# On recurrence for $\mathbb{Z}^{d}$-nilsystems

Sebastián Donoso  Departamento de Ingeniería Matemática and Centro de
Modelamiento Matemático, Universidad de Chile and IRL-CNRS 2807, Beauchef 851,
Santiago, Chile<EMAIL_ADDRESS>,  Felipe Hernández  Department of
Mathematics, Ecole polytechnique fédérale de Lausanne, Rte Cantonale, 1015
Lausanne, Switzerland<EMAIL_ADDRESS>and  Alejandro Maass
Departamento de Ingeniería Matemática, Centro de Modelamiento Matemático and
Millennium Institute Center for Genome Regulation, Universidad de Chile and
IRL-CNRS 2807, Beauchef 851, Santiago, Chile<EMAIL_ADDRESS>
###### Abstract.

We study the topological recurrence phenomenon of actions of locally compact
abelian groups on compact metric spaces. We address the popular Katznelson’s
question in this context and extend recurrence lifting results, known for
$\mathbb{Z}$-actions, to this more general context. In the case
$H=\mathbb{Z}^{d}$, we develop new techniques to analyze Bohr recurrence sets,
and show that Bohr recurrence implies recurrence for a
$\mathbb{Z}^{d}$-nilsystem $(G/\Gamma,\mathbb{Z}^{d})$, under the assumption
that the connected component of the identity in $G$ is abelian. This family
encompasses, for example, all $\mathbb{Z}^{d}$-affine nilsystems.

###### 2020 Mathematics Subject Classification:

37B05, 37B20 (Primary), 77A46 (Secondary)

The first and third authors were partially funded by Centro de Modelamiento
Matemático (CMM) FB210005, BASAL funds for centers of excellence from ANID-
Chile. The first author was partially funded by ANID/Fondecyt/1200897. The
third author was funded by grant ICN2021-044 from the ANID Millennium Science
Initiative. All authors are part of the ECOS-ANID grant C21E04 (ECOS210033)

##  1\. Introduction

The study of the phenomenon of recurrence in ergodic theory has a rich
history. Its origins can be traced back to H. Poincaré’s work in the late 19th
century on the three-body problem, where he discovered that trajectories in
certain dynamical systems tend to return to their starting points even in
chaotic or unpredictable scenarios. Poincaré’s Recurrence Theorem formalizes
this observation, stating that for a measure-preserving transformation $T$ on
a probability space $(X,\mathcal{X},\mu)$ and a set $A\in\mathcal{X}$ with
$\mu(A)>0$, for almost any point $x\in A$ there exists a sequence
$(n_{i})_{i\in\mathbb{N}}$ such that $T^{n_{i}}x\in A$ for all
$i\in\mathbb{N}$.

The recurrence phenomenon has served to prove various deep results in
different areas of mathematics. Since the works of Furstenberg [8], and
Furstenberg and Weiss [9], there has been a profound and fruitful connection
between dynamical systems and number theory through recurrence properties,
and, more specifically, the notion of set of recurrence (see 3.1) playing a
central role in both areas. Within this context, the class of nilsystems has
proved to be particularly important. Nilsystems have been used, for example,
in the ergodic context of multiple convergence along arithmetic progressions
[14] and in the topological context to construct explicit examples of sets
with multiple recurrence [6, 17].

In our study, we focus on sets of Bohr recurrence, which are sets that exhibit
recurrence for all rotations111Rotations are a subclass of nilsystems, see
Section 2.2. Since rotations represent only a narrow subclass of topological
dynamical systems, it is reasonable to expect that sets of topological
recurrence form a narrow subclass of sets of Bohr recurrence. The extent to
which this expectation holds remains an important open problem, known as
Katznelson’s question. Katznelson’s question, popularized by Katznelson [18]
for $\mathbb{Z}$-actions, asks whether every set of Bohr recurrence is also a
set of topological recurrence.

The question has deep historical and mathematical connections to combinatorics
and harmonic analysis. Currently, there is no clear consensus among experts
regarding a positive or negative answer. For a negative answer, only a few
potential examples have been identified, such as sets known to be sets of Bohr
recurrence but with unknown dynamical recurrence properties [11, 7]. For a
positive answer, possible strategies include the use of general structure
theorems in topological dynamics and separate the analysis for factors with
specific properties (such as equicontinuity, weakly mixing, etc). For
instance, every minimal system can be obtained through proximal and
equicontinuous extensions from a strictly PI system (for background see [2]
Chapter 14 or [4] Chapter 6), or by passing through maximal $d$-step
nilsystems and their maximal distal factor (see [12] and [15] for background).
These decomposition methods have shown promising results for $\mathbb{Z}$,
such as lifting Bohr recurrence to arbitrary minimal
$\mathbb{Z}$-pronilsystems [16]. Unfortunately, Glasscock, Koutsogiannis and
Richter constructed in [10, Theorem B] an example of a system and a set that
is of recurrence for its maximal equicontinous factor, but not of recurrence
for the system itself. So, one cannot always reduce the study of recurrence to
a special factor. 222We warn the reader that the aformentioned set is not of
Bohr recurrence. It is of recurrence only for the particular factor
considered.

The main purpose of this article is to study Katznelson’s question outside the
context of $\mathbb{Z}$-actions. For general locally compact abelian group
actions, we begin in Section 3 defining Bohr recurrence for abelian locally
compact groups, using theory of duality and proving that this definition is
equivalent to the usual definition using equicontinuous systems. Then, we
state Katznelson’s questions and we finish Section 3 proving that inverse
limits and proximal extensions lift recurrence (a result that was already
proved for $H=\mathbb{Z}$ by Host, Kra, and Maass in [16]).

In Section 4 we study sets of Bohr recurrence for $\mathbb{Z}^{d}$ group
actions, introducing the notion of essential sets of Bohr recurrence, which
are sets of Bohr recurrence in which every element has no zero coordinates
(see 4.2). We give examples and prove many properties of such sets. Then, we
introduce the notion of Bohr correlations (see 4.4), with which we develop
techniques to classify sets of $\mathbb{Z}^{d}$-Bohr recurrence. In
particular, a subclass of sets of great importance is the class of sets of
Bohr recurrence with the property of complete independence that corresponds to
the case in which the coordinates of a set of Bohr recurrence are correlated
either by $0$ or by an irrational number (see 4.6).

Our main result comes in Section 5, where we generalize the theorem of Host,
Kra, and Maass on minimal $\mathbb{Z}$-nilsystems to $\mathbb{Z}^{d}$-quasi-
affine nilsystems, which are nilsystems $(X=G/\Gamma,T_{1},\ldots,T_{d})$
whose connected component of the identity in $G$ is abelian.

######  Theorem A.

Let $R\subseteq\mathbb{Z}^{d}$ be a set of $\mathbb{Z}^{d}$-Bohr recurrence.
Then, for every integer $s\geq 1$, $R$ is a set of recurrence for every
minimal $s$-step $\mathbb{Z}^{d}$-quasi-affine nilsystem.

The key ingredient to prove A is the following result of independent interest
in sets of Bohr recurrence. The proof, together with the necessary
definitions, is given in Section 4.

######  Theorem B.

Let $R$ be an essential and ordered set of $\mathbb{Z}^{d}$-Bohr recurrence
with the property of complete independence. Then, for all $\epsilon>0$ there
exists a set of Bohr recurrence $R_{\epsilon}\subseteq R$ such that for any
$n\in R_{\epsilon}$ and $w_{1},\ldots,w_{d}\in\mathbb{T}^{r}$ there is
$y\in\mathbb{R}^{r}$ such that

$\displaystyle||y||_{\mathbb{R}^{r}}<\epsilon\leavevmode\nobreak\
\leavevmode\nobreak\ \text{ and }\leavevmode\nobreak\ \leavevmode\nobreak\
||(n_{1}y,\ldots,n_{d}y)-(w_{1},\ldots,w_{d})||_{\mathbb{T}^{d\cdot
r}}<\epsilon.$

This result is key to approximate fibers in $\mathbb{Z}^{d}$-affine $s$-step
nilsystems, allowing us to lift recurrence from lower-step nilfactors.

While our method works well within the $\mathbb{Z}^{d}$-quasi-affine nilsystem
family, extending it to general nilsystems remains a challenge. Currently, we
are exploring a direct proof using spectral methods for the general case. To
be more precise, a generalization of Theorem 1.4 from [1] to
$\mathbb{Z}^{d}$-actions would be key for a measurable proof of the general
case.

##  2\. Notation, terminology, and prerequisites

###  2.1. Topological Dynamical Systems

A topological dynamical system, or just a $H$-system, is a pair $(X,H)$ where
$X$ is a compact metric space and $H$ is a locally compact group of
homeomorphisms of the space $X$ into itself, which for our purposes we will
suppose abelian. The action of an element $g\in H$ on a point $x\in X$ is
denoted by $g(x)$ or $gx$.

A topological dynamical system $(X,H)$ is transitive if for every nonempty
open sets $U,V\subseteq X$, there is $h\in H$ such that $U\cap
h^{-1}V\neq\emptyset$. The orbit of a point $x$ under the action of $H$ is
represented by $\mathcal{O}_{H}(x)=\\{hx:h\in H\\}$ (or simply
$\mathcal{O}(x)$ if the acting group is clear).

A stronger condition than transitivity is minimality. A system $(X,H)$ is said
to be minimal if every point in $X$ is transitive. In other words, there are
no nonempty closed $H$-invariant proper subsets of $X$. A subset $Y\subseteq
X$ is $H$-invariant if $HY=Y$.

We say that $(Y,H)$ is a subsystem of $(X,H)$ if $Y$ is a nonempty closed
$H$-invariant subset of $X$. An essential fact is that every dynamical system
has a minimal subsystem.

A system $(X,H)$ is equicontinuous if for all $\epsilon>0$ there exists
$\delta>0$ such that for every $x,y\in X$, $d_{X}(x,y)<\delta$ implies
$d_{X}(hx,hy)<\epsilon$, for all $h\in H$ ($d_{X}$ denotes a metric in $X$). A
pair $(x,y)\in X\times X$ is called proximal if there exists a sequence
$(h_{n})_{n\in\mathbb{N}}\subseteq H$ such that $d(h_{n}x,h_{n}y)\to 0$ as $n$
goes to infinity. If there is no such sequence, the pair $(x,y)$ is said to be
distal. A system is distal if it does not have any nontrivial proximal pairs.

A factor map between topological dynamical systems $(X,H)$ and $(Y,H)$ is an
onto continuous map $\pi\colon X\to Y$ such that $\pi\circ h=h\circ\pi$ for
all $h\in H$. In this case, $X$ is called an extension of $Y$ and $Y$ a factor
of $X$. If $\pi$ is also injective, we say that $(X,H)$ and $(Y,H)$ are
topologically conjugate (or just conjugate), and $\pi$ is called a conjugacy.

Let $\\{(X_{n},H)\\}_{n\in\mathbb{N}}$ be a sequence of topological dynamical
systems such that for each $n\in\mathbb{N}$ there is a factor map
$\pi_{n+1}:(X_{n+1},H)\to(X_{n},H)$. The inverse limit of
$\\{(X_{n},H)\\}_{n\in\mathbb{N}}$ of these factors is defined as the set

$\Bigl{\\{}(x_{n})_{n\in\mathbb{N}}\in\prod_{n\in\mathbb{N}}X_{n}\leavevmode\nobreak\
:\leavevmode\nobreak\ \pi_{n+1}(x_{n+1})=x_{n}\leavevmode\nobreak\ \text{ for
all }n\in\mathbb{N}\Bigr{\\}},$

equipped with the product topology and the diagonal action of $H$ given by
$h(x_{n})_{n\in\mathbb{N}}=(hx_{n})_{n\in\mathbb{N}}$ for every $h\in H$.

A measure preserving system (m.p.s. for short) $(X,\mathcal{X},\mu,H)$
consists of a probability space $(X,\mathcal{X},\mu)$ endowed with a
$\mathcal{X}$-$\mathcal{X}$-measurable and measure preserving action of $H$.
That is, for any $A\in\mathcal{X}$ and $h\in H$, $hA\in\mathcal{X}$ and
$\mu(hA)=\mu(A)$. A set $A\in\mathcal{X}$ is invariant if $\mu(A\Delta hA)=0$
for all $h\in H$. A m.p.s. $(X,\mu,H)$ is called ergodic if there are no
nontrivial invariant sets, meaning that if $A\in\mathcal{X}$ and $\mu(A\Delta
hA)=0$ for all $h\in H$, then $\mu(A)\in\\{0,1\\}$.

###  2.2. Nilsystems

Let $G$ be a group. For $a,b\in G$ we define their commutator as
$[a,b]=aba^{-1}b^{-1}$. Given subgroups $A,B\subseteq G$, we write $[A,B]$ to
denote the subgroup of $G$ generated by $\\{[a,b]:a\in A,b\in B\\}$. We define
recursively the commutator subgroups $G_{j}$ of $G$ as:

$G_{1}=G,\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
G_{j+1}=[G,G_{j}],\leavevmode\nobreak\ \leavevmode\nobreak\ j\geq 1.$

We say that a group $G$ is nilpotent of order $s$ or $s$-nilpotent if
$G_{s+1}=\\{e_{G}\\}$, where $e_{G}$ is the identity of $G$.

Let $G$ be a $s$-nilpotent Lie group and $\Gamma$ a discrete cocompact
subgroup of $G$ (i.e., $\Gamma$ is countable and $G/\Gamma$ is compact). Then,
we say that the compact nilmanifold $X=G/\Gamma$ is an $s$-step nilmanifold.
Let $H\subseteq G$ be a subgroup and consider the action of $H$ given by left
translations. Then $(X,H)$ is a $H$-system, also called a $s$-step nilsystem,
or to be more precise, a $s$-step $H$-nilsystem. We will always consider the
Haar measure $\mu$ of $(X,H)$ (the unique invariant measure for the action of
left translations).

######  Example 2.1 (Rotations).

Let $X$ be a compact abelian group, $H$ be a group, and $\varphi\colon H\to X$
be a homomorphism. Consider the action of $H$ on $X$ induced by $\varphi$,
that is, $h\cdot x=\varphi(h)+x$, for all $h\in H$ and $x\in X$, where the
group operation of $X$ is written in additive notation. Then $(X,H)$ is a
$1$-step nilsystem, and this system is usually called a rotation.

For a nilpotent Lie group $G$ we denote by $G_{0}$ the connected component of
the identity $e_{G}$ in $G$. Then, $G_{0}$ is an open normal subgroup of $G$
(see [16, Section 4.1]). Various representations of nilsystems are available,
some of them specifically tailored to address specific problems and contexts.
One possible choice allows us to assume that $G$ is simply connected, which
means that $G_{0}$ is simply connected (see [15, Chapter 10, Theorem 13]).
Another property of nilsystems is that they are all distal ([3, Chapter 4,
Theorem 3]).

Now we give some important structural properties of nilsystems that will be
useful in the next chapters.

######  Theorem 2.2 ([21]).

Let $(X=G/\Gamma,H)$ be a $s$-step nilsystem.

  1. (1)

The groups $G_{j}$, $j\geq 2$, are connected. In particular, $G_{j}\subseteq
G_{0}$.

  2. (2)

Let $N=\langle G_{0},H\rangle$ and $Z=X/[N,N]$. Then, the action of $H$ is
ergodic with respect to its unique invariant measure if and only if $X$ is
minimal with respect to the action of $H$, and if and only if $Z$ is minimal
with respect to the action of $H$.

######  Proposition 2.3.

[15] Let $(X=G/\Gamma,H)$ be a $s$-step nilsystem. Then the canonical factor
$\pi\colon X\to X/G_{s}$ is open.

######  Corollary 2.4 ([20]).

Let $(X=G/\Gamma,H)$ be a $s$-step nilsystem with $X$ connected. Then, the
action of $H$ is ergodic with respect to its unique invariant measure on $X$
if and only if it is ergodic with respect to its unique invariant measure on
the maximal torus factor $X/[G_{0},G_{0}]$ of $X$.

The next theorem is a well-known result in the field of nilsystems and it is a
direct consequence of 2.2 (2).

######  Theorem 2.5.

A nilsystem $(X,H)$ is ergodic with respect to its unique invariant measure if
and only if it is minimal.

###  2.3. Quasi-Affine Systems

In this section, we introduce a central object of this work, the quasi-affine
systems. We begin by presenting a motivating proposition that lays the
groundwork for our subsequent definition.

Recall that an action of the additive group $\mathbb{Z}^{d}$ is given by $d$
commuting homeomorphisms of the space. A $s$-step $\mathbb{Z}^{d}$-nilsystem
$(X,T_{1},\ldots,T_{d})$ is said to be $\mathbb{Z}^{d}$-affine if
$X=\mathbb{T}^{r}$ for some $r\geq 1$ and the transformations
$T_{1},\ldots,T_{d}$ are defined by $T_{i}(x)=A_{i}x+\alpha_{i}$, where
$A_{1},\ldots,A_{d}$ are commuting unipotent matrices and
$\alpha_{1},\ldots,\alpha_{d}\in\mathbb{T}^{r}$. Frantzikinakis and Kra showed
in [5] that $\mathbb{Z}^{d}$-affine nilsystems can be characterized by the
fact that $G_{0}$ is abelian.

######  Proposition 2.6 ([5]).

Let $X=G/\Gamma$ be a connected nilmanifold such that $G_{0}$ is abelian.
Then, any nilrotation $T_{a}(x)=ax$ on $X$ is topologically conjugate to a
unipotent affine transformation on some finite dimensional torus.

This motivates the following definition.

######  Definition 2.1.

Let $(X=G/\Gamma,T_{1},\ldots,T_{d})$ be a $s$-step $\mathbb{Z}^{d}$-nilsystem
(with $X$ non necessarily connected). We say that
$(X=G/\Gamma,T_{1},\ldots,T_{d})$ is quasi-affine if $G_{0}$ is abelian.

We will observe in subsequent sections that any minimal $s$-step
$\mathbb{Z}^{d}$-nilsystem can be regarded as a union of connected $s$-step
$\mathbb{Z}^{d}$-nilsystems. Consequently, $\mathbb{Z}^{d}$-quasi-affine
nilsystems are finite unions of $\mathbb{Z}^{d}$-affine nilsystems.

Furthermore, it is a known fact that for any $s$-step nilsystem $(X,H)$, the
closed orbit of any point $x\in X$ remains a $s$-step nilsystem (see for
instance [22, Theorem 1.3] or [15, Section 3.2, Chapter 10]). This property is
also true for quasi-affine nilsystems. To be precise, if
$(X=G/\Gamma,T_{1},\ldots,T_{d})$ is an $s$-step quasi-affine nilsystem and
$(Y=G^{Y}/\Gamma^{Y},T_{1},\ldots,T_{d})$ is an orbit of $X$, then in the
proof of Theorem 1.3 in [15] it is shown that $G_{0}^{Y}$ is a subgroup of
$G_{0}$ and therefore $G_{0}^{Y}$ is also abelian.

##  3\. Bohr Recurrence for locally compact abelian groups

This section focuses on the concept of recurrence for systems and families of
systems, together with the notion of Bohr recurrence, which is established for
locally compact abelian groups using Pontryagin duality. We extend many known
properties for $G=\mathbb{Z}$. Although these properties might be considered
classical, we were unable to find a proper reference in the literature, so we
include them for the sake of completeness. Finally, we state Katznelson’s
question in this context and generalize the fact that proximal extensions and
inverse limits lift up recurrence.

###  3.1. Bohr Recurrence

For a system $(X,H)$ and sets $U,V\subseteq X$ we denote $N_{H}(V,U)=\\{h\in
H:V\cap h^{-1}U\neq\emptyset\\}.$ In the case $U=V$ we just write
$N_{H}(U)=N_{H}(U,U)$, and in the case $V=\\{x\\}$ we write
$N_{H}(x,U)=\\{h\in H:hx\in U\\}.$

######  Definition 3.1.

A set $R\subseteq H$ is a set of recurrence for a system $(X,H)$ if for any
nonempty open set $U\subseteq X$, $R\cap N_{H}(U)\neq\emptyset$. Additionally,
if $\mathcal{F}$ is a family of $H$-systems, a set $R\subseteq H$ is a set of
recurrence for the family $\mathcal{F}$ if for any minimal system $(X,H)$ in
the family $\mathcal{F}$, $R$ is a set of recurrence for $(X,H).$

When we say that $R\subseteq H$ is a $H$-set of recurrence without specifying
any particular family, we will interpret that $R$ is a set or recurrence for
all $H$-systems.

######  Remark 3.1.

If $R$ is a set of recurrence for a system $(X,H)$, then there exist $x\in X$
and a sequence $(h_{n})_{n\in\mathbb{N}}\subseteq R$ such that $h_{n}x\to x$
as $n$ goes to $\infty$. Indeed, for every $n\in\mathbb{N}$, the set
$A_{n}=\\{x\in X:\inf_{h\in H}d(hx,x)<1/n\\}$ is open and dense, and therefore
any point in the $G_{\delta}$-denset set $\cap_{n\in\mathbb{N}}A_{n}$
satisfies the aforementioned property.

Now, we introduce the generalities for studying sets of recurrence in the
family of rotations. We begin by discussing duality in locally compact abelian
groups in order to define Bohr recurrence. For the purposes of this
discussion, we will denote the unit circle by
$\mathbb{S}^{1}=\\{z\in\mathbb{C}:|z|=1\\}$.

Given a locally compact abelian group $H$, a character of $H$ is a continuous
homomorphism $\chi\colon H\to\mathbb{S}^{1}$, and the set $\widehat{H}$ of
characters of $H$ is an abelian group under pointwise multiplication. For a
compact subset $F\subseteq H$ and $\epsilon>0$, let
$P(F,\epsilon)=\\{\chi\in\widehat{H}:|\chi(h)-1|<\epsilon,\forall h\in F\\}$.
The sets $P(F,\epsilon)$, where $F$ and $\epsilon$ range over all compact sets
and positive numbers respectively, is an open basis at the identity
$e_{\widehat{H}}$, and $\widehat{H}$ is a locally compact abelian topological
group ([13, Theorem 23.15]).

Let $H$ be a locally compact abelian group and consider
$\chi_{1},\ldots,\chi_{d}\in\widehat{H}$. The Bohr neighborhood of $e_{H}$ or
just $0$ in $H$ determined by $\\{\chi_{1},\ldots,\chi_{d}\\}$ and radius
$\epsilon>0$ is given by

$\text{Bohr}(\chi_{1},\ldots,\chi_{d};\epsilon)=\\{h\in
H:|\chi_{i}(h)-1|<\epsilon,\leavevmode\nobreak\ \leavevmode\nobreak\ \forall
i\in\\{1,\ldots,d\\}\\}.$

######  Remark 3.2.

If $H$ is compact, then as $\widehat{\widehat{H}}$ can be identified with $H$
(see [13, Theorem 24.3]), the topology generated by the basis of $e_{H}$ given
by the sets $Bohr(\chi_{1},\ldots,\chi_{d};\epsilon)$, for
$\chi_{1},\ldots,\chi_{d}\in\widehat{H}$ and $\epsilon>0$, coincides with the
topology in $H$.

Let $H$ be a locally compact abelian group. We say that $V\subseteq H$ is a
Bohr0 set if $V$ contains a Bohr neighborhood of $0$. Additionally, we say
that $W$ is a set of Bohr recurrence (denoted as $W\in$Bohr${}_{0}^{*}$) if
for every $V\in\text{Bohr}_{0}$, $W\cap V\neq\emptyset$. The family of Bohr0
sets is a filter, which means that it is upward closed and closed under
intersections. Additionally, the family of Bohr${}_{0}^{*}$ sets is partition
regular, this is, for every $A\in\text{Bohr}_{0}^{*}$, if we partition $A$
into $N$ sets $(A_{n})_{n=1}^{N}$, then there is a $n\in\\{1,\ldots,N\\}$ such
that $A_{n}\in\text{Bohr}_{0}^{*}$. This is well-known for the case
$G=\mathbb{Z}$ (see for example [10]) but the proof is identical for the
general case.

A classic example of a Bohr0 set which is not trivial is the following.

######  Proposition 3.3.

Let $(X,H)$ be a rotation and consider $x\in X$ and $U\subseteq X$ a
neighborhood of $x$. Then, $N(x,U)$ is a Bohr0 set.

###### Proof.

Without loss of generality we may assume that $x=e_{X}$, given that every
neighborhood of $x$ is a translation of a neighborhood of $e_{X}$. Let
$\varphi\colon H\to X$ be the homomorphism associated to the action of $H$ on
$X$. For an open neighborhood $U$ of $e_{X}$ we have to prove that
$N(e_{X},U)=\\{h\in H:\varphi(h)\in U\\}=\varphi^{-1}(U)$ is a Bohr0 set. By
3.2, there exist $\chi_{1},\ldots,\chi_{d}\in\widehat{X}$ and $\epsilon>0$ are
such that $Bohr(\chi_{1},\ldots,\chi_{d};\epsilon)\subseteq U$ from which we
deduce
$Bohr(\chi_{1}\circ\varphi,\ldots,\chi_{d}\circ\varphi;\epsilon)\subseteq\varphi^{-1}(U)$.
This means that $N(e_{X},U)$ is a Bohr0 set and we conclude. ∎

As any orbit in an equicontinuous system is, modulo conjugacy, a minimal
rotation, we can extend 3.3 to any equicontinuous system. In particular, we
have that sets of Bohr recurrence are precisely the family of sets of
recurrence for equicontinuous systems and rotations.

###  3.2. Katznelson’s question

In this section, we state Katznelson’s question regarding general group
actions.

Katznelson’s question: Given an action $H$, is every set of Bohr recurrence a
set of (topological) $H$-recurrence?

Notice that in the context of nonabelian groups, one could still formulate
Katznelson’s problem replacing Bohr recurrence by equicontinuous recurrence.
That is, say that $S\subseteq H$ is a set of equicontinuous recurrence if for
any minimal equicontinous system $(X,H)$ and nonempty open set $U\subseteq X$,
there exists $g\in S$ such that $U\cap g^{-1}U\neq\emptyset$. For abelian
groups, as we have seen, equicontinuous recurrence is equivalent to Bohr
recurrence. However, one key advantage of equicontinuous recurrence is that it
does not rely on the theory of characters, which is not readily available for
nonabelian groups. The Katznelson’s problem in this context would be asking if
sets of equicontinuous recurrence are necessarily sets of recurrence. It is
easy to construct groups for which this formulation of Kaztnelson problem is
false, as the following example shows for instance.

######  Example 3.4.

Consider a nonrecurrent map $T$ on the circle (e.g., a north-south map) and
let $S$ be an irrational rotation. We have that $(X,\langle T,S\rangle)$ is a
minimal system and the set $\mathbb{N}\times\\{0\\}$ (here we identify this
set with $\\{T^{n}:n\in\mathbb{N}\\}$) is a set of equicontinous recurrence
but not of recurrence for the system $(X,\langle T,S\rangle)$.

For that reason we restrict ourselves to abelian groups and focus our
attention on $\mathbb{Z}^{d}$, $d\geq 2$ later on. As we pointed out in the
introduction, a strategy to give a positive answer to Katznelson’s question
for $\mathbb{Z}$-actions is to prove that Bohr recurrence can be lifted up
through certain types of extensions for particular classes of systems. This
idea also holds for $H$-actions in our context, and the following theorems
show that recurrence can be lifted up through inverse limits and proximal
extensions.

######  Proposition 3.5.

Let $\\{(X_{n},H)\\}_{n\in\mathbb{N}}$ be a collection of minimal topological
dynamical systems with factor maps $\pi_{n+1}\colon(X_{n+1},H)\to(X_{n},H)$
for $n\in\mathbb{N}$. Suppose that $R$ is a set of recurrence for $(X_{n},H)$
for all $n\in\mathbb{N}$. Then, it is also a set of recurrence for the inverse
limit of these systems with respect to the factors.

######  Proposition 3.6.

Let $\pi\colon(X,H)\to(Y,H)$ be a proximal extension between minimal systems
and let $R$ be a set of recurrence for $(Y,H)$. Then, $R$ is a set of
recurrence for $(X,H)$.

As the proofs of these theorems are highly similar to the original proofs in
[16], we defer them to Appendix A.

Unlike proximal extensions and inverse limits, it is not easy to generalize
the fact that sets of $\mathbb{Z}$-Bohr recurrence are set of recurrence for
$s$-step $\mathbb{Z}$-nilsystems. In the next sections, we focus on doing this
for the family of $\mathbb{Z}^{d}$-quasi affine nilsystems by developing
certain techniques over sets of $\mathbb{Z}^{d}$-Bohr recurrence.

##  4\. Bohr Recurrence for $\mathbb{Z}^{d}$-actions

In this section we establish several properties of sets of
$\mathbb{Z}^{d}$-Bohr recurrence, introducing the notion of Bohr correlations.

###  4.1. Basic properties of sets of $\mathbb{Z}^{d}$-Bohr recurrence

For $d\in\mathbb{N}$, we denote by $[d]=\\{1,\ldots,d\\}$ and for
$x\in\mathbb{R}$ we denote the maximum integer $z$ such that $z\leq x$ by
$\lfloor x\rfloor$, the decimal part of $x$ by $\\{x\\}=x-\lfloor
x\rfloor\in[0,1)$, and the torus norm of $x$ by $\left\lVert
x\right\rVert_{\mathbb{T}}=\min\\{\\{x\\},1-\\{x\\}\\}$, which is the distance
of $x$ to the nearest integer. We generalize this notation for
$x\in\mathbb{R}^{r}$ as follows:

$\lfloor x\rfloor=(\lfloor x_{i}\rfloor)_{i\in[r]},\leavevmode\nobreak\
\leavevmode\nobreak\ \\{x\\}=(x_{i}-\lfloor
x_{i}\rfloor)_{i\in[r]}\in[0,1)^{r},\leavevmode\nobreak\ \leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ \leavevmode\nobreak\ \left\lVert
x\right\rVert_{\mathbb{T}^{r}}=\sum_{i=1}^{r}\left\lVert
x_{i}\right\rVert_{\mathbb{T}}.$

When it is clear, we will simply use $\left\lVert x\right\rVert$ instead of
$\left\lVert x\right\rVert_{\mathbb{T}^{r}}$. With this notation we observe
that $V\subseteq\mathbb{Z}^{d}$ is a Bohr0 set if there exist $\epsilon>0$,
$d\in\mathbb{N}$ and $\alpha_{1},\ldots,\alpha_{d}\in\mathbb{T}^{d}$ such that

$\\{n\in\mathbb{Z}^{d}:\left\lVert
n_{i}\alpha_{i}\right\rVert<\epsilon,\forall i\in[d]\\}\subseteq V.$

######  Definition 4.1 (Ramsey property).

A property is Ramsey if for any set $R\subseteq\mathbb{Z}^{d}$ that has this
property and any partition $R=A\cup B$, at least one of $A$ or $B$ has this
property.

The Ramsey property is just the combinatorial name that receives the quality
of a family being partition regular. Therefore, we have the following
proposition regarding the Ramsey property in the family of sets of
$\mathbb{Z}^{d}$-Bohr recurrence.

######  Proposition 4.1.

The sets of $\mathbb{Z}^{d}$-Bohr recurrence have the Ramsey property.

Analogously to the one dimensional case where sets of recurrence are usually
considered in $\mathbb{N}$ or $\mathbb{Z}\setminus\\{0\\}$ to avoid $0$, we
introduce the following definition.

######  Definition 4.2 (Essentiality).

Let $R\subseteq\mathbb{Z}^{d}$ be a set of $\mathbb{Z}^{d}$-Bohr recurrence.
We say that $R$ is essential if $\forall n\in R$, $\forall j\in[d]$,
$n_{j}\neq 0$.

From now one we make the abuse of notation of denote
$\pi:\mathbb{Z}^{d}\to\mathbb{Z}^{d}$ the permutation of coordinates induced
by a permutation $\pi:[d]\to[d]$.

The name “essential” is just to emphasize that this is the only relevant study
case. In fact, the next proposition shows that every relevant set of
$\mathbb{Z}^{d}$-Bohr recurrence can be reduced to an essential set of
$\mathbb{Z}^{d}$-Bohr recurrence.

######  Proposition 4.2.

Let $R\subseteq\mathbb{Z}^{d}\setminus\\{0\\}$ be a set of
$\mathbb{Z}^{d}$-Bohr recurrence. Then, there exist $d^{\prime}\leq d$, a
permutation $\pi\colon[d]\to[d]$, and an essential set of
$\mathbb{Z}^{d^{\prime}}$-Bohr recurrence
$R^{\prime}\subseteq\mathbb{Z}^{d^{\prime}}$ such that
$R^{\prime}\times\\{0\\}^{d-d^{\prime}}\subseteq\pi(R)$.

###### Proof.

For $J\subseteq[d]$ denote $R_{J}=\\{(n_{j})_{j\in[d]}\in R:n_{j}\neq 0\iff
j\in J\\}.$ Notice that, as $0\notin R$, we have that

$R=\bigcup_{\begin{subarray}{c}J\subseteq[d]\\\
J\neq\emptyset\end{subarray}}R_{J}.$

By the Ramsey property, there exists $J\subseteq[d]$ nonempty such that
$R_{J}$ is a set of Bohr $\mathbb{Z}^{d^{\prime}}$-recurrence, where
$d^{\prime}=|J|$. Now, consider a permutation
$\pi\colon\mathbb{Z}^{d}\to\mathbb{Z}^{d}$ taking the coordinates $J$ to
$\\{1,\ldots,d^{\prime}\\}$ maintaining their order and the set
$R^{\prime}=p(R_{J})$, where $p\colon\mathbb{Z}^{d}\to\mathbb{Z}^{J}$ is the
canonical projection. Then,
$R^{\prime}\times\\{0\\}^{d-d^{\prime}}\subseteq\pi(R)$ and $R^{\prime}$ is an
essential set of $\mathbb{Z}^{d^{\prime}}$-Bohr recurrence. ∎

In particular, we can always reduce to the case in which $R$ is essential when
studying recurrence in the family of nilsystems.

######  Corollary 4.3.

Every set of $\mathbb{Z}^{d}$-Bohr recurrence is a set of recurrence for the
family of $\mathbb{Z}^{d}$-nilsystems if and only if every essential set of
$\mathbb{Z}^{d}$-Bohr recurrence is a set of recurrence for the family of
$\mathbb{Z}^{d}$-nilsystems.

###### Proof.

We prove the nontrivial direction. Let $\mathscr{X}=(X,T_{1},\ldots,T_{d})$ be
a minimal nilsystem and $R\subseteq\mathbb{Z}^{d}$ be a set of
$\mathbb{Z}^{d}$-Bohr recurrence. We may assume $0\notin R$, otherwise the
result is immediate. By 4.2 there exist $d^{\prime}\leq d$, a permutation
$\pi\colon[d]\to[d]$ and an essential set of $\mathbb{Z}^{d^{\prime}}$-Bohr
recurrence $R^{\prime}\subseteq\mathbb{Z}^{d^{\prime}}$ such that
$R^{\prime}\times\\{0\\}^{d-d^{\prime}}\subseteq\pi(R)$. Consider the minimal
$s$-step nilsystem

$\mathscr{Y}=(\overline{O(e_{G})},T_{\pi(1)},\ldots,T_{\pi(d^{\prime})}).$

Notice that for all $n\in R^{\prime}$ if
$m\in\\{n\\}\times\\{0\\}^{d-d^{\prime}}\subseteq\pi(R)$ we have that

$T_{\pi(1)}^{n_{1}}\cdots
T_{\pi(d^{\prime})}^{n_{d^{\prime}}}=T_{1}^{m_{\pi^{-1}(1)}}\cdots
T_{d}^{m_{\pi^{-1}(d)}}.$

In this way, if $R^{\prime}$ is a set of $\mathbb{Z}^{d^{\prime}}$-Bohr
recurrence for $\mathscr{Y}$ then $R$ is a set of $\mathbb{Z}^{d}$-Bohr
recurrence for $\mathscr{X}$. ∎

The following property enables us to consider coordinates of an essential set
of $\mathbb{Z}^{d}$-Bohr recurrence with arbitrary magnitudes. This is the
intuitive generalization of the fact for sets of $\mathbb{Z}-$Bohr recurrence
we can always eliminate a finite amount of elements without losing the
property of Bohr recurrence.

######  Proposition 4.4 (Bands Property).

Let $R\subseteq\mathbb{Z}^{d}$ be a set of $\mathbb{Z}^{d}$-Bohr recurrence,
$k\in\mathbb{Z}\setminus\\{0\\}$ and $i\in[d]$. Set
$B_{k}^{i}=\\{\vec{n}\in\mathbb{Z}^{d}:n_{i}=k\\}$, then the set
$R_{0}=R\setminus B_{k}^{i}$ is a set of $\mathbb{Z}^{d}$-Bohr recurrence.

###### Proof.

We may assume $i=1$, since all the other cases are analogous. It suffices to
show that $\\{k\\}\times\mathbb{Z}^{d-1}$ is not a set of recurrence. Consider
the system $(X=\mathbb{Z}_{|k|+1},T_{1},\ldots,T_{d})$, where $T_{1}(x)=x+1$,
and $T_{j}={\rm id}$ for $j\geq 2$. It is immediate to check that
$\\{k\\}\times\mathbb{Z}^{d-1}$ it is not a set of recurrence for this system.
The Ramsey property allows us to conclude that $R_{0}$ is a set of recurrence,
and its essentiality is direct from that of $R$. ∎

######  Remark 4.5.

We remark that in the previous proof one can also construct systems where all
the transformations act in a non-trivial way.

Next, we prove that the property of being a set of $\mathbb{Z}^{d}$-Bohr
recurrence is invariant under isomorphisms of $\mathbb{Z}^{d}$.

######  Proposition 4.6.

Let $R\subseteq\mathbb{Z}^{d}$ be a set of $\mathbb{Z}^{d}$-Bohr recurrence
and $M$ an integer invertible matrix of dimension $d$. Then, the set defined
by

$M^{-1}R=\\{n=(n_{1},\ldots,n_{d})^{T}\in\mathbb{Z}^{d}:Mn\in R\\},$

is a set of $\mathbb{Z}^{d}$-Bohr recurrence.

###### Proof.

Consider a torus rotation $(\mathbb{T}^{r},T_{1},\ldots,T_{d})$, where
$T_{1},\ldots,T_{d}$ are the rotations associated to elements
$\alpha^{1},\ldots,\alpha^{d}\in\mathbb{R}^{r}$ respectively. Define
$\beta=M^{-T}(\alpha^{1},\ldots,\alpha^{d})^{T}\in\mathbb{R}^{d\times r}$.
Observe that

$V:=\\{m\in\mathbb{Z}^{d}\mid||m^{T}\beta||\leq\epsilon\leavevmode\nobreak\
\text{ and }\leavevmode\nobreak\ M^{-1}m\in\mathbb{Z}^{d}\\}$

defines a Bohr neighborhood of $0$, given that $M^{-1}$ is a matrix with
rational coefficients. Take $m\in R\cap V$ and define
$n=M^{-1}m\in\mathbb{Z}^{d}$. Then,

$m^{T}\beta=m^{T}M^{-T}(\alpha^{1},\ldots,\alpha^{d})^{T}=n^{T}(\alpha^{1},\ldots,\alpha^{d})^{T}=\sum_{i=1}^{d}n_{i}\alpha^{i}.$

In this way, we found $n\in M^{-1}R$ such that $||T_{1}^{n_{1}}\cdots
T_{d}^{n_{d}}x-x||=\left\lVert\sum_{i=1}^{d}n_{i}\alpha^{i}\right\rVert\leq\epsilon,$
concluding that $M^{-1}R$ is a set of $\mathbb{Z}^{d}$-Bohr recurrence. ∎

When dealing with sets of $\mathbb{Z}^{d}$-Bohr recurrence, it is common to
encounter situations where some coordinates are rationally dependent. For
instance, it is not difficult to demonstrate that if
$R\subseteq\mathbb{Z}^{2}$ is a set of Bohr recurrence, then so is
$R^{\prime}=\\{(n_{1},n_{2},n_{1}+n_{2})\in\mathbb{Z}^{3}\mid(n_{1},n_{2})\in
R\\}$. Naturally, the intrinsic recurrence behavior resides in the first two
coordinates, rendering the third coordinate seemingly redundant. The following
definition formalizes this concept, which will be important in the subsequent
section for the purpose of eliminating certain cases.

######  Definition 4.3 (Redundancy).

Let $R\subseteq\mathbb{Z}^{d}$ be a set of $\mathbb{Z}^{d}$-Bohr recurrence.
We say that $R$ is redundant if there is $v\in\mathbb{Q}^{d}\setminus\\{0\\}$
such that $R_{v}=\\{n\in R:v^{T}\cdot n=0\\}$ is a set of
$\mathbb{Z}^{d}$-Bohr recurrence.

######  Remark 4.7.

Notice that is equivalent to take $v\in\mathbb{Z}^{d}\setminus\\{0\\}$ in the
definition of redundancy.

To end this subsection, we prove that the property of being non-redundant is
preserved under the transformation previously described in 4.6.

######  Proposition 4.8.

Let $R\subseteq\mathbb{Z}^{d}$ be a non redundant set of $\mathbb{Z}^{d}$-Bohr
recurrence and $M$ be an invertible matrix with rational coefficients of
dimension $d$. Then, $M^{-1}R$ is non redundant as well.

###### Proof.

By contradiction, suppose that $M^{-1}R$ is redundant, then there exists
$v\in\mathbb{Q}^{d}\setminus\\{0\\}$ such that

$(M^{-1}R)_{v}=\\{n\in M^{-1}R:v^{T}\cdot n=0\\},$

is a set of $\mathbb{Z}^{d}$-Bohr recurrence. Hence, by 4.6, $M(M^{-1}R)_{v}$
is a set of $\mathbb{Z}^{d}$-Bohr recurrence. If we define

$R_{M^{-T}v}=\\{n\in R:(M^{-T}v)^{T}n=0\\},$

then $M(M^{-1}R)_{v}\subseteq R_{M^{-T}v}$ and therefore $R_{M^{-T}v}$ is a
set of $\mathbb{Z}^{d}$-Bohr recurrence. As
$M^{-T}v\in\mathbb{Q}^{d}\setminus\\{0\\}$, we have that $R$ is redundant,
which is a contradiction. ∎

###  4.2. Bohr Correlations

In this section we introduce the notion of Bohr correlation, which in simple
words is a slope of the line around which an specific set of Bohr recurrence
accumulates. To illustrate this notion first we provide some examples.

######  Example 4.9.

Let $R\subseteq\mathbb{Z}$ be a set of $\mathbb{Z}$-Bohr recurrence, then for
every $\alpha_{1},\ldots,\alpha_{d}\in\mathbb{R}$ we have that

$\tilde{R}=\\{(\lfloor n\alpha_{1}+1/2\rfloor,\ldots,\lfloor
n\alpha_{d}+1/2\rfloor)\in\mathbb{Z}^{d}:n\in R\\},$

is a set of $\mathbb{Z}^{d}$-Bohr recurrence.

In 4.9 the set of $\mathbb{Z}^{d}$-Bohr recurrence $\tilde{R}$ is
approximately near the line of slope $(\alpha_{1},\ldots,\alpha_{d})$, in the
sense that for $n\in\tilde{R}$ and $i,j\in[d]$ we have that
$n_{i}/n_{j}\approx\alpha_{i}/\alpha_{j}$ as $\lim_{n\to\infty}\lfloor
n\alpha_{i}+1/2\rfloor/\lfloor n\alpha_{j}+1/2\rfloor=\alpha_{i}/\alpha_{j}$.

###### Proof of 4.9.

Let $R\subseteq\mathbb{Z}$ be a set of $\mathbb{Z}$-Bohr recurrence and
consider $\alpha_{1},\ldots,\alpha_{d}\in\mathbb{R}$. Define

$\tilde{R}=\\{(\lfloor n\alpha_{1}+1/2\rfloor,\ldots,\lfloor
n\alpha_{d}+1/2\rfloor)\in\mathbb{Z}^{d}:n\in R\\}.$

Let $r\in\mathbb{N},$ $\epsilon\in(0,1)$ and
$\beta_{1},\ldots,\beta_{d}\in[0,1)^{r}$. Consider the Bohr neighborhood of
$0$

$B=\\{n\in\mathbb{Z}:\leavevmode\nobreak\ \left\lVert
n\alpha_{i}\right\rVert\leq\epsilon/2,\leavevmode\nobreak\ \left\lVert
n(\alpha_{i}\beta_{i})\right\rVert\leq\epsilon/2,\leavevmode\nobreak\ \forall
i\in[d]\\}.$

As the family of Bohr neighborhood of $0$ is a filter, we have that $R\cap B$
is a set of $\mathbb{Z}^{d}$-Bohr recurrence. Taking $n\in R\cap B$ we have
that for every $i\in[d]$

$\left\lVert\lfloor n\alpha_{i}+1/2\rfloor\beta_{i}\right\rVert=\left\lVert
n\alpha_{i}\beta_{i}+\beta_{i}/2-\\{n\alpha_{i}\\}\beta_{i}\right\rVert\leq\left\lVert
n\alpha_{i}\beta_{i}\right\rVert+\left\lVert
n\alpha_{i}\right\rVert\cdot|\beta_{i}|\leq\epsilon/2+\epsilon/2=\epsilon.$

Therefore $\tilde{R}$ is a set of $\mathbb{Z}^{d}$-Bohr recurrence. ∎

For the next example, we need the following proposition.

######  Proposition 4.10.

Let $R\subseteq\mathbb{Z}^{2}$. Denote $R_{1}=\\{n\in\mathbb{Z}:\exists
m\in\mathbb{Z},(n,m)\in R\\}$ and for $n\in R_{1}$ denote by
$R(n,\bullet)=\\{m\in\mathbb{Z}:(n,m)\in R\\}.$ Suppose
$R_{1}\subseteq\mathbb{Z}\setminus\\{0\\}$ is a set of $\mathbb{Z}$-Bohr
recurrence. For $n\in R_{1}$ denote $L_{n}$ the length of the largest interval
contained in $R(n,\bullet)$. If $L_{n}\to\infty$ as $n$ goes to infinity, then
$R$ is a set of Bohr recurrence.

###### Proof.

If $(X,g_{1},g_{2})$ is a minimal rotation, we can find an arbitrarily large
$n_{1}\in R_{1}$ such that $d(g_{1}^{n_{1}},e_{X})\leq\epsilon/2$. Observe
that the set $N_{g_{2}}(e_{G},B(e_{G},\epsilon/2))$ is syndetic (i.e., it has
bounded gaps) by the fact that the system $(X,g_{2})$ is distal, so
$(\overline{O_{g_{2}}(e_{G})},g_{2})$ is minimal. Taking $n_{1}$ such that
$L_{n_{1}}>L$ (with $L$ the syndetic constant associated to
$B(e_{G},\epsilon/2)$), we have that there exists $n_{2}\in R(n_{1},\bullet)$
such that $d(g_{2}^{n_{2}},e_{X})\leq\epsilon/2$. Thus, there exists
$(n_{1},n_{2})\in R$ such that
$d(g_{1}^{n_{1}}g_{2}^{n_{2}},e_{X})\leq\epsilon$. ∎

######  Example 4.11.

From 4.10, the set $R=\\{(n_{1},n_{2})\in\mathbb{N}^{2}:n_{1}^{2}\leq
n_{2}\leq 2n_{1}^{2}\\}$ is a set of Bohr recurrence such that is
approximately near a line of slope $\infty$, in the sense that for every
$\epsilon>0$, there is $M>0$ such that for all $(n_{1},n_{2})\in R\cap
B_{\mathbb{R}^{2}}(0,M)^{c}$ we have that $|n_{1}/n_{2}|<\epsilon.$

To formalize the notion illustrated in the examples, first we show that we can
reduce to the case in which the coordinates of $R$ are ordered. This is not
strictly necessary, but it makes the notation significantly easier to
manipulate.

We call $R\subseteq\mathbb{Z}^{d}$ an ordered set of $\mathbb{Z}^{d}$-Bohr
recurrence if $R$ is a set of $\mathbb{Z}^{d}$-Bohr recurrence such that

$\forall n=(n_{1},\ldots,n_{d})\in R,\leavevmode\nobreak\ \leavevmode\nobreak\
|n_{1}|\geq\cdots\geq|n_{d}|.$

######  Remark 4.12.

Let $R\subseteq\mathbb{Z}^{d}$ be a set of $\mathbb{Z}^{d}$-Bohr recurrence
and $(X,T_{1},\ldots,T_{d})$ a system. Consider $S_{d}$ the symmetric group
and notice that

$R=\bigcup_{\varphi\in
S_{d}}R\cap\\{(n_{1},\ldots,n_{d})\in\mathbb{Z}^{d}:|n_{\varphi(1)}|\geq\cdots\geq|n_{\varphi(d)}|\\}.$

By the Ramsey property, there is $\varphi\in S_{d}$ such that

$R^{\prime}=R\cap\\{(n_{1},\ldots,n_{d})\in\mathbb{Z}^{d}:|n_{\varphi(1)}|\geq\cdots\geq|n_{\varphi(d)}|\\},$

is a set of $\mathbb{Z}^{d}$-Bohr recurrence. Note that recurrence in
$(X,T_{1},\ldots,T_{d})$ with $R^{\prime}$ is equivalent to recurrence in
$(X,T_{\varphi(1)},\ldots,T_{\varphi(d)})$ with

$R^{\prime\prime}=\\{(n_{\varphi(1)},\ldots,n_{\varphi(d)}):(n_{1},\ldots,n_{d})\in
R^{\prime}\\},$

which is an ordered set of $\mathbb{Z}^{d}$-Bohr recurrence. Additionally,
notice that if $R$ is essential, then so is $R^{\prime\prime}$.

######  Definition 4.4 (Bohr Correlations).

Let $R\subseteq\mathbb{Z}^{d}$ be an essential and ordered set of
$\mathbb{Z}^{d}$-Bohr recurrence. We define a vector of Bohr correlations for
$R$ as $P=(P_{i,j})_{j\geq i}\in[-1,1]^{d\times(d+1)/2}$ such that
$\forall\epsilon>0$

$R_{P,\epsilon}=\Bigl{\\{}n\in R\leavevmode\nobreak\ :\leavevmode\nobreak\
\Bigl{|}\frac{n_{j}}{n_{i}}-P_{i,j}\Bigr{|}\leq\epsilon,\leavevmode\nobreak\
\forall j\geq i\Bigr{\\}},$

is a set of $\mathbb{Z}^{d}$-Bohr recurrence. We denote by $\mathcal{BC}(R)$
the set of Bohr correlations of $R$.

For example, for $d=2$, a vector of Bohr correlations is just the slope of a
line for which many points (a set of Bohr recurrence amount of points) are
inside a cone of angle $\epsilon$ around such line. Notice that it is not
always possible to take $\epsilon=0$ (for instance, this is not possible in
the of recurrence from 4.11).

The existence of Bohr correlations is not direct from the definition, so we
prove it now.

######  Proposition 4.13.

If $R\subseteq\mathbb{Z}^{d}$ is an essential and ordered set of
$\mathbb{Z}^{d}$-Bohr recurrence, then $\mathcal{BC}(R)\neq\emptyset$.

###### Proof.

Notice that $(\frac{n_{j}}{n_{i}})_{j\geq i}\in[-1,1]^{d(d+1)/2}$, $\forall
n\in R$. For $m\in\mathbb{N}$, we can cover $[-1,1]^{d(d+1)/2}$ with finitely
many intervals of the form $\prod_{j\geq
i}(P_{i,j}-\frac{1}{m},P_{i,j}+\frac{1}{m})$, for $(P_{i,j})_{j\geq
i}\in([-1,1]\cap\mathbb{Q})^{d(d+1)/2}$. As $R$ is essential and ordered, for
every $n\in R$ we have that $(\frac{n_{j}}{n_{i}})_{j\geq
i}\subseteq[-1,1]^{d(d+1)/2}$. Hence, using the Ramsey property (after turning
the aforementioned covering into a partition), we obtain
$P^{m}=(P_{i,j}^{m})_{j\geq i}\in([-1,1]\cap\mathbb{Q})^{d(d+1)/2}$ such that

$R\cap\Bigl{\\{}n\in\mathbb{Z}^{d}\leavevmode\nobreak\ :\leavevmode\nobreak\
\Bigl{|}\frac{n_{j}}{n_{i}}-P_{i,j}^{m}\Bigr{|}<\frac{1}{m},\leavevmode\nobreak\
\forall j\geq i\Bigr{\\}},$

is a set of $\mathbb{Z}^{d}$-Bohr recurrence. Passing to a subsequence, we can
suppose that $(P^{m_{l}})_{l\geq 1}$ converges to $P\in[-1,1]^{d(d+1)/2}$. We
claim that

$R_{P,\epsilon}=R\cap\Bigr{\\{}n\in\mathbb{Z}^{d}\leavevmode\nobreak\
:\leavevmode\nobreak\
\Bigl{|}\frac{n_{j}}{n_{i}}-P_{i,j}\Bigl{|}<\epsilon,\leavevmode\nobreak\
\forall j\geq i\Bigl{\\}},$

is a set of $\mathbb{Z}^{d}$-Bohr recurrence for all $\epsilon>0$. Indeed, as
$P^{m_{l}}\to P$, take $l$ large enough so that
$(P^{m_{l}}_{i,j}-1/m_{l},P^{m_{l}}_{i,j}+1/m_{l})\subseteq(P_{i,j}-\epsilon,P_{i,j}+\epsilon),\forall
j\geq i.$ Therefore,

$R\cap\Bigl{\\{}n\in\mathbb{Z}^{d}\leavevmode\nobreak\ :\leavevmode\nobreak\
\Bigl{|}\frac{n_{j}}{n_{i}}-P_{i,j}^{m_{l}}\Bigr{|}<\frac{1}{m_{l}},\leavevmode\nobreak\
\forall j\geq i\Bigr{\\}}\subseteq
R\cap\Bigl{\\{}n\in\mathbb{Z}^{d}\leavevmode\nobreak\ :\leavevmode\nobreak\
\Bigl{|}\frac{n_{j}}{n_{i}}-P_{i,j}\Bigl{|}<\epsilon,\leavevmode\nobreak\
\forall j\geq i\Bigr{\\}},$

and since the left-hand side set is a set of $\mathbb{Z}^{d}$-Bohr recurrence,
so is the right-hand side. We conclude that $P\in\mathcal{BC}(R)$. ∎

Now we prove some properties of Bohr correlations. First, we show that Bohr
correlations are consistent, in the sense that if $1\leq i\leq j\leq l\leq d$,
then the correlation between the $i$-th coordinate and the $l$-th coordinate
is the product of the correlations between the $i$-th coordinate and the
$j$-th coordinate, and the correlation between the $j$-th coordinate and the
$l$-coordinate.

######  Lemma 4.14.

Let $R\subseteq\mathbb{Z}^{d}$ be an essential and ordered set of
$\mathbb{Z}^{d}$-Bohr recurrence, and $P\in\mathcal{BC}(R)$. Then, for all
$i,j,l\in[d]$ with $i\leq j\leq l$, $P_{i,l}=P_{i,j}P_{j,l}.$

###### Proof.

Let $\epsilon>0$, as $R_{P,\epsilon}$ is a set of $\mathbb{Z}^{d}$-Bohr
recurrence we have that $\forall n\in R_{P,\epsilon}$

$\displaystyle|P_{i,l}-P_{i,j}P_{j,l}|$
$\displaystyle\leq\left|\frac{n_{l}}{n_{i}}-P_{i,l}\right|+\left|\frac{n_{j}}{n_{i}}\frac{n_{l}}{n_{j}}-P_{i,j}P_{j,l}\right|$
$\displaystyle\leq\epsilon+\left|\frac{n_{j}}{n_{i}}\frac{n_{l}}{n_{j}}-P_{i,j}\frac{n_{l}}{n_{j}}\right|+\left|P_{i,j}\frac{n_{l}}{n_{j}}-P_{i,j}P_{j,l}\right|$
$\displaystyle\leq\epsilon+\left|\frac{n_{l}}{n_{j}}\right|\epsilon+\left|P_{i,j}\right|\epsilon\leq
3\epsilon.$

Letting $\epsilon$ tend to $0$ we get that $P_{i,l}=P_{i,j}P_{j,l}$. ∎

In what follows, we define the notion of partial vector of Bohr correlations
and then we will show that such a partial vector can be extended to a vector
of Bohr correlations.

######  Definition 4.5.

Let $R\subseteq\mathbb{Z}^{d}$ be an essential and ordered set of
$\mathbb{Z}^{d}$-Bohr recurrence and $I\subseteq\\{(i,j)\in[d]^{2}:j\geq
i\\}$. We say that $P=(P_{i,j})_{(i,j)\in I}\in\mathbb{R}^{I}$ are $I$-Bohr
correlations if $\forall\epsilon>0$,

$R_{I,\epsilon}=\Bigl{\\{}n\in
R:\left|\frac{n_{j}}{n_{i}}-P_{i,j}\right|\leq\epsilon,\leavevmode\nobreak\
\forall(i,j)\in I\Bigr{\\}},$

is a set of $\mathbb{Z}^{d}$-Bohr recurrence.

######  Lemma 4.15.

Let $R\subseteq\mathbb{Z}^{d}$ be an essential and ordered set of
$\mathbb{Z}^{d}$-Bohr recurrence. Consider $I\subseteq\\{(i,j)\in[d]^{2}:j\geq
i\\}$ and $P=(P_{i,j})_{(i,j)\in I}\in[-1,1]^{I}$ an $I$-Bohr correlations.
Then, there exists $\tilde{P}\in\mathcal{BC}(R)$ that extends
$(P_{i,j})_{(i,j)\in I}$, in the sense that $\tilde{P}_{i,j}=P_{i,j}$,
$\forall(i,j)\in I$.

###### Proof.

For every $k\in\mathbb{N}$, take $P^{k}\in\mathcal{BC}(R_{I,\frac{1}{k}})$. As
$(P^{k})_{k\in\mathbb{N}}\subseteq[-1,1]^{d(d+1)/2}$, we can take a convergent
subsequence $(P^{k_{l}})_{l\in\mathbb{N}}$ to a vector
$\tilde{P}\in[-1,1]^{d(d+1)/2}$. We claim that $\tilde{P}\in\mathcal{BC}(R)$
and that $\tilde{P}_{i,j}=P_{i,j}$, $\forall(i,j)\in I$. Indeed, let
$\epsilon>0$ and take $l\in\mathbb{N}$ large enough so that
$|P_{i,j}^{k_{l}}-\tilde{P}_{i,j}|\leq\epsilon/2$, $\forall j\geq i$, and such
that $\frac{1}{k_{l}}\leq\epsilon$. Then, we have that for all
$n\in(R_{I,\frac{1}{k_{l}}})_{P^{k_{l}},\epsilon/2}\subseteq R$

$\left|\frac{n_{j}}{n_{i}}-\tilde{P}_{i,j}\right|\leq\left|\frac{n_{j}}{n_{i}}-P_{i,j}^{k_{l}}\right|+\left|P_{i,j}^{k_{l}}-\tilde{P}_{i,j}\right|\leq\epsilon,\leavevmode\nobreak\
\leavevmode\nobreak\ \forall j\geq i.$

Then, $(R_{I,\frac{1}{k_{l}}})_{P^{k_{l}},\epsilon/2}\subseteq
R_{\tilde{P},\epsilon}$, and as
$(R_{I,\frac{1}{k_{l}}})_{P^{k_{l}},\epsilon/2}$ is a set of
$\mathbb{Z}^{d}$-Bohr recurrence, we conclude that
$\tilde{P}\in\mathcal{BC}(R)$. In addition, for all
$n\in(R_{I,\frac{1}{k_{l}}})_{P^{k_{l}},\epsilon/2}\subseteq R$, and $(i,j)\in
I$,

$|P_{i,j}-\tilde{P}_{i,j}|\leq\left|\frac{n_{j}}{n_{i}}-P_{i,j}\right|+\left|\frac{n_{j}}{n_{i}}-\tilde{P}_{i,j}\right|\leq\frac{3}{2}\epsilon.$

Taking $\epsilon\to 0$ yields $P_{i,j}=\tilde{P}_{i,j}$, $\forall(i,j)\in I$.
This proves the lemma. ∎

Now, we define the property of complete independence, which comprises the case
where the correlations are not rational.

######  Definition 4.6.

Let $R\subseteq\mathbb{Z}^{d}$ be an essential and ordered set of
$\mathbb{Z}^{d}$-Bohr recurrence. We say that $R$ has the property of complete
independence, if there is $P\in\mathcal{BC}(R)$ such that for all $i\in[d]$,
$\\{P_{i,j}:j\geq i,P_{i,j}\neq 0\\},$ are rationally independent.

The main consequence of the property of complete independence is that it
allows us to approximate multiple coordinates with just one variable, which is
the statement of B. For the readers’ convenience, we state B again below.

######  Theorem 4.16.

Let $R$ be an essential and ordered set of $\mathbb{Z}^{d}$-Bohr recurrence
with the property of complete independence. Then, for all $\epsilon>0$ there
exists a set of Bohr recurrence $R_{\epsilon}\subseteq R$ such that for any
$n\in R_{\epsilon}$ and $w_{1},\ldots,w_{d}\in\mathbb{T}^{r}$ there is
$y\in\mathbb{R}^{r}$ such that

$\displaystyle||y||_{\mathbb{R}^{r}}<\epsilon\leavevmode\nobreak\
\leavevmode\nobreak\ \text{ and }\leavevmode\nobreak\ \leavevmode\nobreak\
||(n_{1}y,\ldots,n_{d}y)-(w_{1},\ldots,w_{d})||_{\mathbb{T}^{d\cdot
r}}<\epsilon.$

###### Proof.

Let $\epsilon>0$ and for $l\in[d]$ denote $I_{l}=\\{j\in[d]:j>l,P_{l,j}\neq
0\\}$. By the hypothesis of complete independence, $\\{P_{l,j}\\}_{j\in
I_{l}\cup\\{l\\}}$ are rationally independent, so we can find $N\geq r$ large
enough such that for all $l\in[d]$, $\\{(nP_{l,j})_{j\in
I_{l}}):n\in[-N,N]\\}$, is $\epsilon$-dense in $\mathbb{T}^{|I_{l}|}$.

Let $M>N/\epsilon$ and set $R_{\epsilon}=R_{P,\epsilon/M}\cap B(0,M)^{c}$.
Take $n\in R_{\epsilon}$ and $w\in(\mathbb{T}^{r})^{d}$. Define
$\varphi:\mathbb{R}^{r}\to\mathbb{R}^{d\cdot r}$ by
$\varphi(x)=(n_{1}x,\ldots,n_{d}x)$ and consider
$y=\sum_{i=1}^{d}\frac{y_{i}}{n_{i}},$ where we define $\\{y_{i}\\}_{i=1}^{d}$
inductively as follows: first we define $y_{d}=w_{d},$ and note that
$||\frac{y_{d}}{n_{d}}||_{\mathbb{R}^{r}}\leq|\frac{1}{n_{d}}|\leq\epsilon.$
Next, suppose we have defined for $1\leq l<d$:
$y_{l+1},\ldots,y_{d}\in\mathbb{R}^{r}\cap B(0,C)$, where $C=N+d+1$. Take
$k_{l}\in[-N,N]$ such that for all $j>l$

$||k_{l}P_{l,j}+P_{l,j}(w_{l}-\sum_{i=l+1}^{d}\\{\frac{n_{l}}{n_{i}}y_{i}\\})||\leq\epsilon.$
(1)

The existence of $k_{l}$ is guaranteed by the definition of $N$ and the fact
that such condition is trivial for $j\notin I_{l}$. Now, denoting
$\vec{k_{l}}=(k_{l},...,k_{l})$, we define

$y_{l}=w_{l}+\vec{k_{l}}-\sum_{i=l+1}^{d}\\{\frac{n_{l}}{n_{i}}y_{i}\\}\in\mathbb{R}^{r}\cap
B(0,C).$

In this way, we have that $\left\lVert
y_{l}/n_{l}\right\rVert_{\mathbb{R}^{r}}\leq
C/|n_{l}|\leq(1+\frac{d+1}{N})\epsilon\leq(d+2)\epsilon.$ We claim that

$\left\lVert\varphi(y)-w\right\rVert=\left\lVert\sum_{i=1}^{d}\sum_{j>i}\frac{n_{j}}{n_{i}}e_{j}y_{i}\right\rVert.$

For this we will prove by induction that for every $l=0,...,d$ we have that

$\displaystyle\left\lVert\varphi(y)-w\right\rVert=\left\lVert\sum_{j=1}^{d}n_{j}e_{j}(\sum_{i=1}^{l}\frac{y_{i}}{n_{i}})+\sum_{i=l+1}^{d}\sum_{j>i}\frac{n_{j}}{n_{i}}e_{j}(y_{i})+\sum_{j=1}^{l}\sum_{i=l+1}^{d}\frac{n_{j}}{n_{i}}e_{j}y_{i}-\sum_{j=1}^{l}e_{j}w_{j}\right\rVert.$

Indeed, we start with $l=d$ for which by definition we have that

$\displaystyle\left\lVert\varphi(y)-w\right\rVert=\left\lVert\sum_{j=1}^{d}n_{j}e_{j}(\sum_{i=1}^{d}\frac{y_{i}}{n_{i}})-\sum_{j=1}^{d}e_{j}w_{j}\right\rVert.$

Now assume that for $1\leq l\leq d$ we have proved that

$\displaystyle\left\lVert\varphi(y)-w\right\rVert=\left\lVert\sum_{j=1}^{d}n_{j}e_{j}(\sum_{i=1}^{l}\frac{y_{i}}{n_{i}})+\sum_{i=l+1}^{d}\sum_{j>i}\frac{n_{j}}{n_{i}}e_{j}(y_{i})+\sum_{j=1}^{l}\sum_{i=l+1}^{d}\frac{n_{j}}{n_{i}}e_{j}y_{i}-\sum_{j=1}^{l}e_{j}w_{j}\right\rVert.$

Using the definition of $y_{l}$ we obtain that

$\displaystyle\left\lVert\sum_{j=1}^{d}n_{j}e_{j}(\sum_{i=1}^{l}\frac{y_{i}}{n_{i}})+\sum_{i=l+1}^{d}\sum_{j>i}\frac{n_{j}}{n_{i}}e_{j}y_{i}+\sum_{j=1}^{l}\sum_{i=l+1}^{d}\frac{n_{j}}{n_{i}}e_{j}y_{i}-\sum_{j=1}^{l}e_{j}w_{j}\right\rVert$
$\displaystyle=\left\lVert\sum_{j=1}^{d}n_{j}e_{j}(\sum_{i=1}^{l-1}\frac{y_{i}}{n_{i}})+\sum_{i=l+1}^{d}\sum_{j>i}\frac{n_{j}}{n_{i}}e_{j}y_{i}+\sum_{j=1}^{l-1}\sum_{i=l+1}^{d}\frac{n_{j}}{n_{i}}e_{j}y_{i}-\sum_{j=1}^{l-1}w_{j}+\sum_{\begin{subarray}{c}j=1\\\
j\neq l\end{subarray}}^{d}\frac{n_{j}}{n_{l}}e_{j}y_{l}\right\rVert$
$\displaystyle=\left\lVert\sum_{j=1}^{d}n_{j}e_{j}(\sum_{i=1}^{l-1}\frac{y_{i}}{n_{i}})+\sum_{i=l}^{d}\sum_{j>i}\frac{n_{j}}{n_{i}}e_{j}y_{i}+\sum_{j=1}^{l-1}\sum_{i=l}^{d}\frac{n_{j}}{n_{i}}e_{j}y_{i}-\sum_{j=1}^{l-1}w_{j}\right\rVert,$

in where the last equality comes from merging the second and the last
summation, namely:

$\sum_{i=l+1}^{d}\sum_{j>i}\frac{n_{j}}{n_{i}}e_{j}y_{i}+\sum_{\begin{subarray}{c}j=1\\\
j\neq
l\end{subarray}}^{d}\frac{n_{j}}{n_{l}}e_{j}y_{l}=\sum_{i=l}^{d}\sum_{j>i}\frac{n_{j}}{n_{i}}e_{j}y_{i}.$

This concludes the induction, and using the case $l=0$ we get that

$\left\lVert\varphi(y)-w\right\rVert=\left\lVert\sum_{i=1}^{d}\sum_{j>i}\frac{n_{j}}{n_{i}}e_{j}y_{i}\right\rVert.$

Moreover, by (1) we have that

$\displaystyle||P_{i,j}y_{i}||$
$\displaystyle=\Big{\|}\big{(}P_{i,j}w_{l}+P_{i,j}k_{i}-P_{i,j}\sum_{i=l+1}^{d}\\{\frac{n_{l}}{n_{i}}y_{i}\\}\big{)}\Big{\|}\leq\epsilon.$

Therefore, we found $y\in B_{\mathbb{R}^{r}}(0,\epsilon d(d+2))$ such that

$\displaystyle\left\lVert\varphi(y)-w\right\rVert=\left\lVert\sum_{i=1}^{d}\sum_{j>i}\frac{n_{j}}{n_{i}}e_{j}y_{i}\right\rVert$
$\displaystyle\leq\left\lVert\sum_{i=1}^{d}\sum_{j>i}P_{i,j}y_{i}\right\rVert+\sum_{i=1}^{d}\sum_{j>i}\left\lVert(\frac{n_{j}}{n_{i}}-P_{i,j})e_{j}y_{i}\right\rVert$
$\displaystyle\leq\left\lVert\sum_{i=1}^{d}\sum_{j>i}P_{i,j}y_{i}\right\rVert+\frac{\epsilon
Cd^{2}}{M}$ $\displaystyle\leq
d^{2}\epsilon+d^{2}\epsilon^{2}(d+2)\leq(d+3)^{3}\epsilon.$

To conclude, take $\epsilon=\epsilon^{\prime}/(d+3)^{3}$ and
$M^{\prime}=M(d+2)^{3}$, and replace $(\epsilon,M)$ by
$(\epsilon^{\prime},M^{\prime})$ noting that

$R_{\epsilon^{\prime}}=R_{P,\epsilon^{\prime}/M^{\prime}}\cap
B(0,M^{\prime})^{c}\subseteq R_{P,\epsilon/M}\cap
B(0,M)^{c}=R_{\epsilon},\leavevmode\nobreak\ \text{ and }\leavevmode\nobreak\
B(0,\epsilon d(d+2))\subseteq B(0,\epsilon^{\prime}).$

∎

##  5\. Recurrence in $\mathbb{Z}^{d}$-Quasi Affine Nilsystems

In this section, our objective is to lift $\mathbb{Z}^{d}$-Bohr recurrence to
the broader class of $\mathbb{Z}^{d}$-quasi-affine nilsystems. First, we begin
with a brief outline of the method. Then, we introduce the concept of total
ergodicity for actions of $\mathbb{Z}^{d}$-groups. Notably, this property is
characterized within nilsystems through connectedness. Subsequently, we
demonstrate that when extending Bohr recurrence to nilsystems, we can
consistently simplify the problem by focusing on scenarios where the set of
Bohr recurrence has the property of complete independence. Finally, we use
this reduction to establish that sets characterized by $\mathbb{Z}^{d}$-Bohr
recurrence are sets of recurrence for $\mathbb{Z}^{d}$-quasi-affine
nilsystems.

###  5.1. Outline of the method

In 2016 Host, Kra and Maass established that sets of $\mathbb{Z}$-Bohr
recurrence are sets of recurrence for all minimal $\mathbb{Z}$-nilsystems
(Theorem 4.1 [16]). We manage to generalize this theorem to the
multidimensional setting in the case of quasi-affine nilsystems.

######  Theorem 5.1.

Let $R\subseteq\mathbb{Z}^{d}$ be a set of $\mathbb{Z}^{d}$-Bohr recurrence.
Then, for every integer $s\geq 1$, $R$ is a set of recurrence for every
minimal $s$-step $\mathbb{Z}^{d}$-quasi-affine nilsystem.

In order to establish 5.1, a pivotal technical lemma comes into play.
Specifically, when dealing with a set of Bohr recurrence with the property of
complete independence, this lemma facilitates the approximation of the final
commutator group within a nilspace. This approximation is achieved by applying
the powers of commutators from the dynamics, all of which are drawn from the
aforementioned set of Bohr recurrence, to a single variable.

######  Theorem 5.2.

Let $(X,T_{1},\ldots,T_{d})$ be a minimal $s$-step $\mathbb{Z}^{d}$-quasi-
affine nilsystem, with $X$ connected and $G_{0}$ simply connected, and $R$ an
essential and ordered set of $\mathbb{Z}^{d}$-Bohr recurrence with the
property of complete independence. Then, for all $\epsilon>0$, there is $C>0$
and $R_{\epsilon}\subseteq R$ a set of $\mathbb{Z}^{d}$-Bohr recurrence such
that for all $h_{1},\ldots,h_{d}\in G_{s-1}\cap B(e_{G},C)$ and all $n\in
R_{\epsilon}$ there exist $h\in G_{s-1}$ and $\theta\in G_{s}\cap\Gamma$ such
that

$d_{G}(h,e_{G})<\epsilon,\text{ and
}d_{G}([h,\tau_{1}^{n_{1}}]\cdots[h,\tau_{d}^{n_{d}}],[h_{1},\tau_{1}]\cdots[h_{d},\tau_{d}]\theta)<\epsilon.$
(2)

5.2 allows us to prove 5.1 in the case where $R$ is an essential and ordered
set of $\mathbb{Z}^{d}$-Bohr recurrence with the property of complete
independence and $X$ is connected. In what follows, we will show how to reduce
to such case, prove 5.2, and finally show how it can be used to prove 5.1.

###  5.2. Connectedness in Nilsystems

Notice that in $(X=G/\Gamma,T_{1},\ldots,T_{d})$ we can always assume that $G$
is spanned by $G_{0}$ and the elements $\tau_{1},\ldots,\tau_{d}$ defining the
dynamics. Indeed, setting $G^{\prime}=\langle
G_{0},\tau_{1},\ldots,\tau_{d}\rangle$, we have that $G^{\prime}$ is an open
subgroup of $G$, and since for all $a\in X$ the map $g\to g\cdot a$ is open,
the sets $G^{\prime}\cdot a$, $a\in X$, are open subsets of $X$ that are
pairwise equal or disjoint. Since these sets cover $X$, they are closed in
$X$, hence compact. Moreover, given that $X$ is compact, there exist
$a_{1},\ldots,a_{k}\in X$ such that $\\{X_{i}=G^{\prime}\cdot
a_{i}\\}_{i=1}^{k}$ covers $X$. For $i=1,\ldots,k$, let $\Gamma_{i}$ denote
the stabilizer of $a_{i}$ in $G$, that is, $\Gamma_{i}=g_{i}\Gamma
g_{i}^{-1},$ where $g_{i}\in G$ is an element such that $g_{i}\cdot
e_{X}=a_{i}$. Note that $\Gamma_{i}\cap G^{\prime}$ represents the stabilizer
of $a_{i}$ in $G^{\prime}$, and $X_{i}$ can be viewed as the nilmanifold
$G^{\prime}/(\Gamma_{i}\cap G^{\prime})$. Since $\tau_{1},\ldots,\tau_{d}\in
G^{\prime}$, $X_{i}$ is a $\langle\tau_{1},\ldots,\tau_{d}\rangle$-invariant
set and $(X_{i},T_{1},\ldots,T_{d})$ is a nilsystem. In this light, we obtain
a partition of $X$ into finitely many nilsystems, and each one can be studied
separately. Thus, without loss of generality, we can substitute $G$ by
$G^{\prime}$ and assume that $G$ is spanned by $G_{0}$ and
$\tau_{1},\ldots,\tau_{d}$.

In order to characterize connectedness in nilsystems, we will first need the
following result that characterizes the ergodicity in a $\mathbb{Z}^{d}$-torus
rotation.

######  Theorem 5.3.

Let $(\mathbb{T}^{N},\alpha_{1},\ldots,\alpha_{d})$ be a
$\mathbb{Z}^{d}$-rotation. Then, the following are equivalent:

  1. (1)

$(\mathbb{T}^{N},\alpha_{1},\ldots,\alpha_{d})$ is ergodic,

  2. (2)

$\forall k\in\mathbb{Z}^{N}\setminus\\{0\\}$, $\exists i\in[d]$ such that
$k\cdot\alpha_{i}\notin\mathbb{Z}$,

  3. (3)

$\forall k\in\mathbb{Z}^{N}\setminus\\{0\\}$,
$\exists(t_{i})_{i=1}^{d}\in\mathbb{Z}^{d}$,
$k\cdot(\sum_{i=1}^{d}t_{i}\alpha_{i})\notin\mathbb{Z}$.

###### Proof.

((1) $\Longrightarrow$ (2)) We prove the contrapositive assertion. Suppose
that $\exists k\in\mathbb{Z}^{N}\setminus\\{0\\}$ such that $\forall i\in[d]$
we have $k\cdot\alpha_{i}\in\mathbb{Z}$. Thus, $f(x)=e^{2\pi ik\cdot x}$ is an
invariant function in $L^{\infty}(\mathbb{T}^{N})$ that is not constant, and
thus $(\mathbb{T}^{N},\alpha_{1},\ldots,\alpha_{d})$ is not ergodic.

((2) $\Longrightarrow$ (3)) This implication is direct.

((3) $\Longrightarrow$ (1)) Suppose that
$(\mathbb{T}^{N},\alpha_{1},\ldots,\alpha_{d})$ is not ergodic. Then, there
exists a $\langle\alpha_{1},\ldots,\alpha_{d}\rangle$-invariant function $f\in
L^{\infty}(\mathbb{T}^{N})$ which is not constant. Writing $f$ in the classic
basis of $L^{2}(\mathbb{T}^{N})$

$f(x)=\sum_{k\in\mathbb{Z}}c_{k}e^{2\pi ik\cdot x},$

and using the continuity of the operation $x\to gx$ ,$\forall
g\in\langle\alpha_{1},\ldots,\alpha_{d}\rangle$, we have that

$f(gx)=\sum_{k\in\mathbb{Z}}c_{k}e^{2\pi ik\cdot g}e^{2\pi ik\cdot x}.$

By the invariance of $f$, we conclude that $\forall k\in\mathbb{Z}^{N}$,
$c_{k}=c_{k}e^{2\pi ik\cdot g}.$ As $f$ is not constant, it exists
$k\in\mathbb{Z}^{N}\setminus\\{0\\}$ such that $c_{k}\neq 0$, and therefore
$k\cdot g\in\mathbb{Z}$, $\forall
g\in\langle\alpha_{1},\ldots,\alpha_{d}\rangle$ which contradicts the
hypothesis. ∎

In what follows, we recall total ergodicity and show that
$\mathbb{Z}^{d}$-torus rotations are totally ergodic.

######  Definition 5.1.

A dynamical system $(Y,T_{1},\ldots,T_{d})$ is totally ergodic if for all
$m\in\mathbb{N}^{d}$ the system $(Y,T_{1}^{m_{1}},\ldots,T_{d}^{m_{d}})$ is
ergodic. Equivalently, if the action of any finite index subgroup of
$\mathbb{Z}^{d}$ is ergodic.

5.3 immediately gives:

######  Proposition 5.4.

An ergodic $\mathbb{Z}^{d}$-rotation
$(\mathbb{T}^{N},\alpha_{1},\ldots,\alpha_{d})$ is totally ergodic.

###### Proof.

Let $m\in\mathbb{N}^{d}$. Suppose that
$(\mathbb{T}^{N},m_{1}\alpha_{1},\ldots,m_{d}\alpha_{d})$ is not ergodic.
Then, by 5.3, there is $k\in\mathbb{Z}^{N}\setminus\\{0\\}$ such that $\forall
i\in[d],\leavevmode\nobreak\ \leavevmode\nobreak\ k\cdot
m_{i}\alpha_{i}\in\mathbb{Z}$. In particular,

$\forall i\in[d],\leavevmode\nobreak\ \leavevmode\nobreak\
(k\prod_{j=1}^{d}m_{j})\cdot\alpha_{i}\in\mathbb{Z},$

and by 5.3, $(\mathbb{T}^{N},\alpha_{1},\ldots,\alpha_{d})$ is not ergodic,
which is a contradiction. ∎

We will prove that total ergodicity is equivalent to connectedness in
nilsystems. We need the following lemma.

######  Lemma 5.5.

Let $(X=G/\Gamma,T_{1},\ldots,T_{d})$ be a $\mathbb{Z}^{d}$ -nilsystem and
$X_{0}$ be the connected component of $e_{X}$. Then, there exist
$p_{1},\ldots,p_{d}\in\mathbb{N}$ such that $X_{0}$ is a
$(T_{1}^{p_{1}},\ldots,T_{d}^{p_{d}})$-invariant clopen subset of $X$.

###### Proof.

Let $\mu$ be the Haar measure on $X$, which has full support. As $X_{0}$ is
open, we have that for each $i\in[d]$, there exists a minimal period
$p_{i}\in\mathbb{N}$ for which $\tau_{i}^{p_{i}}X_{0}=X_{0}$, otherwise the
sets $(\tau_{i}^{k}X_{0})_{k\in\mathbb{N}}$ are pairwise disjoint and with the
same measure $\mu(X_{0})$, which is a contradiction. Given that
$\\{\tau_{i}^{j}X_{0}\\}_{j=1}^{p_{i}}$ partitions $X$, and that $X_{0}$ is
open, the set $X_{0}$ is also closed. Finally, as
$\tau_{i}^{p_{i}}X_{0}=X_{0}$, we have the invariance, and we conclude. ∎

######  Remark 5.6.

Notice that $(X_{0},T_{1}^{p_{1}},\ldots,T_{d}^{p_{d}})$ is a connected
$\mathbb{Z}^{d}$-nilsystem. Furthermore, $X$ is the union of finitely many
isomorphic copies of $(X_{0},T_{1}^{p_{1}},\ldots,T_{d}^{p_{d}})$, which are
permuted under the dynamics.

If the original nilsystem is quasi-affine, then all these connected subsystems
are affine (given that $X_{0}=G_{0}\Gamma/\Gamma$), then, by 2.6,
$\mathbb{Z}^{d}$-quasi-affine nilsystems are finite unions of
$\mathbb{Z}^{d}$-affine nilsystems.

Finally, we have the following characterization of total ergodicity in a
$\mathbb{Z}^{d}$-nilsystem.

######  Corollary 5.7.

Let $(X,T_{1},\ldots,T_{d})$ be an ergodic $s$-step $\mathbb{Z}^{d}$-nilsystem
for the Haar measure $\mu$. Then, $X$ is connected if and only if
$(X,T_{1},\ldots,T_{d})$ is totally ergodic for $\mu$.

###### Proof.

If $X$ is connected, then by 2.4 we have that for
$k_{1},\ldots,k_{d}\in\mathbb{N}$, $(X,T_{1}^{k_{1}},\ldots,T_{d}^{k_{d}})$ is
ergodic for $\mu$ if and only if the rotations by
$\tau_{1}^{k_{1}},\ldots,\tau_{d}^{k_{d}}$ are ergodic in the maximal factor
torus, which is always true given that such system is totally ergodic for
$\mu$ by 5.4. On the other hand, if the system is totally ergodic for $\mu$,
by 5.5, we derive that $X_{0}$ is a
$(T_{1}^{p_{1}},\ldots,T_{d}^{p_{d}})$-invariant closed subset of $X$. But, as
$(X,T_{1}^{p_{1}},\ldots,T_{d}^{p_{d}})$ is ergodic for $\mu$, thus minimal by
2.5, we have that $X=X_{0}$, so $X$ is connected. ∎

The following lemma will be useful in subsequent proofs.

######  Lemma 5.8.

Let $(X,T_{1},\ldots,T_{d})$ be an $s$-step $\mathbb{Z}^{d}$-nilsystem and
$K\in\mathbb{Z}^{d\times d}$ a matrix. Define
$S_{i}=T_{1}^{K_{i,1}}\circ\cdots\circ T_{d}^{K_{i,d}},$ for each $i\in[d]$.
Then, $(X,S_{1},\ldots,S_{d})$ is an $s$-step $\mathbb{Z}^{d}$-nilsystem. In
addition, if $K$ is invertible and $(X,T_{1},\ldots,T_{d})$ is connected and
minimal, then $(X,S_{1},\ldots,S_{d})$ is connected and minimal as well.

###### Proof.

Given a matrix $K\in\mathbb{Z}^{d\times d}$ and $(X,T_{1},\ldots,T_{d})$ an
$s$-step $\mathbb{Z}^{d}$-nilsystem, it is clear that
$(X,(T_{1}^{K_{i,1}}\circ\cdots\circ T_{d}^{K_{i,d}})_{i=1}^{d})$ is still an
$s$-step $\mathbb{Z}^{d}$-nilsystem, as $X$ is an $s$-step nilmanifold and the
transformations $(T_{1}^{K_{i,1}}\circ\cdots\circ T_{d}^{K_{i,d}})_{i=1}^{d}$
are given by the corresponding rotations. For the second part of the
statement, suppose that $(X,T_{1},\ldots,T_{d})$ is connected and minimal, and
that $K$ is invertible. Let $N\in\mathbb{N}$ be such that $N\cdot
K^{-T}\in\mathbb{Z}^{d\times d}$. Then, notice that the action of $S_{i}$
corresponds to the action generated by the vector $s_{i}=K^{M}e_{i}$ of
$\mathbb{Z}^{d}$, so $N\cdot e_{i}=N\cdot K^{-T}s_{i}\in\mathbb{Z}^{d}$.
Hence,

$\langle S_{1},\ldots,S_{d}\rangle=\langle(T_{1}^{K_{i,1}}\circ\cdots\circ
T_{d}^{K_{i,d}})_{i=1}^{d}\rangle=\langle T_{1}^{N},\cdots,T_{d}^{N}\rangle,$

and thus the action of $(S_{1},\ldots,S_{d})$ is minimal, given that the
system $(X,T_{1},\ldots,T_{d})$ is totally ergodic by 5.7. ∎

To finish this subsection, we prove the proposition that allows us to reduce
to the connected case in general.

######  Proposition 5.9.

Let $(X,T_{1},\ldots,T_{d})$ be a minimal $s$-step $\mathbb{Z}^{d}$-nilsystem.
Then, there exists an invertible matrix $K\in\mathbb{N}^{d\times d}$ such that
$(X_{0},(T_{1}^{K_{i,1}}\circ\cdots\circ T_{d}^{K_{i,d}})_{i=1}^{d})$ is a
connected minimal $s$-step $\mathbb{Z}^{d}$-nilsystem, where $X_{0}$ is the
connected component of $e_{X}$.

###### Proof.

By 5.5 and 5.8, there are $p_{1},\ldots,p_{d}\in\mathbb{N}$ such that
$(X_{0},T_{1}^{p_{1}},\ldots,T_{d}^{p_{d}})$ is a nilsystem. However,
$(X_{0},T_{1}^{p_{1}},\ldots,T_{d}^{p_{d}})$ is not necessarily minimal. To
fix this, we will define an upper triangular matrix $K\in\mathbb{R}^{d\times
d}$ as follows. For $j\geq i$, we suppose that we have defined $K_{i,l}$ for
every $l<j$ and we define $K_{i,j}$ by

$K_{i,j}=\min\\{k_{j}\in[p_{j}]:\exists k_{j},\ldots
k_{d}\in\mathbb{Z},\tau_{i}^{K_{i,i}}\cdots\tau_{j-1}^{K_{i,j-1}}\tau_{j}^{k_{j}}\cdots\tau_{d}^{k_{d}}X_{0}=X_{0}\\}.$

We claim that $(X_{0},(T_{1}^{K_{i,1}}\circ\cdots\circ
T_{d}^{K_{i,d}})_{i=1}^{d})$ is a minimal nilsystem. As
$(X,T_{1},\ldots,T_{d})$ is a minimal nilsystem, we only need to prove that

$\\{\tau_{1}^{l_{1}}\cdots\tau_{d}^{l_{d}}:l_{1},\ldots,l_{d}\in\mathbb{Z},\tau_{1}^{l_{1}}\cdots\tau_{d}^{l_{d}}X_{0}=X_{0}\\}\subseteq
H=\langle(\tau_{1}^{K_{i,1}}\circ\cdots\circ\tau_{d}^{K_{i,d}})_{i=1}^{d}\rangle.$

We affirm that for all $i\in[d]$, there are
$l_{i,i},l_{i,i+1},\ldots,l_{i,d}\in\mathbb{Z}$ such that

$\tau_{1}^{l_{1}}\cdots\tau_{d}^{l_{d}}\equiv\tau_{i}^{l_{i,i}}\cdots\tau_{d}^{l_{i,d}}\leavevmode\nobreak\
\text{ mod }H.$

Indeed, we assume that this holds for $i<d$, and let $m\in\mathbb{Z}$ and
$r\in\\{0,\ldots,K_{i,i}-1\\}$ such that $l_{i,i}=mK_{i,i}+r$. We define
$l_{i+1,j}=l_{i,j}-mK_{i,j}$ for all $j\geq i+1$ and, as $K_{i,j}=0$ for all
$j<i$, we have that

$\tau_{i}^{l_{i,i}}\cdots\tau_{d}^{l_{i,d}}\equiv\tau_{i}^{r}\tau_{i+1}^{l_{i+1,i+1}}\cdots\tau_{i+1}^{l_{i+1,d}}\leavevmode\nobreak\
\text{ mod }H.$

Note that
$\tau_{i}^{r}\tau_{i+1}^{l_{i+1,i+1}}\cdots\tau_{i+1}^{l_{i+1,d}}X_{0}=X_{0}$,
hence, by the minimality of $K_{i,i}$, we obtain that $r=0$, concluding the
induction. In this light

$\tau_{1}^{l_{1}}\cdots\tau_{d}^{l_{d}}\equiv 0\leavevmode\nobreak\ \text{ mod
}H,$

which implies that $\tau_{1}^{l_{1}}\cdots\tau_{d}^{l_{d}}\in H$. ∎

######  Remark 5.10.

The previous proof is based in the idea that there is a parallelepiped that
generates all the mesh associated to the dynamics of $\\{T_{i}\\}_{i=1}^{d}$
in $X_{0}$ (that parallelepiped repeats itself periodically in
$\mathbb{Z}^{d}$).

With this and 4.6, it will be possible to reduce to the case in that
$(X,T_{1},\ldots,T_{d})$ is an $s$-step nilsystem with $X$ connected.

###  5.3. Quasi-affine nilsystems and the property of complete independence

This section is devoted to showing how we reduce to the case where the set of
Bohr recurrence has a vector of Bohr correlations with the property of
complete independence. First, we prove that if we have a redundant set of Bohr
recurrence we can reduce the dimensionality of the system.

######  Proposition 5.11.

Let $(X,T_{1},\ldots,T_{d})$ be a minimal $s$-step $\mathbb{Z}^{d}$-nilsystem
and let $R\subseteq\mathbb{Z}^{d}$ be a set of $\mathbb{Z}^{d}$-Bohr
recurrence. If $R$ is redundant, then there exist
$R^{\prime}\subseteq\mathbb{Z}^{d-1}$, a set of $\mathbb{Z}^{d-1}$-Bohr
recurrence, and a minimal $s$-step $\mathbb{Z}^{d-1}$-nilsystem
$(Y,T^{\prime}_{1},\ldots,T^{\prime}_{d-1})$ such that if $R^{\prime}$ is a
set of recurrence for $(Y,T^{\prime}_{1},\ldots,T^{\prime}_{d-1})$, then $R$
is a set of recurrence for $(X,T_{1},\ldots,T_{d})$.

###### Proof.

Suppose $R\subseteq\mathbb{Z}^{d}$ is redundant and let
$v=(v^{1},\ldots,v^{d})\in\mathbb{Z}^{d}\setminus\\{0\\}$ given by the
definition of redundancy. Replacing $R$ by $R_{v}=\\{(n_{1},\ldots,n_{d})\in
R:v^{T}\cdot n=0\\}$, we may assume without loss of generality that for all
$n\in R$, $v^{T}\cdot n=0$. Without loss of generality we assume $k=d$ and
$v^{d}=1$. Define the integer matrix $M\in\mathbb{Z}^{d\times d}$ by

$M_{i,j}=\begin{cases}1&\text{ if }i=j\\\ v^{j}&\text{ if }i=d\text{ and
}j\neq d\\\ 0&\text{ else }\\\ \end{cases}.$

Clearly, $M$ is invertible. Notice that $M^{-1}R$ is a set of
$\mathbb{Z}^{d}$-Bohr recurrence by 4.6. Set
$R^{\prime}=\\{(m_{i})_{i\in[d-1]}\in\mathbb{Z}^{d-1}:m\in M^{-1}R\\}$,

$S_{i}=\begin{cases}T_{i}T_{k}^{-v^{i}}&\text{ if }i<d\\\ id_{X}&\text{ if
}i=d\end{cases},$

and $Y=\overline{O_{S_{1},\ldots,S_{d-1}}(e_{X})}$. Clearly,
$(Y,S_{1},\ldots,S_{d-1})$ is a minimal $s$-step $\mathbb{Z}^{d}$-nilsystem.
Finally, since, for all $n\in R$ such that $m=M^{-1}n\in\mathbb{Z}^{d}$

$T_{1}^{n_{1}}\cdots T_{d}^{n_{d}}=S_{1}^{m_{1}}\cdots S_{d}^{m_{d}},$

we have that if $R^{\prime}$ is a set of recurrence for
$(Y,S_{1},\ldots,S_{d-1})$ then $R$ is a set of recurrence for
$(X,T_{1},\ldots,T_{d})$. ∎

######  Remark 5.12.

If $(X=G/\Gamma,T_{1},\ldots,T_{d})$ is a $\mathbb{Z}^{d}$-quasi-affine
nilsystem, then $(Y=G^{Y}/\Gamma^{Y},T^{\prime}_{1},\ldots,T^{\prime}_{d-1})$
is a $\mathbb{Z}^{d}$-quasi-affine nilsystem, given that $G^{Y}_{0}$ is also
abelian.

######  Corollary 5.13.

Let $(X,T_{1},\ldots,T_{d})$ be a minimal $s$-step $\mathbb{Z}^{d}$-nilsystem
and $R\subseteq\mathbb{Z}^{d}$ be a set of $\mathbb{Z}^{d}$-Bohr recurrence.
Then, there exist $d^{\prime}\leq d$,
$R^{\prime}\subseteq\mathbb{Z}^{d^{\prime}}$ a non-redundant set of
$\mathbb{Z}^{d^{\prime}}$-Bohr recurrence and
$(Y,T^{\prime}_{1},\ldots,T^{\prime}_{d^{\prime}})$ a minimal $s$-step
$\mathbb{Z}^{d^{\prime}}$-nilsystem such that if $R^{\prime}$ is a set of
recurrence for $(Y,T^{\prime}_{1},\ldots,T^{\prime}_{d^{\prime}})$ then $R$ is
of recurrence for $(X,T_{1},\ldots,T_{d})$.

###### Proof.

By applying 5.11 iteratively, we get $d^{\prime}\geq 1$ such that there are
$R^{\prime}\subseteq\mathbb{Z}^{d^{\prime}}$ a non-redundant set of
$\mathbb{Z}^{d^{\prime}}$-Bohr recurrence and
$(Y,T^{\prime}_{1},\ldots,T^{\prime}_{d^{\prime}})$ a minimal $s$-step
$\mathbb{Z}^{d^{\prime}}$-nilsystem such that if $R^{\prime}$ is a set of
recurrence for $(Y,T^{\prime}_{1},\ldots,T^{\prime}_{d^{\prime}})$ then $R$ is
of recurrence for $(X,T_{1},\ldots,T_{d})$. This is possible by the fact that
the process stops at most in $d^{\prime}=1$. In this condition every set of
Bohr recurrence is non-redundant. ∎

######  Remark 5.14.

Given 5.12, we have that if the system $(X,T_{1},\ldots,T_{d})$ is quasi-
affine, then $(Y,T^{\prime}_{1},\ldots,T^{\prime}_{d^{\prime}})$ is also
quasi-affine. In particular, by 5.6, $Y$ is a finite union of affine
nilsystems.

Now we present the result which allows us to reduce to the case in which we
have a set of Bohr recurrence $R\subseteq\mathbb{Z}^{d}$ with a Bohr
correlation $P\in\mathcal{BC}(R)$ with the property of complete independence.

######  Theorem 5.15.

Let $R\subseteq\mathbb{Z}^{d}$ be a non-redundant set of $\mathbb{Z}^{d}$-Bohr
recurrence and $(X,T_{1},\ldots,T_{d})$ be a connected minimal $s$-step
$\mathbb{Z}^{d}$-nilsystem. Then, there exist a non-redundant, essential, and
ordered set of $\mathbb{Z}^{d}$-Bohr recurrence
$\tilde{R}\subseteq\mathbb{Z}^{d}$, a connected minimal $s$-step
$\mathbb{Z}^{d}$-nilsystem $(X,S_{1},\ldots,S_{d})$, and
$P\in\mathcal{BC}(\tilde{R})$ with the property of complete independence, such
that if $\tilde{R}$ is a set of recurrence for $(X,S_{1},\ldots,S_{d})$, then
$R$ is a set of recurrence for $(X,T_{1},\ldots,T_{d})$.

###### Proof.

Using 4.3, and possibly replacing $R$ by a subset of it, we may assume that
$R$ is essential. We also assume that $R$ is ordered by 4.12. Let
$P\in\mathcal{BC}(R)$, we will follow an induction process, changing
$(R,P,(X,T_{1},\ldots,T_{d}))$ iteratively. We start with $R^{1}=R$ and
$P^{1}=P$, $T_{i,1}=T_{i}$ for all $i\in[d]$. For $N<d$ suppose that we have
$(R^{N},P^{N},(X,T_{1,N},\ldots,T_{d,N}))$ such that $R^{N}$ is a non-
redundant, essential and ordered set of $\mathbb{Z}^{d}$-Bohr recurrence, and
$P^{N}\in\mathcal{BC}(R)$ is such that $(R^{N},P^{N})$ verifies

$\forall i\in[N],\\{P_{i,j}^{N}:P_{i,j}^{N}\neq 0,\leavevmode\nobreak\ \forall
i\leq j\leq N\\}\text{ are rationally independent}.$ (3)

In addition, suppose that if $R^{N}$ is a set of recurrence for
$(X,T_{1,N},\ldots,T_{d,N})$, then $R$ is a set of recurrence for
$(X,T_{1},\ldots,T_{d})$.

We extend this to $N+1$ as follows: define $l_{N}=\min\\{l\leq N:P_{l,N}\neq
0\\}$ and $C_{N}=\\{P_{l_{N},j}:P_{l_{N},j}^{N}\neq 0,\forall j\geq l_{N}\\}.$
Step 1: If $C_{N}$ is rationally independent, we stop step 1. If not, defining
$I_{N}=\\{j\in[d]:P_{l_{N},j}^{N}\neq 0\\}$ we have that there exists
$v\in\mathbb{Q}^{d}\setminus\\{0\\}$ with $v^{j}=0$ for all $j\in[d]\setminus
I_{N}$, such that

$\sum_{j\in I_{N}}v^{j}P_{l_{N},j}^{N}=0.$ (4)

Write $v^{i}=v^{i}_{1}/v^{i}_{2}$ for $v^{i}_{1}\in\mathbb{Z}$ and
$v^{i}_{2}\in\mathbb{Z}\setminus\\{0\\}$ for each $i\in I_{N}$, considering
$v^{i}_{2}=1$ if $v^{i}_{1}=0$. Fix $k\in I_{N}\setminus[N]$ with $v^{k}\neq
0$ and assume that $v^{k}=1$. Define the integer matrix
$M\in\mathbb{Z}^{d\times d}$ by

$M_{i,j}=\begin{cases}1&\text{ if }i=j\text{ and }j\notin I_{N}\\\
v^{j}_{2}&\text{ if }i=j\text{ and }j\in I_{N}\\\ v^{j}_{1}&\text{ if
}i=k,\leavevmode\nobreak\ j\neq k,\text{ and }j\in I_{N}\\\ 0&\text{ else }\\\
\end{cases}.$

Clearly, $M$ is invertible. Notice that $R^{\prime N}:=M^{-1}R$ is a non-
redundant set of $\mathbb{Z}^{d}$-Bohr recurrence by 4.6 and 4.8. Therefore,
we may assume that it is essential without loss of generality. Remark that
$M^{-1}R$ may not be ordered.

We also define

$T_{j,N}^{\prime}=\begin{cases}T_{i}^{v^{i}_{2}}T_{k}^{v^{i}_{1}}&\text{if
}j\in I_{N}\setminus\\{k\\}\\\ T_{j}&\text{ else}\end{cases}.$

By 5.8, $(X,T_{1,N}^{\prime},\ldots,T_{d,N}^{\prime})$ is a connected minimal
$s$-step nilsystem. We also have that for $n\in R^{N}$ and
$m=M^{-1}n\in\mathbb{Z}^{d}$,

$T_{1,N}^{n_{1}}\cdots T_{d,N}^{n_{d}}=T_{1,N}^{\prime m_{1}}\cdots
T_{d,N}^{\prime m_{d}}.$ (5)

Additionally, we define for $j,l\in[N]\cup I_{N}$, with $j\leq l$,

$P_{j,l}^{\prime
N}=\begin{cases}\frac{v^{j}_{2}}{v^{l}_{2}}P_{j,l}^{N}&\text{if }(j,l)\in
I_{N}\times I_{N}\\\ P_{j,l}&\text{if }j\notin I_{i}\lor l\notin I_{i}\\\
1&\text{if }j=l\end{cases}.$

Now, observe that $R^{\prime N}$ and $P^{\prime N}$ have the property
described in (3) and that reordering coordinates in $I_{N}\cap[N]$ this
property still holds. So, we can assume $R^{\prime N}$ ordered by rearranging
the coordinates of $R^{\prime N}$, $P^{\prime N}$ and
$(X,T_{1,N}^{\prime},\ldots,T_{d,N}^{\prime})$ (possibly reducing to a subset
of $R^{\prime N}$). Then, using 4.15 we can extent $P^{\prime N}$. Finally, if
$R^{\prime N}$ is a set of recurrence for
$(X,T_{1,N}^{\prime},\ldots,T_{d,N}^{\prime}),$ then $R^{N}$ is a set of
recurrence for $(X,T_{1,N},\ldots,T_{d,N})$ by (5). Hence, we can replace
$(R^{N},P^{N},(X,T_{1,N},\ldots,T_{d,N}))$ by $(R^{\prime N},P^{\prime
N},(X,T_{1,N}^{\prime},\ldots,T_{d,N}^{\prime}))$. We note that
$|C_{N}^{\prime}|<|C_{N}|$ given that $P^{\prime N}_{j,k}=0$ for all $j\in[N]$
by Eq. 4. Then we repeat step 1 for a maximum number of cycles since $|C_{n}|$
is initially at most $d$.

Step 2: If $C_{N}$ is rationally independent, we set $P^{N+1}=P^{N}$,
$R^{N}=R^{N+1}$, and
$(X,T_{1,N},\ldots,T_{d,N})=(X,T_{1,N+1},\ldots,T_{d,N+1})$, extending to
$N+1$ the hypotheses.

Hence, after finitely many steps the process ends with the desired conclusion.
∎

######  Remark 5.16.

If the original system $(X,T_{1},\ldots,T_{d})$ in 5.15 is quasi-affine, then
the resultant system $(X,S_{1},\ldots,S_{d^{\prime}})$ is still quasi-affine,
given that $G_{0}$ is abelian in both systems and $X$ is connected.

###  5.4. The Main Theorem

In this section we prove 5.2, which is the main ingredient to conclude 5.1. In
what follows, [3, 21, 19] are sources of background. Given an $s$-step
$\mathbb{Z}^{d}$-nilsystem, by changing the representation of $X=G/\Gamma$ as
mentioned in Section 2.2, we can assume without loss of generality that
$G_{0}$ is simply connected. Set $\Gamma_{0}=\Gamma\cap G_{0}$. Since $G_{0}$
can be endowed with a Malcev basis, there exist $p,r,q\in\mathbb{N}$ such that

$G_{0}/G_{2}\cong\mathbb{R}^{p},\leavevmode\nobreak\ \leavevmode\nobreak\
G_{s}\cong\mathbb{R}^{r},\leavevmode\nobreak\
G_{s-1}/G_{s}\cong\mathbb{R}^{q}$

and

$\Gamma_{0}/(\Gamma_{0}\cap G_{2})\cong\mathbb{Z}^{p},\leavevmode\nobreak\
\leavevmode\nobreak\ \Gamma\cap G_{s}\cong\mathbb{Z}^{r},\leavevmode\nobreak\
\leavevmode\nobreak\ (\Gamma\cap G_{s-1})/(\Gamma\cap
G_{s})\cong\mathbb{Z}^{q}.$

Moreover, the distance $d_{G}$ in $G$ can be chosen such that the
identifications are isometries when the quotient groups are endowed with the
quotient distances and $\mathbb{R}^{p}$, $\mathbb{R}^{r}$ and $\mathbb{R}^{q}$
are endowed with the Euclidean distances.

Our initial step involves extending the scope of Lemma 4.3 from [16]. This
lemma represents the initial foray into approximating $G_{s}$ through
commutators coming from $G_{s-1}$. Unlike 5.2, the set of Bohr recurrence is
not in play, rather, we need to use a different variable in each commutator to
compensate.

######  Proposition 5.17.

Let $(X,T_{1},\ldots,T_{d})$ be a minimal $s$-step nilsystem and assume that
$X$ is connected and $G_{0}$ is simply connected. Then, for every $\epsilon>0$
there exists $C>0$ such that for every $w\in G_{s}$ we can find
$h_{1},\ldots,h_{d}\in G_{s-1}$ and $\gamma\in\Gamma\cap G_{s}$ with

$d_{G}(h_{i},e_{G})<C,\leavevmode\nobreak\ \leavevmode\nobreak\ \forall
i\in[d];\leavevmode\nobreak\ \leavevmode\nobreak\
d_{G}([h_{1},\tau_{1}]\cdots[h_{d},\tau_{d}],w\gamma)<\epsilon.$

As the proof of 5.17 is highly similar, but no identical, to Lemma 4.3 from
[16], we postpone it to Appendix B: Proof of 5.17.

As mentioned previously, 5.17 needs $d$ variables $h_{1},\ldots,h_{d}$ to
approximate $G_{s}$ through commutators. This cannot be improved without
putting into play the set of Bohr recurrence. Besides, some hypotheses over
$R$ are necessary in 5.2, as the following example shows.

######  Example 5.18.

Consider $R^{\prime}\subseteq\mathbb{Z}$ a set of Bohr recurrence and
$(X=\mathbb{T}^{4},T_{1},T_{2})$ a $3$-step affine nilsystem with
$T_{i}x=M_{i}x+\alpha_{i}$, for $i=1,2$, where the matrices are

$M_{1}=\begin{pmatrix}1&1&0&0\\\ 0&1&1&0\\\ 0&0&1&0\\\
0&0&0&1\end{pmatrix},\leavevmode\nobreak\ \leavevmode\nobreak\
\leavevmode\nobreak\ M_{2}=\begin{pmatrix}1&-1&0&0\\\ 0&1&-1&0\\\ 0&0&1&0\\\
0&0&0&1\end{pmatrix},$

and $R=\\{(n,n):n\in R^{\prime}\\}$. Observe that in this case $X=G/\Gamma$
with $G$ the group generated by the transformation
$\\{M_{1}^{n_{1}},M_{2}^{n_{2}}\\}_{n_{1},n_{2}\in\mathbb{Z}}$ and the
rotation by elements of $\mathbb{T}^{4}$, and
$\Gamma=\langle\\{M_{1}^{n_{1}},M_{2}^{n_{2}}\\}_{n_{1},n_{2}\in\mathbb{Z}}\rangle$.

Notice that for $y\in G$

$[y,\tau_{1}^{n}][y,\tau_{2}^{n}]=n(M_{1}-I)y+n(M_{2}-I)y=0.$

Meanwhile, for $h_{1},h_{2}\in G_{s-1}=\mathbb{R}^{2}\times\\{0\\}^{2}$,

$[h_{1},\tau_{1}][h_{2},\tau_{2}]=(M_{1}-I)h_{1}+(M_{2}-I)h_{2},$

takes any value in $G_{s}=\\{0\\}\times\mathbb{R}\times\\{0\\}^{2}$, and thus
the image of $y\to[y,\tau_{1}^{n}][y,\tau_{2}^{n}]$ is not dense in
$G_{s}/\Gamma\cap G_{s}$. Therefore, in 5.17 the condition $h_{1}=h_{2}$
cannot be required without adding extra hypotheses, even taking into account
$(n,n)\in R$.

###### Proof of 5.2.

We recall that since $X$ is connected, there exist $p,r,q\in\mathbb{N}$ such
that

$G_{s}=\mathbb{R}^{r},G_{s}\cap\Gamma=\mathbb{Z}^{r},G_{s-1}/G_{s}=\mathbb{R}^{q},\text{
and }\Gamma\cap G_{s-1}/(\Gamma\cap G_{s})=\mathbb{Z}^{q},$

and these identifications can be taken as isometries. We will use the
properties and notation developed in 5.17. Denote by $\Psi\colon
G_{s-1}/G_{s}\times\cdots\times G_{s-1}/G_{s}\to G_{s}/(G_{s}\cap\Gamma)$ the
map defined by

$\Psi(h_{1},\ldots,h_{d})=\sum_{i=1}^{d}\pi_{s}\circ\Psi^{i}(h_{i}),$

with $\pi_{s}\colon G_{s}\to G_{s}/(\Gamma\cap G_{s})$ the quotient map and
$\Psi^{i}\colon G_{s-1}/G_{s}\to G_{s}$ the map induced by $g\to[g,\tau_{i}]$.
Notice that for every $i\in[d]$ and $hG_{s}\in G_{s-1}/G_{s}$,

$[h,\tau_{i}]=[h,\tau_{i}^{0}]\cdot[h,\gamma_{i}]=[h,\gamma_{i}],$

with $\tau_{i}=\tau_{i}^{0}\gamma_{i}$, where we used that
$[G_{s-1},\tau_{i}^{0}]\subseteq[G_{0},G_{0}]=\\{e_{G}\\}$ given that $G_{0}$
is abelian and 2.2 (1). This implies that $\Psi^{i}$ takes
$(G_{s-1}\cap\Gamma)/(G_{s}\cap\Gamma)$ to $G_{s}\cap\Gamma$. In this way,
$\Psi$ is a (linear) map between $(\mathbb{R}^{q})^{d}$ and $\mathbb{T}^{r}$,
which takes

$\Big{(}(G_{s-1}\cap\Gamma)/(G_{s}\cap\Gamma)\Big{)}^{d}=(\mathbb{Z}^{q})^{d},$

to $e_{G}(\Gamma\cap G_{s})=0+\mathbb{Z}^{r}$. Therefore, we have that the
restriction $\tilde{\Psi}\colon(\mathbb{T}^{q})^{d}\to\mathbb{T}^{r}$ is a
continuous morphism. As $(\mathbb{T}^{q})^{d}$ is compact, $\tilde{\Psi}$ is
uniformly continuous. We note that for $h\in G_{s-1}$,

$\displaystyle\pi_{s}([h,\tau_{1}^{n_{1}}]\cdots[h,\tau_{d}^{n_{d}}])=\pi_{s}([h^{n_{1}},\tau_{1}]\cdots[h^{n_{d}},\tau_{d}])=\pi_{s}(\sum_{i=1}^{d}\Psi^{i}(n_{i}h))=\Psi(n_{1}h,\ldots,n_{d}h),$

and similarly:

$\displaystyle\pi_{s}([h_{1},\tau_{1}]\cdots[h_{d},\tau_{d}])=\pi_{s}(\sum_{i=1}^{d}\Psi^{i}(h_{i}))=\Psi(h_{1},\ldots,h_{d}),$

where we abuse of notation viewing $h,h_{1},\ldots,h_{d}$ as elements of
$\mathbb{R}^{q}=G_{s-1}/G_{s}$.

Let $\delta\in(0,\epsilon)$ be a constant of uniform continuity for
$\Psi_{1},\ldots,\Psi_{d}$ for $\epsilon/d$. Using 4.16 with $\delta$, we find
$R_{\epsilon}\subseteq R$ such that for every $n\in R_{\epsilon}$ there exists
$h\in\mathbb{R}^{q}\cap B(0,\epsilon)$ satisfying

$\left\lVert(n_{1}h,\ldots,n_{d}h)-(h_{1},\ldots,h_{d})\right\rVert_{\mathbb{T}^{q}}<\delta.$

Therefore, we get that

$\left\lVert\tilde{\Psi}(n_{1}h,\ldots,n_{d}h)-\tilde{\Psi}(h_{1},\ldots,h_{d})\right\rVert_{\mathbb{T}^{r}}<\epsilon.$

Lifting this property to $G_{s}=\mathbb{R}^{r}$, we can find
$\theta\in\mathbb{Z}^{r}=\Gamma\cap G_{s}$ such that

$d_{G}([h,\tau_{1}^{n_{1}}]\cdots[h,\tau_{d}^{n_{d}}],[h_{1},\tau_{1}]\cdots[h_{d},\tau_{d}]\theta)<\epsilon.$

This allows us to conclude. ∎

From now on, we assume $s\geq 2$. For an $s$-step $\mathbb{Z}^{d}$-nilsystem
$(X,T_{1},\ldots,T_{d})$ denote by

$\tilde{G}=G/G_{s},\leavevmode\nobreak\ \leavevmode\nobreak\
\tilde{\Gamma}=\Gamma/(\Gamma\cap G_{s})\leavevmode\nobreak\
\leavevmode\nobreak\ \text{ and }\leavevmode\nobreak\
\tilde{X}=\tilde{G}/\tilde{\Gamma}.$ (6)

Then, $\tilde{G}$ is an $(s-1)$-nilpotent Lie group, $\tilde{\Gamma}$ is a
discrete cocompact subgroup, $\tilde{X}$ is an $(s-1)$-nilmanifold and the
quotient map $G\to\tilde{G}$ induces a projection $\pi\colon X\to\tilde{X}$.
Therefore, we can view $\tilde{X}$ as the quotient of $X$ under the action of
$G_{s}$. Let $\tilde{\tau_{1}},\ldots,\tilde{\tau_{d}}$ be the image of
$\tau_{1},\ldots,\tau_{d}$ in $\tilde{G}$ and
$\tilde{T}_{1},\ldots,\tilde{T}_{d}$ be the rotations by
$\tilde{\tau_{1}},\ldots,\tilde{\tau_{d}}$ in $\tilde{X}$. Then,
$(\tilde{X},\tilde{T_{1}},\ldots,\tilde{T_{d}})$ is an $(s-1)$-nilsystem and
$\pi\colon X\to\tilde{X}$ is a factor map.

Finally, we are able to prove the main theorem of this section.

###### Proof of A.

We proceed by induction on $s$. If $s=1$, the result is direct. Henceforth, we
assume that $s\geq 2$, and that the statement holds for $(s-1)$-step
$\mathbb{Z}^{d}$-quasi-affine nilsystems. Let $R\subseteq\mathbb{Z}^{d}$ be a
set of $\mathbb{Z}^{d}$-Bohr recurrence and let
$(X=G/\Gamma,T_{1},\ldots,T_{d})$ be a minimal $s$-step
$\mathbb{Z}^{d}$-quasi-affine nilsystem. First, we can assume that $R$ is non-
redundant, by 5.13, and replacing the pair $(R,(X,T_{1},\ldots,T_{d}))$ by
another pair $(R^{\prime},(Y,T^{\prime}_{1},\ldots,T^{\prime}_{d^{\prime}}))$,
where $d^{\prime}<d$, $R^{\prime}\subseteq\mathbb{Z}^{d^{\prime}}$ is a set of
$\mathbb{Z}^{d^{\prime}}$-Bohr recurrence and
$(Y,T^{\prime}_{1},\ldots,T^{\prime}_{d^{\prime}})$ is a minimal $s$-step
$\mathbb{Z}^{d^{\prime}}$-quasi-affine nilsystem.

Next, we assume without loss of generality that $X$ is connected and $G_{0}$
is simply connected. Indeed, by 5.9 we have that there exists an invertible
matrix $K\subseteq\mathbb{N}^{d\times d}$ such that
$(X_{0},(T_{1}^{K_{i,1}}\circ\cdots\circ T_{d}^{K_{i,d}})_{i=1}^{d})$ is a
minimal affine nilsystem. On the other hand, by 4.6 the set
$R_{0}=\\{n\in\mathbb{Z}^{d}:K\cdot n\in R\\}$ is a non-redundant set of
$\mathbb{Z}^{d}$-Bohr recurrence. Substituting $X$ by $X_{0}$ and $R$ by
$R_{0}$, we reduce to the case where $X$ is connected. We can assume without
loss of generality that $G_{0}$ is simply connected as well.

Thanks to 5.15, we can also assume that $R$ is essential and ordered with
$P\in\mathcal{BC}(R)$ satisfying the complete independence property, changing
the system $(X=G/\Gamma,T_{1},\ldots,T_{d})$ by another connected minimal
$s$-step $\mathbb{Z}^{d}$-affine nilsystem.

Let $U$ be a nonempty open subset of $X$. Without loss of generality we can
assume that $U$ is the open ball $B(e_{X},3\epsilon)$ for some $\epsilon>0$.
Let $\pi\colon X\to\tilde{X}=X/G_{s}$ be the canonical factor map and
$(\tilde{X},\tilde{T}_{1},\ldots,\tilde{T}_{d})$ the $(s-1)$-nilsystem
discussed at the beginning of Section 5.3. Since
$(\tilde{X},\tilde{T}_{1},\ldots,\tilde{T}_{d})$ is an $(s-1)$-step affine
nilsystem, it follows from the induction hypothesis, 4.4, and the fact that
$\pi$ is open by 2.3, that there exist arbitrarily large elements $n\in
R_{\epsilon}$ with
$\pi(B(e_{X},\epsilon))\cap\tilde{T}_{1}^{-n_{1}}\cdots\tilde{T}_{d}^{-n_{d}}\pi(B(e_{X},\epsilon))\neq\emptyset$,
where $R_{\epsilon}$ is given by 5.2. It follows that there exist $x\in X$ and
$v\in G_{s}$ with $d_{X}(x,e_{X})<\epsilon$ and $d_{X}(T_{1}^{n_{1}}\cdots
T_{d}^{n_{d}}x,v\cdot e_{X})<\epsilon$. Lifting $x$ to $G$, we obtain $g\in G$
and $\gamma\in\Gamma$ with

$d_{G}(g,e_{G})<\epsilon\text{ and
}d_{G}(\tau_{1}^{n_{1}}\cdots\tau_{d}^{n_{d}}g,v\gamma)<\epsilon.$

By 5.17, there exists $C>0$ such that for $w=v^{-1}\in G_{s}$ we can find
$h_{1},\ldots,h_{d}\in G_{s-1}$ and $\gamma\in\Gamma\cap G_{s}$ with

$d_{G}(h_{i},e_{G})<C,\forall i\in[d],\text{ and
}d_{G}([h_{1},\tau_{1}]\cdots[h_{d},\tau_{d}],v^{-1}\gamma)<\epsilon.$ (7)

5.2 and the fact that $n\in R_{\epsilon}$ yield the existence of $h\in
G_{s-1}$ and $\theta\in G_{s}\cap\Gamma$ such that

$d_{G}(h,e_{G})<\epsilon\text{ and
}d_{G}([h^{-1},\tau_{1}^{n_{1}}]\cdots[h^{-1},\tau_{d}^{n_{d}}],v^{-1}\theta)<2\epsilon.$
(8)

Write $y=h\cdot x$, and note that $y$ is the projection of $hg$ in $X$ and

$d_{X}(y,e_{X})\leq d_{G}(hg,e_{G})\leq
d_{G}(h,e_{G})+d_{G}(g,e_{G})<2\epsilon.$

Moreover, using the right invariance of the distance $d_{G}$ and the fact that
for $w\in G$, $[G_{s-1},w]\subseteq G_{s}$, we obtain

$\displaystyle d_{X}(T_{1}^{n_{1}}\cdots T_{d}^{n_{d}}y,e_{X})$
$\displaystyle\leq
d_{G}(\tau_{1}^{n_{1}}\cdots\tau_{d}^{n_{d}}hg,\theta\gamma)$
$\displaystyle=d_{G}(h[h^{-1},\tau_{1}^{n_{1}}]\cdots[h^{-1},\tau_{d}^{n_{d}}]\tau_{1}^{n_{1}}\cdots\tau_{d}^{n_{d}}g,\theta\gamma)$
$\displaystyle\leq\epsilon+d_{G}(\tau_{1}^{n_{1}}\cdots\tau_{d}^{n_{d}}g[h^{-1},\tau_{1}^{n_{1}}]\cdots[h^{-1},\tau_{d}^{n_{d}}],\theta\gamma)$
$\displaystyle\leq
2\epsilon+d_{G}([h^{-1},\tau_{1}^{n_{1}}]\cdots[h^{-1},\tau_{d}^{n_{d}}],v^{-1}\theta)<4\epsilon,$

and we are done. ∎

##  Appendix A Proof of 3.5 and 3.6

For convenience, we repeat the statement of 3.5 and, later on, of 3.6.

######  Theorem.

Let $\\{(X_{n},H)\\}_{n\in\mathbb{N}}$ be a collection of minimal topological
dynamical systems with factor maps $\pi_{n+1}:(X_{n+1},H)\to(X_{n},H)$ for
$n\in\mathbb{N}$. Suppose $R$ is a set of recurrence for $(X_{n},H)$ for all
$n\in\mathbb{N}$. Then, it is also a set of recurrence for the inverse limit
of these systems with respect to the factors.

###### Proof.

Denote $X$ the inverse limit and let $U\subseteq X$ be a nonempty open set.
Without loss of generality, we can assume that there is $d\in\mathbb{N}$ and
$U_{0},\ldots,U_{d}$ nonempty open sets of $X_{0},\ldots,X_{d}$ respectively,
such that

$U=\Big{(}\prod_{n=0}^{d}U_{n}\times\prod_{n\geq d+1}X_{n}\Big{)}\cap
X,\leavevmode\nobreak\ \text{ and }\pi_{n+1}(U_{n+1})\subseteq
U_{n},\leavevmode\nobreak\ \leavevmode\nobreak\ \forall
n\in\\{0,\ldots,d-1\\}.$

This last assumption is possible by replacing iteratively $U_{n}$ by
$U_{n}\cap\pi_{n+1}^{-1}(U_{n})$ for $n\in\\{0,\ldots,d-1\\}$. Now, by
hypothesis, there exists $h\in R$ and $x_{d}\in U_{d}$ such that $hx_{d}\in
U_{d}$. We define inductively $x_{n}=\pi_{n+1}(x_{n+1})$ for
$n\in\\{0,\ldots,d-1\\}$. Notice that if $x_{n+1}\in U_{n+1}$ and $hx_{n+1}\in
U_{n+1}$, then $x_{n},hx_{n}\in U_{n}$. Indeed,
$x_{n}=\pi_{n+1}(x_{n+1})\in\pi(U_{n+1})\subseteq U_{n}$ and
$hx_{n}=h\pi_{n+1}(x_{n+1})=\pi_{n+1}(hx_{n+1})\in\pi(U_{n+1})\subseteq
U_{n}$. Hence, we have that $x_{n},hx_{n}\in U_{n}$, $\forall
n\in\\{0,\ldots,d\\}.$ We extend $(x_{n})_{n=0}^{d}$ to a point
$x=(x_{n})_{n\in\mathbb{N}}\in X$. Given the previous computation, we have
that

$x\in\Big{(}\prod_{n=0}^{d}U_{n}\times\prod_{n\geq d+1}X_{n}\Big{)}\cap
X\subseteq U\text{ and }hx\in\Big{(}\prod_{n=0}^{d}U_{n}\times\prod_{n\geq
d+1}X_{n}\Big{)}\cap X\subseteq U.$

Therefore, $U\cap h^{-1}U\neq\emptyset,$ concluding that $R$ is a set of
recurrence for $X$. ∎

In the following, we will focus on proving that proximal extensions preserve
Bohr recurrence. A extension $\pi\colon X\to Y$ is called proximal if the
fiber $\pi^{-1}(\\{y_{0}\\})$ of every $y_{0}\in Y$ is proximal (meaning that
every pair of points in $\pi^{-1}(\\{y_{0}\\})$ are proximal). We start with a
lemma.

######  Lemma A.1.

Let $\pi:(X,H)\to(Y,H)$ be a proximal extension between minimal systems. Then,
for all $x_{0},x_{1}\in X$ lying in the same fiber of $\pi$, there exists a
sequence $(h_{n})_{n\in\mathbb{N}}\subseteq H$ such that
$(h_{n}x_{0})_{n\in\mathbb{N}}\text{ and }(h_{n}x_{1})_{n\in\mathbb{N}}$
converge to $x_{0}$.

###### Proof.

Let $y_{0}\in Y$, $x_{0},x_{1}\in\pi^{-1}(\\{y_{0}\\})$ and $m\in\mathbb{N}$.
By proximality and compactness, there exists a sequence
$(g_{n})_{n\in\mathbb{N}}\subseteq H$ such that the sequences
$(g_{n}x_{0})_{n\in\mathbb{N}}$ and $(g_{n}x_{1})_{n\in\mathbb{N}}$ converge
to the same point $x\in X$. Also, by minimality of $(X,H)$, there exists $g\in
H$ such that $d_{X}(gx,x_{0})<1/(2m)$, and, by continuity of the action, we
have that for $n$ large enough $d_{X}(gg_{n}x_{0},gx)<1/(2m)$ and
$d_{X}(gg_{n}x_{1},gx)<1/(2m)$. Thus, $d_{X}(gg_{n}x_{0},x_{0})<1/m$ and
$d_{X}(gg_{n}x_{1},x_{0})<1/m$. Taking $h_{m}=gg_{n}$ the result follows. ∎

Now we present the proof of 3.6.

######  Theorem.

Let $\pi\colon(X,H)\to(Y,H)$ be a proximal extension between minimal systems
and let $R$ be a set of recurrence for $(Y,H)$. Then, $R$ is a set of
recurrence for $(X,H)$.

###### Proof.

Let $\epsilon>0$. As $R$ is a set of recurrence for $(Y,H)$, there exists a
sequence $(h_{n})_{n\in\mathbb{N}}\subseteq R$ such that $h_{n}y_{0}\to y_{0}$
as $n$ goes to $\infty$. Let $x_{0}\in X$ with $\pi(x_{0})=y_{0}$. Without
loss of generality we can assume by compactness that
$(h_{n}x_{0})_{n\in\mathbb{N}}$ converges to $x_{1}\in X$. The points
$x_{0},x_{1}$ belongs to $\pi^{-1}(\\{y_{0}\\})$ and this fiber is proximal by
hypothesis. Then, by A.1, there exists a sequence
$(\bar{h}_{n})_{n\in\mathbb{N}}\subseteq H$ such that
$(\bar{h}_{n}x_{0})_{n\in\mathbb{N}}$ and
$(\bar{h}_{n}x_{1})_{n\in\mathbb{N}}$ converge to $x_{0}$. Fix
$n\in\mathbb{N}$ such that $d_{X}(\bar{h}_{n}x_{0},x_{0})<\epsilon$ and
$d_{X}(\bar{h}_{n}x_{1},x_{0})<\epsilon$, and let $\delta>0$ be such that
$d_{X}(x,x^{\prime})<\delta\Longrightarrow
d_{X}(\bar{h}_{n}x,\bar{h}_{n}x^{\prime})<\epsilon.$ Now, consider
$m\in\mathbb{N}$ such that $d(h_{m}x_{0},x_{1})<\delta$. We have that
$d(\bar{h}_{n}h_{m}x_{0},\bar{h}_{n}x_{1})<\epsilon$ and thus
$d(\bar{h}_{n}h_{m}x_{0},x_{0})<2\epsilon$. Letting $z=\bar{h}_{n}x_{0}$, we
have that $d(z,x_{0})<\epsilon$ and $d(h_{m}z,x_{0})<2\epsilon$. Therefore $R$
is a set of recurrence for $X$. ∎

## Appendix B: Proof of 5.17

Once again, we repeat the statement of 5.17.

######  Proposition.

Let $(X,T_{1},\ldots,T_{d})$ be a minimal $s$-step nilsystem and assume that
$X$ is connected and $G_{0}$ is simply connected. Then, for every $\epsilon>0$
there exists $C>0$ such that for every $w\in G_{s}$ we can find
$h_{1},\ldots,h_{d}\in G_{s-1}$ and $\gamma\in\Gamma\cap G_{s}$ with

$d_{G}(h_{i},e_{G})<C,\leavevmode\nobreak\ \leavevmode\nobreak\ \forall
i\in[d];\leavevmode\nobreak\ \leavevmode\nobreak\
d_{G}([h_{1},\tau_{1}]\cdots[h_{d},\tau_{d}],w\gamma)<\epsilon.$

###### Proof.

Since $X$ is connected, it follows that $G=\langle G_{0},\Gamma\rangle$ and
there exist $\tau_{1}^{0},\ldots,\tau_{d}^{0}\in G_{0}$ and
$\gamma_{1}^{0},\ldots,\gamma_{d}^{0}\in\Gamma$ such that
$\tau_{i}=\tau_{i}^{0}\gamma_{i}^{0}$, where the $\tau_{i}$’s in $G$ are the
elements defining the action of the $T_{i}$’s. Recall also that $G=\langle
G_{0},\tau_{1},\ldots,\tau_{d}\rangle$, so we have that $G=\langle
G_{0},\gamma_{1}^{0},\ldots,\gamma_{d}^{0}\rangle$ and thus
$\Gamma=\langle\Gamma_{0},\gamma_{1}^{0},\ldots,\gamma_{d}^{0}\rangle$ with
$\Gamma_{0}=\Gamma\cap G_{0}$. As stated before, as $X$ is connected, there
exist $p,r,q\in\mathbb{N}$ such that

$G_{s}=\mathbb{R}^{r},G_{s}\cap\Gamma=\mathbb{Z}^{r},G_{s-1}/G_{s}=\mathbb{R}^{q}\text{
and }\Gamma\cap G_{s-1}/(\Gamma\cap G_{s})=\mathbb{Z}^{q},$

and these identifications are isometries. We also recall that
$Z=G/(G_{2}\Gamma)=G_{0}/(G_{2}\Gamma_{0})=\mathbb{T}^{p}$, and that the image
$\alpha=(\alpha_{i})_{i=1}^{d}$ of $\tau=(\tau_{i})_{i=1}^{d}$ in $Z$ induces
a minimal system. Let $(\beta_{i})_{i=1}^{d}$ be the projection of
$(\tau_{i}^{0})_{i=1}^{d}$ to $G_{0}/G_{2}=\mathbb{R}^{p}$. Then, the
projection of $(\beta_{i})_{i=1}^{d}$ in $G_{0}/(G_{2}\Gamma_{0})$ is equal to
the projection $\alpha$ of $\tau$ in $G/(G_{2}\Gamma)$. It follows by 5.3 that

$\text{ for all }N\in\mathbb{Z}^{p}\setminus\\{0\\}\text{ there exists
}i\in[d]\text{ such that }N\cdot\beta_{i}\notin\mathbb{Z}.$ (9)

We will denote $\beta_{i,k}$ the $k$-th coordinate of $\beta_{i}$, for
$i\in[d]$ and $k\in[p]$.

Let $\pi_{s}\colon G_{s}\to G_{s}/(\Gamma\cap G_{s})$ be the quotient map. We
claim that the map $f\colon G_{s-1}\times\cdots\times G_{s-1}\to
G_{s}/(\Gamma\cap G_{s}),$ defined by
$f(h_{1},\ldots,h_{d})=\pi_{s}([h_{1},\tau_{1}]\cdots[h_{d},\tau_{d}])$, has
dense image. Assuming the claim, there exists $C>0$ such that

$f\Big{(}(B_{G}(e_{G},C)\times\cdots\times
B_{G}(e_{G},C))\cap(G_{s-1}\times\cdots\times G_{s-1})\Big{)},$

is $\epsilon$-dense in $G_{s}/(\Gamma\cap G_{s})$, which corresponds to the
statement of the proposition.

To prove the claim, we assume by contradiction that it is false. First note
that for every $i\in[d]$ the map $g\to[g,\gamma_{i}^{0}]$ induces a group
homomorphism $F^{i}:G_{s-1}/G_{s}\to G_{s}$. Using additive notation and
writing in coordinates,

$\text{ for }1\leq l\leq r,\leavevmode\nobreak\
(F^{i}(x))_{l}=\sum_{j=1}^{q}F^{i}_{l,j}x_{j},$

and since $[G_{s-1}\cap\Gamma,\gamma_{i}^{0}]\subseteq G_{s}\cap\Gamma,$
$F^{i}$ maps $(G_{s-1}\cap\Gamma)/(G_{s}\cap\Gamma)$ to $G_{s}\cap\Gamma$, so
we have that the coefficients $F^{i}_{l,j}$ are integers.

The commutator map $G_{s-1}\times G_{0}\to G_{s}$ induces a homomorphism
$\Phi\colon G_{s-1}/G_{s}\times G_{0}/G_{2}\to G_{s}$. Using additive notation
and writing in coordinates,

$\text{ for }1\leq l\leq
r,(\Phi(x,y))_{l}=\sum_{j=1}^{q}\sum_{k=1}^{p}\Phi_{l,j,k}x_{j}y_{k},$

where the coefficients $\Phi_{l,j,k}$ are integers since $\Phi$ maps
$(G_{s-1}\cap\Gamma)\times\Gamma_{0}$ to $G_{s}\cap\Gamma$. For each
$i\in[d]$, the commutator map $[\bullet,\tau_{i}]:G_{s-1}\to G_{s}$ induces a
homomorphism $\Psi^{i}:G_{s-1}/G_{s}\to G_{s}$. Using multiplicative notation,
we have that $\Psi^{i}(x)=\Phi(x,\tau_{i}^{0})F^{i}(x)$. In coordinates, with
additive notation,

$\text{ for }1\leq l\leq
r,(\Psi^{i}(x))_{l}=\sum_{j=1}^{q}(F^{i}_{l,j}+\sum_{k=1}^{p}\Phi_{l,j,k}\beta_{l,k})x_{j}.$
(10)

Note that $f(G_{s-1}\times\cdots\times G_{s-1})$ is the range of
$\pi_{s}\circ(\sum_{i=1}^{d}\Psi^{i})$, and if this range is not dense in
$\mathbb{T}^{r}$, then it is included in a proper subtorus. Thus, there exist
integers $\lambda_{1},\ldots,\lambda_{r}$, not all equal to $0$, such that the
range of $\sum_{i=1}^{d}\Psi^{i}$ is included in the group $K$ defined by the
relation

$z\in K\text{ if and only if }\sum_{l=1}^{r}\lambda_{l}z_{l}\in\mathbb{Z}.$

In coordinates,

$\text{ for every
}(x_{i})_{i=1}^{d}\in(\mathbb{R}^{q})^{d},\leavevmode\nobreak\
\leavevmode\nobreak\
\sum_{l=1}^{r}\lambda_{l}\sum_{i=1}^{d}\sum_{j=1}^{q}(F^{i}_{l,j}+\sum_{k=1}^{p}\Phi_{l,j,k}\beta_{i,k})x_{i,j}\in\mathbb{Z},$

and thus

$\text{ for }1\leq j\leq q,\leavevmode\nobreak\ \leavevmode\nobreak\ 1\leq
i\leq d,\leavevmode\nobreak\ \leavevmode\nobreak\
\sum_{l=1}^{r}(F^{i}_{l,j}+\sum_{k=1}^{p}\Phi_{l,j,k}\beta_{i,k})\lambda_{l}=0.$
(11)

Since the coefficients $F^{i}_{l,j}$ are integers,

$\text{ for }1\leq j\leq q,\leavevmode\nobreak\ \leavevmode\nobreak\ 1\leq
i\leq d,\leavevmode\nobreak\ \leavevmode\nobreak\
\sum_{k=1}^{p}\Bigl{(}\sum_{l=1}^{r}\Phi_{l,j,k}\lambda_{i}\Bigr{)}\beta_{i,k}\in\mathbb{Z}.$
(12)

Thus, we have that

$\text{for }1\leq j\leq q,\leavevmode\nobreak\ \leavevmode\nobreak\
(\sum_{l=1}^{r}\lambda_{l}\Phi_{l,j,k})_{k=1}^{p}=0,$ (13)

otherwise (9) contradicts (12). This implies that the range of $\Phi$ is
included in the proper closed subgroup $K$ of $G_{s}=\mathbb{R}^{r}$, and thus
$[G_{0},G_{s-1}]\subseteq K$. Moreover, inserting (13) into (11), we get that

$\text{ for }1\leq j\leq q,\leavevmode\nobreak\ \leavevmode\nobreak\ 1\leq
i\leq d,\leavevmode\nobreak\ \leavevmode\nobreak\
\sum_{l=1}^{r}F^{i}_{l,j}\lambda_{l}=0.$

This means that the range of $F^{i}$ is included in $K$. In particular,
$[\gamma_{1}^{0}\cdots\gamma_{d}^{0},G_{s-1}]\subseteq K$. As
$G=\langle\Gamma_{0},G_{0}\rangle$ and for every $x\in G_{s-1}$ the map
$g\to[g,x]$ is a group homomorphism, then

$G_{s}=[G,G_{s-1}]=[G_{0},G_{s-1}][\gamma_{1}^{0}\cdots\gamma_{d}^{0},G_{s-1}]\subseteq
K,$

which is a contradiction, concluding the claim and finishing the proof. ∎

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