rely on the joint ergodicity results from [5]). Therefore, in
order to prove Theorem 1.3, we have to tackle the problem of convergence in
nilmanifolds. Besides, we can prove the stronger result of pointwise
convergence when the system $(X,\mu,T)$ is a nilsystem. We remark here that
the optimal restrictions on the functions $a_{1},...,a_{k}$ in Theorem 1.3 are
conjectured to be that the functions are good for convergence when the system
$(X,\mu,T)$ is a rotation on some torus $\mathbb{T}^{d}$. More precisely, we
have the following:

######  Conjecture 1.

Let $a_{1},...,a_{k}$ be functions in $\mathcal{H}$ such that for all real
numbers $t_{1},...,t_{k}\in[0,1)$, the averages

$\frac{1}{N}\sum_{n=1}^{N}e(t_{1}\lfloor a_{1}(n)\rfloor+\dots+t_{k}\lfloor
a_{k}(n)\rfloor)$

converge. Then, for any measure preserving system $(X,\mu,T)$ and functions
$f_{1},...,f_{k}\in L^{\infty}(\mu)$, the averages

(6) $\frac{1}{N}\sum_{n=1}^{N}T^{\lfloor a_{1}(n)\rfloor}f_{1}\dots T^{\lfloor
a_{k}(n)\rfloor}f_{k}$

converge in $L^{2}(\mu)$ and, if $(X,\mu,T)$ is a nilsystem, then those
averages converge pointwise.

######  Remark.

It can be shown that the above condition on the exponentials of the involved
sequences is not sufficient, if we substitute the Hardy sequences by other
more general sequences.

###  1.3. Short overview of the proof and additional remarks

The main idea of the proof is that functions in $\mathcal{H}$ of polynomial
growth can be approximated sufficiently well by polynomials in small
intervals. The equidistribution properties of polynomial sequences in
nilmanifolds, even on small intervals, are well understood from [7]. We use
these quantitative results of Green-Tao (see Theorem F in Section 2) to show
that the averages over small intervals are equal to the integral of a
continuous function in our nilmanifold. The above approach was used in [3] to
show that (following the notation of Theorem 1.2) the sequence $b^{a(n)}$ is
uniformly distributed for all $b\in G$ and any function $a(n)$ satisfying (P).
In the case that we need to cover here, there are a few more technical
difficulties in the proof, since we have to find polynomial expansions for
several functions simultaneously, which also tend to be of increased
complexity. Our argument differs quite a bit from the methods used in [16],
which relied on applications of the van der Corput inequality. In simplistic
terms, this replaces the problem of studying equidistribution for a function
$a(t)\in\mathcal{H}$ by the problem of studying equidistribution properties
for the derivatives $a^{\prime},a^{\prime\prime},a^{\prime\prime\prime}$ and
so on. This cannot be used to cover, for example, functions in the range
$t\prec a(t)\prec t\log t$, because we can easily see that the derivative
$a^{\prime}$ must satisfy $a^{\prime}(t)\prec\log t$, which does not have good
equidistribution properties even on the 1-dimensional torus $\mathbb{T}$.
While we can sidestep this problem, our argument also has limitations in that
we do not cover functions that grow very slowly, such as $\log^{c}t$ for
$c>1$. The main reason is that when we consider averages on small intervals,
these functions become approximately equal to a constant and our method of
using the Taylor expansion for approximation breaks down. That explains the
existence of the function $t^{\varepsilon}$ in (3) instead of the term $\log
t$, which would be optimal.

In addition, we do not cover the case where the functions $a_{i}(t)$ are real
polynomials, because the reduction to a statement on connected simply
connected Lie groups becomes a lot more complicated. For example, consider a
nilmanifold $X=G/\Gamma$ with $G$ connected and simply connected and elements
$b_{1},b_{2}\in G$ that commute. It may be possible to describe explicitly the
orbit of the sequence $(b_{1}^{n^{3/2}}\Gamma,b_{2}^{n^{2}}\Gamma)$ on
$X\times X$. On the one hand, invoking Leibman’s theorem on polynomial orbits
(see Section 2), we deduce that the sequence $b_{2}^{n^{2}}\Gamma$ visits
cyclically subnilmanifolds of the form $Hx_{1},...,Hx_{r}$, where $H$ is a
closed (and, thus, simply connected) connected rational subgroup of $G$ and
$x_{1},...,x_{r}\in X$. We can also reduce to the case that $H$ contains the
elements of the polynomial sequence $b_{2}^{n^{2}}$, using a factorization
theorem implicit in [7]. On the other hand, the nilmanifold
$\overline{b_{1}^{\mathbb{R}}\Gamma}$ has a representation as
$H^{\prime}/(H^{\prime}\cap\Gamma)$, where $H^{\prime}$ is again a connected,
simply connected Lie subgroup of $G$ that contains all elements of the form
$b_{1}^{s}$. Thus, the sequence $(b_{1}^{n^{3/2}},b_{2}^{n^{2}})$ takes values
on the product Lie group $H^{\prime}\times H$ (after passing to appropriate
arithmetic progressions) and we can apply Theorem F as we do in the main proof
without many additional complications. However, assume we now want to study
the orbit of the sequence $b_{1}^{n^{3/2}}b_{2}^{n^{2}}\Gamma$. We expect it
to be a union of subnilmanifolds of $X$ of the form $G^{\prime}x_{i}$ for some
rational subgroup $G^{\prime}$ of $G$, but it is not clear what the group
$G^{\prime}$ should be in this case.

### Notational conventions

Throughout this article, we denote by $\mathbb{N}=\\{1,2,...\\}$ the set of
natural numbers and $\mathbb{Z}^{+}=\mathbb{N}\cup\\{0\\}$. We denote
$\mathbb{T}^{d}=\mathbb{R}^{d}/\mathbb{Z}^{d}$, $e(t)=e(2\pi it)$, while
$\lVert x\rVert_{\mathbb{T}}=d(x,\mathbb{Z})$ and $\\{x\\}$ denote the
distance of $x$ from the nearest integer and the fractional part of $x$
respectively. For an element ${\bf x}=(x_{1},...,x_{k})$ on $\mathbb{R}^{k}$,
we denote $|{\bf x}|=\max\\{|x_{1}|,...,|x_{k}|\\}$.

For two sequences $a_{n},b_{n}$, we say that $b_{n}$ dominates $a_{n}$ and
write $a_{n}\prec b_{n}$ or $a_{n}=o(b_{n})$, when $a_{n}/b_{n}$ goes to 0, as
$n\to+\infty$. In addition, we write $a_{n}\ll b_{n}$ or $a_{n}=O(b_{n})$, if
there exists a positive constant $C$ such that $|a_{n}|\leq C|b_{n}|$ for
large enough $n$. When we want to denote dependence of this constant on some
parameters $h_{1},...,h_{k}$, we will use the notation
$a_{n}=O_{h_{1},...,h_{k}}(b_{n})$. We use similar notation for asymptotic
relations between functions on some real variable $t$, where we understand
that these hold when we take $t\to+\infty$.

Finally, we also use the symbol $\mathbb{E}$ to denote averages (over some
range that will be implicit by the corresponding subscripts each time).

###  1.4. Acknowledgement

I would like to thank my advisor Nikos Frantzikinakis for helpful discussions.

##  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
and 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 $f\in\mathcal{H}$. An example of a Hardy field is the field
$\mathcal{LE}$ of logarithmico-exponential functions, which contains the
functions defined by a finite combination of the operations $+,-,\cdot,\div,\
\exp$, $\log$ and composition of functions acting on a real variable $t$
(which takes values on some half-line $[x,+\infty)$) and real constants.

Any two functions $f,g\in\mathcal{H}$ with $g$ not identicaly zero are
comparable, that is the limit

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

exists and thus it makes sense to compare their growth rates. In addition,
every non-constant function in $\mathcal{H}$ is eventually monotone (and,
therefore, has constant sign eventually). In the next section, we have
gathered some lemmas regarding growth rates of functions in $\mathcal{H}$,
which will be used repeatedly throughout the proofs. The proofs of these
lemmas can be found in [17], so we shall omit them.

A function $f\in\mathcal{H}$ is sub-linear if $f(t)\prec t$, while we call it
sub-fractional when $f(t)\prec t^{\varepsilon}$, for all $\varepsilon>0$. In
addition, we call $f$ strongly non-polynomial if there exists a non-negative
integer $k$, such that $t^{k}\prec f(t)\prec t^{k+1}$. We also include the
functions $f\in\mathcal{H}$ that satisfy $\lim\limits_{t\to+\infty}f(t)=0$ in
the strongly non-polynomial functions in order to avoid extra casework in the
proofs of the main theorems.

###  2.2. Measure preserving systems and Host-Kra structure theorem

A measure preserving system is a quadruple $(X,\mathcal{X},\mu,T)$, where
$(X,\mathcal{X},\mu)$ is a Lebesgue probability space and $T$ is an invertible
measure preserving map, that is $\mu(T^{-1}(A))=\mu(A)$ for all
$A\in\mathcal{X}$. It is called ergodic, if all the $T$-invariant functions
are constant. For the purposes of this article, a factor of the system
$(X,\mathcal{X},\mu,T)$ is a $T$-invariant sub-$\sigma$-algebra of
$\mathcal{X}$. However, when there is no confusion, we will omit the
$\sigma$-algebra $\mathcal{X}$ from the quadruple $(X,\mathcal{X},\mu,T)$.

Let $(X,\mu,T)$ be a 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 $s\in\mathbb{Z}^{+}$,

(7)
$\lvert\\!|\\!|f|\\!|\\!\rvert_{s+1,T}^{2^{s+1}}=\lim\limits_{H\to\infty}\underset{0\leq
h\leq H}{\mathbb{E}}\lvert\\!|\\!|\bar{f}\cdot
T^{h}f|\\!|\\!\rvert_{s,T}^{2^{s}}.$

In the ergodic case, the existence of these limits and the fact that these
quantities are indeed seminorms was established in [10]. In the same article,
it was shown that the above seminorms define a factor $\mathcal{Z}_{s-1}(X)$
of $X$ for every $s\geq 1$, which is characterized by the following relation:

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

The importance of these factors hinges on the following remarkable structure
theorem of Host-Kra [10]:

######  Theorem D (Host-Kra).

Let $(X,\mu,T)$ be an ergodic system. Then, the factor $\mathcal{Z}_{s}(X)$ is
an inverse limit of $s$-step nilsystems.

The last property means that there exist $T$-invariant sub-$\sigma$-algebras
$\mathcal{Z}_{s}(n),n\in\mathbb{N}$ that span $\mathcal{Z}_{s}$, such that the
factor $\mathcal{Z}_{s}(n)$ is isomorphic as a system to an $s$-step
nilsystem.

###  2.3. Nilmanifolds and equidistribution

####  2.3.1. Basics of nilsystems and polynomial sequences

Let $G$ be a $k$-step nilpotent Lie group and let $\Gamma$ be a co-compact
discrete subgroup222A subgroup $H$ of a topological group $G$ is called
discrete, if the quotient topology on the space $G/H$ is the discrete
topology. It is called co-compact, if the quotient topology makes $G/H$ a
compact space. We call a subgroup with both of the above properties uniform.
of $G$. Then, the space $X=G/\Gamma$ is called a $k$-step nilmanifold. The
space $X$ may have several representations of the form $G/\Gamma$ (with
possible variance in the degree of nilpotency). Let $G^{\circ}$ be the
connected component of $e_{G}$ in $G$. If we assume that $G/G^{\circ}$ is
finitely generated333Without loss of generality we can assume that, because
our results deal with the action of $G$ on finitely many elements of $X$.,
then, by passing to the universal cover of $G$, it can be shown that $X$ has a
representation $G/\Gamma$ where $G$ is simply connected. In addition, we can
argue as in [12, Section 1.11] to deduce that $X$ can be embedded as a
subnilmanifold in some nilmanifold $G^{\prime}/\Gamma^{\prime}$, where
$G^{\prime}$ is a connected and simply connected nilpotent Lie group and every
translation on $X$ has a representation in $G^{\prime}/\Gamma^{\prime}$.

Let $b$ be any element on $G$. Then, $b$ acts on $G$ by left multiplication.
Let $m_{X}$ be the image of the Haar measure of $G$ on $X$ under the natural
projection map. Then, $m_{X}$ is invariant under the action of the element $b$
(and therefore the action of $G$). If we set $T(g\Gamma)=(bg)\Gamma$, then the
transformation $T$ is called a nilrotation and $(X,m_{X},T)$ is called a
nilsystem. If the transformation $T$ is ergodic, we say that $b$ acts
ergodically on the nilmanifold $X$. It can be shown that $b$ acts ergodically
on $X$ if and only the sequence $(b^{n}x)_{n\in\mathbb{N}}$ is dense on $X$
for all $x\in X$.

We now fix a nilmanifold $G/\Gamma$. A rational subgroup of $H\leq G$ is a
subgroup of $G$ such that $H\cdot e_{X}$ is a closed subset of $X$, where
$e_{X}$ is the identity element of $X$. Equivalently, $H\Gamma$ is a closed
subset of the space $G$. A subnilmanifold of $X$ is a set $Y\subset X$ of the
form $H\cdot x$, where $x$ is an element of $X$ and $H$ is a rational subgroup
of $G$.

A polynomial sequence in a group $G$ is a sequence of the form

$a_{1}^{p_{1}(n)}\cdot...\cdot a_{m}^{p_{m}(n)}$

where $a_{i}\in G$ for all $1\leq i\leq m$ and $p_{i}$ are integer
polynomials. In the case that $G$ is a connected, simply connected Lie group,
then we can allow the polynomials $p_{i}$ to have real coefficients. We also
call the projected sequence $v(n)\Gamma$ on $X$ a polynomial sequence on $X$.

####  2.3.2. Nilorbits and Ratner’s theorem

Let $G$ be a connected and simply-connected Lie group. It is well known that
the exponential map $\exp$ from the Lie algebra of $G$ to $G$ is a
diffeomorphism. In particular, it is a bijection between $G$ and its Lie
algebra $\mathcal{B}$. For $g\in G$ and $s\in\mathbb{R}$ we can then define
the element $b^{s}$ as the unique element of $G$ satisfying $b^{s}=\exp(sX)$,
where $\exp(X)=b$. As a corollary of Ratner’s theorem [15], we get the
following:

######  Lemma 2.1.

Let $G/\Gamma$ be a nilmanifold with $G$ connected and simply-connected. For
any elements $b_{1},...,b_{k}\in\Gamma$, we have that the set

$\overline{b_{1}^{\mathbb{R}}\cdots
b_{k}^{\mathbb{R}}\Gamma}=\overline{\\{b_{1}^{t_{1}}\cdots
b_{k}^{t_{k}}\Gamma{:}\;t_{1},...,t_{k}\in\mathbb{R}\\}}$

is a subnilmanifold of $X$ with a representation $H/\Delta$, for some closed
connected and rational subgroup $H$ of $G$ that contains the elements
$b_{1}^{s},...,b_{k}^{s}$ for all $s\in\mathbb{R}$ and $\Delta$ is a uniform
subgroup of $H$.

We call the set $\\{\overline{b^{s}\Gamma{:}\;s\in\mathbb{R}}\\}$ the nil-
orbit of the element $b$. We will similarly denote by
$\overline{b^{\mathbb{Z}}\Gamma}$ the set
$\\{\overline{b^{n}\Gamma{:}\;n\in\mathbb{Z}}\\}$.

For every $b\in G$, we have that the sequence $b^{n}\Gamma$ is equidistributed
in the set $\\{\overline{b^{n}\Gamma{:}\;n\in\mathbb{Z}}\\}$. Therefore, if
$g$ is any other element in $G$, we have that the sequence $b^{n}g\Gamma$ is
equidistributed in the nilmanifold
$g\overline{\\{(g^{-1}bg)^{n}\Gamma,n\in\mathbb{N}\\}}$. This follows by
noting that $b^{n}g=g(g^{-1}bg)^{n}$. A similar relation holds for the
elements of the set $(b^{s}g)_{s\in\mathbb{R}}$, assuming $G$ is connected,
simply connected. This trick, which is called the change of base point trick,
will be used when we want to show that some sequence $v(n)x$ is
equidistributed (on some specific nilmanifold depending on $x$) in order to
change the base point $x$ to $\Gamma$.

####  2.3.3. Horizontal torus and equidistribution

Assume $X=G/\Gamma$ is a $s$-step nilmanifold with $G$ connected and simply
connected. Let $G_{2}=[G,G]$ and let $Z=G/(G_{2}\Gamma)$. This is called the
horizontal torus of $X$. We observe that $Z$ is connected, simply connected,
compact Abelian Lie group, and thus isomorphic to some torus $\mathbb{T}^{d}$.
For a $b\in G$, it can be shown that the nilrotation induced by $b$ is
ergodic, if and only if the induced action of $b$ on $Z$ is ergodic.

A horizontal character $\chi$ is a continuous group morphism
$\chi:G\to\mathbb{C}$, such that $\chi(g\gamma)=\chi(g)$ for all
$\gamma\in\Gamma$. We observe that $\chi$ also annihilates $G_{2}$ and
therefore descends to the horizontal torus $Z$. Thus, $\chi$ becomes a
character on some torus $\mathbb{T}^{d}$ and can be written in the form $t\to
e(k\cdot t)$ where $k\in\mathbb{Z}^{d}$ and $t\in\mathbb{R}^{d}$ (here the
operation $\cdot$ denotes the usual inner product on $\mathbb{R}^{d}$).

Let $b_{n}$ be a sequence of elements on $G/\Gamma$. We say that $b_{n}$ is
uniformly distributed on $X=G/\Gamma$ if and only if for every continuous
function $F:X\to\mathbb{C}$, we have

$\lim\limits_{N\to+\infty}\underset{1\leq n\leq
N}{\mathbb{E}}F(b_{n}\Gamma)=\int_{X}Fdm_{X}$

where $m_{X}$ is the (normalized) Haar measure of $X$. When $b_{n}$ is a
polynomial sequence on $X$, we have the following qualitative equidistribution
result due to Leibman [12, Theorem B,Theorem C].

######  Theorem E (Leibman).

a) Let $v(n)\Gamma$ be a polynomial sequence on a nilmanifold $X$. Then, there
exist a positive integer $q$, a closed connected subgroup $H\leq G$ and
elements $x_{0},...,x_{q-1}\in X$, such that $Hx_{j}$ are closed
subnilmanifolds of $X$ that are either disjoint or equal,

$\bigcup_{j=0}^{q-1}Hx_{j}=\overline{\\{v(n)\Gamma,n\in\mathbb{N}\\}}$

and, in addition, the sequence $v(qn+r)$ is uniformly distributed on the
subnilmanifold $Hx_{r}$ for every $r\in\\{0,1,...,q-1\\}$.
b) Let $v(n)\Gamma$ be a polynomial sequence on a nilmanifold $X=G/\Gamma$,
with $G$ connected and simply connected. Then, $v(n)\Gamma$ is equidistributed
on $X$ if and only if the projection of the sequence $v(n)\Gamma$ on the
horizontal torus $Z$ of $X$ is dense on $Z$.

######  Remark.

In the case that $G$ is not connected, then part b) can be modified as
follows: the polynomial sequence $v(n)\Gamma$ on $X$ is equidistributed if and
only if the (natural) projection of the sequence on the nilmanifold
$G/[G_{0},G_{0}]\Gamma$ is dense there. The nilmanifold
$G/[G_{0},G_{0}]\Gamma$ is sometimes called the maximal factor torus of $X$ in
the bibliography. However, we will always reduce to the case that the group
$G$ is connected in this article.

The above theorem has an analog for polynomials with several variables [13].
We use this to establish the following lemma.

######  Lemma 2.2.

Let $G/\Gamma$ be a nilmanifold with $G$ connected and simply-connected and
let be $b_{1},...,b_{k}\in\Gamma$ be any pairwise commuting elements. Then,
there exists a real number $s$ such that

$\overline{b_{1}^{\mathbb{R}}\dots
b_{k}^{\mathbb{R}}\Gamma}=\overline{\\{b_{1}^{n_{1}s}\dots
b_{k}^{n_{k}s}\Gamma{:}\;n_{1},...,n_{k}\in\mathbb{Z}\\}}.$

###### Proof.

In order for the (multivariable) polynomial sequence $b_{1}^{n_{1}s}\dots
b_{k}^{n_{k}s}$ to be equidistributed on the nilmanifold
$X=\overline{b_{1}^{\mathbb{R}}\dots b_{k}^{\mathbb{R}}\Gamma}$, it is
sufficient to check whether it is dense on the horizontal torus $Z$ of $X$
(recall that $X$ has a representation as $H/\Delta$, where $H$ is connected
and simply connected), which is homeomorphic to some torus $\mathbb{T}^{d}$.
If we denote by $(b_{i,1},...,b_{i,d})$ the projection of the element
$b_{i}\Gamma$ on $Z$, then we want to check whether the sequence

$\big{(}s(n_{1}b_{i,1}+\dots+n_{k}b_{k,1}),...,s(n_{1}b_{1,d}+\dots+n_{k}b_{k,d})\big{)}$

is dense on $\mathbb{T}^{d}$. It suffices to choose $s$ so that $1/s$ is
rationally independent of any integer combination of the coordinates
$b_{i,j}$. This concludes the proof. ∎

####  2.3.4. Quantitative equidistribution

Assume that $p(t)$ is a polynomial. Then, $p(n)$ can be expressed uniquely in
the form

$p(n)=\sum_{i=0}^{d}\binom{n}{i}a_{i}$

for some real numbers $a_{i}$ and $d\in\mathbb{N}$. For $N\in\mathbb{N}$, we
define

$\lVert e(p(n))\rVert_{C^{\infty}[N]}=\max\limits_{1\leq i\leq d}(N^{i}\lVert
a_{i}\rVert_{\mathbb{R}/\mathbb{Z}})$

where $\lVert a\rVert_{\mathbb{R}/\mathbb{Z}}$ (or $\lVert
a\rVert_{\mathbb{T}}$) denotes the distance of $a$ from the nearest integer.

The sequence $(g(n)\Gamma)_{1\leq n\leq N}$ is said to be
$\delta$-equidistributed (on the nilmanifold $X=G/\Gamma$) if and only if for
any Lipschitz function $F:X\to\mathbb{C}$, we have that

$\big{|}\underset{1\leq n\leq
N}{\mathbb{E}}F(g(n)\Gamma)-\int_{X}Fd\mu_{X}\big{|}\leq\delta\lVert
F\rVert_{\text{Lip}(X)}$

where

$\lVert F\rVert_{\text{Lip}(X)}=\lVert F\rVert_{\infty}+\sup_{x,y\in X,\ x\neq
y}\frac{|F(x)-F(y)|}{d_{X}(x,y)}$

for some specific metric $d_{X}$ on $X$ (see [7] for the definition of this
metric).

The main tool that we shall use is the following theorem of Green-Tao that
describes the orbits of polynomial sequences in finite intervals:

######  Theorem F.

[7, Theorem 2.9] Let $X=G/\Gamma$ be a nilmanifold with $G$ connected and
simply connected, and $d\in\mathbb{N}$. Then for every small enough $\delta>0$
there exist a positive constant $M=M(X,d,\delta)$ with the following property:
For every $N\in\mathbb{N}$, if $g:\mathbb{Z}\to G$ is a polynomial sequence of
degree at most $d$ such that the finite sequence $(g(n)\Gamma)_{1\leq n\leq
N}$ is not $\delta$-equidistributed, then for some non-trivial horizontal
character $\chi$ with $\lVert\chi\rVert\leq M$ we have

$\lVert\chi(g(n))\rVert_{C^{\infty}(N)}\leq M$

( where $\chi$ is thought of as a character of the horizontal torus $Z$ and
g(n) as a polynomial sequence in $Z$).

######  Remark.

If the underlying Lie group $G$ is not connected, then we have to study the
equidistribution properties of polynomial sequences through their projections
on the maximal factor torus, namely the nilmanifold $G/([G_{0},G_{0}]\Gamma)$
(recall that $G_{0}$ denotes the connected component of the identity), a
situation that is substantially more complex. Fortunately, for the purposes of
this article, we can reduce our problem to the case when $G$ is connected and
simply connected.

##  3\. Preparations for the proof

###  3.1. An example of convergence

Assume $X=G/\Gamma$ is a nilmanifold with $G$ connected and simply connected.
We will show that the averages

$\underset{1\leq n\leq N}{\mathbb{E}}f(b_{1}^{n^{3/2}}x)\cdot g(b_{2}^{n\log
n}x)$

converge for any $x\in X$ to the product $\int_{Y_{1}}f\
dm_{Y_{1}}\cdot\int_{Y_{2}}g\ dm_{Y_{2}}$, where $b_{1},b_{2}\in G$ and
$Y_{1},Y_{2}$ are the nilmanifolds
$\overline{b_{1}^{R}\Gamma},\overline{b_{2}^{\mathbb{R}}\Gamma}$ respectively.

It is straightforward to see that, if we show that the averages

$\underset{N\leq n\leq N+L(N)}{\mathbb{E}}f(b_{1}^{n^{3/2}}x)\cdot
g(b_{2}^{n\log n}x)$

converge, for some sub-linear function $L(t)$, then our result follows (see
also [3, Lemma 3.3]). Passing to the nilmanifold $X\times X$, we can easily
see that our problem reduces to showing that the averages

$\underset{N\leq n\leq N+L(N)}{\mathbb{E}}F(b_{1}^{n^{3/2}}b_{2}^{n\log n}x)$

converge for any nilmanifold $X=G/\Gamma$, commuting elements444When we pass
to the product $X\times X$, we have to study the actions of the elements
$(b_{1},e_{G})$ and $(e_{G},b_{2})$, which obviously commute. $b_{1},b_{2}\in
G$ and function $F\in C(X)$. Due to density, we can actually pick
$F\in\text{Lip}(X)$.

Let $X^{\prime}$ denote the subnilmanifold
$\overline{b_{1}^{\mathbb{R}}b_{2}^{\mathbb{R}}\Gamma}$ of $X$. As we have
seen in the previous section, this set is indeed a subnilmanifold of $X$ and
has a representation as $H/\Delta$, with $H$ connected, simply connected and
containing all elements $b_{1}^{s}$ and $b_{2}^{s}$ for any $s\in\mathbb{R}$.
In this example, we will also assume that
$X^{\prime}=\overline{b_{1}^{\mathbb{Z}}b_{2}^{\mathbb{Z}}\Gamma}$. In the
main proof, we will use Lemma 2.2 to reduce the general case of the theorem to
this one.

Using the Taylor expansion around the point $N$, we can write

$(N+h)^{3/2}=N^{3/2}+\frac{3}{2}hN^{1/2}+\frac{3h^{2}}{8N^{1/2}}-\frac{h^{3}}{16N^{3/2}}+\frac{3h^{4}}{128\xi_{h}^{5/2}},\
\ \text{ for some }\ \ \xi_{h}\in[N,N+h]$

for every $0\leq h\leq L(N)$. If we choose $L(t)\in\mathcal{H}$ to satisfy

$t^{1/2}\prec L(t)\prec t^{5/8}$

then the last term in the above expansion is smaller than $o_{N}(1)$, while
the second to last term can be shown to be unbounded. Similarly, we can write

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

If we choose again $L(t)$ to satisfy

$t^{1/2}\prec L(t)\prec t^{2/3},$

we can show that the last term is $o_{N}(1)$, while the $h^{2}$ term is
unbounded. For instance, we can choose $L(t)=t^{3/5}$ and all the precious
conditions we imposed are satisfied.

######  Remark.

In general the function $L(t)$ will depend on the initial functions in
$\mathcal{H}$. For example, if we had the two functions $t^{1/2}$ and
$t^{5/2}$ in the beginning, then for $L(t)=t^{0.51}$, we can write

$(N+h)^{1/2}=c_{1}\frac{h}{N^{1/2}}+\ \text{ lower order terms }\ +o_{N}(1)$

and

$(N+h)^{5/2}=c_{2}\frac{h^{5}}{N^{5/2}}+\ \text{ lower order terms }\
+o_{N}(1),$

where $0\leq h\leq N^{0.51}$ (here $c_{1},c_{2}$ are some real numbers). Then,
the argument proceeds as below. The fact that we can always find such a
function $L(t)$ is discussed in the following subsection.

Since the function $F$ is continuous, we can disregard the highest order terms
in the above expansion, since they are both $o_{N}(1)$. Our problem reduces to
showing that the averages

$\underset{0\leq h\leq
L(N)}{\mathbb{E}}F(b_{1}^{N^{3/2}+\frac{3}{2}hN^{1/2}+\frac{3h^{2}}{8N^{1/2}}-\frac{h^{3}}{16N^{3/2}}}b_{2}^{N\log
N+h(\log N+1)+\frac{h^{2}}{2N}}x)$

converge. For the sake of simplicity, we will show that the averages

$\underset{0\leq h\leq
L(N)}{\mathbb{E}}F(b_{1}^{\frac{h^{3}}{N^{3/2}}}b_{2}^{\frac{h^{2}}{N}}x)$

converge, since both of these statements follow from the same arguments. For
convenience, we will also assume that $x=\Gamma$.

Let $\delta>0$. We consider the finite sequence

$(a(h)\Gamma)_{0\leq h\leq
L(N)}=\Big{(}b_{1}^{\frac{h^{3}}{N^{3/2}}}b_{2}^{\frac{h^{2}}{N}}\Gamma\Big{)}_{0\leq
h\leq L(N)}$

and we show that, if $N$ is large enough, then it is $\delta$-equidistributed
on the subnilmanifold
$X^{\prime}=\overline{b_{1}^{\mathbb{R}}b_{2}^{\mathbb{R}}\Gamma}$ of $X$. It
is obvious that $a(n)\Gamma$ is a polynomial sequence in $X^{\prime}$ (of
fixed degree). If the given sequence is not $\delta$-equidistributed, we can
invoke Theorem F to find a positive constant $M=M(X^{\prime},\delta)$ such
that, for infinitely many numbers $N$, there exists a non-trivial horizontal
character $\chi_{N}$ of magnitude at most $M$, such that

$\lVert\chi_{N}(a(h)\Gamma)\rVert_{C^{\infty}[L(N)]}\leq M.$

The horizontal torus $Z$ of $X^{\prime}$ is isomorphic to some
$\mathbb{T}^{d}$ ($d\in\mathbb{N}$) and we also let $p$ denote the projection
map from $X^{\prime}$ to $Z$. Since there are only finitely many characters of
magnitude at most $M$ on $\mathbb{T}^{d}$, we conclude that there exists a
character $\chi$ such that
$\lVert\chi(p(a(h))\Gamma)\rVert_{C^{\infty}[L(N)]}\leq M$ for infinitely many
$N\in\mathbb{N}$. Suppose $\chi$ corresponds to the character
$(t_{1},...,t_{d})\to e(k_{1}t_{1}+\dots+k_{d}t_{d})$ for integers
$k_{1},...,k_{d}$ and we also write $p(b_{1}\Gamma)=(x_{1},...,x_{d})$ and
$p(b_{2}\Gamma)=(y_{1},...,y_{d})$. Then, the last inequality implies that

$\lVert
e\big{(}\frac{h^{3}}{N^{3/2}}\sum_{i=1}^{d}k_{i}x_{i}+\frac{h^{2}}{N}\sum_{i=1}^{d}k_{i}y_{i}\big{)}\rVert_{C^{\infty}[L(N)]}\leq
M.$

The definition of the $C^{\infty}[L(N)]$ norms implies the relations

$L(N)^{3}\big{\lVert}\frac{\sum_{i=1}^{d}k_{i}x_{i}}{N^{3/2}}\big{\rVert}_{\mathbb{T}}\leq
M$

and

$L(N)^{2}\big{\lVert}\frac{\sum_{i=1}^{d}k_{i}y_{i}}{N}\big{\rVert}_{\mathbb{T}}\leq
M.$

for infinitely many $N$. Due to our choice of the function $L(N)$, the above
relations fail for $N$ sufficiently large unless

$\sum_{i=1}^{d}k_{i}x_{i}\in\mathbb{Z}\ \ \text{ and
}\sum_{i=1}^{d}k_{i}y_{i}\in\mathbb{Z}.$

This implies that $\chi\circ p(b_{1}\Gamma)=x\circ p(b_{2}\Gamma)=0$ and we
can easily see that this implies that $\chi\circ
p(b_{1}^{m}b_{2}^{n}\Gamma)=0$ for any $m,n\in\mathbb{Z}$. Since elements of
this form are dense in $\overline{b_{1}^{\mathbb{R}}b_{2}^{\mathbb{R}}\Gamma}$
by our initial hypothesis, we get that $\chi$ must be the trivial character,
which is a contradiction.

######  Remark.

We explain briefly here why we have to use the $t^{\varepsilon}$ term in (3)
instead of the conjectured optimal term $\log t$. Assuming we had the
functions $\log^{2}t$ and $t\log t$ in this example, then for any choice of
the sub-linear function $L(t)$ that would give a good polynomial approximation
for the function $t\log t$, we would have

$\max_{0\leq h\leq L(N)}|\log^{2}(N+h)-\log^{2}N|=o_{N}(1),$

which means that sequence $\log^{2}n$ is essentially constant in the small
intervals $[N,N+L(N)]$. If we proceed exactly as in the above argument, we can
actually show that

$\big{|}\underset{N\leq n\leq L(N)}{\mathbb{E}}F(b_{1}^{\log^{2}n}b_{2}^{n\log
n}\Gamma)-\int_{Y_{2}}F(b_{1}^{\log^{2}N}y)\
dm_{Y_{2}}(y)\big{|}\leq\delta\lVert
F(b_{1}^{\log^{2}N}\cdot)\rVert_{\text{Lip}(Y_{2})}$

for large enough $N$, where $Y_{2}=\overline{b_{2}^{\mathbb{R}}\Gamma}$ and
$F(b_{1}^{\log^{2}n}\cdot)$ denotes the function $y\to F(b_{1}^{\log^{2}n}y)$
in the nilmanifold $Y_{2}$. However, the Lipschitz norm above is of the order
$\log^{2}N\lVert F\rVert_{\text{Lip}(X)}$, which diverges as $N\to+\infty$, so
the above bound cannot be useful for any purposes.

###  3.2. Some properties of Hardy field functions

We recall here that we are working with a Hardy field $\mathcal{H}$ that
contains the logarithmico-exponential functions. All of the results presented
below where proven in [17], and thus we omit their proofs. The main purpose of
this subsection is to establish that we can always find a common polynomial
expansion in the proofs below, as we did in the example above for the
functions $t\log t$ and $t^{3/2}$.

######  Proposition 3.1.

Let $f\in\mathcal{H}$ have polynomial growth. Then, for any natural number, we
have

$f^{(k)}(t)\ll\frac{f(t)}{t^{k}}.$

In addition, if $t^{\delta}\prec f(t)$ for some $\delta>0$, we have

$f^{\prime}(t)\sim\frac{f(t)}{t}.$

######  Proposition 3.2.

Let $f\in\mathcal{H}$ be strongly non-polynomial with $f(t)\succ\log t$. Then,
for $k$ sufficiently large, we have

$1\prec|f^{(k)}(t)|^{-1/k}\prec|f^{(k+1)}(t)|^{-1/(k+1)}\prec t.$

Let $f\in\mathcal{H}$ be a strongly non-polynomial function that satisfies
$f(t)\succ\log t$. If $k$ is large enough, we can find a function
$L(t)\in\mathcal{H}$ such that

(8) $|f^{(k)}(t)|^{-1/k}\prec L(t)\prec|f^{(k+1)}(t)|^{-1/(k+1)}.$

We restrict the function $f$ to intervals of the form $[N,N+L(N)]$. Observe
that if $0\leq h\leq L(N)$, then we have

$f(N+h)=f(N)+...+\frac{h^{k}f^{(k)}(N)}{k!}+\frac{h^{k+1}f^{(k+1)}(\xi_{h,N})}{(k+1)!}$

for some $\xi_{h,N}\in[N,N+h]$. Using Proposition 3.2, we deduce that
$|f^{(k+1)}(t)|\to 0$ monotonically (the monotonicity follows from the fact
that the function $f^{(k+1)}(t)$ belongs to $\mathcal{H}$). Then, for $N$
sufficiently large,

$\Big{|}\frac{h^{k+1}f^{(k+1)}(\xi_{h,N})}{(k+1)!}\Big{|}\leq\Big{|}\frac{L(N)^{k+1}f^{(k+1)}(N)}{(k+1)!}\Big{|}\prec
1,$

because of the choice of $L(t)$. Similarly, we can prove that

$\Big{|}\frac{L(N)^{k}a^{(k)}(N+L(N))}{k!}\Big{|}\to+\infty.$

Indeed, since $L$ is a sub-linear function, we can easily check that the
functions $a^{(k)}(t+L(t))$ and $a^{(k)}(t)$ have the same growth rate and
thus we only need to prove that

(9) $\Big{|}\frac{L(N)^{k}a^{(k)}(N)}{k!}\Big{|}\to+\infty.$

This follows similarly as above555This last relation shows that the sequence
$a(n)$ has ”good” equidistribution properties when restricted to the interval
$[N,N+L(N)]$. More specifically, $a(n)$ becomes a polynomial sequence on this
interval and we will see that it will satisfy the conditions of Theorem F for
$N$ large enough. .

In conclusion, functions that satisfy (8) have the following characteristic
property: the sequence $f(n)$, when restricted to intervals of the form
$[N,N+L(N)]$, is asymptotically equal to a polynomial sequence (that depends
on $N$) of degree exactly $k$. This motivates us to study the properties of
functions that satisfy (8). The lemmas below will be used to establish that
the above approximations can be done in tandem for any finite collection of
functions $f_{1},...,f_{k}\in\mathcal{H}$.

###  3.3. The sub-classes $S(f,k)$

Let $f\in\mathcal{H}$ be a strongly non-polynomial function such, that
$f(t)\gg t^{\delta}$, for some $\delta>0$. For $k\in\mathbb{N}$ sufficiently
large (it is only required that $f^{(k)}(t)\to 0$), we define the subclass
$S(f,k)$ of $\mathcal{H}$ as

$S(f,k)=\\{g\ {:}\;g(t)\prec t\ \text{ and }\ \
|f^{(k)}(t)|^{-\frac{1}{k}}\preceq
g(t)\prec|f^{(k+1)}(t)|^{-\frac{1}{k+1}}\\},$

where the notation $g(t)\preceq f(t)$ means that the limit
$\lim\limits_{t\to\infty}|f(t)/g(t)|$ is non-zero. We have the following
lemma.

######  Lemma 3.3.

Let $f\in\mathcal{H}$ be a strongly non-polynomial function with $f(t)\gg
t^{\delta}$, for some $\delta>0$.
i) The class $S(f,k)$ is non-empty, for $k$ sufficiently large.
ii) For any $0<c<1$ sufficiently close to 1, there exists
$k_{0}\in\mathbb{N}$, such that the function $t\to t^{c}$ belongs to
$S(f,k_{0})$.
iii) The class $S(f,k)$ does not contain all functions of the form $t\to
t^{c}$, for $c$ sufficiently close to 1.

The following Proposition relates the behavior of the subclasses $S(f,k)$ and
$S(g,\ell)$ for different functions $f,g\in\mathcal{H}$.

######  Proposition 3.4.

For any two functions $f,g\in\mathcal{H}$ as above that also satisfy $g(t)\ll
f(t)$, we have the following:

i) The relation $S(f,k)=S(g,k)$ holds for some $k\in\mathbb{N}$ if and only if
$f\sim g$.

ii) There exist infinitely many pairs of integers $(k,\ell)$, such that
$S(f,k)\cap S(g,\ell)\neq\emptyset$.

######  Remark.

It is straightforward to generalize the second statement of the above
proposition to the case of $k$ distinct functions $f_{1},...,f_{k}$ in
$\mathcal{H}$. We will use this observation in the following sections to find
a function $L$ in the intersection of these classes. Note that our previous
discussion implies that for such a function $L$, all the involved functions
$f_{1},...,f_{k}$ will have a polynomial expansion on intervals of the form
$[N,N+L(N)]$.

As an example, let us consider a fractional power $t^{\delta}$ with
$0<\delta<1$ and two functions $f,g\in\mathcal{H}$ such that $f(t)\gg
t^{\varepsilon}$ for some $\varepsilon>0$, while $\log t\prec g(t)\ll
t^{\delta}$ for all $\delta>0$. If $\delta$ is close enough to 1, then the
function $t^{\delta}$ will belong to $S(f,k)$ for some $k\in\mathbb{N}$. Using
the approximations after Proposition 3.2, we can see that the sequence $f(n)$
is approximated by a polynomial sequence of degree $k$ on intervals of the
form $[N,N+N^{\delta}]$. On the other hand, the sequence $g(n)$, restricted to
the same interval, is $o_{N}(1)$ close to the value $g(N)$, which means that
it is approximately equal to a constant on this interval.

###  3.4. Removing the integer parts

In this part, we show that part a) of Theorem 1.2 follows from part b) of the
same theorem. This is a consequence of Lemma 3.5 below. The fact that part a)
of Theorem 1.1 follows from part b) of the same theorem is precisely the
statement of [3, Lemma 5.1], which is proven using very similar arguments to
the proof of Lemma 3.5. First of all, we give some definitions.

If a collection of sequences of integers has the property that the averages

(10) $\underset{1\leq n\leq N}{\mathbb{E}}\
f_{1}(b_{1}^{{a_{1}(n)}}x)\cdot...\cdot f_{k}(b_{k}^{{a_{k}(n)}}x)$

converge for all nilmanifolds $X=G/\Gamma$, elements $b_{1},...,b_{k}\in G$
and continuous functions $f_{1},...,f_{k}$, we will say that this collection
is pointwise good for nilsystems. We will use the same definition for
sequences of real numbers, but with the underlying assumption that the
nilmanifolds in which convergence takes place arise from Lie groups that are
connected and simply connected. The following lemma is similar to [3, Lemma
5.1] and is proven below using the same arguments.

######  Lemma 3.5.

Let $a_{1}(n),...,a_{k}(n)$ be sequences of real numbers that satisfy the
following:
a) The collection $a_{1}(n),...,a_{k}(n)$ is pointwise good for nilsystems.
b) For every $1\leq i\leq k$, we have that the sequence
$(a_{i}(n)\mathbb{Z})_{n\in\mathbb{N}}$ satisfies one of the following:

  1. (1)

It is uniformly distributed in $\mathbb{T}$.

  2. (2)

It converges to some $c=c(i)\in\mathbb{T}$ different from $0$.

  3. (3)

It converges to 0 and the sequence $\\{a_{i}(n)\\}-\frac{1}{2}$ has a constant
sign eventually.

Then, the sequences $\lfloor a_{1}(n)\rfloor,...,\lfloor a_{k}(n)\rfloor$ are
pointwise good for nilsystems.

######  Remark.

Instead of the number $\frac{1}{2}$ in the third condition, we could have used
any $\alpha\in(0,1)$. We mainly use this condition in the following manner: if
the function $f(t)$ converges monotonically to $k\in\mathbb{Z}$ as
$t\to+\infty$, then we obviously have $\\{a_{i}(t)\\}\to 0$ and, in addition,
the sequence $\\{a_{i}(n)\\}$ does not not oscillate between intervals of the
form $[0,\varepsilon]$ and $[1-\varepsilon,1)$. Thus, the sequence
$\\{a_{i}(n)\\}-\frac{1}{2}$ will indeed have a constant sign (positive if $f$
increases to $k$ and negative otherwise).

###### Proof.

Let $X=G/\Gamma$ be a nilmanifold and $b_{1},...,b_{k}\in G$. Let
$f_{1},..,f_{k}\in C(X)$. Under our hypotheses, we want to show that the
averages

$\underset{1\leq n\leq N}{\mathbb{E}}f_{1}(b_{1}^{\lfloor
a_{1}(n)\rfloor}x\Gamma)\cdot...\cdot f_{k}(b_{k}^{\lfloor a_{k}(n)\rfloor}x)$

converge for all $x\in X$. Using the lifting argument in the previous
sections, we may assume that $G$ is connected and simply connected. Now,
passing to the product $X^{k}$, we want to prove that

(11) $\underset{1\leq n\leq N}{\mathbb{E}}F(b_{1}^{\lfloor
a_{1}(n)\rfloor}\cdot...\cdot b_{k}^{\lfloor a_{k}(n)\rfloor}x)$

converges for all $x$ in the nilmanfold $X=G/\Gamma$, where $G$ is connected,
simply connected, $F\in C(X)$ and $b_{1},...,b_{k}$ are commuting elements.

Fix some $i\in\\{1,2,...,k\\}$. If the sequence $a_{i}(n)$ satisfies the
second condition, that is $a_{i}(n)\mathbb{Z}$ converges to $c\mathbb{Z}$
($c\neq 0$), then, for $n$ sufficiently large, we have $\lfloor
a_{i}(n)\rfloor=a_{i}(n)-\\{c\\}+o_{n}(1)$. This implies that $b_{i}^{\lfloor
a_{i}(n)\rfloor}=b_{i}^{-\\{c\\}}b_{i}^{a_{i}(n)+o_{n}(1)}$. Since the
function $F$ is continuous, we can disregard the contribution of the
$o_{n}(1)$ term, while the $b_{i}^{-\\{c\\}}$ can be absorbed by the point
$x$. Therefore, we see that in this case we can remove the integer part for
the sequence $a_{i}(n)$. An entirely similar argument shows that the same
holds if $a_{i}(n)$ satisfies the third condition.

In order to finish the proof, we will consider below the case that each of the
sequences $a_{i}(n)\mathbb{Z}$ is equidistributed on $\mathbb{T}$ for
convenience (namely, they satisfy the first condition). It is straightforward
to adapt our argument to cover the general case, since we can easily remove
the integer parts for those sequences that satisfy the second or third
condition as we did above. We can also assume that $x=\Gamma$, since the
general case is the same up to changes in notation.

We rewrite the averages in (11) as

$\displaystyle\underset{1\leq n\leq
N}{\mathbb{E}}{F}(b_{1}^{-\\{a_{1}(n)\\}}b_{1}^{{a_{1}(n)}}\cdot...\cdot
b_{k}^{-\\{a_{k}(n)\\}}b_{k}^{{a_{k}(n)}}\Gamma)=$
$\displaystyle\underset{1\leq n\leq
N}{\mathbb{E}}\widetilde{F}(a_{1}(n)\mathbb{Z},b_{1}^{a_{1}(n)}\Gamma,...,a_{k}(n)\mathbb{Z},b_{k}^{a_{k}(n)}\Gamma)$

where $\widetilde{F}:\mathbb{T}^{k}\times X^{k}\to\mathbb{C}$ is the function
defined by the relation

$\widetilde{F}(x_{1}\mathbb{Z},y_{1}\Gamma,....,x_{k}\mathbb{Z},y_{k}\Gamma)=F(b_{1}^{-\\{x_{1}\\}}y_{1}\cdot...b_{k}^{-\\{x_{k}\\}}y_{k}\Gamma).$

Let $v(n)$ be the sequence
$(a_{1}(n)\mathbb{Z},b_{1}^{a_{1}(n)}\Gamma,...,a_{k}(n)\mathbb{Z},b_{k}^{a_{k}(n)}\Gamma)$.
By our hypothesis, for any continuous function $f$ on
$\widetilde{X}=\mathbb{T}^{k}\times X^{k}$, the averages of the sequence
$f(v(n))$ converge. However, note that the function $\widetilde{F}$ may have
discontinuities at points where one of the $x_{1},...,x_{k}$ is an integer.
Our goal is to show that this property of continuous functions also holds for
the function $\widetilde{F}$.

Let $\varepsilon>0$. For every $1\leq i\leq k$, we define the set

$A_{i}(\varepsilon)=\\{n\in\mathbb{N}{:}\;\lVert
a_{i}(n)\rVert_{\mathbb{T}}\geq\varepsilon\\}.$

Observe that this set has natural density $1-2\varepsilon$, since the sequence
$a_{i}(n)$ is uniformly distributed $(mod\ 1)$. Let $I_{\varepsilon}$ be the
subset $[\varepsilon,1-\varepsilon]\times...\times[\varepsilon,1-\varepsilon]$
of $\mathbb{T}^{k}$. We calculate the upper density of the set
$B_{\varepsilon}=\\{n\in\mathbb{N}{:}\;(a_{1}(n),...,a_{k}(n))\notin
I_{\varepsilon}\\}$. We have

$d^{*}(B_{\varepsilon})=d^{*}(A_{1}^{c}(\varepsilon)\cup...\cup
A_{k}^{c}(\varepsilon))\leq\sum_{i=1}^{k}d^{*}(A_{i}^{c}(\varepsilon))=\sum_{i=1}^{k}d(A_{i}^{c}(\varepsilon))=2k\varepsilon.$

Now, for every $\varepsilon>0$, we define a continuous function
$F_{\varepsilon}$ that agrees everywhere with $\widetilde{F}$ on
$I_{\varepsilon}\times X^{k}$ and such that all the $F_{\varepsilon}$ are
bounded uniformly by $2\lVert\widetilde{F}\rVert_{\infty}$.

By our assumption, we get that the sequence $v(n)$ takes values in
$I_{\varepsilon}\times X^{k}$ for all $n\in\mathbb{N}$ outside a set of upper
density at most $2k\varepsilon$. Therefore, we get

$\big{|}\underset{1\leq n\leq
N}{\mathbb{E}}\widetilde{F}(v(n))-\underset{1\leq n\leq
N}{\mathbb{E}}{F_{\varepsilon}}(v(n))\big{|}\leq 4\lVert
F\rVert_{\infty}k\varepsilon.$

Since the averages $\underset{1\leq n\leq
N}{\mathbb{E}}{F_{\varepsilon}}(v(n))$ converge as $N\to\infty$
($F_{\varepsilon}$ is continuous), we easily deduce that the averages
corresponding to the function $\widetilde{F}$ form a Cauchy sequence and,
therefore, converge. The conclusion follows. ∎

Using the previous lemma, we can now prove that the first part of Theorem 1.2
follows from the second part.

###### Proof of part a) of Theorem 1.2.

We will have to check whether the conditions of Lemma 3.5 are satisfied. Let
$a_{1},...,a_{k}\in\mathcal{H}$ be as in the statement of Theorem 1.2.
Condition a) of Lemma 3.5 is satisfied by our hypothesis. Now, we verify the
second condition.

Fix some $i\in\\{1,2,..,k\\}$. We consider two cases:
i) Assume that the function $a_{i}(t)$ is such that $|a_{i}(t)-q(t)|\succ
t^{\varepsilon}$ for all polynomials $q(t)$ with rational coefficients. Then,
the sequence $a_{i}(n)\mathbb{Z}$ is uniformly distributed on $\mathbb{T}$
(satisfying condition (1)), due to the equidistribution results in [2].
ii) Assume that the function $a_{i}(t)$ is such that
$\lim\limits_{t\to+\infty}a_{i}(t)=c\notin\mathbb{Z}$. Then, the sequence
$a_{i}(n)$ satisfies condition (2) of Lemma 3.5.
iii) Assume that neither of the above conditions hold. Since $a_{i}(t)$ must
satisfy (4), we deduce that $a_{i}(t)$ converges to some integer $c$. However,
since $a_{i}(t)$ converges to $c$ monotonically (functions in $\mathcal{H}$
are eventually monotone), we easily see that condition (3) of Lemma 3.5 is
satisfied and we are done. ∎

##  4\. Proofs of main theorems

The main tool we are going to use in our proof is the quantitative Green-Tao
theorem on polynomial orbits (Theorem F). A technical obstruction in our proof
is that among the functions $a_{1},...,a_{k}$ in the statement of Theorem 1.2,
we must separate the polynomial functions and the functions that satisfy
$t^{d}\prec a_{i}(t)\prec t^{d+1}$. We start with an elementary lemma.

Let $a$ be a function in $\mathcal{H}$. We partition $\mathcal{H}$ into
equivalence classes by the relation $f\sim g$ if and only if the limit of
$f(t)/g(t)$ as $t\to+\infty$ is a non-zero real number. Thus, $f\sim g$ if and
only if the functions $f,g$ have the same growth rate. The zero function in
$\mathcal{H}$ forms its own equivalence class.

We will define the non-polynomial growth rate of a function $a\in\mathcal{H}$
as follows:
i) If $a$ is strongly non-polynomial (recall the definition in Section 2), we
define it as the corresponding equivalence class of $a$.
ii) If $a$ is written in the form $p(t)+x(t)$, where $x(t)$ is strongly non-
polynomial (or the zero function), we define it as the equivalence class of
the function $x\in\mathcal{H}$.

It is straightforward to see that the non-polynomial growth rate is well
defined. We also say that $a\in\mathcal{H}$ has trivial growth rate, if
$\lim\limits_{t\to+\infty}a(t)=0$. Recall that we also included these
functions when we defined the strongly non-polynomial functions. We will need
the following lemma.

######  Lemma 4.1.

Let $a_{1},...,a_{k}\in\mathcal{H}$ be arbitrary functions. Then, the set
$\mathcal{L}(a_{1},...,a_{k})$ of non-trivial linear combinations has a basis
$(g_{1},...,g_{m},u_{1},...,u_{\ell})$666We can possibly have $m=0$ or
$\ell=0$., such that the functions $u_{1},...,u_{\ell}$ have the form
$p_{i}(t)+o_{t}(1)$, where $p_{i}$ is a real polynomial for every $1\leq
i\leq\ell$ and $g_{1},...,g_{m}$ have distinct and non-trivial non-polynomial
growth rates.

###### Proof.

We may write the functions $a_{1},..,a_{k}$ in the form $p_{i}(t)+x_{i}(t)$
where $p_{i}$ are real polynomials and $x_{i}(t)$ are strongly non-polynomial
functions (or identically zero). Without loss of generality, assume that

$x_{1}(t)\gg x_{2}(t)\gg\dots\gg x_{k}(t).$

We also consider the number $l\in\\{0,1,...,k\\}$ for which
$x_{l+1}(t)=o_{t}(1)$.

We induct on $k$. For $k=1$, we have nothing to prove. Assume the claim holds
for $k-1$. We consider two cases.
i) If the functions $x_{1},...,x_{l}$ have distinct growth rates, then we are
done.
ii) Assume two of the functions among $x_{1},...,x_{l}$ (say $x_{1},x_{2}$ for
simplicity) have the same growth rate. In particular, let $m\geq 2$ be the
largest number such that $x_{1}\sim x_{2}\sim\dots\sim x_{m}$. Then, we can
write $x_{i}(t)=x_{1}(t)+y_{i}(t)$, where $y_{i}(t)\prec x_{1}(t)$. We have
$a_{1}(t)=p_{1}(t)+x_{1}(t)$ and
$a_{i}(t)=(p_{1}(t)+x_{1}(t))+(p_{i}(t)-p_{1}(t)+y_{i}(t))$ for $i\leq m$. The
claim then follows from the induction hypothesis on the functions
$p_{2}(t)-p_{1}(t)+y_{2}(t),...,p_{m}(t)-p_{1}(t)+y_{m}(t),p_{m+1}(t)+x_{m}(t),...,p_{k}(t)+x_{k}(t)$.
∎

###### Proof of Theorem 1.2.

As we have established in the previous section, we only need to prove the
second part. Let $a_{1},...,a_{k}\in\mathcal{H}$ be functions whose linear
combinations satisfy either (3) or (4). Passing to the product nilmanifold (as
we did in the example in the previous section), we easily see that our problem
follows from the following statement:

If $X=G/\Gamma$ is a nilmanifold with $G$ connected and simply connected and
$b_{1},...,b_{k}\in G$ are commuting elements and all linear combinations of
the functions $a_{1},...,a_{k}\in\mathcal{H}$ satisfy (3) or (4), then for any
$x\in X$ and continuous function $F:X\to\mathbb{C}$, we have that the averages

$\underset{1\leq n\leq N}{\mathbb{E}}F(b_{1}^{a_{1}(n)}\dots
b_{k}^{a_{k}(n)}x)$

converge.

### Step 1: Some more reductions

Applying Lemma 4.1, we can find a basis ${g_{1},...,g_{s}}$ for the set
$\mathcal{L}(a_{1},...,a_{k})$, such that the functions $g_{1},...,g_{s}$
satisfy the conditions of the aforementioned lemma. If we write

(12) $a_{i}(t)=\sum_{j=1}^{s}c_{i,j}g_{j}(t),$

then we can rewrite the above average as

(13) $\underset{1\leq n\leq N}{\mathbb{E}}F(u_{1}^{g_{1}(n)}\dots
u_{s}^{g_{s}(n)}x)$

for some commuting elements $u_{1},...,u_{s}\in G$.

If one of the functions $g_{1},...,g_{s}$ is such that the limit
$\lim\limits_{t\to+\infty}g_{i}(t)$ is a real number (which can be the case
when a linear combination of the original functions satisfies (4)), we can
invoke the continuity of $F$ to eliminate the term $u_{i}^{g_{i}(n)}$.
Therefore, we may assume that all of the functions $g_{1}(t),...,g_{s}(t)$ go
to $\pm\infty$, as $t\to+\infty$. We denote by $v(n)$ the sequence
$u_{1}^{g_{1}(n)}\dots u_{s}^{g_{s}(n)}x$. In order to establish the result,
it suffices to show that the sequence $v(n)x$ is uniformly distributed on the
nilmanifold $X^{\prime}=\overline{u_{1}^{\mathbb{R}}\dots
u_{s}^{\mathbb{R}}x}$. We will prove this in the case $x=\Gamma$, since the
general case follows from this using the change of base point trick that we
discussed in Section 2. In addition, we can invoke Lemma 2.2 to find a real
number $s_{0}$, such that
$X^{\prime}=\overline{(u_{1}^{s_{0}})^{\mathbb{Z}}...(u_{s}^{s_{0}})^{\mathbb{Z}}\Gamma}$.
Replacing the functions $g_{i}(t)$ by the functions $g_{i}(t)/s_{0}$ (the
assumptions on the linear combinations of the functions $g_{i}$ remain
unchanged), we can reduce our problem to the case that
$X^{\prime}=\overline{u_{1}^{\mathbb{Z}}...u_{s}^{\mathbb{Z}}\Gamma}$.

We want to show that for any continuous function $F$ from
$X^{\prime}=G^{\prime}/\Gamma^{\prime}$ ($G^{\prime}$ is connected, simply
connected and $\Gamma^{\prime}$ is a uniform subgroup), the averages

$\underset{1\leq n\leq N}{\mathbb{E}}F(v(n)\Gamma^{\prime})$

converge to the integral $\int_{X^{\prime}}F\ dm_{X^{\prime}}$. Since
Lipschitz functions are dense in the space $C(X^{\prime})$, we may assume that
$F$ is Lipschitz continuous. In addition, we may assume after rescaling that
$\lVert F\rVert_{\text{Lip}(X^{\prime})}\leq 1$.

The statement of Lemma 4.1 implies that the functions $g_{1},...,g_{s}$ can be
written in the form $p_{i}(t)+x_{i}(t)$, such that

$x_{1}(t)\gg\dots\gg x_{s}(t)$

and, if $l$ is such that $x_{l}(t)\succ 1$ and $x_{l+1}(t)\prec 1$, then the
functions $x_{1}(t),...,x_{l}(t)$ have distinct growth rates. Let $0\leq m\leq
l$ be a natural number such that $x_{m}(t)\gg t^{\varepsilon}$ for some
$\varepsilon>0$, while $x_{m+1}$ is a sub-fractional function. Due to our
hypothesis (namely (3)), every non-trivial linear combination of the
polynomials $p_{m+1}(t),...,p_{s}(t)$ must be a polynomial that does not
belong to $\mathbb{Z}[t]$. We rewrite the sequence $v(n)\Gamma^{\prime}$ in
the form

$v(n)\Gamma^{\prime}=\prod_{i=1}^{m}u_{i}^{p_{i}(n)+x_{i}(n)}\cdot\prod_{i=m+1}^{s}u_{i}^{p_{i}(n)+x_{i}(n)}\Gamma^{\prime}.$

### Step 2: Approximating by polynomials

Let $L(t)$ be a sub-linear function with
$\lim\limits_{t\to+\infty}L(t)=+\infty$ that we will choose later. It suffices
to show that the sequence of the averages

(14) $\underset{N\leq n\leq N+L(N)}{\mathbb{E}}F(v(n)\Gamma)$

converges to $\int_{X^{\prime}}F\ dm_{X^{\prime}}$.

Let $r$ be a very large natural number compared to the degrees of the
polynomials $p_{i}$ and the functions $x_{i}(t)$. For such a choice of $r$, we
have that $x_{i}^{(r)}(t)=o_{t}(1)$. We choose the natural numbers
$k_{i},i\in\\{1,...,m\\}$, so that the sub-class $S(x_{i},k_{i})$ (see the
definitions in Subsection 3.2) contains the function $(x_{1}^{(r)}(t))^{-1/r}$
(obviously $k_{1}=r$). If we choose $r$ large enough, we can ensure that the
numbers $k_{i}$ are also very large compared to the degrees of the polynomials
$p_{i}$. We also choose a function $L(t)$ in the intersection of the classes
$S(x_{i},k_{i})$, which is non-empty.

We use the Taylor expansion for the functions $x_{1}(t),...,x_{m}(t)$ to write
(see the discussion after Proposition 3.2)

(15)
$x_{i}(N+h)=x_{i}(N)+\dots+\frac{x_{i}^{k_{i}}(N)h^{k}}{k_{i}!}+o_{N}(1)=q_{i,N}(h)+o_{N}(1)$

for $0\leq h\leq L(N)$. If, on the other hand, we have $i>m$ (in this case,
the function $x_{i}$ is sub-fractional), then

(16) $\max_{0\leq h\leq L(N)}|x_{i}(N+h)-x_{i}(N)|=o_{N}(1).$

In addition, we denote $p_{i,N}(h)=p_{i}(N+h)$. Thus, we rewrite the quantity
in (14) as

$\underset{0\leq h\leq
L(N)}{\mathbb{E}}F(w_{N}\prod_{i=1}^{m}u_{i}^{q_{i,N}(h)+p_{i,N}(h)}\prod_{i=m+1}^{s}u_{i}^{p_{i,N}(h)}\Gamma^{\prime})$

where we discarded the $o_{N}(1)$ terms, because $F$ is continuous. Here,
$w_{N}=\prod_{i=m+1}^{s}u_{i}^{x_{i}(N)}$.

### Step 3: Using the quantitative equidistribution theorem

Let $Z\cong\mathbb{T}^{d}$ be the horizontal torus of the nilmanifold
$X^{\prime}=\overline{u_{1}^{\mathbb{R}}\dots u_{s}^{\mathbb{R}}\Gamma}$ and
let $\pi:X^{\prime}\to Z$ denote the projection map. Let $\delta>0$. We claim
that the finite polynomial sequence

(17)
$\Big{(}\prod_{i=1}^{m}u_{i}^{q_{i,N}(h)+p_{i,N}(h)}\prod_{i=m+1}^{s}u_{i}^{p_{i,N}(h)}\Gamma^{\prime}\Big{)}_{0\leq
h\leq L(N)}$

is $\delta$-equidistributed on the nilmanifold $X^{\prime}$ for $N$
sufficiently large. If the claim does not hold for a natural number $N$, then
by Theorem F, there exists a real number $M>0$ and a non-trivial horizontal
character $\chi_{N}$ of modulus $\leq M$ such, that

$\big{\lVert}\chi_{N}\circ\pi(\prod_{i=1}^{m}u_{i}^{q_{i,N}(h)+p_{i,N}(h)}\prod_{i=m+1}^{s}u_{i}^{p_{i,N}(h)}\Gamma^{\prime})\big{\rVert}_{C^{\infty}[L(N)]}\leq
M.$

Since there are only finitely many characters of modulus at most $M$, we have
that, if our claim fails, there exists a horizontal character $\chi$, so that

(18)
$\big{\lVert}\chi\circ\pi(\prod_{i=1}^{m}u_{i}^{q_{i,N}(h)+p_{i,N}(h)}\prod_{i=m+1}^{s}u_{i}^{p_{i,N}(h)}\Gamma^{\prime})\big{\rVert}_{C^{\infty}[L(N)]}\leq
M$

holds for infinitely many $N\in\mathbb{N}$.

The function $\chi\circ\pi$ is a character on $\mathbb{T}^{d}$ of modulus
$\leq M$ and, thus, has the form $(t_{1},...,t_{d})\to
e(k_{1}t_{1}+...k_{d}t_{d})$ for $k_{i}\in\mathbb{Z}\cap[-M,M]$. We also write
$\pi(u_{i})=(u_{i,1},...,u_{i,d})$. Then, the relation (18) can be rewritten
as

(19)
$\big{\lVert}\sum_{i=1}^{m}(q_{i,N}(h)+p_{i,N}(h))(k_{1}u_{i,1}+\dots+k_{d}u_{i,d})+\sum_{i=m+1}^{s}(p_{i,N}(h))(k_{1}u_{i,1}+\dots+k_{d}u_{i,d})\big{\rVert}_{C^{\infty}[L(N)]}\leq
M.$

Assume that not all numbers $k_{1}u_{i,1}+\dots+k_{d}u_{i,d}$ are zero. We
consider two cases:

i) Firstly, suppose that $m=0$. Then, (19) implies that

(20)
$\big{\lVert}\sum_{i=1}^{s}(p_{i,N}(h))(k_{1}u_{i,1}+\dots+k_{d}u_{i,d})\big{\rVert}_{C^{\infty}[L(N)]}\leq
M.$

Let $R(t)=\sum_{i=1}^{s}(p_{i}(t))(k_{1}u_{i,1}+\dots+k_{d}u_{i,d})$. This is
a linear combination of the polynomials $p_{i}(t)$. However, this linear
combination is not a polynomial in $\mathbb{Q}[t]$ due to our assumptions on
the polynomials $p_{i}(n)$, unless, of course, all the coefficients
$k_{1}u_{i,1}+\dots+k_{d}u_{i,d}$ (for $1\leq i\leq s$) in this combination
are zero, which we have assumed to not be the case. Thus, $R(t)$ has at least
one non-constant coefficient irrational and is uniformly distributed in
$\mathbb{T}$. The relation (20) implies that $\lVert
R(N+h)\rVert_{C^{\infty}[L(N)]}\leq M$ for infinitely many $N$. It is
straightforward to see that this fails for $N$ large enough for any polynomial
with at least one non-constant coefficient irrational.

ii) Suppose now that $m>0$. We work in the case when not all of the numbers
$k_{1}u_{i,1}+\dots k_{d}u_{i,d}$ with $1\leq i\leq m$ are zero, because
otherwise we can use the same argument as in the previous case to reach a
contradiction.

If the positive integer $\ell$ is larger than the degrees of the polynomials
$p_{i,N}(h)$ and smaller than the numbers $k_{i}$ (which we have chosen to be
very large compared to the degrees of the polynomials $p_{i,N}$), then the
coefficient of the term $h^{\ell}$ in the polynomial appearing in (19) is
equal to

$\frac{1}{\ell!}\sum_{i=1}^{m}x_{i}^{(\ell)}(N)(k_{1}u_{i,1}+\dots+k_{d}u_{i,d})$

and, thus, we must have

$L(N)^{\ell}\big{\lVert}\frac{1}{\ell!}\sum_{i=1}^{m}x_{i}^{(\ell)}(N)(k_{1}u_{i,1}+\dots+k_{d}u_{i,d})\big{\rVert}_{\mathbb{T}}\ll
M$

for infinitely many $N$. We can choose $\ell$ so that $x_{i}^{\ell}(t)\prec 1$
for every $1\leq i\leq m$, since we have chosen the $k_{i}$ to be very large
compared to the degrees of the functions $x_{i}(t)$ as well. The last
inequality is equivalent to

$L(N)^{\ell}\big{|}\sum_{i=1}^{m}x_{i}^{(\ell)}(N)(k_{1}u_{i,1}+\dots+k_{d}u_{i,d})\big{|}\ll
M.$

However, the Hardy field function inside the absolute value above has the same
growth rate as the function $x_{1}^{(\ell)}(t)$, since the functions
$x_{1},...,x_{m}$ are strongly non-polynomial and have distinct growth rates,
unless, of course, $k_{1}u_{1,1}+\dots+k_{d}u_{1,d}=0$. Therefore,

$L(N)^{\ell}\big{|}x_{1}^{(\ell)}(N)(k_{1}u_{1,1}+\dots+k_{d}u_{1,d})\big{|}\ll
M$

for infinitely many $N$, which contradicts the fact that $L(t)\in
S(x_{1},k_{1})$. Thus, we must have $k_{1}u_{1,1}+\dots+k_{d}u_{1,d}=0$.
Working inductively, we also get $k_{1}u_{i,1}+\dots+k_{d}u_{i,d}$=0 for all
$1\leq i\leq m$, which is a contradiction.

To summarize, we have that if the sequence in (17) is not
$\delta$-equidistributed, then all the numbers $k_{1}u_{i,1}+\dots
k_{d}u_{i,d}$ are zero. Equivalently, we have $\chi\circ\pi(u_{i})=0$ for all
$1\leq i\leq s$. This implies that the character $\chi$ is the trivial
character on $X^{\prime}$. Indeed, the character $\chi$ annihilates all
elements $u_{1}^{n_{1}}\cdots u_{s}^{n_{s}}\Gamma$, where
$n_{1},...,n_{s}\in\mathbb{Z}$ and by density of those elements on
$X^{\prime}$ (recall our assumption that $X^{\prime}$ is also equal to
nilmanifold $\overline{u_{1}^{\mathbb{Z}}...u_{s}^{\mathbb{Z}}\Gamma}$),
$\chi$ is zero everywhere. This is a contradiction (the horizontal characters
appearing when we applied Theorem F are assumed to be non-trivial).

In conclusion, we have that the finite polynomial sequence in (17) is
$\delta$-equidistributed. Thus, we deduce that the averages in (14) are
$\delta\lVert F(w_{N}\cdot)\rVert_{\text{Lip}(X^{\prime})}=\delta\lVert
F\rVert_{\text{Lip}(X^{\prime})}$ close to the quantity
$\int_{X^{\prime}}F(w_{N}x)\ dm_{X^{\prime}}(x)$. Taking $\delta\to 0$ and
also utilizing the fact that the action of $w_{N}$ on $X^{\prime}$ preserves
the Haar measure of $X^{\prime}$, we get that the desired limit is equal to
$\int_{X^{\prime}}F(x)\ dm_{X^{\prime}}(x)$ and we are done. ∎

###### Proof of Theorem 1.1.

As we explained in the previous section (before the statement of Lemma 3.5),
the first part follows from the second part and, in turn, this second part
follows using similar arguments as in the proof of Theorem 1.2. We only
highlight the main differences here:
a) In (13), all the functions $g_{1},...,g_{s}$ satisfy (3) (there are no
functions $g_{i}$ with $g_{i}(t)=o_{t}(1)$). We also have $k=s$.
b) The limit of the averages is again $\int_{X^{\prime}}F(x)\
dm_{X^{\prime}}(x)$, and we want to show that the limit is equal to
$\int_{X}Fdm_{X}$, where $X$ is the nilmanifold
$\overline{b_{1}^{\mathbb{R}}...b_{k}^{\mathbb{R}}\Gamma}$. Recall that each
$u_{i}$ is equal to $b_{1}^{c_{i,1}}\cdots b_{k}^{c_{i,k}}$ (by (12)) and the
numbers $c_{i,j}$ form an invertible $k\times k$ matrix (due to the linear
independence assumption on the original functions $a_{1},...,a_{k}$). Thus, we
can also write $b_{i}=\prod_{j=1}^{k}u_{i}^{c^{\prime}_{i,j}}$ for some
numbers $c^{\prime}_{i,j}$ (here, we use that the elements $b_{i}$ are
pairwise commuting). Combining the above, we have that
$b_{1}^{\mathbb{R}}...b_{k}^{\mathbb{R}}=u_{1}^{\mathbb{R}}...u_{k}^{\mathbb{R}}$
and thus the closures of their projections on $G/\Gamma$ define the same
subnilmanifold. ∎

###  4.1. Proof of Theorem 1.3

Finally, we give a proof of Theorem 1.3 here. We use the following proposition
from [17]. Although it will not be used in the proof, we have to assume below
that the Hardy field $\mathcal{H}$ that we work with is closed under
composition and compositional inversion of functions, since the following
proposition was proven under this assumption.

######  Proposition 4.2.

Assume that the functions $a_{1},,...,a_{k}\in\mathcal{H}$ 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 $s$, 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

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

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

###### Proof of Theorem 1.3.

Using a standard ergodic decomposition argument, we may assume that the system
$(X,\mu,T)$ is ergodic. Our first objective is to show that the Host-Kra
factors are characteristic for the averages in (5). If the functions
$a_{1},...,a_{k}$ are such that the conditions of Proposition 4.2 are
satisfied, then the result follows using a standard telescoping argument. If
this does not hold, then we can consider two cases:
a) Assume that one of the functions (say $a_{1}$) has growth rate smaller than
or equal to $\log t$. Then, using (3) and (4), we deduce that $a_{1}$
converges monotonically to some real number $c$ and the integer part of
$a_{1}(n)$ becomes a constant. Thus, a standard argument shows that we only
need to show that the averages

$\underset{1\leq n\leq N}{\mathbb{E}}T^{\lfloor a_{2}(n)\rfloor}f_{2}\dots
T^{\lfloor a_{k}(n)\rfloor}f_{k}$

converge.
b) Assume that two of the functions (say $a_{1},a_{2}$) are such that their
difference is dominated by $\log t$. We will assume that the function
$a_{1}(t)$ goes to $\pm\infty$ as $t\to+\infty$, because otherwise we can
repeat the above argument to eliminate both $a_{1}$ and $a_{2}$. In that case,
the function $a_{1}(t)$ satisfies (3) and by Theorem A, the sequence
$a_{1}(n)$ is equidistributed $(mod\ 1)$. Observe that due to (3) and (4)
again, we must have $a_{2}(t)=a_{1}(t)+c+x(t)$, where the function
$x(t)\in\mathcal{H}$ converges to $0$ monotonically and $c$ is a real number.
Thus, for $t\in\mathbb{R}$ sufficiently large we have

(22) $\lfloor a_{2}(t)\rfloor=\lfloor a_{1}(t)+x(t)+c\rfloor=\lfloor
a_{1}(t)\rfloor+\lfloor c\rfloor+e(t)$

where $e(t)\in\\{0,\pm 1,\pm 2\\}$ and the value of $e(t)$ depends on whether
the inequalities

$\\{a_{1}(t)+c\\}+\\{x(t)\\}\leq 1$

and

$\\{a_{1}(t)\\}+\\{c\\}\leq 1$

hold or not, as well as whether $x(t)$ is eventually positive or negative.

Define $A_{z}=\\{t\in\mathbb{R},e(t)=z\\}$ for $z\in\\{0,\pm 1,\pm 2\\}$.
Then, we see that our multiple averages are equal to the sum

$\sum_{z\in\\{0,\pm 1,\pm 2\\}}\underset{1\leq n\leq N}{\mathbb{E}}{\bf
1}_{A_{z}}(n)\ T^{\lfloor a_{1}(n)\rfloor}(f_{1}\cdot T^{\lfloor
c\rfloor+z}f_{2})\cdot T^{\lfloor a_{3}(n)\rfloor}f_{3}\dots T^{\lfloor
a_{k}(n)\rfloor}f_{k}.$

For a fixed $z$, we want to show that the corresponding average converges. For
$n\in\mathbb{N}$ large enough, we will approximate the sequence ${\bf
1}_{A_{z}}(n)$ by sequences of the form $F(\\{a_{1}(n)\\})$, where $F$ is a
continuous function. Indeed, let us show this for $z=0$ (the other cases
follow similarly). Assume that $x(t)$ decreases to 0 (the other case) is
similar, which means that $x(t)$ is eventually positive and also
$\\{x(t)\\}=x(t)$ eventually. In addition, we can also assume that $c$ is
positive. Observe that for $t\in A_{0}$, we have

$\lfloor a_{2}(t)\rfloor=\lfloor a_{1}(t)\rfloor+\lfloor c\rfloor$

by definition. This is equivalent to the inequality

$\\{a_{1}(t)\\}+\\{x(t)\\}\leq 1-\\{c\\}.$

If we define the set $A=\\{(x,y)\in\mathbb{T}^{2}{:}\;x+y\leq 1-\\{c\\}\\}$,
then ${\bf 1}_{A_{0}}(t)=1_{A}(\\{a_{1}(t)\\},\\{x(t)\\})$. Observe that the
function ${\bf 1}_{A}$ has discontinuities only on the line $x+y=1-\\{c\\}$.

Let $0<\varepsilon<\frac{1}{2}$. We can define a continuous function
$F_{\varepsilon}:\mathbb{T}^{2}\to\mathbb{C}$ with $\lVert
F_{\varepsilon}\rVert_{\infty}\leq 1$ such that $F_{\varepsilon}$ agrees
everywhere with ${\bf 1}_{A}$ on the set

$B=\\{(x,y)\in\mathbb{T}^{2}{:}\;x+y\leq
1-\\{c\\}-\varepsilon\\}\cup\\{(x,y)\in\mathbb{T}^{2}{:}\;x+y\geq
1-\\{c\\}+\varepsilon\\}$

and, in addition, $\lVert F_{\varepsilon}-{\bf 1}_{A}\rVert_{\infty}\leq 2$.
We also consider the sets

$C=\\{n\in\mathbb{N}{:}\;\\{a_{1}(n)\\}\in[0,1-\\{c\\}-2\varepsilon]\cup[1-\\{c\\}+2\varepsilon,1)\\}\text{
and }D=\\{n\in\mathbb{N}{:}\;\\{x_{1}(n)\\}<\varepsilon\\}.$

Obviously, if $n\in C\cup D$, then we have $(\\{a_{1}(n)\\},\\{x(n)\\})\in B$.
In addition, we have $n\in D$ for every $n$ sufficiently large. Finally, we
also observe that the set $C$ has natural density $1-4\varepsilon$, since
$a_{1}(n)$ is equidistributed $(mod\ 1)$. We also define the continuous
function $G_{\varepsilon}:\mathbb{T}\to\mathbb{C}$ by
$G_{\varepsilon}(x)=F_{\varepsilon}(x,0)$.

We combine the above to deduce that

$\big{\lVert}\underset{1\leq n\leq N}{\mathbb{E}}{\bf 1}_{A_{0}}(n)\
T^{\lfloor a_{1}(n)\rfloor}(f_{1}\cdot T^{\lfloor c\rfloor+z}f_{2})\cdot
T^{\lfloor a_{3}(n)\rfloor}f_{3}\dots T^{\lfloor a_{k}(n)\rfloor}f_{k}-\\\
\underset{1\leq n\leq
N}{\mathbb{E}}F_{\varepsilon}(\\{a_{1}(n)\\},\\{x(n)\\})\ T^{\lfloor
a_{1}(n)\rfloor}(f_{1}\cdot T^{\lfloor c\rfloor+z}f_{2})\cdot T^{\lfloor
a_{3}(n)\rfloor}f_{3}\dots T^{\lfloor
a_{k}(n)\rfloor}f_{k}\big{\rVert}_{L^{2}(\mu)}\leq\\\ \underset{1\leq n\leq
N}{\mathbb{E}}\big{|}{\bf
1}_{A}(\\{a_{1}(n)\\},\\{x(n)\\})-F(\\{a_{1}(n)\\},\\{x(n)\\})\big{|}$

by the triangle inequality. For $n\in C\cup D$, the absolute value appearing
in the last quantity becomes 0, while the complement of $C\cup D$ has natural
density at most $4\varepsilon$ and, using the bound $\lVert
F_{\varepsilon}-{\bf 1}_{A}\rVert_{\infty}\leq 2$ for these exceptional
values, we deduce that

$\big{\lVert}\underset{1\leq n\leq N}{\mathbb{E}}{\bf 1}_{A_{0}}(n)\
T^{\lfloor a_{1}(n)\rfloor}(f_{1}\cdot T^{\lfloor c\rfloor+z}f_{2})\cdot
T^{\lfloor a_{3}(n)\rfloor}f_{3}\dots T^{\lfloor a_{k}(n)\rfloor}f_{k}-\\\
\underset{1\leq n\leq
N}{\mathbb{E}}F_{\varepsilon}(\\{a_{1}(n)\\},\\{x(n)\\})\ T^{\lfloor
a_{1}(n)\rfloor}(f_{1}\cdot T^{\lfloor c\rfloor+z}f_{2})\cdot T^{\lfloor
a_{3}(n)\rfloor}f_{3}\dots T^{\lfloor
a_{k}(n)\rfloor}f_{k}\big{\rVert}_{L^{2}(\mu)}\leq 8\varepsilon+o_{N}(1).$

In addition, we have
$F_{\varepsilon}(\\{a_{1}(n)\\},\\{x(n)\\})=G_{\varepsilon}(\\{a_{1}(n)\\})$
for $n\in C\cup D$ and, thus a similar argument yields

$\big{\lVert}\underset{1\leq n\leq N}{\mathbb{E}}{\bf 1}_{A_{0}}(n)\
T^{\lfloor a_{1}(n)\rfloor}(f_{1}\cdot T^{\lfloor c\rfloor+z}f_{2})\cdot
T^{\lfloor a_{3}(n)\rfloor}f_{3}\dots T^{\lfloor a_{k}(n)\rfloor}f_{k}-\\\
\underset{1\leq n\leq N}{\mathbb{E}}G_{\varepsilon}(\\{a_{1}(n)\\})T^{\lfloor
a_{1}(n)\rfloor}(f_{1}\cdot T^{\lfloor c\rfloor+z}f_{2})\cdot T^{\lfloor
a_{3}(n)\rfloor}f_{3}\dots T^{\lfloor
a_{k}(n)\rfloor}f_{k}\big{\rVert}_{L^{2}(\mu)}\leq 16\varepsilon+o_{N}(1).$

If we take $\varepsilon\to 0$, we deduce that it suffices to establish that
the averages

$\underset{1\leq n\leq N}{\mathbb{E}}F(\\{a_{1}(n)\\})\ T^{\lfloor
a_{1}(n)\rfloor}(f_{1}\cdot T^{\lfloor c\rfloor+z}f_{2})\dots T^{\lfloor
a_{k}(n)\rfloor}f_{k}$

converge for any continuous function $F$ on $\mathbb{T}$. After approximating
$F$ by trigonometric polynomials, it suffices to show that the averages

$\underset{1\leq n\leq N}{\mathbb{E}}e(k_{1}a_{1}(n))\ T^{\lfloor
a_{1}(n)\rfloor}(f_{1}\cdot T^{\lfloor c\rfloor+z}f_{2})\dots T^{\lfloor
a_{k}(n)\rfloor}f_{k}$

converge in norm for any $k_{1}\in\mathbb{Z}$.

Proceeding inductively, our problem reduces to the following: let
$a_{1},...,a_{k}$ satisfy (3) or (4) and let $b_{1},...,b_{m}$ be a subset of
$\\{a_{1},...,a_{k}\\}$, so that the functions $b_{1},...,b_{m}$ satisfy the
hypothesis of Proposition 4.2. Then, for any integers $l_{1},...l_{k}$, the
averages

$\underset{1\leq n\leq N}{\mathbb{E}}e(l_{1}a_{1}(n)+\dots+l_{k}a_{k}(n))\
T^{\lfloor b_{1}(n)\rfloor}f_{1}\dots T^{\lfloor b_{m}(n)\rfloor}f_{m}$

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

Now we can apply Proposition 4.2 and use a standard telescopic argument to
show that the limiting behavior of the above averages does not change if we
replace the functions $f_{i}$ by their projections to the factor $Z_{s}(X)$
(the number $s$ is the one given by Proposition 4.2). However, by Theorem D,
the factors $Z_{s}(X)$ are inverse limits of $s$-step nilsystems. Thus, by
another standard limiting argument, we may reduce to the case that the space
$X$ is a nilmanifold and $\mu$ is its Haar measure, while the transformation
$T$ is the action (by left multiplication) of an element $g$ on $X$. Finally,
we can approximate the functions $f_{i}$ by continuous functions and reduce
our problem to the following:

If $X=G/\Gamma$ is a nilmanifold with $g\in G$ and the functions
$a_{1},...,a_{k},b_{1},...,b_{m}\in\mathcal{H}$ are as above, then for any
continuous functions $f_{1},...,f_{m}$ the averages

$\underset{1\leq n\leq N}{\mathbb{E}}e(l_{1}a_{1}(n)+\dots+l_{k}a_{k}(n))\
f_{1}(g^{\lfloor b_{1}(n)\rfloor}x)\dots f_{k}(g^{\lfloor b_{m}(n)\rfloor}x)$

converge in mean.

We can show that these averages converge pointwise for every $x\in X$. First
of all, it suffices to show that the averages

$\underset{1\leq n\leq N}{\mathbb{E}}e(l_{1}a_{1}(n)+\dots+l_{k}a_{k}(n))\
f_{1}(g^{{b_{1}(n)}}x)\dots f_{k}(g^{{b_{m}(n)}}x)$

converge pointwise, where $X=G/\Gamma$ is such that $G$ is connected, simply
connected nilpotent Lie group. This follows by standard modifications in the
argument of Lemma 3.5, so we omit the details.

Now, observe that we can write the above averages in the form

$\underset{1\leq n\leq
N}{\mathbb{E}}F_{0}(g_{0}^{l_{1}a_{1}(n)+\dots+l_{k}a_{k}(n)}\tilde{x})\
F_{1}(\tilde{g}^{{b_{1}(n)}}\tilde{x})\dots
F_{k}(\tilde{g}^{{b_{m}(n)}}\tilde{x}),$

where $g_{0}=(1_{\mathbb{T}},e_{G})$ and $\tilde{g}=(1_{\mathbb{T}},g)$ act on
the product nilmanifold $\mathbb{T}\times X$, the point $\tilde{x}$ is just
$(\mathbb{Z},x)$ and the functions $F_{i}$ are defined by

$F_{0}(y\mathbb{Z},a\Gamma)=e(y)\ \text{ and
}F_{i}(y\mathbb{Z},a\Gamma)=f_{i}(a\Gamma)\ \text{for }i\geq 1.$

These are continuous functions on $\mathbb{T}\times X$. The functions
$l_{1}a_{1}(t)+\dots+l_{k}a_{k}(t),b_{1}(t),...,b_{m}(t)$ satisfy the
hypotheses of Theorem 1.2 and the result follows. ∎

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