# Topological mild mixing of all orders along polynomials

Yang Cao  and  Song Shao  CAS Wu Wen-Tsun Key Laboratory of Mathematics, and
Department of Mathematics, University of Science and Technology of China,
Hefei, Anhui, 230026, P.R. China<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

A minimal system $(X,T)$ is topologically mildly mixing if all non-empty open
subsets $U,V$, $\\{n\in\mathbb{Z}:U\cap T^{-n}V\neq\emptyset\\}$ is an
IP∗-set. In this paper we show that if a minimal system is topologically
mildly mixing, then it is mild mixing of all orders along polynomials. That
is, suppose that $(X,T)$ is a topologically mildly mixing minimal system,
$d\in\mathbb{N}$, $p_{1}(n),\ldots,p_{d}(n)$ are integral polynomials with no
$p_{i}$ and no $p_{i}-p_{j}$ constant, $1\leq i\neq j\leq d$, then for all
non-empty open subsets $U,V_{1},\ldots,V_{d}$, $\\{n\in\mathbb{Z}:U\cap
T^{-p_{1}(n)}V_{1}\cap T^{-p_{2}(n)}V_{2}\cap\ldots\cap
T^{-p_{d}(n)}V_{d}\neq\emptyset\\}$ is an IP∗-set. We also give the theorem
for systems under abelian group actions.

###### 2010 Mathematics Subject Classification:

Primary: 37B20, Secondary: 37B05, 37A25

This research is supported by NNSF of China (11971455, 11571335).

##  1\. Introduction

###  1.1. Background

IP-systems were introduced and studied in topological dynamics by Furstenberg
and Weiss in [7], in ergodic theory by Furstenberg in [8, 9]. Let $S$ be an
abelian semigroup and $\mathbb{N}=\\{1,2,\ldots\\}$ be the set of natural
numbers. An IP-set of $S$ consists of a sequence of (not necessarily distinct)
elements $s_{i}\in S,i\in\mathbb{N}$, together with all products
$s_{i_{1}}s_{i_{2}}\ldots s_{i_{k}}$ that can be formed using distinct indices
$i_{1}<i_{1}<\ldots<i_{k}$. Let $\mathcal{F}$ denote the family of all non-
empty finite subsets of $\mathbb{N}$. An $\mathcal{F}$-sequence in $S$ is a
sequence $\\{s_{\alpha}\\}_{\alpha\in\mathcal{F}}$ indexed by the set
$\mathcal{F}$. If set $s_{\alpha}=s_{i_{1}}s_{i_{2}}\cdots s_{i_{k}}$ for any
$\alpha=\\{i_{1},i_{2},\cdots,i_{k}\\}\in\mathcal{F}$, then any IP-set has
form $\\{s_{\alpha}\\}_{\alpha\in\mathcal{F}}$. Note that if
$\alpha\cap\beta=\emptyset$, then $s_{\alpha\cup\beta}=s_{\alpha}s_{\beta}$.
In general $s_{\alpha}s_{\beta}$ need not belong to the set and so an IP-set
need not be a semigroup. On the other hand every countable semigroup is an IP-
set. If each generator $s_{i}$ of an IP-set occurs infinitely often in the
sequence $\\{s_{i}\\}_{i=1}^{\infty}$, then the IP-set generated is a
semigroup. Thus IP-sets should be viewed as generalized semigroups.

For example, let $\\{n_{i}\\}_{i=1}^{\infty}\subseteq\mathbb{Z}$, where
$\mathbb{Z}$ is the set of integers. An IP-set generated by the sequence
$\\{n_{i}\\}_{i=1}^{\infty}$ is the set

$FS(\\{n_{i}\\}_{i=1}^{\infty}):=\\{n_{i_{1}}+n_{i_{2}}+\cdots+n_{i_{k}}:i_{1}<i_{2}<\cdots<i_{k}\
\text{for some $k\in\mathbb{N}$}\\}.$

If we denote $n_{\alpha}=\sum_{i\in\alpha}n_{i}$, then
$FS(\\{n_{i}\\}_{i=1}^{\infty})=\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}.$ A
set is called an IP∗-set if it intersects any IP-set. An important result
about IP-sets is the following Hindman theorem [13]: For any finite partition
of $\mathbb{N}$, one of the cells of the partition contains an IP-set.

An IP-set $\\{T_{\alpha}\\}_{\alpha\in\mathcal{F}}$ of commuting
transformations of a space $X$ will be called an IP-system. The IP-version of
van der Waerden’s theorem was given by Furstenberg and Weiss in [7], and an
ergodic Szemerédi theorem for IP-systems was built by Furstenberg and
Katznelson in [10]. One of mixing properties related to IP-system is mild
mixing. The notion of mild mixing was introduced by Furstenberg and Weiss [6]
(this property was studied independently by Walters [19] from the different
viewpoint). Mild mixing is stronger than weak mixing, and weaker than strong
mixing [19, 6].

A measure preserving system $(X,\mathcal{B},\mu,T)$ is mildly mixing if for
any $A,B\in\mathcal{B}$ and any $\epsilon>0$,

$\\{n\in\mathbb{Z}:|\mu(A\cap T^{-n}B)-\mu(A)\mu(B)|<\epsilon\\}$

is an IP∗-set. Furstenebrg showed that mild mixing implies mild mixing of all
orders:

Theorem (Furstenberg) [8, 9] Let $(X,\mathcal{B},\mu,T)$ be a mildly mixing
system and $d\in\mathbb{N}$. Let $A_{0},A_{1},\ldots,A_{d}\in\mathcal{B}$ and
$a_{1},a_{2},\ldots,a_{d}$ be distinct non-zero integers. Then for any
$\epsilon>0$,

$\\{n\in\mathbb{Z}:|\mu(A_{0}\cap T^{-a_{1}n}A_{1}\cap\ldots\cap
T^{-a_{d}n}A_{d})-\mu(A_{0})\mu(A_{1})\ldots\mu(A_{d})|<\epsilon\\}$

is an IP∗-set.

An integral polynomial is a polynomial taking integer values at the integers.
Bergelson gave a polynomial ergodic theorem for mildly mixing systems.

Theorem (Bergelson)[1] Let $(X,\mathcal{B},\mu,T)$ be a mildly mixing system
and $d\in\mathbb{N}$. Let $p_{1}(n),\ldots,p_{d}(n)$ be integral polynomials
with no $p_{i}$ and no $p_{i}-p_{j}$ constant, $1\leq i\neq j\leq d$. Then for
all $A_{0},A_{1},\ldots,A_{d}\in\mathcal{B}$ and all $\epsilon>0$,

$\\{n\in\mathbb{Z}:|\mu(A_{0}\cap T^{-p_{1}(n)}A_{1}\cap\ldots\cap
T^{-p_{d}(n)}A_{d})-\mu(A_{0})\mu(A_{1})\ldots\mu(A_{d})|<\epsilon\\}$

is an IP∗-set.

Theorem above was generalized to commuting transformations by Bergelson and
McCutcheon [4]. The following result is a special case of [4, Theorem 0.14].
Note that in [4, Theorem 0.14] polynomials there are from $\mathbb{Z}^{m}$ to
$\mathbb{Z}$.

Theorem (Bergelson-McCutcheon)[4] Let $(X,\mathcal{B},\mu,\Gamma)$ be a
measure preserving system, where $\Gamma$ is an abelian group such that for
each $T\in\Gamma$, $T\neq e_{\Gamma}$111$e_{\Gamma}$ is the unit of $\Gamma$.,
is mildly mixing. For $d,k\in\mathbb{N}$, let $T_{1},\ldots,T_{d}\in\Gamma$,
and $p_{i,j},1\leq i\leq k,1\leq j\leq d$ be integral polynomials such that
the expressions $g_{i}(n)$

$g_{i}(n)=T_{1}^{p_{i,1}(n)}\cdots T_{d}^{p_{i,d}(n)},\quad i=1,2,\ldots,k,$

and the expressions

$g_{i}(n)g_{j}(n)^{-1}=T_{1}^{p_{i,1}(n)-p_{j,1}(n)}\cdots
T_{d}^{p_{i,d}(n)-p_{j,d}(n)},\quad i\neq j\in\\{1,2,\ldots,k\\},$

depend nontrivially on $n$222We say that $g(n)$ depends nontrivially on $n$,
if $g(n)$ is a nonconstant mapping of $\mathbb{Z}$ into $\Gamma$.. Then for
all $A_{0},A_{1},\ldots,A_{k}\in\mathcal{B}$ and any $\epsilon>0$,

$\\{n\in\mathbb{Z}:|\mu(A_{0}\cap g_{1}(n)^{-1}A_{1}\cap\ldots\cap
g_{k}(n)^{-1}A_{k})-\mu(A_{0})\mu(A_{1})\ldots\mu(A_{k})|<\epsilon\\}$

is an IP∗-set.

The purpose of this paper is to give the topological version of theorems
above.

###  1.2. Main results

The notion of topological mild mixing was introduced by Glasner and Weiss
[12], and independently by Huang and Ye [15]. A minimal system $(X,T)$ is
topologically mildly mixing if all non-empty open subsets $U,V$,

$N(U,V)=\\{n\in\mathbb{Z}:U\cap T^{-n}V\neq\emptyset\\}$

is an IP∗-set.333This is not the original definition for topological mild
mixing in [12, 15]. There a topological system $(X,T)$ is said to be
topologically mildly mixing if the product system with any transitive system
is still transitive. But for minimal systems, the definitions are consistent
[15].

First corresponding to Furstenberg’s theorem, we have

######  Theorem 1.1.

Let $(X,T)$ be a topologically mildly mixing minimal system and
$d\in\mathbb{N}$. Let $U,V_{1},\ldots,V_{d}$ be non-empty open subsets and
$a_{1},a_{2},\ldots,a_{d}$ be distinct non-zero integers. Then

$\\{n\in\mathbb{Z}:U\cap T^{-a_{1}n}V_{1}\cap T^{-a_{2}n}V_{2}\cap\ldots\cap
T^{-a_{d}n}V_{d}\neq\emptyset\\}$

is an IP∗-set.

As corollary of Theorem 1.1, we have:

######  Corollary 1.2.

Let $(X,T)$ be a topologically mildly mixing minimal system. Then for any
$d\in\mathbb{N}$, any distinct
$a_{1},\ldots,a_{d}\in\mathbb{Z}\setminus\\{0\\}$, and every IP-set $A$, there
is a dense $G_{\delta}$ subset $X_{0}$ such that for any $x\in X_{0}$,

$\overline{\\{(T^{a_{1}n}x,\ldots,T^{a_{d}n}x):n\in A\\}}=X^{d}.$

Then we have the following result corresponding to Bergelson’s theorem:

######  Theorem 1.3.

Let $(X,T)$ be a topologically mildly mixing minimal system and
$d\in\mathbb{N}$. Let $p_{1}(n),\ldots,p_{d}(n)$ be integral polynomials with
no $p_{i}$ and no $p_{i}-p_{j}$ constant, $1\leq i\neq j\leq d$. Then for all
non-empty open subsets $U,V_{1},\ldots,V_{d}$,

$\\{n\in\mathbb{Z}:U\cap T^{-p_{1}(n)}V_{1}\cap
T^{-p_{2}(n)}V_{2}\cap\ldots\cap T^{-p_{d}(n)}V_{d}\neq\emptyset\\}$

is an IP∗-set.

By Theorem 1.3, we have the following corollaries.

######  Corollary 1.4.

Let $(X,T)$ be a topologically mildly mixing minimal system and
$d\in\mathbb{N}$. Let $p_{1}(n),\ldots,p_{d}(n)$ be integral polynomials with
no $p_{i}$ and no $p_{i}-p_{j}$ constant, $1\leq i\neq j\leq d$. Then for
every IP-set $A$, there is a dense $G_{\delta}$ subset $X_{0}$ such that for
any $x\in X_{0}$,

$\\{(T^{p_{1}(n)}x,\ldots,T^{p_{d}(n)}x):n\in A\\}$

is dense in $X^{d}$.

######  Corollary 1.5.

Let $(X,T)$ be a topologically mildly mixing minimal system, and let $p(n)$ be
a nontrivial integral polynomial. Then for every IP-set $A$, there is a dense
$G_{\delta}$ subset $X_{0}$ such that for any $x\in X_{0}$, $\\{T^{p(n)}x:n\in
A\\}$ is dense in $X$.

Finally we have a general result for minimal systems under abelian group
actions.

######  Theorem 1.6.

Let $(X,\Gamma)$ be a minimal system, where $\Gamma$ is an abelian group such
that for each $T\in\Gamma$, $T\neq e_{\Gamma}$, is topologically mildly mixing
minimal. For $d,k\in\mathbb{N}$, let $T_{1},\ldots,T_{d}\in\Gamma$, and
$p_{i,j},1\leq i\leq k,1\leq j\leq d$ be integral polynomials such that the
expressions

$g_{i}(n)=T_{1}^{p_{i,1}(n)}\cdots T_{d}^{p_{i,d}(n)},\quad i=1,2,\ldots,k,$

and the expressions

$g_{i}(n)g_{j}(n)^{-1}=T_{1}^{p_{i,1}(n)-p_{j,1}(n)}\cdots
T_{d}^{p_{i,d}(n)-p_{j,d}(n)},\quad i\neq j\in\\{1,2,\ldots,k\\},$

depend nontrivially on $n$. Then for all non-empty open subsets
$U,V_{1},\ldots,V_{k}$,

$\\{n\in\mathbb{Z}:U\cap g_{1}(n)^{-1}V_{1}\cap
g_{2}(n)^{-1}V_{2}\cap\ldots\cap g_{k}^{-1}V_{k}\neq\emptyset\\}$

is an IP∗-set.

As a corollary, we have:

######  Corollary 1.7.

Under assumptions of Theorem 1.6, for any IP-set $A$ there is a dense
$G_{\delta}$ set $X_{0}$ of $X$ such that for all $x\in X_{0}$,

$\\{(g_{1}(n)x,\ldots,g_{k}(n)x):n\in A\\}$

is dense in $X^{k}$.

As in [4, 9, 10], we will deal with mixing for IP-systems, and get results for
mild mixing as corollaries. Here we will not repeat the statements of results
for mixing IP-systems, and see Section 3 for details.

###  1.3. The organization of the paper

We organize the paper as follows. In Section 2, we introduce some basic
notions and results needed in the paper. In Section 3, we give the statements
of the results of the paper. In Section 4 we give a proof for the linear case
to warm up. In Section 5, we give examples to show how to use PET-induction to
prove the results, and in Section 6 we give the complete proof for the main
result Theorem 3.2.

##  2\. Preliminaries

###  2.1. Topological transformation groups

A topological dynamical system (t.d.s. for short) is a triple
$\mathcal{X}=(X,\Gamma,\Pi)$, where $X$ is a compact metric space with metric
$\rho$, $\Gamma$ is a Hausdorff topological group with the unit $e_{\Gamma}$
and $\Pi:\Gamma\times X\rightarrow X$ is a continuous map such that
$\Pi(e_{\Gamma},x)=x$ and $\Pi(s,\Pi(t,x))=\Pi(st,x)$. We shall fix $\Gamma$
and suppress the action symbol. Then for any $t\in\Gamma$, $t:X\rightarrow X$
is a homeomorphism, and $e_{\Gamma}={\rm Id}_{X}$ is the identity map.

Let $(X,\Gamma)$ be a t.d.s. and $x\in X$, then
$\mathcal{O}(x,\Gamma)=\\{tx:t\in\Gamma\\}$ denotes the orbit of $x$. A subset
$A\subseteq X$ is called invariant if $ta\subseteq A$ for all $a\in A$ and
$t\in\Gamma$. When $Y\subseteq X$ is a closed and $\Gamma$-invariant subset of
the system $(X,\Gamma)$, we say that the system $(Y,\Gamma)$ is a subsystem of
$(X,\Gamma)$. If $(X,\Gamma)$ and $(Y,\Gamma)$ are two dynamical systems,
their product system is the system $(X\times Y,\Gamma)$, where
$t(x,y)=(tx,ty)$.

A system $(X,\Gamma)$ is called minimal if $X$ contains no proper non-empty
closed invariant subsets. $(X,\Gamma)$ is called transitive if for all non-
empty open sets $U,V$, there is some $t\in\Gamma$ such that $U\cap
t^{-1}V\neq\emptyset$. It is easy to verify that a system is minimal iff every
orbit is dense. A point $x\in X$ is called a minimal point if
$(\overline{\mathcal{O}(x,\Gamma)},\Gamma)$ is a minimal subsystem. A system
$(X,\Gamma)$ is weakly mixing if the product system $(X\times X,\Gamma)$ is
transitive.

When $\Gamma=\mathbb{Z}$, $(X,\Gamma)$ is determined by a homeomorphism $T$,
i.e. $T$ is the transformation corresponding to $1$ of $\mathbb{Z}$. In this
case, we usually denote $(X,\mathbb{Z})$ by $(X,T)$.

###  2.2. Furstenberg families

####  2.2.1. Furstenberg families

A collection $\mathcal{G}$ of subsets of $\mathbb{Z}$ is a (Furstenberg)
family if it is hereditary upward, i.e. $B_{1}\subset B_{2}$ and
$B_{1}\in\mathcal{G}$ imply $B_{2}\in\mathcal{G}$. A family $\mathcal{G}$ is
called proper if it is neither empty nor the entire power set of $\mathbb{Z}$,
or, equivalently if $\mathbb{Z}\in\mathcal{G}$ and
$\emptyset\not\in\mathcal{G}$. Any nonempty collection $\mathcal{A}$ of
subsets of $\mathbb{Z}$ generates a family
$\mathcal{G}(\mathcal{A}):=\\{B\subset\mathbb{Z}:B\supset A$ for some
$A\in\mathcal{A}\\}$.

For a family $\mathcal{G}$, its dual is the family
$\mathcal{G}^{\ast}:=\\{B\subset\mathbb{Z}:B\cap A\neq\emptyset\ \text{for
all}\ A\in\mathcal{G}\\}$. It is not hard to see that
$\mathcal{G}^{*}=\\{B\subset\mathbb{Z}:\mathbb{Z}\setminus
B\not\in\mathcal{G}\\}$, from which we have that if $\mathcal{G}$ is a family,
then $(\mathcal{G}^{*})^{*}=\mathcal{G}.$

####  2.2.2. Filter and Ramsey property

If a family $\mathcal{G}$ is closed under finite intersections and is proper,
then it is called a filter. A family $\mathcal{G}$ has the Ramsey property if
$A=A_{1}\cup A_{2}\in\mathcal{G}$ implies that $A_{1}\in\mathcal{G}$ or
$A_{2}\in\mathcal{G}$. It is well known that

######  Lemma 2.1.

[8, Lemma 9.4, Lemma 9.5] A proper family has the Ramsey property if and only
if its dual $\mathcal{G}^{*}$ is a filter. Let $\mathcal{G}$ be a family
having Ramsey property, then for all $A\in\mathcal{G}$ and
$B\in\mathcal{G}^{*}$, we have $A\cap B\in\mathcal{G}$.

###  2.3. IP systems

Let $\mathcal{F}$ denote the family of all non-empty finite subsets of
$\mathbb{N}$, i.e. $\alpha\in\mathcal{F}$ iff
$\alpha=\\{i_{1},i_{2},\cdots,i_{k}\\}\subset\mathbb{N}$,
$i_{1}<i_{2}<\cdots<i_{k}$ for some $k$. For $\alpha,\beta\in\mathcal{F}$,
$\alpha<\beta$ (or $\beta>\alpha$) if $\max\alpha<\min\beta$. A subset of
$\mathcal{F}$

$\mathcal{F}^{(1)}=FU(\\{\alpha_{i}\\}_{i=1}^{\infty}):=\\{\bigcup_{i\in\beta}\alpha_{i}:\
\beta\in\mathcal{F}\\}$

is called IP-ring, where $\alpha_{1}<\alpha_{2}<\cdots<\alpha_{n}<\cdots$.

Let $\\{n_{i}\\}_{i=1}^{\infty}\subseteq\mathbb{Z}$. An IP-set generated by
the sequence $\\{n_{i}\\}_{i=1}^{\infty}$ is the set

$FS(\\{n_{i}\\}_{i=1}^{\infty})=\\{n_{i_{1}}+n_{i_{2}}+\cdots+n_{i_{k}}:i_{1}<i_{2}<\cdots<i_{k}\
\text{for some $k\in\mathbb{N}$}\\}.$

If we denote $n_{\alpha}=\sum_{i\in\alpha}n_{i}$, then

$FS(\\{n_{i}\\}_{i=1}^{\infty})=\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}.$

Note that we do not require the elements of $\\{n_{i}\\}$ are distinct. A set
is called an IP∗-set if it intersects any IP-set. We denote by
$\mathcal{F}_{IP}$ and $\mathcal{F}_{IP}^{*}$ the families generated by all
IP-sets and IP∗-sets respectively.

Here are some equivalent versions of Hindman theorem [13].

######  Theorem 2.2.

[8, Theorem 8.12] For any finite partition
$\mathcal{F}=\bigcup_{i=1}^{r}C_{i}$, one of $C_{i}$ contains an $IP$-ring.

######  Theorem 2.3.

[8, Proposition 9.6] $\mathcal{F}_{IP}$ has Ramsey property.

Let $A=\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$ be an IP-set. A subset $B$ of
$A$ is an IP-subset of $A$ if it is an IP-set itself, i.e. there is some
sequence $\\{m_{i}\\}_{i=1}^{\infty}\subset A$ such that
$B=\\{m_{\alpha}\\}_{\alpha\in\mathcal{F}}\subset A$. It is easy to see that
$B$ is an IP-subset of $A$ if and only if there is some IP-ring
$\mathcal{F}^{(1)}$ such that
$B=\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}^{(1)}}$ [9].

######  Definition 2.4.

Let $A=\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$ be an IP-set. The family
generated by all IP-subset of $\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$ is
denoted by $\mathcal{G}_{A}$, that is

$\mathcal{G}_{A}=\\{B\subset\mathbb{Z}:B\supset\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}^{(1)}},\mathcal{F}^{(1)}\
\text{is an IP-ring}\\}.$

######  Remark 2.5.

  1. (1)

By Hindman theorem, for any IP-set
$A=\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$, $\mathcal{G}_{A}$ has Ramsey
property and its dual $\mathcal{G}^{*}_{A}$ is a filter. By Lemma 2.1, if
$A\in\mathcal{G}_{A}$ and $B\in\mathcal{G}_{A}^{*}$, then $A\cap
B\in\mathcal{G}_{A}$.

  2. (2)

When $A=\mathbb{Z}$, then $\mathcal{G}_{A}=\mathcal{F}_{IP}$ and
$\mathcal{G}_{A}^{*}=\mathcal{F}_{IP}^{*}$.

###  2.4. IP-mixing and mild mixing

######  Definition 2.6.

Let $(X,\\{T_{\alpha}\\}_{\alpha\in\mathcal{F}})$ be a IP-system.
$(X,\\{T_{\alpha}\\}_{\alpha\in\mathcal{F}})$ is called mixing if for all non-
empty open subsets $U,V$ of $X$, and any IP-ring $\mathcal{F}^{(1)}$, there is
some $\alpha\in\mathcal{F}^{(1)}$,

$U\cap T^{-1}_{\alpha}V\neq\emptyset.$

Let $(X,T)$ be a t.d.s. and let $\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$ be
an IP-set of $\mathbb{Z}$. Let $T_{\alpha}=T^{n_{\alpha}}$ for
$\alpha\in\mathcal{F}$. Then $\\{T_{\alpha}\\}_{\alpha\in\mathcal{F}}$ is an
IP-system. If $\\{T_{\alpha}\\}_{\alpha\in\mathcal{F}}$ is mixing as an IP-
system, we also say that it is mixing along
$\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$. To be precise, we have

######  Definition 2.7.

Let $(X,T)$ be a t.d.s. and let $\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$ be
an IP-set. $(X,T)$ is said to be mixing along
$\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$ if for all non-empty open subsets
$U,V$ of $X$, and any IP-ring $\mathcal{F}^{(1)}$, there is some
$\alpha\in\mathcal{F}^{(1)}$,

$U\cap T^{-n_{\alpha}}V\neq\emptyset.$

That is,

$N(U,V)=\\{n\in\mathbb{Z}:U\cap
T^{-n}V\neq\emptyset\\}\in\mathcal{G}_{A}^{*}.$

######  Remark 2.8.

When the IP-set $\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$ is $\mathbb{Z}$, we
cover the definition of topological mild mixing. That is, a minimal system
$(X,T)$ is topologically mildly mixing if all non-empty open subsets $U,V$,
$N(U,V)=\\{n\in\mathbb{Z}:U\cap
T^{-n}V\neq\emptyset\\}\in\mathcal{F}_{IP}^{*}.$

Since $\mathcal{G}_{A}^{*}$ is a filter, we have the following important
observation.

######  Proposition 2.9.

Let $A=\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$ be an IP-set and
$d\in\mathbb{N}$. If $(X,T_{1}),\ldots,(X,T_{d})$ are mixing along
$\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$ systems, then
$(X_{1}\times\ldots\times X_{d},T_{1}\times\ldots\times T_{d})$ is mixing
along $\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$.

In particular, if $(X,T_{1}),\ldots,(X,T_{d})$ are topologically mildly mixing
minimal systems. Then $(X_{1}\times\ldots\times X_{d},T_{1}\times\ldots\times
T_{d})$ is topologically mildly mixing.

For an IP set $\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$, if $(X,T)$ is mixing
along $\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$ and
$k\in\mathbb{Z}\setminus\\{0\\}$, we do not know if $(X,T^{k})$ is mixing
along $\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$. But it is true for a mildly
mixing system.

######  Proposition 2.10.

Let $k\in\mathbb{Z}\setminus\\{0\\}$. If $(X,T)$ is topologically mildly
mixing, then so does $(X,T^{k})$.

###### Proof.

Let $A=FS(\\{n_{i}\\}_{i=1}^{\infty})$ be an IP-set. Then
$kA:=FS(\\{kn_{i}\\}_{i=1}^{\infty})$ is also an IP-set. Since $(X,T)$ is
topologically mildly mixing, we have

$N(U,V)=\\{n\in\mathbb{Z}:U\cap T^{-n}V\neq\emptyset\\}\cap kA\neq\emptyset.$

It follows that

$\\{n\in\mathbb{Z}:U\cap T^{-kn}V\neq\emptyset\\}\cap A\neq\emptyset.$

Thus $(X,T^{k})$ is topologically mildly mixing. ∎

###  2.5. Bergelson-Leibman theorem

We will need the following Bergelson-Leibman’ theorem.

######  Theorem 2.11 (Bergelson-Leibman).

[5] Let $(X,\Gamma)$ be a t.d.s. with $\Gamma$ an abelian group, and let
$d,k\in\mathbb{N}$. Let $T_{1},\ldots,T_{d}\in\Gamma$, and let $p_{i,j}$ be
integral polynomials with $p_{i,j}(0)=0$, $i=1,2,\ldots,k,j=1,2,\ldots,d$.
Then, for any positive $\epsilon$, there exist $x\in X$ and $n\in\mathbb{N}$
such that

(2.1) $\rho(T_{1}^{p_{i,1}(n)}T_{2}^{p_{i,2}(n)}\ldots
T_{d}^{p_{i,d}(n)}x,x)<\epsilon$

for all $i=1,\ldots,k$ simultaneously. Moreover, the set

$\\{n\in\mathbb{Z}:\forall\epsilon>0,\exists x\in X\ \text{ such that}\
\forall i\in\\{1,2,\ldots,k\\},\eqref{a11}\ \text{is satisfied}\\}$

is an IP∗-set.

###  2.6. $\\{g_{1},\ldots,g_{k}\\}$-transitive along
$\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$

Let $({X},\Gamma)$ be a t.d.s., and
$A=\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$ be an IP-set. For
$d,k\in\mathbb{N}$, let $T_{1},\ldots,T_{d}\in\Gamma$, and $p_{i,j}(n)$ be
integral polynomials, $1\leq i\leq k,1\leq j\leq d$ and

$g_{i}(n)=T_{1}^{p_{i,1}(n)}\cdots T_{d}^{p_{i,d}(n)},\quad i=1,2,\ldots,k.$

######  Definition 2.12.

We call $(X,\Gamma)$ $\\{g_{1},\ldots,g_{k}\\}$-transitive along
$\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$ if for all non-empty open sets
$U_{1},\ldots,U_{d},V_{1},\ldots,V_{d},\alpha_{0}\in\mathcal{F}$,and any IP-
ring $\mathcal{F}^{(1)}$, there is some $\alpha\in\mathcal{F}^{(1)}$ such that
$\alpha>\alpha_{0}$ and

$U_{1}\times\ldots\times U_{k}\cap g_{1}^{-1}(n_{\alpha})\times\ldots\times
g_{k}^{-1}(n_{\alpha})(V_{1}\times\ldots\times V_{d})\neq\emptyset.$

The following lemma is a generalization of Lemma 3 of [17]. For completeness,
we include a proof.

######  Lemma 2.13.

Let $(X,\Gamma)$ be a t.d.s. and $T\in\Gamma$. Let $g_{1},\ldots,g_{k}$ be as
above. If $({X},\Gamma)$ is $\\{g_{1},\ldots,g_{k}\\}$-transitive along
$\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$, then for all non-empty open sets
$V_{1},\ldots,V_{k}$ of $X$, and any IP-ring $\mathcal{F}^{(1)}$, there is an
increasing sequence
$\\{\alpha_{n}\\}_{n=0}^{\infty}\subseteq\mathcal{F}^{(1)}$ such that
$|n_{\alpha_{n}}|-n>0$ for all $n$, and for each $i\in\\{1,2,\ldots,k\\}$,
there is a descending sequence $\\{V_{i}^{(n)}\\}_{n=0}^{\infty}$ of open
subsets of $V_{i}$ such that for each $n\geq 0$ one has that

$g_{i}(n_{\alpha_{j}})T^{-j}V_{i}^{(n)}\subseteq V_{i},\quad\text{for
all}\quad 0\leq j\leq n.$

###### Proof.

Let $V_{1},\ldots,V_{k}$ be non-empty open subsets of $X$ and let
$\mathcal{F}^{(1)}$ be an IP-ring. Since $({X},\Gamma)$ is
$\\{g_{1},\ldots,g_{k}\\}$-transitive along
$\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$, there is some
$\alpha_{0}\in\mathcal{F}^{(1)}$ such that

$V_{1}\times\ldots\times V_{k}\cap
g_{1}^{-1}(n_{\alpha_{0}})\times\ldots\times
g_{k}^{-1}(n_{\alpha_{0}})(V_{1}\times\ldots\times V_{k})\neq\emptyset.$

That is, $g_{i}^{-1}(n_{\alpha_{0}})V_{i}\cap V_{i}\neq\emptyset$ for all
$i=1,\ldots,k$. Put $V_{i}^{(0)}=g_{i}^{-1}(n_{\alpha_{0}})V_{i}\cap V_{i}$
for all $i=1,\ldots,k$ to complete the base step.

Now assume that for $n\geq 1$ we have found a sequence
$\alpha_{0}<\alpha_{1}<\ldots<\alpha_{n-1}$ of $\mathcal{F}^{(1)}$ and for
each $i=1,\ldots,k$, we have non-empty open subsets $V_{i}\supseteq
V_{i}^{(0)}\supseteq V_{i}^{(1)}\ldots\supseteq V_{i}^{(n-1)}$ such that for
each $m=0,1,\ldots,n-1$ one has that $|n_{\alpha_{m}}|-m>0$ and

(2.2) $g_{i}(n_{\alpha_{j}})T^{-j}V_{i}^{(m)}\subseteq V_{i},\quad\text{for
all}\quad 0\leq j\leq m.$

For $i=1,\ldots,k$, let $U_{i}=T^{-n}(V_{i}^{(n-1)})$. Since $({X},\Gamma)$ is
$\\{g_{1},\ldots,g_{k}\\}$-transitive along
$\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$, there is some
$\alpha_{n}\in\mathcal{F}^{(1)}$such that $\alpha_{n}>\alpha_{n-1}$,
$|n_{\alpha_{n}}|-n>0$, and

$U_{1}\times\ldots\times U_{k}\cap
g_{1}^{-1}(n_{\alpha_{n}})\times\ldots\times
g_{k}^{-1}(n_{\alpha_{n}})(V_{1}\times\ldots\times V_{k})\neq\emptyset.$

That is, $g_{i}^{-1}(n_{\alpha_{n}})V_{i}\cap U_{i}\neq\emptyset$ for all
$i=1,\ldots,k$.

Then for $i=1,\ldots,k$,

$g_{i}(n_{\alpha_{n}})U_{i}\cap
V_{i}=g_{i}(n_{\alpha_{n}})T^{-n}V_{i}^{(n-1)}\cap V_{i}\not=\emptyset.$

Let

$V_{i}^{(n)}=V_{i}^{(n-1)}\cap\big{(}g_{i}(n_{\alpha_{n}})T^{-n}\big{)}^{-1}V_{i}.$

Then $V_{i}^{(n)}\subseteq V_{i}^{(n-1)}$ is non-empty open set and clearly

$g_{i}(n_{\alpha_{n}})T^{-n}V_{i}^{(n)}\subseteq V_{i}.$

Since $V_{i}^{(n)}\subseteq V_{i}^{(n-1)}$, (2.2) still holds for
$V_{i}^{(n)}$. Hence we finish our induction. The proof of the lemma is
complete. ∎

##  3\. Statements of Main results

###  3.1. Main Results

Let $({X},\Gamma)$ be a t.d.s., and
$A=\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$ be an IP-set. For
$d,k\in\mathbb{N}$, let $T_{1},\ldots,T_{d}\in\Gamma$, and $p_{i,j}(n)$ be
integral polynomials, $1\leq i\leq k,1\leq j\leq d$ and

(3.1) $g_{i}(n)=T_{1}^{p_{i,1}(n)}\cdots T_{d}^{p_{i,d}(n)},\quad
i=1,2,\ldots,k.$

######  Definition 3.1.

Let $A=\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$ be an IP-set, and let
$(X,\Gamma)$ and $g_{1},\ldots,g_{k}$ be defined as above. We say that
$(X,\Gamma)$ is
$\\{g_{1},\ldots,g_{k}\\}_{\Delta}$-$\mathcal{G}_{A}^{*}$-transitive if for
all given non-empty open sets $U,V_{1},\ldots,V_{k}$,
$\alpha_{0}\in\mathcal{F}$ and IP-ring $\mathcal{F}^{(1)}$, there is
$\alpha\in\mathcal{F}^{(1)}$ such that $\alpha>\alpha_{0}$ and

$U\cap(g_{1}(n_{\alpha})^{-1}V_{1}\cap\ldots\cap
g_{k}(n_{\alpha})^{-1}V_{k})\not=\emptyset,$

equivalently,

$\\{n\in\mathbb{Z}:U\cap(g_{1}(n)^{-1}V_{1}\cap\ldots\cap
g_{k}(n)^{-1}V_{k})\not=\emptyset\\}\in\mathcal{G}_{A}^{*}.$

The main result of this paper is the following:

######  Theorem 3.2.

Let $A=\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$ be an IP-set and let
$(X,\Gamma)$ be a topological system, where $\Gamma$ is an abelian group such
that for each $T\in\Gamma$, $T\neq e_{\Gamma}$, is mixing along
$A=\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$ and minimal. For
$d,k\in\mathbb{N}$, let $T_{1},\ldots,T_{d}\in\Gamma$,
$\\{p_{i,j}(n)\\}_{1\leq i\leq k,1\leq j\leq d}$ be integral polynomials such
that the expressions

$g_{i}(n)=T_{1}^{p_{i,1}(n)}\cdots T_{d}^{p_{i,d}(n)},\quad i=1,2,\ldots,k,$

depend nontrivially on $n$ for $i=1,2,\dots,k$, and for all $i\neq
j\in\\{1,2,\ldots,k\\}$ the expressions $g_{i}(n)g_{j}(n)^{-1}$ depend
nontrivially on $n$. Then $(X,\Gamma)$ is
$\\{g_{1},\ldots,g_{k}\\}_{\Delta}$-$\mathcal{G}_{A}^{*}$-transitive.

An immediate consequence of Theorem 3.2 is as follow.

######  Theorem 3.3.

Let $A=\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$ be an IP-set,
$d\in\mathbb{N}$, and let $p_{1}(n),\ldots,p_{d}(n)$ be integral polynomials
with no $p_{i}$ and no $p_{i}-p_{j}$ constant, $1\leq i\neq j\leq d$.444In
[1], the polynomials $p(n),q(n)$ are essentially distinct if
$p(n)-q(n)\not\equiv{\rm const.}$ If we use this terminology, the condition in
the theorem becomes that $p_{1},\ldots,p_{d}$ are pairwise essentially
distinct integral polynomials. Suppose that $(X,T)$ is a minimal system such
that $(X,T^{k})$ are mixing along $\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$
for all $k\in\mathbb{Z}\setminus\\{0\\}$. Then $(X,T)$ is
$\\{T^{p_{1}(n)},\ldots,T^{p_{d}(n)}\\}_{\Delta}$-$\mathcal{G}_{A}^{*}$-transitive.

When $\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$ is taken as $\mathbb{Z}$, we
have Theorem 1.6 and Theorem 1.3 respectively.

###  3.2. A Lemma

Before going on, we need the following easy observation.

######  Lemma 3.4.

Let $(X,\Gamma)$ be a t.d.s., $\Gamma$ be a group, and $d,k\in\mathbb{N}$. Let
$T_{1},\ldots,T_{d}\in\Gamma$, $\\{p_{i,j}(n)\\}_{1\leq i\leq k,1\leq j\leq
d}$ be integral polynomials, and let

$g_{i}(n)=T_{1}^{p_{i,1}(n)}\cdots T_{d}^{p_{i,d}(n)},\quad i=1,2,\ldots,k.$

Let $A\subseteq\mathbb{Z}$ be a sequence. Then there is a dense $G_{\delta}$
set $X_{0}$ of $X$ such that for every $x\in X_{0}$

$\\{(g_{1}(n)x,g_{2}(n)x,\ldots,g_{k}(n)x)\in X^{k}:n\in A\\}$

is dense in $X^{k}$ if and only if for all given non-empty open sets
$U,V_{1},\ldots,V_{k}$ there is $n\in A$ such that

$U\cap(g_{1}(n)^{-1}V_{1}\cap\ldots\cap g_{k}(n)^{-1}V_{k})\not=\emptyset.$

###### Proof.

One direction is obvious. Now assume that for all given non-empty open sets
$U,V_{1},\ldots,V_{k}$ of $X$, there is $n\in A$ such that
$U\cap(g_{1}(n)^{-1}V_{1}\cap\ldots\cap g_{k}(n)^{-1}V_{k})\not=\emptyset.$

Let $\mathcal{U}$ be a countable base of $X$, and let

$X_{0}=\bigcap_{V_{1},\ldots,V_{k}\in\mathcal{U}}\bigcup_{n\in
A}g_{1}(n)^{-1}V_{1}\cap\ldots\cap g_{k}(n)^{-1}V_{k}.$

Then it is easy to see that the dense $G_{\delta}$ subset $X_{0}$ is what we
need. ∎

###  3.3. Corollaries

By Lemma 3.4, we have following corollaries.

######  Corollary 3.5.

Let $A=\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$ be an IP-set and let
$(X,\Gamma)$, $g_{1},\ldots,g_{k}$ be as in Theorem 3.2. Then for any IP-ring
$\mathcal{F}^{(1)}$, there is a dense $G_{\delta}$ set $X_{0}$ of $X$ such
that for any $x\in X_{0}$

$\\{(g_{1}(n_{\alpha})x,\ldots,g_{k}(n_{\alpha})x):\alpha\in\mathcal{F}^{(1)}\\}$

is dense in $X^{k}$.

######  Corollary 3.6.

Let $A=\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$ be an IP-set,
$d\in\mathbb{N}$, and let $p_{1}(n),\ldots,p_{d}(n)$ be integral polynomials
with no $p_{i}$ and no $p_{i}-p_{j}$ constant, $1\leq i\neq j\leq d$. Suppose
that $(X,T)$ is a minimal system such that $(X,T^{k})$ are mixing along
$\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$ for all
$k\in\mathbb{Z}\setminus\\{0\\}$. Then for any IP-ring $\mathcal{F}^{(1)}$,
there is a dense $G_{\delta}$-set $X_{0}$ of $X$ such that for any $x\in
X_{0}$

$\\{(T^{p_{1}(n_{\alpha})}x,\ldots,T^{p_{d}(n_{\alpha})}x):\alpha\in\mathcal{F}^{(1)}\\}$

is dense in $X^{d}$.

Thus Corollary 1.4 and Corollary 1.7 follow from Corollary 3.5, Corollary 3.6
and Proposition 2.10.

##  4\. Linear case for the commutative actions

In this section we prove the linear case of Theorem 3.2. There are two reasons
for doing this. One is to show some basic ideas of the proof of Thereom 3.2 by
this special case, and another is that the conditions in the result of this
section is a little weaker than that of Theorem 3.2.

Now we have the main result of this section.

######  Theorem 4.1.

Let $(X,\Gamma)$ be a minimal t.d.s with $\Gamma$ abelian, and
$d\in\mathbb{N}$. Let $\\{T_{\alpha}^{(1)}\\}_{\alpha\in\mathcal{F}}$,
$\\{T_{\alpha}^{(2)}\\}_{\alpha\in\mathcal{F}}$, $\ldots$ ,
$\\{T_{\alpha}^{(d)}\\}_{\alpha\in\mathcal{F}}$ be mixing IP-systems in
$\Gamma$ such that
$\\{T^{(i)}_{\alpha}(T^{(j)}_{\alpha})^{-1}\\}_{\alpha\in\mathcal{F}}$ is
mixing, $i\neq j\in\\{1,2,\ldots,d\\}$ . Then for all non-empty open sets
$U,V_{1},\ldots,V_{d}$ of $X$, and any IP-ring $\mathcal{F}^{(1)}$, there is
some $\alpha\in\mathcal{F}^{(1)}$

$U\cap(T_{\alpha}^{(1)})^{-1}V_{1}\cap\ldots\cap(T_{\alpha}^{(d)})^{-1}V_{d}\neq\emptyset.$

The proof of Theorem 4.1 is the same to the following special case.

######  Theorem 4.2.

Let $X$ be a compact metric space and $d\in\mathbb{N}$. Let
$T_{1},T_{2},\ldots,T_{d}:X\rightarrow X$ be homeomorphisms with
$T_{i}T_{j}=T_{j}T_{i}$ for all $i,j\in\\{1,\ldots,d\\}$. Let
$A=\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$ be an IP-set. If for all $1\leq
i\leq d$ and $j\neq k\in\\{1,\ldots,d\\}$, $(X,T_{i})$ and
$(X,T_{j}T^{-1}_{k})$ are mixing along
$\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$ minimal systems, then for all non-
empty open sets $U,V_{1},\ldots,V_{d}$ of $X$

$\\{n\in\mathbb{Z}:U\cap T_{1}^{-n}V_{1}\cap\ldots\cap
T_{d}^{-n}V_{d}\neq\emptyset\\}\in\mathcal{G}_{A}^{*}.$

###### Proof.

We will prove by induction on $d$ that for all non-empty open sets
$U,V_{1},\ldots,V_{d}$ of $X$, any IP-ring $\mathcal{F}^{(1)}$, and
$\gamma\in\mathcal{F}$, there exists $\alpha\in\mathcal{F}^{(1)}$,
$\alpha>\gamma$, such that

$U\cap T_{1}^{-n_{\alpha}}V_{1}\cap\ldots\cap
T_{d}^{-n_{\alpha}}V_{d}\neq\emptyset.$

$d=1$ is trivial. Now we assume that the result holds for $d\geq 1$. Let
$U,V_{1},\ldots,V_{d},V_{d+1}$ be non-empty open subsets of $X$. We will show
that for any $\gamma\in\mathcal{F}$ there is some $\alpha\in\mathcal{F}^{(1)}$
such that $\alpha>\gamma$ and

$U\cap T_{1}^{-n_{\alpha}}V_{1}\cap\ldots\cap T_{d}^{-n_{\alpha}}V_{d}\cap
T_{d+1}^{-n_{\alpha}}V_{d+1}\neq\emptyset.$

Since $(X,T_{1})$ is minimal, there is some $N\in\mathbb{N}$ such that
$X=\bigcup_{j=0}^{N}T_{1}^{-j}U$. By Proposition 2.9
$(X^{d+1},T_{1}\times\ldots\times T_{d+1})$ is mixing along
$\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$, and hence it is
$\\{T_{1},\ldots,T_{d+1}\\}$-transitive along
$\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$. By Lemma 2.13, there are non-empty
subsets $V_{1}^{(N)},\ldots,V_{d+1}^{(N)}$ and
$\gamma<\alpha_{0}<\alpha_{1}<\ldots<\alpha_{N}$ with
$\alpha_{i}\in\mathcal{F}^{(1)}$ for $0\leq i\leq N$ such that for each
$i=1,2,\ldots,d+1$, one has that $|n_{\alpha_{j}}|>j$ and

$T_{i}^{n_{\alpha_{j}}}T_{1}^{-j}V_{i}^{(N)}\subseteq V_{i},\quad\text{for
all}\quad 0\leq j\leq N.$

Now applying the induction hypothesis to the system $(X,\langle
T_{2}T_{1}^{-1},\ldots,T_{d+1}T_{1}^{-1}\rangle)$ and non-empty subsets
$V_{1}^{(N)},\ldots,V_{d+1}^{(N)}$, there is some $\beta\in\mathcal{F}^{(1)}$
such that $\beta>\alpha_{N}$ and

$V_{1}^{(N)}\cap(T_{2}T_{1}^{-1})^{-n_{\beta}}V_{2}^{(N)}\cap\ldots\cap(T_{d}T_{1}^{-1})^{-n_{\beta}}V_{d}^{(N)}\cap(T_{d+1}T_{1}^{-1})^{-n_{\beta}}V_{d+1}^{(N)}\neq\emptyset.$

Hence there is some $x\in V_{1}^{(N)}$ such that
$(T_{i}T_{1}^{-1})^{n_{\beta}}x\in V_{i}^{(N)}$ for $i=2,\ldots,d+1$. Clearly,
there is some $y\in X$ such that $T_{1}^{n_{\beta}}y=x$. Since
$X=\bigcup_{j=0}^{N}T_{1}^{-j}U$, there is some $j\in\\{0,1,\ldots,N\\}$ such
that $T_{1}^{j}z=y$ for some $z\in U$. Thus for each $i=1,2,\ldots,d+1$

$\begin{split}T_{i}^{n_{\beta}+n_{\alpha_{j}}}z&=T_{i}^{n_{\beta}+n_{\alpha_{j}}}T_{1}^{-j}y=T_{i}^{n_{\beta}+n_{\alpha_{j}}}T_{1}^{-j}T_{1}^{-n_{\beta}}x\\\
&=T_{i}^{n_{\alpha_{j}}}T_{1}^{-j}(T_{i}T_{1}^{-1})^{n_{\beta}}x\in
T_{i}^{n_{\alpha_{j}}}T_{1}^{-j}V_{i}^{(N)}\subseteq V_{i}.\end{split}$

That is,

$z\in U\cap T_{1}^{-n_{\alpha}}V_{1}\cap\ldots\cap
T_{d}^{-n_{\alpha}}V_{d}\cap T_{d+1}^{-n_{\alpha}}V_{d+1},$

where $\alpha=\beta\cup\alpha_{j}\in\mathcal{F}^{(1)}$ as
$\beta\cap\alpha_{j}=\emptyset$. The proof is complete. ∎

As a corollary, one has the following:

######  Corollary 4.3.

Let $A=\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$ be an IP-set and let $(X,T)$
be a minimal system. Let $U,V_{1},\ldots,V_{d}$ be non-empty open subsets and
$a_{1},a_{2},\ldots,a_{d}$ be distinct non-zero integers. If for all
$k\in\mathbb{Z}\setminus\\{0\\}$, $(X,T^{k})$ are mixing along $A$, then

$\\{n\in\mathbb{Z}:U\cap T^{-a_{1}n}V_{1}\cap T^{-a_{2}n}V_{2}\cap\ldots\cap
T^{-a_{d}n}V_{d}\neq\emptyset\\}\in\mathcal{G}_{A}^{*}.$

##  5\. The PET-induction and some examples

We will use the PET-induction to prove Theorem 3.2. The PET-induction was
introduced by Bergelson in [1], where PET stands for polynomial ergodic
theorem or polynomial exhaustion technique (see [1, 4]). See also [2, 3] for
more on PET, and [18] for PET of nilpotent group actions.

###  5.1. Notations

In the sequal, let $(X,\Gamma)$ be a topological system, where $\Gamma$ is a
abelian group generated by $T_{1},\ldots,T_{d}$, $d\in\mathbb{N}$.

An integral polynomial is a polynomial taking integer values at the integers.
Let the $\mathcal{P}$ be the collection of all integral polynomials, i.e. all
polynomials taking integer values on the integers and $\mathcal{P}_{0}$ be the
collection of elements $p$ of $\mathcal{P}$ with $p(0)=0$.

In rest of the paper, we assume that all the polynomials involved have zero
constant term, that is, all polynomials are in $\mathcal{P}_{0}$. It is clear
that this assumption can be made without the loss of generality.

We will fix the above notations in the rest of the paper.

###  5.2. The $\Gamma$-polynomial group

A $\Gamma$-polynomial $g$ is the one which is represented in the form

(5.1)
$g(n)=\prod_{j=1}^{d}T_{j}^{p_{j}(n)}=T_{1}^{p_{1}(n)}T_{2}^{p_{2}(n)}\cdots
T_{d}^{p_{d}(n)},$

where $p_{1},\ldots,p_{d}$ are integral polynomials. For $\Gamma$-polynomials
$g(n)=T_{1}^{p_{1}(n)}T_{2}^{p_{2}(n)}\cdots T_{d}^{p_{d}(n)}$ and
$h(n)=T_{1}^{q_{1}(n)}T_{2}^{q_{2}(n)}\cdots T_{d}^{q_{d}(n)}$, we define the
product and the inverse by

$gh(n)=T_{1}^{p_{1}(n)+q_{1}(n)}T_{2}^{p_{2}(n)+q_{2}(n)}\cdots
T_{d}^{p_{d}(n)+q_{d}(n)},$

and

$g^{-1}(n)=T_{1}^{-p_{1}(n)}T_{2}^{-p_{2}(n)}\cdots T_{d}^{-p_{d}(n)}.$

Then the set of $\Gamma$-polynomials is a group, and it is denoted by ${{\bf
P}\Gamma}$. For $T_{1}^{a_{1}}T_{2}^{a_{2}}\ldots
T_{d}^{a_{d}}\in\Gamma,(a_{1},a_{2},\ldots,a_{d})\in\mathbb{Z}^{d}$, its
corresponding $\Gamma$-polynomial is $g(n)=T_{1}^{a_{1}n}T_{2}^{a_{2}n}\ldots
T_{d}^{a_{d}n}$. And thus $\Gamma$ itself is a subgroup of ${{\bf P}\Gamma}$
and $e_{\Gamma}={\rm Id}_{X}$.

Let

${{\bf P}\Gamma}_{0}=\\{g\in{{\bf
P}\Gamma}:g(n)=T_{1}^{p_{1}(n)}T_{2}^{p_{2}(n)}\cdots
T_{d}^{p_{d}(n)},p_{1},p_{2},\ldots,p_{d}\in\mathcal{P}_{0}\\}.$

And let ${{\bf P}\Gamma}_{0}^{*}={{\bf P}\Gamma}_{0}\setminus\\{e_{\Gamma}\\}$

###  5.3. The weight of $\Gamma$-polynomials

Let $\mathbb{Z}_{+}=\\{0,1,2,\ldots\\}$. The weight, $w(g)$, of a
$\Gamma$-polynomial $g(n)=\prod_{j=1}^{d}T_{j}^{p_{j}(n)}$ is the pair
$(l,k)$, $l\in\\{0,1,\ldots,d\\}$, $k\in\mathbb{Z}_{+}$ for which $p_{j}=0$
for any $j>l$ and, if $l\neq 0$, then $p_{l}\neq 0$ and ${\rm deg}(p_{l})=k$.
A weight $(l,k)$ is greater than a weight $(l^{\prime},k^{\prime})$, denoted
by $(l,k)>(l^{\prime},k^{\prime})$, if $l>l^{\prime}$ or $l=l^{\prime}$,
$k>k^{\prime}$.

For example, $T_{1}^{n},T_{2}^{n},T_{1}^{n}T_{2}^{n^{3}}$ have weights
$(1,1),(2,1),(2,3)$ respectively, and $(2,3)>(2,1)>(1,1)$.

Let us now define an equivalence relation on ${{\bf P}\Gamma}$:
$g(n)=\prod_{j=1}^{d}T_{j}^{p_{j}(n)}$ is equivalent to
$h(n)=\prod_{j=1}^{d}T_{j}^{q_{j}(n)}$, if $w(g)=w(h)$ and, if it is $(l,k)$,
the leading coefficients of the polynomials $p_{l}$ and $q_{l}$ coincide; we
write then $g\sim h$. For example,

$T_{3}^{n^{2}}\sim T_{1}^{n^{6}}T_{3}^{n^{2}+3n}\sim
T_{1}^{n}T_{2}^{n^{3}+3n}T_{3}^{n^{2}+5n}.$

The weight of an equivalence class is the weight of any of its elements.

###  5.4. System and its weight vector

A system $\SS$ is a finite subset of ${{\bf P}\Gamma}$. For a system $\SS$, if
we write $\SS=\\{f_{i}\\}_{i=1}^{v}$ then we require that $f_{i}\neq f_{j}$
for $1\leq i\neq j\leq v$. For every system $\SS$ we define its weight vector
$\phi(A)$ as follows. Let $w_{1}<w_{2}<\ldots<w_{q}$ be the set of the
distinct weights of all equivalence classes appeared in $\SS$. For
$i=1,2,\ldots,q$, let $\phi(w_{i})$ be the number of the equivalence classes
of elements of $\SS$ with the weight $w_{i}$. Let the weight vector
$\phi(\SS)$ be

$\phi(\SS)=(\phi(w_{1})w_{1},\phi(w_{2})w_{2},\ldots,\phi(w_{q})w_{q}).$

For example, let $\SS=\\{T_{1}^{n}$, $T_{1}^{2n}$, $T_{1}^{3n}$,
$T_{1}^{n^{2}}$, $T_{1}^{n^{2}+n}$, $T_{2}^{3n^{2}+2n}$,
$T_{1}^{n^{6}}T_{2}^{3n^{2}+n}$, $T_{1}^{n^{4}+n^{3}+n}T_{2}^{3n^{2}+2n}$,
$T_{1}^{n}T_{3}^{n^{3}}$, $T_{2}^{n^{5}}T_{3}^{2n^{3}+n^{2}}$,
$T_{1}^{n^{2}}T_{2}^{n}T_{3}^{3n^{3}+2n^{2}}\\}$. Then
$\phi(\SS)=\big{(}3(1,1),1(1,2),1(2,2),3(3,3)\big{)}$.

Let $\SS,\SS^{\prime}$ be two systems. We say that $\SS^{\prime}$ precedes a
system $\SS$ if there exists a weight $w$ such that
$\phi(\SS)(w)>\phi(\SS^{\prime})(w)$ and $\phi(\SS)(u)=\phi(\SS^{\prime})(u)$
for all weight $u>w$. We denote it by $\phi(\SS)\succ\phi(\SS^{\prime})$ or
$\phi(\SS^{\prime})\prec\phi(\SS)$.

For example, let $w_{1}<w_{2}<\ldots<w_{q}$ be a sequence of weights, then

$(a_{1}w_{1},\ldots,a_{q}w_{q})\succ(b_{1}w_{1},\ldots,b_{q}w_{q})$

if and only if $(a_{1},\ldots,a_{q})>(b_{1},\ldots,b_{q})$.

###  5.5. The PET-induction

#### 5.5.1.

In order to prove that a result holds for all systems $\SS$, we start with the
system whose weight vector is $\\{1(1,1)\\}$. That is,
$\SS=\\{T_{1}^{a_{1}n}\\}$, where $a_{1}\in\mathbb{Z}\setminus\\{0\\}$. Then
let $\SS\subseteq{{\bf P}\Gamma}_{0}$ be a system whose weight vector is
greater than $\\{1(1,1)\\}$, and assume that for all systems $\SS^{\prime}$
preceding $\SS$, we have that the result holds for $\SS^{\prime}$. Once we
show that the result still holds for $\SS$, we complete the whole proof. This
procedure is called the PET-induction.

#### 5.5.2.

For example, we outline how to use the language of PET-induction to formulate
the proof of Theorem 3.3. Now $\Gamma=\langle
T\rangle=\\{T^{n}:n\in\mathbb{Z}\\}$, and ${{\bf
P}\Gamma}=\\{T^{p(n)}:p\in\mathcal{P}\\}$. For each $T^{p(n)}\in{{\bf
P}\Gamma}$, its weight $w(T^{p(n)})=(1,k)$, where $k$ is the degree of $p(n)$.
A system $\SS$ has the form of
$\\{T^{p_{1}(n)},T^{p_{2}(n)},\ldots,T^{p_{d}(n)}\\}$, where
$p_{1},\ldots,p_{d}\in\mathcal{P}$ are distinct polynomials. Its weight vector
$\phi(\SS)$ has the form of

$\big{(}a_{1}(1,1),a_{2}(1,2),\ldots,a_{k}(1,k)\big{)}.$

Under the order of weight vectors, one has

$\begin{split}&\big{(}1(1,1)\big{)}\prec\big{(}2(1,1)\big{)}\prec\ldots\prec\big{(}m(1,1)\big{)}\prec\ldots\prec\big{(}1(1,2)\big{)}\prec\big{(}1(1,1),1(1,2)\big{)}\prec\ldots\prec\\\
&\big{(}m(1,1),1(1,2)\big{)}\prec\ldots\prec\big{(}2(1,2)\big{)}\prec\big{(}1(1,1),2(1,2)\big{)}\prec\ldots\prec\big{(}m(1,1),2(1,2)\big{)}\prec\ldots\prec\\\
&\big{(}m(1,1),k(1,2)\big{)}\prec\ldots\prec\big{(}1(1,3)\big{)}\prec\big{(}1(1,1),1(1,3)\big{)}\prec\ldots\prec\big{(}m(1,1),k(1,2),1(1,3)\big{)}\\\
&\prec\ldots\prec\big{(}2(1,3)\big{)}\prec\ldots\prec\big{(}a_{1}(1,1),a_{2}(1,2),\ldots,a_{k}(1,k)\big{)}\prec\ldots.\end{split}$

To prove Theorem 3.3, we will use induction on the weight vectors. We start
from the systems with the weight vector $(1(1,1))$, i.e.
$\SS=\\{T^{a_{1}n}\\},a_{1}\in\mathbb{Z}\setminus\\{0\\}$. After that, we
assume that the result holds for all systems whose weight vectors are $\prec$
$\big{(}a_{1}(1,1)$, $a_{2}(1,2)$, $\ldots$, $a_{k}(1,k)$ $\big{)}$. Then we
show that the result also holds for the system with weight vector $\big{(}$
$a_{1}(1,1),$ $a_{2}(1,2),\ldots,a_{k}(1,k)$ $\big{)}$, and hence the proof is
completed.

To illustrate the basic ideas, we show the result for the system
$\\{T^{an^{2}+bn}\\}$, and the system $\SS=\\{T^{n^{2}},T^{2n^{2}}\\}$, whose
weight vectors are $\big{(}1(1,2)\big{)}$ and $\big{(}2(1,2)\big{)}$
respectively.

###  5.6. Example 1: $(X,T)$ is
$\\{T^{an^{2}+bn}\\}_{\Delta}$-$\mathcal{G}_{A}^{*}$-transitive.

Let $A=\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$ be an IP-set, and let
$(X,T^{k})$ be a mixing along $\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$
minimal system for every $k\in\mathbb{Z}\setminus\\{0\\}$. We show that for
all non-empty open subsets $U,V$ of $X$,

$N_{p}(U,V)=\\{n\in\mathbb{Z}:U\cap
T^{-p(n)}V\neq\emptyset\\}\in\mathcal{G}_{A}^{*},$

where $p(n)=an^{2}+bn$. It suffices to show that for all non-empty open
subsets $U,V$ of $X$, IP-rings $\mathcal{F}^{(1)}$, and all
$\alpha_{0}\in\mathcal{F}$, there is some $\alpha\in\mathcal{F}^{(1)}$ such
that $\alpha>\alpha_{0}$ and

$U\cap T^{-an_{\alpha}^{2}-bn_{\alpha}}V\neq\emptyset.$

###### Proof.

Since $(X,T)$ is minimal, there is some $N\in\mathbb{N}$ such that
$X=\bigcup_{i=0}^{N}T^{-i}U$. Let $p(n)=an^{2}+bn$, and

$q(n,m)=p(n+m)-p(m)-p(n)=2anm.$

For each $i=0,1,\ldots,N$, by Theorem 2.11 there are some $y_{i}\in T^{-i}V$
and $\alpha_{i}\in\mathcal{F}^{(1)}$ such that

$T^{p(n_{\alpha_{i}})}y_{i}\in T^{-i}V,\ 0\leq i\leq N.$

We may assume that $\alpha_{0}<\alpha_{1}<\ldots<\alpha_{N}$. Let $V_{i}$ be a
neighborhood of $y_{i}$ such that

$T^{p(n_{\alpha_{i}})}V_{i}\subseteq T^{-i}V,\ 0\leq i\leq N.$

Now apply Corollary 4.3 to
$q(n_{\alpha_{0}},n)=2an_{\alpha_{0}}n,q(n_{\alpha_{1}},n)=2an_{\alpha_{1}}n,\ldots,q(n_{\alpha_{N}},n)=2an_{\alpha_{N}}n$,
and there are some $x$ and $\beta\in\mathcal{F}^{(1)}$ such that
$\beta>\alpha_{N}$ and

$T^{q(n_{\alpha_{i}},n_{\beta})}x=T^{2an_{\alpha_{i}}n_{\beta}}x\in
V_{i},\quad\forall i\in\\{0,\ldots,N\\}.$

Then we have that

$T^{p(n_{\alpha_{i}}+n_{\beta})-p(n_{\beta})}x=T^{a(n_{\alpha_{i}}+n_{\beta})^{2}-an_{\beta}^{2}+bn_{\alpha_{i}}}x=T^{an_{\alpha_{i}}^{2}+bn_{\alpha_{i}}+2an_{\alpha_{i}}n_{\beta}}x\in
T^{p(n_{\alpha_{i}})}V_{i}\subseteq T^{-i}V.$

Hence

$T^{i-p(n_{\beta})}x\in T^{-p(n_{\alpha_{i}}+n_{\beta})}V,\quad\forall
i\in\\{0,1,\ldots,N\\}.$

Since $X=\bigcup_{i=0}^{N}T^{-i}U$, there is some $i_{0}\in\\{0,1,\ldots,N\\}$
such that $T^{i_{0}-p(n_{\beta})}x\in U$, and thus

$U\cap T^{-p(n_{\alpha})}V\neq\emptyset,$

where $\alpha=\alpha_{i_{0}}\cup\beta>\alpha_{0}$. That is,

$N_{p}(U,V)\cap\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}^{(1)}}\neq\emptyset.$

Since $\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}^{(1)}}$ is an arbitrary IP-
subset of $\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$, it follows that
$N_{p}(U,V)$ is an $\mathcal{G}_{A}^{*}$-set. ∎

###  5.7. Example 2: $(X,T)$ is
$\\{T^{n^{2}},T^{2n^{2}}\\}_{\Delta}$-$\mathcal{G}_{A}^{*}$-transitive.

Let $A=\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$ be an IP-set, and let
$(X,T^{k})$ be a mixing along $\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$
minimal system for every $k\in\mathbb{Z}\setminus\\{0\\}$. To show this
example, we need to verify the following cases one by one:

Case 1: When the weight vector is $\big{(}d(1,1)\big{)}$: $(X,T)$ is
$\\{T^{a_{1}n},\ldots,T^{a_{d}n}\\}_{\Delta}$-$\mathcal{G}_{A}^{*}$-transitive,
where $a_{1},\ldots,a_{d}\in\mathbb{Z}\setminus\\{0\\}$ are distinct integers.

Case 2: When the weight vector is $\big{(}1(1,2)\big{)}$:

  1. (1)

$(X,T)$ is $\\{T^{an^{2}+b_{1}n},\ldots,T^{an^{2}+b_{d}n}\\}$-transitive along
$A$,

  2. (2)

$(X,T)$ is
$\\{T^{an^{2}+b_{1}n},\ldots,T^{an^{2}+b_{d}n}\\}_{\Delta}$-$\mathcal{G}_{A}^{*}$-transitive,
where $b_{1},\ldots,b_{d}$ are distinct integers and
$a\in\mathbb{Z}\setminus\\{0\\}$.

Case 3: When the weight vector is $\big{(}r(1,1),1(1,2)\big{)}$:

  1. (1)

$(X,T)$ is
$\\{T^{c_{1}n},\ldots,T^{c_{r}n},T^{an^{2}+b_{1}n},\ldots,T^{an^{2}+b_{d}n}\\}$-transitive
along $A$,

  2. (2)

$(X,T)$ is
$\\{T^{c_{1}n},\ldots,T^{c_{r}n},T^{an^{2}+b_{1}n},\ldots,T^{an^{2}+b_{d}n}\\}_{\Delta}$-$\mathcal{G}_{A}^{*}$-transitive,
where $a\in\mathbb{Z}\setminus\\{0\\}$, $b_{1},\ldots,b_{d}$ are distinct
integers and $c_{1},\ldots,c_{r}$ are distinct non-zero integers.

Case 4: When the weight vector is $\big{(}2(1,2)\big{)}$:

  1. (1)

$(X,T)$ is $\\{T^{n^{2}},T^{2n^{2}}\\}$-transitive along $A$,

  2. (2)

$(X,T)$ is
$\\{T^{n^{2}},T^{2n^{2}}\\}_{\Delta}$-$\mathcal{G}_{A}^{*}$-transitive.

####  5.7.1. Case 1: $(X,T)$ is
$\\{T^{a_{1}n},\ldots,T^{a_{d}n}\\}_{\Delta}$-$\mathcal{G}_{A}^{*}$-transitive,
where $a_{1},\ldots,a_{d}\in\mathbb{Z}\setminus\\{0\\}$ are distinct integers.

###### Proof.

This is Corollary 4.3. ∎

####  5.7.2. Case 2: $(X,T)$ is
$\\{T^{an^{2}+b_{1}n},\ldots,T^{an^{2}+b_{d}n}\\}_{\Delta}$-$\mathcal{G}_{A}^{*}$-transitive,
where $b_{1},\ldots,b_{d}$ are distinct integers and
$a\in\mathbb{Z}\setminus\\{0\\}$.

###### Proof.

Let $p_{i}(n)=an^{2}+b_{i}n$ for $1\leq i\leq d$. First we show that $(X,T)$
is $\\{T^{p_{1}},\ldots,T^{p_{d}}\\}$-transitive along $A$, that is, for all
given open non-empty subsets $U_{1},\ldots,U_{d},V_{1},\ldots,V_{d}$ of $X$,
$\alpha_{0}\in\mathcal{F}$, and all IP-rings $\mathcal{F}^{(1)}$, there is
$\alpha\in\mathcal{F}^{(1)}$ such that $\alpha>\alpha_{0}$ and

$(U_{1}\times\ldots\times
U_{d})\cap(T^{-p_{1}(n_{\alpha})}V_{1}\times\ldots\times
T^{-p_{d}(n_{\alpha})}V_{d})\not=\emptyset.$

By Example 1, for $i\in\\{1,2,\ldots,d\\}$, $N_{p_{i}}(U_{i},V_{i})$ is an
$\mathcal{G}_{A}^{*}$-set. Thus

$\\{n\in\mathbb{Z}:(U_{1}\times\ldots\times
U_{d})\cap(T^{-p_{1}(n)}V_{1}\times\ldots\times
T^{-p_{d}(n)}V_{d})\not=\emptyset\\}=\bigcap_{i=1}^{d}N_{p_{i}}(U_{i},V_{i})$

is an $\mathcal{G}_{A}^{*}$-set as $\mathcal{G}_{A}^{*}$ is a filter. In
particular, there is $\alpha\in\mathcal{F}^{(1)}$ such that
$\alpha>\alpha_{0}$ and

$(U_{1}\times\ldots\times
U_{d})\cap(T^{-p_{1}(n_{\alpha})}V_{1}\times\ldots\times
T^{-p_{d}(n_{\alpha})}V_{d})\not=\emptyset.$

To prove the theorem it remains to show for all given non-empty open sets
$U,V_{1},\ldots,V_{d}$

$N_{\\{p_{1},\ldots,p_{d}\\}}(U,V_{1},\ldots,V_{d})=\\{n:U\cap(T^{-p_{1}(n)}V_{1}\cap\ldots\cap
T^{-p_{d}(n)}V_{d})\not=\emptyset\\}$

is an $\mathcal{G}_{A}^{*}$-set. To do this, we show that for any IP-ring
$\mathcal{F}^{(1)}$, there is $\alpha\in\mathcal{F}^{(1)}$ such that

$U\cap(T^{-p_{1}(n_{\alpha})}V_{1}\cap\ldots\cap
T^{-p_{d}(n_{\alpha})}V_{d})\not=\emptyset.$

Let $\mathcal{F}^{(1)}$ be an IP-ring. And then
$\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}^{(1)}}$ is an IP-subset of
$\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$. Assume that
$\bigcup_{i=0}^{N}T^{-i}U=X$ for some $N\in\mathbb{N}$. By Lemma 2.13, there
are increasing sequence $\\{\alpha_{j}\\}_{j=0}^{N}\subseteq\mathcal{F}^{(1)}$
and $V_{i}^{N}\subset V_{i}$ such that $|n_{\alpha_{j}}|>j$ and

$T^{p_{i}(n_{\alpha_{j}})-j}V_{i}^{(N)}\subset V_{i},\ \ 0\leq j\leq N,1\leq
i\leq d.$

Let

$q_{i}(m,n)=p_{i}(m+n)-p_{i}(m)-p_{1}(n),\quad 1\leq i\leq d.$

Applying Case 1 to $q_{i}(n_{\alpha_{j}},n)$ , i.e.
$2n_{\alpha_{1}}n,\ldots,2n_{\alpha_{N}}n,\ldots,2n_{\alpha_{1}}n+(b_{d}-b_{1})n,\ldots,2n_{\alpha_{N}}n+(b_{d}-b_{1})n$
(since $\\{\alpha_{j}\\}$ is an increasing sequence, we can choose a
subsequence such that the numbers are distinct), then there is $x\in X$ and
$\beta\in\mathcal{F}^{(1)}$ such that $\beta>\alpha_{N}$ and

$T^{q_{1}(n_{\alpha_{j}},n_{\beta})}x\in
V_{1}^{(N)},\ldots,T^{q_{d}(n_{\alpha_{j}},n_{\beta})}x\in V_{d}^{(N)},0\leq
j\leq N.$

Let $y=T^{-p_{1}(n_{\beta})}x$. Since $X=\bigcup_{i=0}^{N}T^{-i}U$, there is
$z\in U$, $0\leq b\leq N$ such that $y=T^{b}z$. Thus
$z=T^{-p_{1}(n_{\beta})-b}x$. We have

$T^{p_{i}(n_{\beta}+n_{\alpha_{b}})}z=T^{q_{i}(n_{\alpha_{b}},n_{\beta})+p_{i}(n_{\alpha_{b}})-b}x\in
T^{p_{i}(n_{\alpha_{b}})-b}V_{i}^{(N)}\subset V_{i},1\leq i\leq d.$

That is

$z\in U\cap(T^{-p_{1}(n_{\alpha})}V_{1}\cap\ldots\cap
T^{-p_{d}(n_{\alpha})}V_{d})\not=\emptyset,$

where $\alpha=\beta\cup\alpha_{b}\in\mathcal{F}^{(1)}$ as
$\beta\cap\alpha_{b}=\emptyset$. Since
$\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}^{(1)}}$ is an arbitrary IP-subset of
$\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$, it follows that
$N_{\\{p_{1},\ldots,p_{d}\\}}(U,V_{1},\ldots,V_{d})$ is an
$\mathcal{G}_{A}^{*}$-set. ∎

####  5.7.3. Case 3: $(X,T)$ is
$\\{T^{c_{1}n},\ldots,T^{c_{r}n},T^{an^{2}+b_{1}n},\ldots,T^{an^{2}+b_{d}n}\\}_{\Delta}$-$\mathcal{G}_{A}^{*}$-transitive,
where $a\in\mathbb{Z}\setminus\\{0\\}$, $b_{1},\ldots,b_{d}$ are distinct
integers and $c_{1},\ldots,c_{r}$ are distinct non-zero integers.

###### Proof.

Let $p_{1}(n)=n^{2}+b_{1}n,\ldots,p_{d}(n)=n^{2}+b_{d}n$ and
$h_{1}(n)=c_{1}n,\ldots,h_{r}(n)=c_{r}n$. First similar to Case 2, we have
that $(X,T)$ is
$\\{T^{p_{1}},\ldots,T^{p_{d}},T^{h_{1}},\ldots,T^{h_{r}}\\}$-transitive along
$A$, that is, for all given open non-empty subsets $U_{1},\ldots,U_{r+d}$,
$W_{1},\ldots,W_{r}$, $V_{1},\ldots,V_{d}$ of $X$, and all IP-rings
$\mathcal{F}^{(1)}$, there is $\alpha\in\mathcal{F}^{(1)}$ such that

$(U_{1}\times\ldots\times
U_{r+d})\bigcap(T^{-h_{1}(n_{\alpha})}W_{1}\times\ldots\times
T^{-h_{r}(n_{\alpha})}W_{r})\times(T^{-p_{1}(n_{\alpha})}V_{1}\times\ldots\times
T^{-p_{d}(n_{\alpha})}V_{d})\not=\emptyset.$

Next we need to show for all given non-empty open sets
$U,W_{1},\ldots,W_{r},V_{1},\ldots,V_{d}$ of $X$, and all IP-rings
$\mathcal{F}^{(1)}$, there is $\alpha\in\mathcal{F}^{(1)}$ such that

$U\cap(T^{-h_{1}(n_{\alpha})}W_{1}\cap\ldots\cap
T^{-h_{r}(n_{\alpha})}W_{r})\cap(T^{-p_{1}(n_{\alpha})}V_{1}\cap\ldots\cap
T^{-p_{d}(n_{\alpha})}V_{d})\not=\emptyset.$

That is,

(5.2)
$\begin{split}&N_{\\{p_{1},\ldots,p_{d},h_{1},\ldots,h_{r}\\}}(U,W_{1},\ldots,W_{r},V_{1},\ldots,V_{d})\\\
=&\\{n:U\cap(T^{-h_{1}(n)}W_{1}\cap\ldots\cap
T^{-h_{r}(n)}W_{r})\cap(T^{-p_{1}(n)}V_{1}\cap\ldots\cap
T^{-p_{d}(n)}V_{d})\not=\emptyset\\}\end{split}$

is an $\mathcal{G}_{A}^{*}$-set.

Let $\mathcal{F}^{(1)}$ be an IP-ring. Then
$\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}^{(1)}}$ is an IP-subset of
$\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$. Since $(X,T)$ is minimal, there is
some $N\in\mathbb{N}$ such that $\bigcup_{i=0}^{N}T^{-i}U=X$. By Lemma 2.13
there is an increasing suquence
$\\{\alpha_{j}\\}_{j=0}^{N}\subseteq\mathcal{F}^{(1)}$, $W_{i}^{(N)}\subset
W_{i}$ and $V_{s}^{(N)}\subset V_{s}$ such that $|n_{\alpha_{j}}|>j$ and

$T^{h_{i}(n_{\alpha_{j}})-j}W_{i}^{(N)}\subset W_{i},\ \text{and}\
T^{p_{s}(n_{\alpha_{j}})-j}V_{s}^{(N)}\subset V_{s}$

for $0\leq j\leq N,1\leq i\leq r,1\leq s\leq d$.

Let

$\widetilde{h}_{i}(n)=h_{i+1}(n)-h_{1}(n),\quad 1\leq i\leq r-1,$

and

$q_{s}(m,n)=p_{s}(n+m)-p_{s}(m)-h_{1}(n),\quad 1\leq s\leq d.$

We will prove by induction. Assume that we have established (5.2) for
$r^{\prime}\leq r-1$. By Case 2 (for $r=1$) or the inductive assumption (for
$r\geq 2$) there are $x\in W_{1}^{(N)}$ and $\beta\in\mathcal{F}^{(1)}$ such
that $\beta>\alpha_{N}$ and

$T^{\widetilde{h}_{1}(n_{\beta})}x\in
W_{2}^{(N)},\ldots,T^{\widetilde{h}_{r^{\prime}}(n_{\beta})}x\in
W_{r^{\prime}+1}^{(N)}$

(we do not have the above line when $r=1$) and

$T^{q_{s}(n_{\alpha_{j}},n_{\beta})}x=T^{p_{s}(n_{\alpha_{j}}+n_{\beta})-p_{s}(n_{\alpha_{j}})-h_{1}(n_{\beta})}x\in
V_{s}^{(N)},\ 0\leq j\leq N,1\leq s\leq d.$

Let $y=T^{-h_{1}(n_{\beta})}x$. Then by $X=\bigcup_{i=0}^{N}T^{-i}U$ there is
$z\in U$ and $0\leq b\leq N$ such that $y=T^{b}z$. Then
$z=T^{-h_{1}(n_{\beta})-b}x$ and we have

$T^{h_{i}(n_{\beta}+n_{\alpha_{b}})}z=T^{h_{i}(n_{\beta}+n_{\alpha_{b}})}T^{-h_{1}(n_{\beta})-b}x=T^{\widetilde{h}_{i-1}(n_{\beta})+h_{i}(n_{\alpha_{b}})-b}x\in
T^{h_{i}(n_{\alpha_{b}})-b}W_{i}^{(N)}\subset W_{i}$

for $1\leq i\leq r^{\prime}+1$ and

$T^{p_{s}(n_{\beta}+n_{\alpha_{b}})}z=T^{q_{s}(n_{\alpha_{b}},n_{\beta})+p_{s}(n_{\alpha_{b}})-b}x\in
T^{p_{s}(n_{\alpha_{b}})-b}V_{s}^{(N)}\subset V_{s}$

for $1\leq s\leq d$. This implies that

$z\in U\cap(T^{-h_{1}(n_{\alpha})}W_{1}\cap\ldots\cap
T^{-h_{r^{\prime}+1}(n_{\alpha})}W_{r^{\prime}+1})\cap(T^{-p_{1}(n_{\alpha})}V_{1}\cap\ldots\cap
T^{-p_{d}(n_{\alpha})}V_{d})$

with $\alpha=\beta\cup\alpha_{b}\in\mathcal{F}^{(1)}$ as
$\beta\cap\alpha_{b}=\emptyset$. Since
$\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}^{(1)}}$ is an arbitrary IP-subset of
$A$, it follows that
$N_{\\{p_{1},\ldots,p_{d},h_{1},\ldots,h_{r}\\}}(U,W_{1},\ldots,W_{r},V_{1},\ldots,V_{d})$
is an $\mathcal{G}_{A}^{*}$-set. ∎

####  5.7.4. Case 4: $(X,T)$ is
$\\{T^{n^{2}},T^{2n^{2}}\\}_{\Delta}$-$\mathcal{G}_{A}^{*}$-transitive.

###### Proof.

Let $p_{1}(n)=n^{2},p_{2}(n)=2n^{2}$. By the same method in the proof of Case
1, $(X,T)$ is $\\{T^{p_{1}},T^{p_{2}}\\}$-transitive along $A$, that is, for
all given open non-empty subsets $U_{1},U_{2},V_{1},V_{2}$ of $X$ and IP-rings
$\mathcal{F}^{(1)}$, there is $\alpha\in\mathcal{F}^{(1)}$ such that

$(U_{1}\times U_{2})\cap(T^{-p_{1}(n_{\alpha})}V_{1}\times
T^{-p_{2}(n_{\alpha})}V_{2})\not=\emptyset.$

To prove the theorem we need to show for all given non-empty open sets
$U,V_{1},V_{2}$ and IP-rings $\mathcal{F}^{(1)}$, there is
$\alpha\in\mathcal{F}^{(1)}$ with

$U\cap(T^{-p_{1}(n_{\alpha})}V_{1}\cap
T^{-p_{2}(n_{\alpha})}V_{2})\not=\emptyset.$

That is,

$N_{\\{p_{1},p_{2}\\}}(U,V_{1},V_{2})=\\{n:U\cap(T^{-p_{1}(n)}V_{1}\cap
T^{-p_{2}(n)}V_{2})\not=\emptyset\\}$

is an $\mathcal{G}_{A}^{*}$-set.

Let $\mathcal{F}^{(1)}$ be an IP-ring. Then
$\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}^{(1)}}$ is an IP-subset of
$\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$. Let $X=\bigcup_{i=0}^{N}T^{-i}U$
for some $N\in\mathbb{N}$. By Lemma 2.13 there is an increasing sequence
$\\{\alpha_{j}\\}_{j=0}^{N}\subseteq\mathcal{F}^{(1)}$, $V_{1}^{(N)}\subset
V_{1}$ and $V_{2}^{(N)}\subset V_{2}$ such that $|n_{\alpha_{j}}|>j$ and

$T^{p_{1}(n_{\alpha_{j}})-j}V_{1}^{(N)}\subset V_{1}\ \text{and}\
T^{p_{2}(n_{\alpha_{j}})-j}V_{2}^{(N)}\subset V_{2},\quad 0\leq j\leq N.$

Let $q_{i}(m,n)=p_{i}(n+m)-p_{i}(m)-p_{1}(n)$ for $n,m\in\mathbb{Z}$ and
$i=1,2$. Since $\\{\alpha_{j}\\}$ is an increasing sequence of
$\mathcal{F}^{(1)}$, we have that all

$q_{i}(n_{\alpha_{j}},n)=\begin{cases}2n_{\alpha_{j}}n\,&\text{ if }i=1\\\
n^{2}+4n_{\alpha_{j}}n\,&\text{ if }i=2\end{cases}$

are distinct non-constant polynomials in $n$ for $0\leq j\leq N,1\leq i\leq
2$.

By Case 3, there are $x\in X$ and $\beta\in\mathcal{F}^{(1)}$ such that
$\beta>\alpha_{N}$ and

$T^{q_{1}(n_{\alpha_{j}},n_{\beta})}x\in V_{1}^{(N)},\quad
T^{q_{2}(n_{\alpha_{j}},n_{\beta})}x\in V_{2}^{(N)},\quad 0\leq j\leq N.$

Let $y=T^{-p_{1}(n_{\beta})}x$. Then by $X=\bigcup_{i=0}^{N}T^{-i}U$ there is
$z\in U$, $0\leq b\leq N$ such that $y=T^{b}z$. Thus
$z=T^{-p_{1}(n_{\beta})-b}x$. We have

$T^{p_{1}(n_{\beta}+n_{\alpha_{b}})}z=T^{q_{1}(n_{\alpha_{b}},n_{\beta})+p_{1}(n_{\alpha_{b}})-b}x\in
T^{p_{1}(n_{\alpha_{b}})-b}V_{1}^{(N)}\subset V_{1}$

and

$T^{p_{2}(n_{\beta}+n_{\alpha_{b}})}z=T^{q_{2}(n_{\alpha_{b}},n_{\beta})+p_{2}(n_{\alpha_{b}})-b}x\in
T^{p_{2}(n_{\alpha_{b}})-b}V_{2}^{(N)}\subset V_{2}.$

That is,

$z\in U\cap(T^{-p_{1}(n_{\alpha})}V_{1}\cap T^{-p_{2}(n_{\alpha})}V_{2})$

with $\alpha=\beta\cup\alpha_{b}\in\mathcal{F}^{(1)}$ as
$\beta\cap\alpha_{b}=\emptyset$. Since
$\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}^{(1)}}$ is an arbitrary IP-subset, it
follows that $N_{\\{p_{1},p_{1}\\}}(U,V_{1},V_{2})$ is an
$\mathcal{G}_{A}^{*}$-set. ∎

##  6\. Proof of Theorem 3.2

In this section, we give a proof of Theorem 3.2.

Let $A=\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$ be an IP-set and let
$(X,\Gamma)$ be a topological system, where $\Gamma$ is an abelian group such
that for each $T\in\Gamma$, $T\neq e_{\Gamma}$, is mixing along
$A=\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$ and minimal. For
$d,k\in\mathbb{N}$, let $T_{1},\ldots,T_{d}\in\Gamma$,
$\\{p_{i,j}(n)\\}_{1\leq i\leq k,1\leq j\leq d}$ be integral polynomials such
that the expressions

$g_{i}(n)=T_{1}^{p_{i,1}(n)}\cdots T_{d}^{p_{i,d}(n)},\quad i=1,2,\ldots,k,$

depend nontrivially on $n$ for $i=1,2,\dots,k$, and for all $i\neq
j\in\\{1,2,\ldots,k\\}$ the expressions $g_{i}(n)g_{j}(n)^{-1}$ depend
nontrivially on $n$.

Using the language introduced in last section, Theorem 3.2 is restated as
follows:

Theorem: For any system $\SS=\\{g_{1},\ldots,g_{k}\\}\subseteq{{\bf
P}\Gamma}_{0}^{*}$, $(X,\Gamma)$ is
$\\{g_{1},\ldots,g_{k}\\}_{\Delta}$-$\mathcal{G}_{A}^{*}$-transitive.

###### Proof.

We will prove Theorem using the PET-induction. We will use the notations in
Section 5 freely.

We start with the system whose weight vector is $\\{d(1,1)\\}$ with
$d\in\mathbb{N}$. That is, $\SS=\\{T_{1}^{c_{1}n},\ldots,T_{1}^{c_{d}n}\\}$,
where $c_{1},\ldots,c_{d}\in\mathbb{Z}$. This case is Corollary 4.3.

Now let $\SS=\\{g_{1},\ldots,g_{k}\\}\subseteq{{\bf P}\Gamma}_{0}^{*}$ be a
system whose weight vector is greater than $\\{d(1,1)\\}$ for all
$d\in\mathbb{N}$, and assume that for all systems $\SS^{\prime}$ preceding
$\SS$, we have $X$ is
$\SS^{\prime}_{\Delta}$-$\mathcal{G}_{A}^{*}$-transitive. Now we show that $X$
is $\SS_{\Delta}$-$\mathcal{G}_{A}^{*}$-transitive.

Step 1. $(X,\Gamma)$ is $\SS=\\{g_{1},\ldots,g_{k}\\}$-transitive along
$\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$

Since $\mathcal{G}_{A}^{*}$ is a filter, it is sufficient to show that for all
$f\in\SS$, and for all given non-empty open subsets $U,V$ of $X$,

$N_{f}(U,V):=\\{n\in\mathbb{Z}:U\cap f(n)^{-1}V\neq\emptyset\\}$

is an $\mathcal{G}_{A}^{*}$-set.

If $f(m+n)=f(n)f(m)$ for all $n,m\in\mathbb{Z}$, then there is
$(a_{1},a_{2},\ldots,a_{d})\in\mathbb{Z}^{d}\setminus\\{(0,0,\ldots,0)\\}$
such that

$f(n)=T^{a_{1}n}_{1}T_{2}^{a_{2}n}\ldots T_{d}^{a_{d}n}.$

In this case $f\in\Gamma$, and by our assumption, $(X,f)$ is mixing along $A$.

Now we assume that $f(m+n)\not\equiv f(n)f(m),n,m\in\mathbb{Z}$. Let
$T\in\Gamma$ be an element of $\Gamma$ with $T\neq e_{\Gamma}$. Since $(X,T)$
is minimal, there is some $N\in\mathbb{N}$ such that
$X=\bigcup_{i=0}^{N}T^{-i}U$. Let $h(m,n)=f(m)^{-1}f(m+n)f(n)^{-1}\in{{\bf
P}\Gamma}_{0}^{*}.$ Let $\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}^{(1)}}$ be an
IP-subset of $\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$.

For each $i=1,2,\ldots,N$, by Theorem 2.11 there are some $y_{i}\in T^{-i}V$
and $\alpha_{i}\in\mathcal{F}^{(1)}$ such that

$f(n_{\alpha_{i}})y_{i}\in T^{-i}V,\ 0\leq i\leq N.$

We may assume that $\alpha_{0}<\alpha_{1}<\ldots<\alpha_{N}$. Let $V_{i}$ be a
neighborhood of $y_{i}$ such that

$f(n_{\alpha_{i}})V_{i}\subseteq T^{-i}V,\ 0\leq i\leq N.$

Let

$\SS^{\prime}=\\{h(n_{\alpha_{i}},n):0\leq i\leq N\\}.$

Then $\SS^{\prime}\subset{{\bf P}\Gamma}_{0}^{*}$ is a system.

Notice that $w(h(n_{\alpha_{i}},n))<w(f)$ for all $i=0,1,\ldots,N$, then we
have $\phi(\SS^{\prime})\prec\phi(\\{f\\})$. On the other hand, since
$f\in\SS$, then $\phi(\\{f\\})\preccurlyeq\phi(\SS)$. Hence $\SS^{\prime}$
precedes $\SS$. By the inductive hypothesis, there are some $x$ and
$\beta\in\mathcal{F}^{(1)}$ such that $\beta>\alpha_{N}$ and

$h(n_{\alpha_{i}},n_{\beta})x\in V_{i},\quad\forall i\in\\{0,1,\ldots,N\\}.$

Then we have that

$f(n_{\alpha_{i}}+n_{\beta})f(n_{\beta})^{-1}x=f(n_{\alpha_{i}})h(n_{\alpha_{i}},n_{\beta})x\in
f(n_{\alpha_{i}})V_{i}\subseteq T^{-i}V.$

Hence

$T^{i}f(n_{\beta})^{-1}x\in f(n_{\alpha_{i}}+n_{\beta})^{-1}V,\quad\forall
i\in\\{0,1,\ldots,N\\}.$

Since $X=\bigcup_{i=0}^{N}T^{-i}U$, there is some $i_{0}\in\\{0,1,\ldots,N\\}$
such that $T^{i_{0}}f(n_{\beta})^{-1}x\in U$, and thus

$U\cap f(n_{\alpha})^{-1}V\neq\emptyset,$

where $\alpha=\alpha_{i_{0}}\cup\beta>\alpha_{0}$ as
$\alpha_{i_{0}}\cap\beta=\emptyset$. That is,

$N_{f}(U,V)\cap\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}^{(1)}}\neq\emptyset.$

Since $\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}^{(1)}}$ is an arbitrary IP-
subset of $\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$, it follows that
$N_{f}(U,V)$ is an $\mathcal{G}_{A}^{*}$-set.

Step 2. $X$ is $\\{g_{1},\ldots,g_{k}\\}_{\Delta}$-$\mathcal{G}_{A}^{*}$
transitive

Let $\SS=\\{g_{1},\ldots,g_{k}\\}$. By Lemma 3.4 we need to show for all given
non-empty open sets $U,V_{1},\ldots,V_{v}$ of $X$ and all IP-rings
$\mathcal{F}^{(1)}$, there is $\alpha\in\mathcal{F}^{(1)}$ such that

(6.1)
$U\cap{g_{1}(n_{\alpha})^{-1}}V_{1}\cap\ldots\cap{g_{k}(n_{\alpha})^{-1}}V_{k}\not=\emptyset.$

That is,

$N_{\SS}(U,V_{1},\ldots,V_{k})=\\{n:U\cap{g_{1}(n_{\alpha})^{-1}}V_{1}\cap\ldots\cap{g_{k}(n_{\alpha})^{-1}}V_{k}\not=\emptyset\\}$

is an $\mathcal{G}_{A}^{*}$-set.

Let $T\in\Gamma$ be an element of $\Gamma$ with $T\neq e_{\Gamma}$. As $(X,T)$
is minimal, there is some $N\in\mathbb{N}$ such that
$X=\bigcup_{i=0}^{N}T^{-i}U$. By Lemma 2.13, there are
$\\{\alpha_{j}\\}_{j=0}^{N}\subseteq\mathcal{F}^{(1)}$ with
$|n_{\alpha_{j}}|>j$ for all $j=0,\ldots,N$, and $V_{t}^{(N)}\subset V_{t}$
for $t=1,\ldots,k$ such that

(6.2) $g_{t}(n_{\alpha_{j}})T^{-j}V_{t}^{(N)}\subset V_{t},\quad\forall\ 0\leq
j\leq N.$

Let $f\in\SS$ be a $\Gamma$-polynomial of weight minimal in $\SS$: $w(f)\leq
w(g_{j})$ for any $j=1,\ldots,k$. Without loss of generality assume that
$f=g_{1}$. Let

$g_{t,j}(n)=g_{t}(n_{\alpha_{j}})^{-1}g_{t}(n+n_{\alpha_{j}})f(n)^{-1}$

and

$\begin{split}\SS^{\prime\prime}=\\{g_{t,j}:1\leq t\leq k,\ 0\leq j\leq
N\\}\setminus\\{e_{\Gamma}\\}.\end{split}$

Since $\\{\alpha_{j}\\}$ is an increasing sequence, we can choose them such
that all elements in $\SS^{\prime\prime}$ are distinct.

If $g_{t}$ is not equivalent to $f$, the $\Gamma$-polynomials
$g_{t,0},\ldots,g_{t,N}\in\SS^{\prime\prime}$ have the same weights as $g_{t}$
itself and their equivalence is preserved, that is, if $g_{t}$ is equivalent
to $g_{s}$ then $g_{t,j}$ is equivalent to $g_{s,i}$ for every
$j,i=0,\ldots,N$. If $g_{t}$ is equivalent to $f$, then the weights of these
$\Gamma$-polynomials decrease: $w(g_{t,j})<w(g_{t})=w(f)$. So, the number of
equivalence classes having weights greater than $w(f)$ does not change,whereas
the number of equivalence classes of $\Gamma$-polynomials having the minimal
weight in $\SS$ decreases by 1 when we pass from $\SS$ to
$\SS^{\prime\prime}$. Then $\SS^{\prime\prime}$ precedes $\SS$. Notice that if
$g_{t,j}=e_{\Gamma}$, then $t=1$. By the inductive hypothesis, $X$ is
$\SS^{\prime\prime}_{\Delta}$-$\mathcal{G}_{A}^{*}$ transitive, and there are
$x\in V_{1}^{(N)}$ and $\beta\in\mathcal{F}^{(1)}$ such that
$\beta>\alpha_{N}$ and

$g_{t,j}(n_{\beta})x=g_{t}(n_{\alpha_{j}})^{-1}g_{t}(n_{\beta}+n_{\alpha_{j}})f(n_{\beta})^{-1}x\in
V_{t}^{(N)},\ 0\leq j\leq N,1\leq t\leq k.$

Let $y={f(n_{\beta})^{-1}}x$. Then by $X=\bigcup_{i=0}^{N}T^{-i}U$ there is
$z\in U$ and $0\leq b\leq N$ such that $y=T^{b}z$. Then
$z=T^{-b}f(n_{\beta})^{-1}x$ and we have for each $1\leq t\leq k$

$\begin{split}{g_{t}(n_{\beta}+n_{\alpha_{b}})}z&=g_{t}(n_{\beta}+n_{\alpha_{b}})T^{-b}f(n_{\beta})^{-1}x\\\
&=g_{t}(n_{\alpha_{b}})T^{-b}\big{(}g_{t}(n_{\alpha_{b}})^{-1}g_{t}(n_{\beta}+n_{\alpha_{b}})f(n_{\beta})^{-1}\big{)}x\\\
&=g_{t}(n_{\alpha_{b}})T^{-b}g_{t,b}(n_{\beta})x\\\ &\in
g_{t}(n_{\alpha_{b}})T^{-b}V_{t}^{(N)}\subset V_{t}.\end{split}$

This implies that

$z\in
U\cap{g_{1}(n_{\alpha})^{-1}}V_{1}\cap\ldots\cap{g_{k}(n_{\alpha})^{-1}}V_{k}$

with $\alpha=\beta\cup\alpha_{b}\in\mathcal{F}^{(1)}$ as
$\beta\cap\alpha_{b}=\emptyset$. Since
$\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}^{(1)}}$ is an arbitrary IP-subset of
$\\{n_{\alpha}\\}_{\alpha\in\mathcal{F}}$, it follows that
$N_{\SS}(U,V_{1},\ldots,V_{k})$ is an $\mathcal{G}_{A}^{*}$-set. Hence the
proof of the theorem is complete. ∎

##  7\. Further discussions

First recall that a subset $S$ of $\mathbb{Z}$ is syndetic if it has a bounded
gap, i.e. there is $N\in\mathbb{N}$ such that $\\{i,i+1,\cdots,i+N\\}\cap
S\neq\emptyset$ for every $i\in{\mathbb{Z}}$. $S$ is thick if it contains
arbitrarily long runs of integers. A subset $S$ of $\mathbb{Z}$ is piecewise
syndetic if it is an intersection of a syndetic set with a thick set. A set
$S$ is called thickly syndetic or replete if for every $N\in\mathbb{N}$ the
positions where length $N$ runs begin form a syndetic set. Note that the set
of all thickly syndetic sets is a filter. In [14, Theorem 4.7.] it is shown
that for a minimal and weakly mixing system $(X,T)$,

$N(U,V)=\\{n\in\mathbb{Z}:U\cap T^{-n}V\neq\emptyset\\}$

is thickly syndetic for all nonempty open subsets $U,V$ of $X$.

Weak mixing of all orders was studied in [11, 16]. One of main result of [16]
is as follows. Let $(X,\Gamma)$ be a topological system, where $\Gamma$ is a
nilpotent group such that for each $T\in\Gamma$, $T\neq e_{\Gamma}$, is weakly
mixing and minimal. For $d,k\in\mathbb{N}$ let $T_{1},\ldots,T_{d}\in\Gamma$,
$\\{p_{i,j}(n)\\}_{1\leq i\leq k,1\leq j\leq d}$ be integral polynomials such
that the expression

$g_{i}(n)=T_{1}^{p_{i,1}(n)}\cdots T_{d}^{p_{i,d}(n)}$

depends nontrivially on $n$ for $i=1,2,\ldots,k$, and for all $i\neq
j\in\\{1,2,\ldots,k\\}$ the expressions $g_{i}(n)g_{j}(n)^{-1}$ depend
nontrivially on $n$. Then for all non-empty open sets $U_{1},\ldots,U_{k}$ and
$V_{1},\ldots,V_{k}$ of $X$

$\\{n\in\mathbb{Z}:U_{1}\times\ldots\times U_{k}\cap
g_{1}(n)^{-1}\times\ldots\times g_{k}(n)^{-1}(V_{1}\times\ldots\times
V_{k})\neq\emptyset\\}$

is a thickly syndetic set, and

$\\{n\in\mathbb{Z}:U\cap(g_{1}(n)^{-1}V_{1}\cap\ldots\cap
g_{k}(n)^{-1}V_{k})\not=\emptyset\\}$

is a syndetic set. We do not know that if the latter is thickly syndetic?

Also a minimal system $(X,T)$ is weak mixing if and only if for all non-empty
open sets $U,V$, $N(U,V)=\\{n\in\mathbb{Z}:U\cap T^{-n}V\neq\emptyset\\}$ is
lower Banach density 1 [15]. Recall that for a subset $S$ of $\mathbb{Z}$, the
upper Banach density and lower Banach density of $S$ are

$BD^{*}(S)=\limsup_{|I|\to\infty}\frac{|S\cap I|}{|I|},\ \text{and}\
BD_{*}=\liminf_{|I|\to\infty}\frac{|S\cap I|}{|I|},$

where $I$ ranges over intervals of $\mathbb{Z}$. We also have the following
question: under conditions above, is

$\\{n\in\mathbb{Z}:U\cap(g_{1}(n)^{-1}V_{1}\cap\ldots\cap
g_{k}(n)^{-1}V_{k})\not=\emptyset\\}$

lower Banach density 1?

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