††* Corresponding author.

# Topologically mildly mixing of higher orders along generalized polynomials

Yang Cao  Department of Mathematics, Nanjing University, Nanjing 210093, P.R.
China<EMAIL_ADDRESS>and  Jianjie Zhao∗ School of Mathematics, Hangzhou
Normal University, Hangzhou 311121, P.R. China<EMAIL_ADDRESS>
###### Abstract.

This paper is devoted to studying the multiple recurrent property of
topologically mildly mixing systems along generalized polynomials. We show
that if a minimal system is topologically mildly mixing, then it is mild
mixing of higher orders along generalized polynomials. Precisely, suppose that
$(X,T)$ is a topologically mildly mixing minimal system, $d\in\mathbb{N}$,
$p_{1},\dots,p_{d}$ are integer-valued generalized polynomials with
$(p_{1},\dots,p_{d})$ non-degenerate. Then for all non-empty open subsets
$U,V_{1},\dots,V_{d}$ of $X$,

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

is an IP∗-set.

###### Key words and phrases:

generalized polynomials; mildly mixing; recurrence

###### 2020 Mathematics Subject Classification:

Primary: 37B20, Secondary: 37B05, 11B83, 37A25

##  1\. Introduction

Throughout this paper a _topological dynamical system_ (or _dynamical system_
, _system_ for short) is a pair $(X,T),$ where $X$ is a non-empty compact
metric space with a metric $\rho$ and $T$ is a homeomorphism from $X$ to
itself.

Mixing is a fundamental notion in qualitative theory of dynamical systems, it
refers to indecomposability of dynamics and synchronization of transfer times
between open sets. In the hierarchy of mixing there is a notion, namely that
of mildly mixing which was introduced by Glasner and Weiss [10], and
independently by Huang and Ye [12]. It is well-known that mild mixing is
stronger than weak mixing, and weaker than strong mixing. A dynamical system
$(X,T)$ is said to be _topologically mildly mixing_ if the product system with
any transitive system is still transitive. As is shown in [12], when $(X,T)$
is minimal, another characterization of mild mixing, via the hitting time set
of two non-empty open subsets, goes as follows: a minimal system $(X,T)$ is
topologically mildly mixing if for all non-empty open subsets $U,V$ of $X,$

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

is an IP∗-set. (A subset $A$ of an additive semigroup $S$ is _IP ∗_ if for any
sequence $\\{n_{i}\\}$ in $S$, there is some finite, non-empty set
$\alpha\subset\mathbb{N}$ such that $\sum_{i\in\alpha}n_{i}\in A$.)

In light of this characterization, it is natural to ask whether one may
obtain, e.g.,

$N(U,V_{1},V_{2})=\\{n\in\mathbb{Z}:U\cap T^{-n}V_{1}\cap
T^{-2n}V_{2}\neq\emptyset\\}$

is an IP∗-set. The answer is yes. In particular, Cao and Shao has shown in [7]
that in mildly mixing minimal systems, for any $d\in\mathbb{N}$ and non-empty
open subsets $U,V_{1},\dots,V_{d}$ of $X,$

(1.1) $\\{n\in\mathbb{Z}:U\cap T^{-n}V_{1}\cap\dots\cap
T^{-dn}V_{d}\neq\emptyset\\}$

is an IP∗-set, which is entitled “Mildly mixing of all orders.” However,
linear orders proved not to be “all”. Indeed, as was shown in [7], for mildly
mixing minimal systems, suppose $p_{i}(x)\in\mathbb{Z}[x],$ $1\leq i\leq d,$
having the property that no $p_{i}$ and no $p_{i}-p_{j}$ is constant, $1\leq
i\neq j\leq d,$ one obtains

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

is an IP∗-set. Thus we see that mildly mixing implies not merely mildly mixing
of higher linear orders but of higher polynomial orders as well. The next
question that arises is this: do such polynomial functions constitute a
suitably “most general class” of integer sequences along which mildly mixing
minimal systems are well behaved? In this paper, we give a negative answer.

The goal of this paper is extending (1.2) to classes of sequences which are
more general than those of the form $p(n),$ where $p$ is a polynomial
satisfying $p(\mathbb{Z})\subset\mathbb{Z}$. For example, we show that for
every mildly mixing minimal system $(X,T)$ and all non-empty open subsets
$U,V_{1},\dots,V_{d}$ of $X,$

(1.3) $\\{n\in\mathbb{Z}:U\cap T^{-\left\lceil\pi
n\left\lceil\sqrt{5}n^{2}+2n\right\rceil\right\rceil}V_{1}\cap\dots\cap
T^{-d\left\lceil\pi
n\left\lceil\sqrt{5}n^{2}+2n\right\rceil\right\rceil}V_{d}\neq\emptyset\\}$

is an IP∗-set, where $\left\lceil a\right\rceil$ ($a\in\mathbb{R}$) means the
minimum integer which is the nearest integer to $a$. More generally, the role
of the sequence $\left\lceil\pi
n\left\lceil\sqrt{5}n^{2}+2n\right\rceil\right\rceil$ in (1.3) can be played
by any integer-valued generalized polynomial under suitable hypotheses. The
set of _integer-valued generalized polynomials_ $\mathbb{Z}\to\mathbb{Z}$ is
the smallest set $\mathcal{G}$ that is a function algebra (i.e., is closed
under sum and products) containing $\mathbb{Z}[x]$ and having the additional
property that for all $k\in\mathbb{N}$, $a_{1},\dots,a_{k}\in\mathbb{R}$ and
$p_{1},\dots,p_{k}\in\mathcal{G},$ the mapping
$n\to\left\lceil\sum_{i=1}^{k}a_{i}p_{i}(n)\right\rceil$ is in $\mathcal{G}.$
Generalized polynomials have been studied extensively, see for example the
remarkable paper by Bergelson and Leibman [6] and references therein. In this
paper we note that our definition of the generalized polynomials is slightly
different from the usual one.

In the other hand, measurable dynamics and topological dynamics are two sister
branches of the theory of dynamical systems that use similar words to describe
different but parallel notions in their respective theories. For a measurable
preserving system $(X,\mathcal{B},\mu,T)$, where $T$ is an invertible
measurable preserving transformation acting on a probability space
$(X,\mathcal{B},\mu),$ which is called _mildly mixing_ if for any
$A,B\in\mathcal{B}$ and any $\epsilon>0$ the set of $n$ for which $|\mu(A\cap
T^{-n}B)-\mu(A)\mu(B)|<\epsilon$ is an IP∗-set. In [8, Section 9.5],
Furstenberg showed that mild mixing implies mild mixing of all orders. Later
on, Bergelson stated in [1, Theorem 4.8], establishing a polynomical extension
of Furstenberg’s, as an unproved corollary to (the proof of) his main result.
In 2009, McCutcheon formulated and proved the following result.

Theorem[15, Theorem C] If $p_{i},1\leq i\leq d$ are generalized polynomials,
such that no $p_{i}$ and no $p_{i}-p_{j},$ $1\leq i<j\leq d,$ is constant on
an IP-set, (A subset $E$ of $\mathbb{Z}$ is IP if there is a sequence
$\\{n_{i}\\}\subset\mathbb{Z}$ such that for any non -empty
$\alpha\subset\mathbb{N}$, $\sum_{i\in\alpha}n_{i}\in E$.) then for any mildly
mixing system $(X,\mathcal{B},\mu,T)$ and any
$A_{0},A_{1},\dots,A_{d}\in\mathcal{B},$ the set
$\\{n\in\mathbb{Z}:|\mu(A_{0}\cap T^{-p_{1}(n)}A_{1}\cap\dots\cap
T^{-p_{d}(n)}A_{d})-\mu(A_{0})\mu(A_{1})\dots\mu(A_{d})|<\epsilon\\}$ is an
IP∗-set.

In this paper we are interested in obtaining the topological version of the
theorem above. The next theorem is the main result of this paper.

######  Theorem 1.1.

Let $(X,T)$ be a topologically mildly mixing minimal system,
$p_{1},\dots,p_{d}$ be integer-valued generalized polynomials with the
property that $(p_{1},\dots,p_{d})$ is non-degenerate (see Definition 2.21).
Then for all non-empty open subsets $U,V_{1},\dots,V_{d}$ of $X$,

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

is an IP∗-set.

As a corollary of Theorem 1.1 (see Lemma 2.11 for the reason), we have:

######  Corollary 1.2.

Let $(X,T)$ be a topologically mildly mixing minimal system and
$d\in\mathbb{N}$. Let $p_{1},\dots,p_{d}$ be integer-valued generalized
polynomials with the property that $(p_{1},\dots,p_{d})$ is non-degenerate.
Then for every IP-set $A$, there is a dense $G_{\delta}$ subset $X_{0}$ of $X$
such that for any $x\in X_{0}$,

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

is dense in $X^{d}$.

The paper is organized as follows. In Section 2, we provide some basic
notations, definitions and results which will be used later. In Section 3, we
prove Theorem 1.1 for integer-valued generalized polynomials of degree $1.$ In
Section 4, we show the generalized polynomial extension of van der Waerden’s
theorem which is an important ingredient of our method. In Section 5, we give
the complete proof of the main result Theorem 1.1.

##  2\. Preliminaries

###  2.1. Subsets of integers

In this paper, the set of integers and the set of positive integers are
denoted by $\mathbb{Z}$ and $\mathbb{N}$ respectively. Let $\mathcal{F}$
denote the family of all non-empty finite subsets of $\mathbb{N},$ i.e.,
$\alpha\in\mathcal{F}$ iff
$\alpha=\\{i_{1},i_{2},\dots,i_{k}\\}\subset\mathbb{N}$,
$i_{1}<i_{2}<\dots<i_{k}$ for some $k\in\mathbb{N}$. For
$\alpha,\beta\in\mathcal{F}$, we denote $\alpha<\beta$ (or $\beta>\alpha$) if
$\max\alpha<\min\beta$.

Let $\\{n_{i}\\}_{i=1}^{\infty}$ be a sequence of $\mathbb{Z}$, define

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

Note that we do not require the elements of $\\{n_{i}\\}_{i=1}^{\infty}$ to be
distinct. We denote $n_{\alpha}=\sum_{i\in\alpha}n_{i}$, then

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

An IP-set is a set containing $\\{n_{\alpha}:\alpha\in\mathcal{F}\\}$ for some
infinite sequence $\\{n_{i}\\}_{i=1}^{\infty}\subseteq\mathbb{Z}.$ Note that
for any $\alpha_{0}\in\mathcal{F}$, the set
$\\{n_{\alpha}:\alpha>\alpha_{0},\alpha\in\mathcal{F}\\}$ is still an IP-set.
A set is called an IP∗-set if it intersects any IP-set. Let $\mathcal{F}_{IP}$
and $\mathcal{F}_{IP}^{*}$ denote the family of IP-sets and IP∗-sets
respectively. It is well known that the following lemmas hold (see, e.g., [8,
9]).

######  Lemma 2.1.

$\mathcal{F}_{IP}^{*}$ is a filter, i.e., $A_{1},A_{2}\in\mathcal{F}_{IP}^{*}$
implies that $A_{1}\cap A_{2}\in\mathcal{F}_{IP}^{*};$ for all
$A\in\mathcal{F}_{IP}$ and $B\in\mathcal{F}_{IP}^{*},$ we have $A\cap
B\in\mathcal{F}_{IP}.$

######  Lemma 2.2.

Let $m\neq 0$ be an integer. Any IP-set contains an IP-set consisting of
integers divisible by $m.$

###  2.2. Partition regular sets

Bergelson, Hindman and Kra [3] introduced the partition regular sets in
$\mathbb{N}$. We now extend it to $\mathbb{Z}$, and generalize some results to
more general cases.

Let $\mathcal{B}$ be a set of subsets of $\mathbb{Z}.$ We say that
$\mathcal{B}$ is _partition regular_ if whenever $A_{1}\cup\dots\cup
A_{n}\in\mathcal{B}$ one has some $A_{i}\in\mathcal{B}$ with
$i\in\\{1,\dots,n\\}.$

The following well known theorem shows that $\mathcal{F}_{IP}$ is partition
regular.

######  Theorem 2.3.

[9, Theorem 1.4] Let $A$ be an IP-set. If $A$ is partitioned into finitely
many sets, $A=C_{1}\cup\dots\cup C_{r},$ then some $C_{j}$ is an IP-set with
$j\in\\{1,\dots,r\\}$.

Our one aim is to generalize [3, Lemma 2.2] (see Lemma 2.5), and for that we
need to recall some notations.

For a real number $a\in\mathbb{R}$, let $||a||=\inf\\{|a-n|:n\in\mathbb{Z}\\}$
and

$\left\lceil a\right\rceil=\inf\\{m\in\mathbb{Z}:|a-m|=||a||\\}.$

We put $\\{a\\}=a-\left\lceil a\right\rceil$, then
$\\{a\\}\in(-\frac{1}{2},\frac{1}{2}]$.

We denote $[a]$ the greatest integer not exceeding $a$ and put
$\\{\\{a\\}\\}=a-[a]$, then $\\{\\{a\\}\\}\in[0,1)$.

We have the following observations:

######  Lemma 2.4.

Let $k\in\mathbb{N}$ and $r_{1},\dots,r_{k}\in\mathbb{R}$. If

  1. (1)

If $-\frac{1}{2}<\\{r_{1}\\}+\dots+\\{r_{k}\\}\leq\frac{1}{2},$ then
$\left\lceil r_{1}+\dots+r_{k}\right\rceil=\left\lceil
r_{1}\right\rceil+\dots+\left\lceil r_{k}\right\rceil.$

  2. (2)

If $0\leq\\{\\{r_{1}\\}\\}+\dots+\\{\\{r_{k}\\}\\}<1,$ then
$[r_{1}+\dots+r_{k}]=[r_{1}]+\dots+[r_{k}].$

######  Lemma 2.5.

Let $\mathcal{B}$ be a set of subsets of $\mathbb{Z}$ such that

  1. (1)

$\mathcal{B}$ is partition regular.

  2. (2)

For each $A\in\mathcal{B}$, there exists $x,y\in A$ with $x+y\in A$.

Then for each $b_{i},\alpha_{i},c_{j},\beta_{j}\in\mathbb{R}$,
$i=1,\dots,t_{1},j=1,\dots,t_{2}$, $t_{1},t_{2}\in\mathbb{N}$, any
$\epsilon>0$ and every $A\in\mathcal{B}$, there exists $B\in\mathcal{B}$ such
that $B\subset A$ and for all $n\in B$,

$\\{b_{i}\left\lceil\alpha_{i}n\right\rceil\\},\\{\alpha_{i}n\\},\\{c_{j}\left\lceil\beta_{j}n\right\rceil\\}\
\text{and}\ \\{\beta_{j}n\\}\in(-\epsilon,\epsilon)$

for all $i=1,\dots,t_{1}$ and $j=1,\dots,t_{2}$.

###### Proof.

Let $b_{i},\alpha_{i},c_{j},\beta_{j}\in\mathbb{R}$,
$i=1,\dots,t_{1},j=1,\dots,t_{2}$, $t_{1},t_{2}\in\mathbb{N}$, $\epsilon>0$
and $A\in\mathcal{B}$ be given. Pick $m\in\mathbb{N}$ such that
$\frac{1}{2m}<\epsilon$ and $m>2$.

For each $i=1,\dots,t_{1}$, put

$C_{1}^{i}=\\{n\in A:-\frac{1}{2m}<\\{\alpha_{i}n\\}\leq\frac{1}{2m}\\},$
$D^{i}_{1}=\\{n\in A:-\frac{1}{2m}<\\{b_{i}\left\lceil\alpha
n_{i}\right\rceil\\}\leq\frac{1}{2m}\\},$ $C^{i}_{k}=\\{n\in
A:\frac{k-1}{2m}<\alpha_{i}n-[\alpha_{i}n]\leq\frac{k}{2m}\\},$
$D^{i}_{k}=\\{n\in
A:\frac{k-1}{2m}<b_{i}\left\lceil\alpha_{i}n\right\rceil-[b_{i}\left\lceil\alpha_{i}n\right\rceil]\leq\frac{k}{2m}\\},$

where $k\in\\{2,3,\dots,2m-1\\}$. And for each $j=1,\dots,t_{2}$, put

$E_{1}^{j}=\\{n\in A:-\frac{1}{2m}<\\{\beta_{j}n\\}\leq\frac{1}{2m}\\},$
$F^{j}_{1}=\\{n\in
A:-\frac{1}{2m}<\\{c_{j}\left\lceil\beta_{j}c_{j}\right\rceil\\}\leq\frac{1}{2m}\\},$
$E^{j}_{k}=\\{n\in
A:\frac{k-1}{2m}<\beta_{j}n-[\beta_{j}n]\leq\frac{k}{2m}\\},$
$F^{j}_{k}=\\{n\in
A:\frac{k-1}{2m}<c_{j}\left\lceil\beta_{j}n\right\rceil-[c_{i}\left\lceil\beta_{i}n\right\rceil]\leq\frac{k}{2m}\\},$

where $k\in\\{2,3,\dots,2m-1\\}$.

Put

$A_{1}=\bigcap_{i=1}^{t_{1}}C_{1}^{i}\cap\bigcap_{i=1}^{t_{1}}D_{1}^{i}\cap\bigcap_{j=1}^{t_{2}}E_{1}^{j}\cap\bigcap_{j=1}^{t_{2}}F_{1}^{j}$

and

$A_{(k^{C},k^{D},k^{E},k^{F})}=\bigcap_{i=1}^{t_{1}}C^{i}_{k^{C}}\cap\bigcap_{i=1}^{t_{1}}D^{i}_{k^{D}}\cap\bigcap_{j=1}^{t_{2}}E^{j}_{k^{E}}\cap\bigcap_{j=1}^{t_{2}}F^{j}_{k^{F}}$

where $k^{C},k^{D},k^{E},k^{F}\in\\{1,\dots,2m-1\\}$, and
$A_{(1,1,1,1)}=A_{1}$.

Clearly,

$A=\bigcup_{k^{C},k^{D},k^{E},k^{F}=1,\dots,2m-1}A_{(k^{C},k^{D},k^{E},k^{F})}.$

By condition (1) we have $A_{(k^{C},k^{D},k^{E},k^{F})}\in\mathcal{B}$ for
some $k^{C},k^{D},k^{E},k^{F}\in\\{1,\dots,2m-1\\}$. We will show that such
$k^{C},k^{D},k^{E},k^{F}$ satisfy that $k^{C}=k^{D}=k^{E}=k^{F}=1$.

######  Claim 1.

$k^{C}=1$ and $k^{E}=1$.

###### Proof.

Firstly, we show that $k^{C}=1.$ Let $x$ and $y$ be two elements of
$A_{(k^{C},k^{D},k^{E},k^{F})}$, then for all $i=1,\dots,t_{1}$, one has
$x,y\in C^{i}_{k^{C}}$.

Fix $i=1,\dots,t_{1}$. If $2\leq k^{C}<m-1$, then $x+y\in C^{i}_{2k^{C}-1}\cup
C^{i}_{2k^{C}}$. If $k^{C}=m$, then $x+y\in C^{i}_{2m-1}\cup C^{i}_{1}$. If
$m+1\leq k^{C}\leq 2m-1$, then $x+y\in C^{i}_{2(k^{C}-m)-1}\cup
C^{i}_{2(k^{C}-m)}$. In any of these cases $x+y\notin C^{i}_{k^{C}},$ so
$x+y\notin A_{(k^{C},k^{D},k^{E},k^{F})}$and hence by condition (2) we must
have $k^{C}=1$.

By the same method, we have $k^{E}=1$.

∎

######  Claim 2.

$k^{D}=1$ and $k^{F}=1$.

###### Proof.

Firstly , we show that $k^{D}=1$. Since $\frac{1}{2m}<\frac{1}{2}$ and by
Claim 1 and Lemma 2.4, then for all $x,y\in A_{(1,k^{D},1,k^{F})}$ we have

(2.1)
$\left\lceil\alpha_{i}x\right\rceil+\left\lceil\alpha_{i}y\right\rceil=\left\lceil\alpha_{i}(x+y)\right\rceil$

for any $i=1,\dots,t_{1}$.

Let $x,y\in A_{(1,k^{D},1,k^{F})}$, then for all $i=1,\dots,t_{1}$, one has
$x,y\in D^{i}_{k^{D}}$.

Now fix $i=1,\dots,t_{1}$. If $2\leq k^{D}\leq m-1$, that is

$\frac{k^{D}-1}{2m}<b_{i}\left\lceil\alpha_{i}x\right\rceil-[b_{i}\left\lceil\alpha_{i}x\right\rceil]\leq\frac{k^{D}}{2m}$

and

$\frac{k^{D}-1}{2m}<b_{i}\left\lceil\alpha_{i}y\right\rceil-[b_{i}\left\lceil\alpha_{i}y\right\rceil]\leq\frac{k^{D}}{2m}.$

Then
$0<\\{\\{b_{i}\left\lceil\alpha_{i}x\right\rceil\\}\\}+\\{\\{b_{i}\left\lceil\alpha_{i}y\right\rceil\\}\\}<1$,
thus by Lemma 2.4 and $(\ref{2-sum})$, we have

$[b_{i}\left\lceil\alpha_{i}x\right\rceil]+[b_{i}\left\lceil\alpha_{i}y\right\rceil]=[b_{i}\left\lceil\alpha_{i}x\right\rceil+b_{i}\left\lceil\alpha_{i}y\right\rceil]=[b_{i}\left\lceil\alpha_{i}(x+y)\right\rceil],$

which shows that $x+y\in D^{i}_{2k^{D}-1}\cup D^{i}_{2k^{D}}$. By the same
discussion, we have if $k^{D}=m$, then $x+y\in D^{i}_{2k^{D}-1}\cup
D_{1}^{i}$; if $m+1\leq k^{D}\leq 2m-1$, then $x+y\in D^{i}_{2(k^{D}-m)-1}\cup
D^{i}_{2(k^{D}-m)}$. In any of these cases $x+y\notin A_{(1,k^{D},1,k^{F})}$,
and hence by condition (2) we must have $k^{D}=1$.

By the same method, we have $k^{F}=1$.

∎

So far, for all $n\in A_{(1,1,1,1)}\in\mathcal{B}$ we have

$\\{b_{i}\left\lceil\alpha_{i}n\right\rceil\\},\\{\alpha_{i}n\\},\\{c_{j}\left\lceil\beta_{j}n\right\rceil\\}\
\text{and}\ \\{\beta_{j}n\\}\in(-\epsilon,\epsilon)$

for all $i=1,\dots,t_{1}$ and $j=1,\dots,t_{2}$.

∎

######  Remark 2.6.

$\mathcal{F}_{IP}$ satisfies the hypotheses of Lemma 2.5.

######  Lemma 2.7.

Let $p(n)=\sum\limits_{i=1}^{\rm{t_{1}}}\left\lceil
b_{i}\left\lceil\alpha_{i}n\right\rceil\right\rceil-\sum\limits_{j=1}^{\rm{t_{2}}}\left\lceil
c_{j}\left\lceil\beta_{j}n\right\rceil\right\rceil$ with
$t_{1},t_{2}\in\mathbb{N},\ b_{i},\alpha_{i},c_{j},\beta_{j}\in\mathbb{R},\
i=1,\dots,t_{1}$, $j=1,\dots,t_{2}$ and

$\sum_{i=1}^{t_{1}}b_{i}\alpha_{i}-\sum_{j=1}^{t_{2}}c_{j}\beta_{j}\neq 0.$

If $A$ is an IP-set, then $\\{p(n):n\in A\\}$ is also an IP-set.

###### Proof.

Let $\epsilon=\frac{1}{4}$. By Lemma 2.5 and Remark 2.6, there exists an IP-
set $B\subset A$ such that for all $n\in B$, we have

$\\{b_{i}\left\lceil\alpha_{i}n\right\rceil\\},\\{\alpha_{i}n\\},\\{c_{j}\left\lceil\beta_{j}n\right\rceil\\}\
\text{and}\ \\{\beta_{j}n\\}\in(-\epsilon,\epsilon)$

for all $i=1,\dots,t_{1}$ and $j=1,\dots,t_{2}$.

Pick the sequence $\\{a_{n}\\}_{n=1}^{\infty}$ of $\mathbb{Z}$ with
$FS(\\{a_{n}\\}_{n=1}^{\infty})\subseteq B$. We show by induction on $|F|$
that for a finite subset $F$ of $\mathbb{N},$

$\displaystyle\sum_{n\in F}\biggl{(}\sum\limits_{i=1}^{\rm{t_{1}}}\left\lceil
b_{i}\left\lceil\alpha_{i}a_{n}\right\rceil\right\rceil-\sum\limits_{j=1}^{\rm{t_{2}}}\left\lceil
c_{j}\left\lceil\beta_{j}a_{n}\right\rceil\right\rceil\biggr{)}$
$\displaystyle=$ $\displaystyle\sum\limits_{i=1}^{\rm{t_{1}}}\left\lceil
b_{i}\left\lceil\alpha_{i}(\sum_{n\in
F}a_{n})\right\rceil\right\rceil-\sum\limits_{j=1}^{\rm{t_{2}}}\left\lceil
c_{j}\left\lceil\beta_{j}(\sum_{n\in F}a_{n})\right\rceil\right\rceil.$

This shows that $FS(\\{\sum\limits_{i=1}^{\rm{t_{1}}}\left\lceil
b_{i}\left\lceil\alpha_{i}a_{n}\right\rceil\right\rceil-\sum\limits_{j=1}^{\rm{t_{2}}}\left\lceil
c_{j}\left\lceil\beta_{j}a_{n}\right\rceil\right\rceil\\}_{n=1}^{\infty})\subset\\{p(n):n\in
A\\}$.

If $|F|=1$, the conclusion is trivial. We now assume that $|F|>1$. Pick $m\in
F$ and let $F^{\prime}=F\backslash\\{m\\}$. By induction, we have

$\displaystyle\sum_{n\in
F^{\prime}}\biggl{(}\sum\limits_{i=1}^{\rm{t_{1}}}\left\lceil
b_{i}\left\lceil\alpha_{i}a_{n}\right\rceil\right\rceil-\sum\limits_{j=1}^{\rm{t_{2}}}\left\lceil
c_{j}\left\lceil\beta_{j}a_{n}\right\rceil\right\rceil\biggr{)}$
$\displaystyle=$ $\displaystyle\sum\limits_{i=1}^{\rm{t_{1}}}\left\lceil
b_{i}\left\lceil\alpha_{i}(\sum_{n\in
F^{\prime}}a_{n})\right\rceil\right\rceil-\sum\limits_{j=1}^{\rm{t_{2}}}\left\lceil
c_{j}\left\lceil\beta_{j}(\sum_{n\in
F^{\prime}}a_{n})\right\rceil\right\rceil.$

Then

$\displaystyle\sum_{n\in F}\biggl{(}\sum\limits_{i=1}^{\rm{t_{1}}}\left\lceil
b_{i}\left\lceil\alpha_{i}a_{n}\right\rceil\right\rceil-\sum\limits_{j=1}^{\rm{t_{2}}}\left\lceil
c_{j}\left\lceil\beta_{j}a_{n}\right\rceil\right\rceil\biggr{)}$
$\displaystyle=$
$\displaystyle\biggl{(}\sum\limits_{i=1}^{\rm{t_{1}}}\left\lceil
b_{i}\left\lceil\alpha_{i}a_{m}\right\rceil\right\rceil-\sum\limits_{j=1}^{\rm{t_{2}}}\left\lceil
c_{j}\left\lceil\beta_{j}a_{m}\right\rceil\right\rceil\biggr{)}+\sum_{n\in
F^{\prime}}\biggl{(}\sum\limits_{i=1}^{\rm{t_{1}}}\left\lceil
b_{i}\left\lceil\alpha_{i}a_{n}\right\rceil\right\rceil-\sum\limits_{j=1}^{\rm{t_{2}}}\left\lceil
c_{j}\left\lceil\beta_{j}a_{n}\right\rceil\right\rceil\biggr{)}$
$\displaystyle=$
$\displaystyle\biggl{(}\sum\limits_{i=1}^{\rm{t_{1}}}\left\lceil
b_{i}\left\lceil\alpha_{i}a_{m}\right\rceil\right\rceil-\sum\limits_{j=1}^{\rm{t_{2}}}\left\lceil
c_{j}\left\lceil\beta_{j}a_{m}\right\rceil\right\rceil\biggr{)}$
$\displaystyle+\biggl{(}\sum\limits_{i=1}^{\rm{t_{1}}}\left\lceil
b_{i}\left\lceil\alpha_{i}(\sum_{n\in
F^{\prime}}a_{n})\right\rceil\right\rceil-\sum\limits_{j=1}^{\rm{t_{2}}}\left\lceil
c_{j}\left\lceil\beta_{j}(\sum_{n\in
F^{\prime}}a_{n})\right\rceil\right\rceil\biggr{)}$ $\displaystyle=$
$\displaystyle\sum\limits_{i=1}^{\rm{t_{1}}}\left\lceil
b_{i}\left\lceil\alpha_{i}(\sum_{n\in
F}a_{n})\right\rceil\right\rceil-\sum\limits_{j=1}^{\rm{t_{2}}}\left\lceil
c_{j}\left\lceil\beta_{j}(\sum_{n\in F}a_{n})\right\rceil\right\rceil.$

The last equation follows from that $a_{m}$, $\sum_{n\in F^{\prime}}a_{n}$ are
in $B$ and Lemma 2.4.

∎

As a consequence of the above lemma, we have

######  Corollary 2.8.

Let $q\in\mathbb{Z}.$ If $A$ is an IP-set, then
$\\{\left\lceil\frac{n}{q}\right\rceil:n\in A\\}$ is an IP-set.

And we have the following useful result.

######  Lemma 2.9.

Let $q\in\mathbb{Z}.$ If $A$ is an IP∗-set, then $qA=\\{qn:n\in A\\}$ is an
IP∗-set.

###### Proof.

For any given IP-set $B,$ we will show that $qA\cap B\neq\emptyset.$ By Lemma
2.2 one sees that there exists $B^{\prime}\subseteq B$ with $B^{\prime}$ is an
IP-set and $B^{\prime}\subseteq q\mathbb{Z}.$ Let
$B^{\prime\prime}=\\{\left\lceil\frac{n}{q}\right\rceil:n\in B^{\prime}\\}.$
By Corollary 2.8, one has $B^{\prime\prime}$ is an IP-set. Pick $m\in
B^{\prime\prime}\cap A$ and pick $n\in B^{\prime}\subseteq B$ such that
$m=\left\lceil\frac{n}{q}\right\rceil.$ Since $q$ divides $n$ we have
$m=\frac{n}{q}$ so $n=qm.$ Thus $n\in qA\cap B.$ The proof is complete. ∎

###  2.3. Topological dynamics

Let $(X,T)$ be a dynamical system. For $x\in X$, we denote the orbit of $x$ by
$\text{orb}(x,T)=\\{T^{n}x\in X:n\in\mathbb{Z}\\}$. A point $x\in X$ is called
a _transitive point_ if the orbit of $x$ is dense in $X$, i.e.,
$\overline{\text{orb}(x,T)}=X$. A dynamical system $(X,T)$ is called _minimal_
if every point $x\in X$ is a transitive point.

Let $U,V\subset X$ be two non-empty open sets, the _hitting time set_ of $U$
and $V$ is denoted by

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

We say that $(X,T)$ is _(topologically) transitive_ if for all non-empty open
sets $U,V\subset X$, $N(U,V)$ is non-empty. Two dynamical systems are _weakly
disjoint_ if their product is transitive. A dynamical system is called _mildly
mixing_ if it is weakly disjoint from any transitive system.

We say that $(X,T)$ is _IP ∗-transitive_ if for all non-empty open sets
$U,V\subset X$, $N(U,V)$ is an IP∗-set. When $(X,T)$ is a minimal system, it
is mildly mixing if and only if it is IP∗-transitive (see [12] for more
details).

Let $p_{i}:\mathbb{Z}\rightarrow\mathbb{Z},i=1,\dots,k$, we say that $(X,T)$
is _$\\{p_{1},\dots,p_{k}\\}$ -IP∗-transitive_ if for any non-empty open sets
$U_{i},V_{i}\subset X,i=1,\dots,k$,

$N(\\{p_{1},\dots,p_{k}\\},U_{1}\times\dots\times U_{k},V_{1}\times\dots\times
V_{k}):=\bigcap_{i=1}^{k}N(p_{i},U_{i},V_{i})$

is an IP∗-set, where $N(p_{i},U_{i},V_{i}):=\\{n\in\mathbb{Z}:U_{i}\cap
T^{-p_{i}(n)}V_{i}\neq\emptyset\\}$, $i=1,\dots,k$. $(X,T)$ is
_$\\{p_{1},\dots,p_{d}\\}_{\Delta}$ -IP∗-transitive_ if for all non-empty open
sets $U,V_{1},\dots,V_{d}\subset X$,

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

is an IP∗-set.

The following lemma is a generalization of [13, Lemma 5]. For completeness, we
include a proof.

######  Lemma 2.10.

Let $(X,T)$ be a dynamical system, $d\in\mathbb{N}$ and maps
$p_{1},\dots,p_{d}:\mathbb{Z}\to\mathbb{Z}$ such that $(X,T)$ is
$\\{p_{1},\dots,p_{d}\\}$-IP∗-transitive. Then for all non-empty open subsets
$V_{1},\dots,V_{d}$ of $X$, for any IP-set
$A=\\{n_{\alpha}:\alpha\in\mathcal{F}\\}$ and any subsequence
$\\{r(n)\\}_{n=0}^{\infty}$ of natural numbers, there is an increasing
sequence $\\{\alpha_{j}\\}_{j=0}^{\infty}\subset\mathcal{F}$ such that
$|n_{\alpha_{0}}|>r(0),|n_{\alpha_{j}}|>|n_{\alpha_{j-1}}|+r(|n_{\alpha_{j-1}}|)$
for all $j\geq 1$, and for each $i\in\\{1,\dots,d\\}$, 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

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

###### Proof.

Let $V_{1},\dots,V_{d}$ be non-empty open subsets of $X$,
$A=\\{n_{\alpha}:\alpha\in\mathcal{F}\\}$ be an IP-set and
$\\{r(n)\\}_{n=0}^{\infty}$ be a subsequence of $\mathbb{N}$. Since $(X,T)$ is
$\\{p_{1},\dots,p_{d}\\}$-IP∗-transitive,
$\bigcap_{i=1}^{d}N(p_{i},V_{i},V_{i})$ is an IP∗-set. Thus there exists
$\alpha_{0}\in\mathcal{F}$ such that

$n_{\alpha_{0}}\in\bigcap_{i=1}^{d}N(p_{i},V_{i},V_{i})\cap A$

and $|n_{\alpha_{0}}|>r(0).$ That is $T^{-p_{i}(n_{\alpha_{0}})}V_{i}\cap
V_{i}\neq\emptyset$ for all $i=1,\dots,d$. Put
$V_{i}^{(0)}=T^{-p_{i}(n_{\alpha_{0}})}V_{i}\cap V_{i}$ for all $i=1,\dots,d$
to complete the base step.

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

$T^{p_{i}(n_{\alpha_{j}})}T^{-j}V_{i}^{(m)}\subset V_{i},\ \ \text{for all}\
0\leq j\leq m.$

For $i=1,\dots,d$, let $U_{i}=T^{-n}V_{i}^{(n-1)}$. Then
$\bigcap_{i=1}^{d}N(p_{i},U_{i},V_{i})$ is an IP∗-set since $(X,T)$ is
$\\{p_{1},\dots,p_{d}\\}$-IP∗-transitive, thus there exists
$\alpha_{n}\in\mathcal{F}$ such that $\alpha_{n}>\alpha_{n-1}$,
$|n_{\alpha_{n}}|>|n_{\alpha_{n-1}}|+r(|n_{\alpha_{n-1}}|)$ , and

$(U_{1}\times\dots\times U_{d})\cap
T^{-p_{1}(n_{\alpha_{n}})}\times\dots\times
T^{-p_{d}(n_{\alpha_{n}})}(V_{1}\times\dots\times V_{d})\neq\emptyset.$

That is

$T^{-p_{i}(\alpha_{n})}V_{i}\cap U_{i}\neq\emptyset,\ \text{for all}\
i=1,\dots,d.$

Then for $i=1,\dots,d$,

$T^{p_{i}(n_{\alpha_{n}})}U_{i}\cap
V_{i}=T^{p_{i}(n_{\alpha_{n}})}T^{-n}V_{i}^{(n-1)}\cap V_{i}\neq\emptyset.$

Let

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

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

$T^{p_{i}(n_{\alpha_{n}})}T^{-n}V_{i}^{(n)}\subset V_{i}.$

Since $V_{i}^{(n)}\subset V_{i}^{(n-1)}$, we have

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

Hence we finish the induction. The proof is completed.

∎

######  Lemma 2.11.

Let $(X,T)$ be a dynamical system, $d\in\mathbb{N}$,
$p_{1},\dots,p_{d}:\mathbb{Z}\to\mathbb{Z}$ and let $A$ be a subset of
$\mathbb{Z}$. Then the following assertions are equivalent.

  1. (1)

If $U,V_{1},\dots,V_{d}\subset X$ are non-empty open sets, then there exists
$n\in A$ such that

$U\cap T^{-p_{1}(n)}V_{1}\cap\dots\cap T^{-p_{d}(n)}V_{d}\neq\emptyset.$

  2. (2)

There exists a dense $G_{\delta}$ subset $X_{0}\subset X$ such that for every
$x\in X_{0}$,

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

is dense in $X^{d}$.

###### Proof.

The proof is similar to the proof in [16]. For completeness, we include a
proof.

$(\ref{Lemma:1open sets})\Rightarrow(\ref{Lemma:2dense})$: Consider a
countable base of open balls $\\{B_{k}:k\in\mathbb{N}\\}$ of $X$. Put

$X_{0}=\bigcap_{(k_{1},\dots,k_{d})\in\mathbb{N}^{d}}\bigcup_{n\in
A}\bigcap_{i=1}^{d}T^{-p_{i}(n)}B_{k_{i}}.$

For any $(k_{1},\dots,k_{d})\in\mathbb{N}^{d}$, the set $\cup_{n\in
A}\cap_{i=1}^{d}T^{-p_{i}(n)}B_{k_{i}}$ is open, and is dense by
$(\ref{Lemma:1open sets})$. Thus by the Baire category theorem, $X_{0}$ is a
dense $G_{\delta}$ subset of $X$. By construction, for every $x\in X_{0}$,

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

is dense in $X^{d}$.

$(\ref{Lemma:2dense})\Rightarrow(\ref{Lemma:1open sets})$: Choose $x\in
X_{0}\cap U$ and $n\in A$ such that

$(T^{p_{1}(n)}x,\dots,T^{p_{d}(n)}x)\in V_{1}\times\dots\times V_{d},$

then $x\in U\cap T^{-p_{1}(n)}V_{1}\cap\dots\cap T^{-p_{d}(n)}V_{d}$. ∎

###  2.4. Generalized polynomials

In [11], Huang, Shao and Ye introduced the notions of $GP_{d}$ and
$\mathcal{F}_{GP_{d}}$, we now recall their definitions.

######  Definition 2.12.

Let $d\in\mathbb{N},$ the _generalized polynomials_ of degree $\leq d$
(denoted by $GP_{d}$) is defined as follows. For $d=1$, $GP_{1}$ is the
smallest collection of functions from $\mathbb{Z}$ to $\mathbb{R}$ containing
$\\{h_{a}:a\in\mathbb{R}\\}$ with $h_{a}(n)=an$ for each $n\in\mathbb{Z}$,
which is closed under taking $\left\lceil{\cdot}\right\rceil$, multiplying by
a constant and finite sums.

Assume that $GP_{i}$ is defined for $i<d$. Then $GP_{d}$ is the smallest
collection of functions from $\mathbb{Z}$ to $\mathbb{R}$ containing $GP_{i}$
with $i<d$, functions of the forms

$a_{0}n^{p_{0}}\left\lceil{f_{1}(n)}\right\rceil\dots\left\lceil{f_{k}(n)}\right\rceil$

(with $a_{0}\in\mathbb{R},p_{0}\geq 0,k\geq 0,f_{l}\in GP_{p_{l}},1\leq l\leq
k$ and $\sum_{l=0}^{k}p_{l}=d$), which is closed under taking
$\left\lceil{\cdot}\right\rceil$, multiplying by a constant and finite sums.
Let $GP=\bigcup_{i=1}^{\infty}GP_{i}$. Note that if $p\in GP$, then $p(0)=0$.

######  Definition 2.13.

Let $\mathcal{F}_{GP_{d}}$ be the family generated by the sets of the form

$\bigcap_{i=1}^{k}\\{n\in\mathbb{Z}:p_{i}(n)\ (\text{mod}\ \mathbb{Z})\
\in(-\varepsilon_{i},\varepsilon_{i})\\},$

where $k\in\mathbb{N}$, $p_{i}\in GP_{d}$, and $\varepsilon_{i}>0,1\leq i\leq
k$. Note that $p_{i}(n)\ (\text{mod}\
\mathbb{Z})\in(-\varepsilon_{i},\varepsilon_{i})$ if and only if
$\\{p_{i}(n)\\}\in(-\varepsilon_{i},\varepsilon_{i})$.

######  Remark 2.14.

$\mathcal{F}_{GP_{d}}$ is a filter, i.e., $A_{1},A_{2}\in\mathcal{F}_{GP_{d}}$
implies that $A_{1}\cap A_{2}\in\mathcal{F}_{GP_{d}}$.

A subset $A\subset\mathbb{Z}$ is a _Nil d Bohr0-set_ if there exists a
$d$-step nilsystem $(X,T)$, $x_{0}\in X$ and an open set $U\subset X$
containing $x_{0}$ such that $N(x_{0},U):=\\{n\in\mathbb{Z}:T^{n}x_{0}\in
U\\}$ is contained in $A$. Denote by $\mathcal{F}_{d,0}$ the family consisting
of all Nild Bohr0-sets. A subset $A\subset\mathbb{Z}$ is called _Nil Bohr
0-set_ if $A\in\mathcal{F}_{d,0}$ for some $d\in\mathbb{N}$. In [11], the
authors proved the following theorem.

######  Theorem 2.15.

[11, Theorem B] Let $d\in\mathbb{N}$. Then
$\mathcal{F}_{d,0}=\mathcal{F}_{GP_{d}}$.

######  Remark 2.16.

Since a nilsystem is distal, it is shown in [8, Theorem 9.11] that every Nild
Bohr0-set is an IP∗-set. Together with Remark 2.14 we know
$\mathcal{F}_{GP_{d}}$ is a filter and any $A\in\mathcal{F}_{GP_{d}}$ is an
IP∗-set.

Zhang and Zhao [17] introduced the notions of integer-valued generalized
polynomials and the special integer-valued generalized polynomials. In this
paper, we adopt their relevant definitions and list some results given by
Zhang and Zhao (see [17] for more details).

######  Definition 2.17.

For $d\in\mathbb{N}$, the integer-valued generalized polynomials of degree
$\leq d$ is defined by

$\widetilde{GP_{d}}=\\{\left\lceil p(n)\right\rceil:p(n)\in GP_{d}\\},$

and the integer-valued generalized polynomials is then defined by

$\mathcal{G}=\bigcup_{i=1}^{\infty}\widetilde{GP_{i}}.$

For $p(n)\in\mathcal{G}$, the least $d\in\mathbb{N}$ such that
$p\in\widetilde{GP_{d}}$ is defined by the _degree_ of $p$, denoted by
$\deg(p)$.

Since the integer-valued generalized polynomials are very complicated, we will
also specify a subclass of the integer-valued generalized polynomials, i.e.
the special integer-valued generalized polynomials (denoted by
$\widetilde{SGP}$), which will be used in the proof of our main results.

We need to recall the defintion of $L(a_{1},a_{2},\dots,a_{l})$ in Defintion
4.2 of [11]. For $a\in\mathbb{R}$, we define $L(a)=a$. For
$a_{1},a_{2}\in\mathbb{R}$, we define $L(a_{1},a_{2})=a_{1}\left\lceil
L(a_{2})\right\rceil$. Inductively, for
$a_{1},a_{2},\dots,a_{l}\in\mathbb{R}(l\geq 2)$,

$L(a_{1},a_{2},\dots,a_{l})=a_{1}\left\lceil
L(a_{2},\dots,a_{l})\right\rceil.$

Before introducing the definition of $\widetilde{SGP}$, we recall the notion
of the simple generalized polynomials.

######  Definition 2.18.

For $d\in\mathbb{N}$, the simple generalized polynomials of degree $\leq d$
(denoted by $\widehat{SGP_{d}}$ ) is defined as follows. $\widehat{SGP_{d}}$
is the smallest collection of generalized polynomials of the forms

$\prod_{i=1}^{k}L(a_{i,1}n^{j_{i,1}},\dots,a_{i,l_{i}}n^{j_{i,l_{i}}}),$

where $k\geq 1$, $1\leq l_{i}\leq d$,
$a_{i,1},a_{i,2},\dots,a_{i,l_{i}}\in\mathbb{R}$,
$j_{i,1},j_{i,2},\dots,j_{i,l_{i}}\geq 0$ and
$\sum_{i=1}^{k}\sum_{t=1}^{l_{i}}j_{i,t}\leq d$.

######  Definition 2.19.

For $d\in\mathbb{N}$, the special integer-valued generalized polynomials of
degree $\leq d$ (denoted by $\widetilde{SGP_{d}}$) is defined as follows.

$\widetilde{SGP_{d}}=\\{\sum_{i=1}^{k}c_{i}\left\lceil
p_{i}(n)\right\rceil:p_{i}(n)\in\widehat{SGP_{d}}\text{ and
}c_{i}\in\mathbb{Z}\\}.$

The special integer-valued generalized polynomials is then defined by

$\widetilde{SGP}=\bigcup_{d=1}^{\infty}\widetilde{SGP_{d}}.$

Clearly $\widetilde{SGP}\subset\mathcal{G}$.

The following lemma shows the relationship between $\widetilde{GP_{d}}$ and
$\widetilde{SGP_{d}}$.

######  Lemma 2.20.

[17, Lemma 2.12] Let $d\in\mathbb{N}$ and $p(n)\in\widetilde{GP_{d}}$. Then
there exists $h(n)\in\widetilde{SGP_{d}}$ and a set

$C=C(\delta,q_{1},\dots,q_{t})=\bigcap_{k=1}^{t}\\{n\in\mathbb{Z}:\\{q_{k}(n)\\}\in(-\delta,\delta)\\}$

such that

$p(n)=h(n),\forall n\in C,$

where $\delta>0$ is small enough and $q_{k}\in\widehat{SGP_{d}},k=1,\dots,t$
for some $t\in\mathbb{N}$.

By Lemma 2.20, every $p(n)\in\widetilde{GP_{d}}$ corresponds to an
$h(n)\in\widetilde{SGP_{d}},$ the maximal-degree components of
$p(n)\in\widetilde{GP_{d}}$ is denoted by the maximal-degree components of the
corresponding $h(n)\in\widetilde{SGP_{d}}.$

######  Definition 2.21.

Let $p(n)\in\mathcal{G}$ and $A(p(n))$ be the sum of the coefficients of the
maximal-degree components of $p(n)$. Let $p_{1},\dots,p_{d}\in\mathcal{G}$, a
tuple $(p_{1},\dots,p_{d})$ is called a non-degenerate tuple if $A(p_{i})\neq
0$ and $A(p_{i}-p_{j})\neq 0$, $1\leq i\neq j\leq d$.

For instance, $A(\left\lceil an^{2}\left\lceil bn\right\rceil+\left\lceil
cn^{3}\right\rceil\right\rceil+dn^{3}+2n^{2})=ab+c+d,$ $A(n+n\left\lceil 2\pi
n-\left\lceil 2\pi n\right\rceil\right\rceil)=2\pi-2\pi=0,$
$(n^{2}+n,n^{2}+\left\lceil\sqrt{3}n\right\rceil)$ is non-degenerate,
$(n\left\lceil 2\pi n\right\rceil+n,\left\lceil 2\pi n^{2}\right\rceil+2n)$ is
not non-degenerate.

######  Definition 2.22.

Let $p(n)\in\widetilde{SGP}$, $m\in\mathbb{Z}$ and $C\subset\mathbb{Z}$. We
say that $p$ is _proper_ with respect to (w.r.t. for short) $m$ and $C$ if for
every $n\in C$,

  * •

if $\deg(p)=1$, $p(n+m)=p(n)+p(m)$.

  * •

if $\deg(p)>1$, $p(n+m)-p(n)-p(m)=q(n),$ where $q(n)\in\widetilde{SGP}$ and
$\deg(q)<\deg(p)$.

######  Definition 2.23.

Let $p(n)\in\widetilde{SGP}$ and $m\in\mathbb{Z}$.

  * •

If $p(n)=\left\lceil L(a_{1}n^{j_{1}},\dots,a_{l}n^{j_{l}})\right\rceil$, we
say $m$ is good w.r.t. $p(n)$ if for any $1\leq t\leq l$,
$\\{L(a_{t}m^{j_{t}},a_{t+1}m^{j_{t+1}},\dots,a_{l}m^{j_{l}})\\}\neq\frac{1}{2}$.

  * •

If $p(n)=\left\lceil\prod_{i=1}^{k}r_{i}(n)\right\rceil$ with
$r_{i}(n)=L(a_{i,1}n^{j_{i,1}},\dots,a_{i,l_{i}}n^{j_{i,l_{i}}})$, we say $m$
is good w.r.t. $p(n)$ if $\\{\prod_{i=1}^{k}r_{i}(m)\\}\neq\frac{1}{2}$ and
$m$ is good w.r.t. $\left\lceil r_{i}(n)\right\rceil$ for each $1\leq i\leq
k$.

  * •

If $p(n)=\sum_{t=1}^{k}c_{t}\left\lceil q_{t}(n)\right\rceil$ with
$c_{t}\in\mathbb{Z}$ and each $q_{t}(n)$ is of the form
$\prod_{i=1}^{k}r_{i}(n)$ with
$r_{i}(n)=L(a_{i,1}n^{j_{i,1}},\dots,a_{i,l_{i}}n^{j_{i,l_{i}}})$, we say $m$
is good w.r.t. $p(n)$ if $m$ is good w.r.t. $\left\lceil q_{t}(n)\right\rceil$
for each $1\leq t\leq k$.

The following lemmas are very useful in our proof.

######  Lemma 2.24.

[17, Lemma 2.16] Let $p(n)\in\widetilde{SGP}$. Then there exist $\delta>0$,
$Q\subset\widehat{SGP_{s}}$ (for some $s\in\mathbb{N}$) and

$C(\delta,Q)=\bigcap_{q(n)\in
Q}\\{n\in\mathbb{Z}:\\{q(n)\\}\in(-\delta,\delta)\\}$

such that for each $m\in C(\delta,Q)$, $m$ is good w.r.t. $p(n)$.

######  Lemma 2.25.

[17, Lemma 2.19] Let $p_{1},\dots,p_{d}\in\widetilde{SGP}$. Let
$l\in\mathbb{N},m_{j}\in\mathbb{Z}$ and $m_{j}$ is good w.r.t. $p_{i}(n)$ for
$1\leq i\leq d,1\leq j\leq l$. Then there exists a Nils Boh0-set $C$ with the
form

$C=\bigcap_{k=1}^{t}\\{n\in\mathbb{Z}:\\{q_{k}(n)\\}\in(-\delta,\delta)\\}$

such that for all $(i,j)\in\\{1,\dots,d\\}\times\\{1,\dots,l\\}$, $p_{i}(n)$
is proper w.r.t. $m_{j}$ and $C$, where $\delta>0$ is a small enough number,
$s=\mathop{\max}_{1\leq i\leq d}\deg(p_{i})$ and
$q_{k}\in\widehat{SGP_{s}},k=1,\dots,t$ for some $t\in\mathbb{N}$.

The Nils Bohr0-set $C$ above is called _associated_ to
$\\{p_{1},\dots,p_{d}\\}$ and $\\{m_{1},\dots,m_{l}\\}$.

##  3\. Proof of Theorem 1.1 for degree $1$ integer-valued generalized
polynomials

In this section, we will prove Theorem 1.1 for degree $1$ integer-valued
generalized polynomials.

The proof of the next lemma is identical with the proof of Lemma 2.7, and we
so omit it.

######  Lemma 3.1.

Let $A$ be an IP-set and $p\in\widetilde{SGP_{1}}$ with $A(p(n))\neq 0$. Then

$\\{p(n)\in\mathbb{Z}:n\in A\\}$

is also an IP-set.

######  Theorem 3.2.

Let $(X,T)$ be a topologically mildly mixing minimal system and
$p\in\widetilde{SGP_{1}}$ with $A(p(n))\neq 0$. Then for any non-empty open
subsets $U,V$ of $X$,

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

is an IP∗-set.

###### Proof.

We may assume that

$p(n)=\sum\limits_{i=1}^{\rm{t_{1}}}\left\lceil
b_{i}\left\lceil\alpha_{i}n\right\rceil\right\rceil-\sum\limits_{j=1}^{\rm{t_{2}}}\left\lceil
c_{j}\left\lceil\beta_{j}n\right\rceil\right\rceil,n\in\mathbb{Z},$

with $t_{1},t_{2}\in\mathbb{N}$, $\alpha_{i},b_{i}\in\mathbb{R},$
$\beta_{j},c_{j}\in\mathbb{R},$ $i=1,\dots,t_{1}$, $j=1,\dots,t_{2}$ and
$\sum_{i=1}^{t_{1}}b_{i}\alpha_{i}-\sum_{j=1}^{t_{2}}c_{j}\beta_{j}\neq 0.$

Since $(X,T)$ is a mildly mixing minimal system, one has
$A=\\{n\in\mathbb{Z}:U\cap T^{-n}V\neq\emptyset\\}$ is an IP∗-set. If
$N(p,U,V)$ is not an IP∗-set, there exists an IP-set $F$ such that $F\cap
N(p,U,V)=\emptyset$. That is for any $m\in F$,

$U\cap T^{-p(m)}V=\emptyset.$

By Lemma 3.1, one has $\\{p(m)\in\mathbb{Z}:m\in F\\}$ is an IP-set. then

$\\{p(m)\in\mathbb{Z}:m\in F\\}\cap A\neq\emptyset.$

Choose $m\in F$ such that $p(m)\in A$, that is $U\cap
T^{-p(m)}V\neq\emptyset,$ so there is a contradiction, thus $N(p,U,V)$ is an
IP∗-set.

∎

First we prove an even more special case.

######  Theorem 3.3.

Let $(X,T)$ be a topologically mildly mixing minimal system, $d\in\mathbb{N}$,
and $p_{1},\dots,p_{d}\in\widetilde{SGP_{1}}$ with $(p_{1},\dots,p_{d})$ non-
degenerate. Then for all non-empty open subsets $U,V_{1},\dots,V_{d}$ of $X$,

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

is an IP∗-set.

###### Proof.

We will prove by induction on $d$ that for all non-empty open sets
$U,V_{1},\dots,V_{d}$ of $X$, any IP-set $A$ there exists $n\in A$ such that

$U\cap T^{-p_{1}(n)}V_{1}\cap\dots\cap T^{-p_{d}(n)}V_{d}\neq\emptyset.$

If $d=1$, by Lemma 3.2, $N(p_{1},U,V_{1})$ is an IP∗-set. Now we assume that
the result holds for $d>1$. Next we will show that the result holds for $d+1$.
Let $U,V_{1},\dots,V_{d},V_{d+1}$ be non-empty open subsets of $X$. We will
show that there exists $n\in A$ such that

$U\cap T^{-p_{1}(n)}V_{1}\cap\dots\cap T^{-p_{d+1}(n)}V_{d+1}\neq\emptyset.$

By Lemma 2.24, there exists $\delta>0,$ $Q\subset\widehat{SGP_{f}}$ (for some
$f\in\mathbb{N}$) and

$C=C(\delta,Q)=\bigcap_{q(n)\in
Q}\\{n\in\mathbb{Z}:\\{q(n)\\}\in(-\delta,\delta)\\}$

such that for each $m\in C$, $m$ is good w.r.t. $p_{i}(n),i=1,\dots,d+1.$ Let
$\tilde{C}=C\cap A$. Since $C$ is an IP∗-set, one has $\tilde{C}$ is an IP-set
(by Lemma 2.1). We may assume that
$\tilde{C}=\\{n_{\alpha}:\alpha\in\mathcal{F}\\}$ for some sequence
$\\{n_{i}\\}_{i=1}^{\infty}\subset\mathbb{Z}$.

Since $(X,T)$ is minimal, there is some $l\in\mathbb{N}$ such that
$X=\cup_{j=0}^{l}T^{j}U$. By Lemma 2.10, there are non-empty open subsets
$V_{1}^{(l)},\dots,V_{d+1}^{(l)}$ of $X$ and
$\alpha_{0}<\alpha_{1}<\dots<\alpha_{l}\in\mathcal{F}$ such that for each
$i=1,\dots,d+1$, one has

$T^{p_{i}(n_{\alpha_{j}})}T^{-j}V_{i}^{(l)}\subset V_{i},\ \text{for all }\
0\leq j\leq l.$

Notice that $n_{\alpha_{j}}\in\tilde{C}\subset C,j=0,1,\dots,l$ is good w.r.t.
$p_{i}(n),i=1,\dots,d+1.$ By Lemma 2.25, there is a Nil1 Bohr0-set $C_{1}$
associated to $\\{p_{1},\dots,p_{d+1}\\}$ and
$\\{n_{\alpha_{0}},\dots,n_{\alpha_{l}}\\}.$ Thus we have for any
$i\in\\{1,\dots,d+1\\}$ and each $j\in\\{0,\dots,l\\},$
$p_{i}(n+n_{\alpha_{j}})=p_{i}(n)+p_{i}(n_{\alpha_{j}})$ when $n\in C_{1}.$

Let $q_{i}=p_{i}-p_{1}\in\widetilde{SGP_{1}}$, $i=2,\dots,d+1$. Now applying
the induction hypothesis, there is some $\beta\in\mathcal{F}$ such that
$\beta>\alpha_{l}$, $n_{\beta}\in
C_{1}\cap\\{n_{\alpha}:\alpha\in\mathcal{F}\\}$ and

$V_{1}^{(l)}\cap T^{-q_{2}(n_{\beta})}V_{2}^{(l)}\cap\dots\cap
T^{-q_{d+1}(n_{\beta})}V_{d+1}^{(l)}\neq\emptyset.$

Hence there is some $x\in V_{1}^{(l)}$ such that
$T^{p_{i}(n_{\beta})-p_{1}(n_{\beta})}x\in V_{i}^{(l)}$ for $i=2,\dots,d+1.$
Clearly, there exists $y\in X$ such that $T^{p_{1}(n_{\beta})}y=x$. Since
$X=\cup_{j=0}^{l}T^{j}U$, there is some $b\in\\{0,1,\dots,l\\}$ such that
$T^{b}z=y$ for some $z\in U$. Thus for each $i=1,\dots,d+1$,

$\displaystyle T^{p_{i}(n_{\beta})+p_{i}(n_{\alpha_{b}})}z$
$\displaystyle=T^{p_{i}(n_{\beta})}T^{p_{i}(n_{\alpha_{b}})}T^{-b}T^{-p_{1}(n_{\beta})}x$
$\displaystyle=T^{p_{i}(n_{\alpha_{b}})}T^{-b}T^{p_{i}(n_{\beta})-p_{1}(n_{\beta})}x$
$\displaystyle\subseteq T^{p_{i}(n_{\alpha_{b}})}T^{-b}V^{(l)}_{i}\subseteq
V_{i}.$

Since $n_{\beta}\in C_{1},$ one has

$p_{i}(n_{\beta}+n_{\alpha_{b}})=p_{i}(n_{\beta})+p_{i}(n_{\alpha_{b}})$

for all $i=1,\dots,d+1$. Hence we have,

$z\in U\cap T^{-p_{1}(n_{\alpha})}V_{1}\cap\dots\cap
T^{-p_{d}(n_{\alpha})}V_{d}\cap T^{-p_{d+1}(n_{\alpha})}V_{d+1},$

where $\alpha=\beta\cup\alpha_{b}$ as $\beta\cap\alpha_{b}=\emptyset$ and
$n_{\alpha}\in A.$ By induction the proof is complete. ∎

Now we can prove our main result for degree $1$ integer-valued generalized
polynomials.

######  Theorem 3.4.

Let $(X,T)$ be a topologically mildly mixing minimal system, $d\in\mathbb{N}$
and $p_{1},\dots,p_{d}\in\widetilde{GP_{1}}$ with $(p_{1},\dots,p_{d})$ non-
degenerate. Then for all non-empty open subsets $U,V_{1},\dots,V_{d}$ of $X$,

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

is an IP∗-set.

###### Proof.

Let $p_{1},\dots,p_{d}\in\widetilde{GP_{1}}$. By Lemma 2.20, there exist
$h_{i}(n)\in\widetilde{SGP_{1}}$, $i=1,\dots,d$ and
$C=C(\delta,q_{1},\dots,q_{k})$ such that

$p_{i}(n)=h_{i}(n),\forall n\in C,i=1,\dots,d,$

where $\delta$ is small enough and $q_{1},\dots,q_{k}\in\widehat{SGP_{1}}$.

Set

$N=\\{n\in\mathbb{Z}:U\cap T^{-h_{1}(n)}V_{1}\cap\dots\cap
T^{-h_{d}(n)}V_{d}\neq\emptyset\\},$

by Theorem 3.4, $N$ is an IP∗-set, and thus $N\cap C$ is also an IP∗-set.
Since for any $n\in N\cap C\subseteq C$, $p_{i}(n)=h_{i}(n),i=1,\dots,d$, we
have

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

that is,

$N\cap C\subseteq\\{n\in\mathbb{Z}:U\cap T^{-p_{1}(n)}V_{1}\cap\dots\cap
T^{-p_{d}(n)}V_{d}\neq\emptyset\\},$

hence

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

is an IP∗-set. ∎

##  4\. The PET-induction and the generalized polynomial extension of van der
Waerden’s theorem

In this section, we will use the PET-induction (Polynomial Exhaustion
Technique) to prove the generalized polynomial form of van der Waerden’s
theorem. The PET-induction was introduced by Bergelson in [1]. See also [4, 5,
14] for more on the PET-induction. We firstly review some notions from [17]
(most of which is similar to that used in [1]).

######  Definition 4.1.

Let $p(n),q(n)\in\widetilde{SGP}$, we denote $p\sim q$ if $\deg(p)=\deg(q)$
and $\deg(p-q)<\deg(p)$. One can easily check that $\sim$ is an equivalence
relation. A _system_ $P$ is a finite subset of $\widetilde{SGP}$. Given a
system $P$, we define its _weight vector_
$\Phi(P)=(\omega_{1},\omega_{2},\dots)$, where $\omega_{i}$ is the number of
equivalent classes under $\sim$ of degree $i$ special integer-valued
generalized polynomials represented in $P$. For distinct weight vectors
$\Phi(P)=(\omega_{1},\omega_{2},\dots)$ and
$\Phi(P^{\prime})=(\upsilon_{1},\upsilon_{2},\dots)$, one writes
$\Phi(P)>\Phi(P^{\prime})$ if $\omega_{d}>\upsilon_{d}$, where $d$ is the
largest $j$ satisfying $\omega_{j}\neq\upsilon_{j}$, then we say that
$P^{\prime}$ _precedes_ $P$. This is a well-ordering of the set of all
possible weight vectors and the PET-induction is simply an induction scheme on
this ordering.

For example, let $P=\\{3n^{2}-\left\lceil\sqrt{5}n\right\rceil,\left\lceil\pi
n^{3}\left\lceil\sqrt{2}n\right\rceil\right\rceil+\left\lceil\frac{1}{5}n^{3}\right\rceil,n\left\lceil\pi
n\right\rceil,\left\lceil\pi
n^{3}\left\lceil\sqrt{2}n\right\rceil\right\rceil+\left\lceil\pi
n^{2}\right\rceil\\}$, then $\Phi(P)=(0,2,0,1,0,\dots)$.

Now we are ready to show the following generalized polynomial extension of van
der Waerden’s theorem, which follows the train of thought established in the
proof of the polynomial van der Waerden’s theorem in [2].

######  Theorem 4.2.

Let $(X,T)$ be a dynamical system with the metric $\rho$ and
$p_{1},\dots,p_{d}\in\widetilde{SGP}.$ Then for any $\epsilon>0$, there exists
$x\in X$ and $n\in\mathbb{N}$ such that

(4.1) $\rho(T^{p_{t}(n)}x,x)<\epsilon$

for all $t=1,\dots,d$ simultaneously. Moreover, the set

$\\{n\in\mathbb{Z}:\forall\epsilon>0,\exists x\in X\ \text{s.t.}\ \forall
t\in\\{1,\dots,d\\},\ \eqref{a11}\ \text{is satisfied}\\}$

is an IP∗-set.

###### Proof.

Without loss of generality, we may assume that $(X,T)$ is minimal.

First of all notice that Theorem 4.2 holds trivially if all $p_{t}(n)$,
$t=1,\dots,d$ are zeros. We shall prove (using PET-induction) that Theorem 4.2
is valid for any system by showing that its validity for an arbitary system
$\SS=\\{p_{1},\dots,p_{d}\\}$ follows from its validity for all the systems
preceding $\SS$.

Given $\epsilon>0$ and let $A=FS(\\{s_{i}\\}_{i\in\mathbb{N}})$ be an IP-set.
For $n\in A,$ $n=s_{i_{1}}+\dots+s_{i_{m}},$ define its support $\sigma(n)$ by
$\sigma(n)=\\{i_{1},\dots,i_{m}\\}$ and let

$A_{n}=FS(\\{s_{i}\\}_{i\in\mathbb{N}\setminus\sigma(n)});$

for $n_{1},\dots,n_{q}\in A$ define

$A_{n_{1},\dots,n_{q}}=FS(\\{s_{i}\\}_{i\in\mathbb{N}\setminus\cup^{q}_{j=1}\sigma(n_{j})}).$

It is clear from the definition for IP-set that for any $n\in A$ and $l\in
A_{n},$ we have $n+l\in A.$

By Lemma 2.24, there exists $\delta>0,$ $Q\subset\widehat{SGP_{a}}$ (for some
$a\in\mathbb{N}$) and

$C=C(\delta,Q)=\bigcap_{q(n)\in
Q}\\{n\in\mathbb{Z}:\\{q(n)\\}\in(-\delta,\delta)\\}$

such that for each $m\in C$, $m$ is good w.r.t. $p_{t}(n)\ \text{for}\ 1\leq
t\leq d.$ Let $\tilde{A}=A\cap C$, then $\tilde{A}$ is also an IP-set. We may
assume that $\tilde{A}=FS(\\{\tilde{s}_{i}\\}_{i\in\mathbb{N}})$. Moreover,
denote

$\tilde{A}_{n_{1},\dots,n_{q}}=FS(\\{\tilde{s}_{i}\\}_{i\in\mathbb{N}\setminus\cup^{q}_{j=1}\sigma(n_{j})})\subseteq\tilde{A}$

with $n_{1},\dots,n_{q}\in\tilde{A}.$ Notice that
$\tilde{A}_{n_{1},\dots,n_{q}}$ are still IP-sets for any
$n_{1},\dots,n_{q}\in\tilde{A}.$

Let $p_{1}$ be of the minimal degree in the system $\SS$, we may assume that
$\SS$ does not contain trivial polynomials, and so $\deg(p_{1})\geq 1.$
Consider the system

$\SS_{0}=\\{p_{2}-p_{1},\ p_{3}-p_{1},\dots,p_{d}-p_{1}\\}$

(if $d=1,$ $\SS_{0}$ is empty).

Notice that the elements of $\SS$ nonequivalent to $p_{1}$ do not change their
degree and the equivalence of one to another after they have been subtracted
by $p_{1}$; on the other hand, the degrees of the elements of $\SS$ which are
equivalent to $p_{1}$ do decrease after these elements have been subtracted by
$p_{1}$. Hence, the number of equivalence classes with the minimal degree in
$\SS$ decreases by $1$ when we pass from $\SS$ to $\SS_{0}$ (although some new
equivalence classes with smaller degrees can arise in $\SS_{0}$). This means
that $\Phi(\SS_{0})<\Phi(\SS)$, and by the PET-induction hypothesis, the
statement of the theorem is valid for $\SS_{0}.$

Therefore, one can choose $y_{0}\in X,$ $n_{1}\in\tilde{A}$ such that

$\rho(T^{p_{t}(n)-p_{1}(n)}y_{0},y_{0})<\frac{\epsilon}{2}\ \text{for}\ t=2\
,\dots,d.$

Denote $x_{0}=y_{0},$ $x_{1}=T^{-p_{1}(n_{1})}y_{0}$, then

$T^{p_{1}(n_{1})}x_{1}=x_{0},\
\rho(T^{p_{t}(n_{1})}x_{1},x_{0})<\frac{\epsilon}{2}\ \text{for}\
t=2,\dots,d.$

We will find a sequence of points $x_{0},x_{1},x_{2},\dots\in X$ and a
sequence of integers $n_{1},n_{2},\dots$ with $n_{1}\in\tilde{A},$
$n_{m+1}\in\tilde{A}_{n_{1},\dots,n_{m}},$ $m\in\mathbb{N}$ such that for
every $l,m,l<m,$ one has

(4.2) $\rho(T^{p_{t}(n_{m}+\dots+n_{l+1})}x_{m},x_{l})<\frac{\epsilon}{2}\
\text{for any}\ t=1,\dots,d.$

The points $x_{0},x_{1}$ and the integer $n_{1}$ have already been chosen,
suppose that $x_{m}\in X,\ n_{m}\in\tilde{A}_{n_{1},\dots,n_{m-1}}$ have been
chosen. The inequality $\eqref{ml}$ holds not only for $x_{m}$ but also for
all the points of the $\epsilon_{m}$-neighborhood of $x_{m}$ for some
$\epsilon_{m},$ $0<\epsilon_{m}<\frac{\epsilon}{2}.$ Since $(X,T)$ is assumed
to be minimal, there is $N\in\mathbb{N}$ such that
$X=\cup_{i=0}^{N}T^{-i}B(x_{m},\frac{\epsilon_{m}}{2})$, where
$B(x_{m},\frac{\epsilon_{m}}{2})=\\{x\in
X:\rho(x,x_{m})<\frac{\epsilon_{m}}{2}\\}$. Choose $\delta_{m}>0$ such that
for any $y,y^{\prime}\in X$ with $\rho(y,y^{\prime})<\delta_{m}$, one has
$\rho(T^{b}y,T^{b}y^{\prime})<\frac{\epsilon_{m}}{2}$, for all $b=0,\dots,N$.
Thus for every $y\in X$, there is some $b,$ $0\leq b\leq N$ such that the
inequality $\rho(y,y^{\prime})<\delta_{m}$ implies

(4.3) $\rho(T^{b}y^{\prime},x_{m})<\epsilon_{m}.$

Let

$\displaystyle\\{p_{t,l}(n)\\}_{1\leq t\leq d,\ 0\leq l\leq m}$
$\displaystyle=$
$\displaystyle\begin{cases}p_{t}(n+n_{m}+\dots+n_{l+1})-p_{t}(n_{m}+\dots+n_{l+1})-p_{1}(n),&\text{if}\
0\leq l\leq m-1,\\\ p_{t}(n)-p_{1}(n),&\text{if}\ l=m.\end{cases}$

(Notice that if $l=m-1,$ then $n_{m}+\dots+n_{l+1}=n_{m}.$)

Since
$n_{1}\in\tilde{A},n_{2}\in\tilde{A}_{n_{1}},\dots,n_{m}\in\tilde{A}_{n_{1},\dots,n_{m-1}},$
one gets

$\\{n_{m},n_{m}+n_{m-1},\dots,n_{m}+\dots+n_{1}\\}\subset\tilde{A}.$

By Lemma 2.25, there is a Nil${}_{h_{1}}$ Bohr0-set $C_{1}$ associated to
$\\{p_{1},\dots,p_{d}\\}$ and
$\\{n_{m},n_{m}+n_{m-1},\dots,n_{m}+\dots+n_{1}\\}$. For any $1\leq t\leq d,\
0\leq l\leq m-1$ and $n\in C_{1}$, put

$D(p_{t}(n),n_{m}+\dots+n_{l+1})=p_{t}(n+n_{m}+\dots+n_{l+1})-p_{t}(n_{m}+\dots+n_{l+1})-p_{t}(n),$

then

$D(p_{t}(n),n_{m}+\dots+n_{l+1})\in\widetilde{SGP}$

with $\deg(D(p_{t}(n),n_{m}+\dots+n_{l+1}))<\deg(p_{t})$.

Let $q_{t,l}(n)=D(p_{t}(n),n_{m}+\dots+n_{l+1})+p_{t}(n)-p_{1}(n)$ if $0\leq
l\leq m-1$ and $q_{t,m}(n)=p_{t}(n)-p_{1}(n)$. Then for all $0\leq l\leq m$,
$q_{t,l}(n)\in\widetilde{SGP}$ and

$p_{t,l}(n)=q_{t,l}(n),\forall n\in C_{1}.$

Denote

$\SS_{m}=\\{q_{t,l}:1\leq t\leq d,\ 0\leq l\leq m\\}.$

If $p_{t}(n)$ is not equivalent to $p_{1}(n)$, the elements
$q_{t,0}(n),\dots,q_{t,m}(n)\in\SS_{m}$ are equivalent to $p_{t}(n)$ and their
equivalence is preseved, that is, if $p_{t}(n)$ is equivalent to $p_{s}(n)$
then $q_{t,j}(n)$ is equivalent to $q_{s,i}(n)$ for every $j,i=0,\dots,m$. If
$p_{t}(n)$ is equivalent to $p_{1}(n)$, then the degrees of these elements
decrease: $\deg(q_{t,j})<\deg(p_{t})=\deg(p_{1}(n)),\ 0\leq j\leq m$. So the
number of equivalence classes having degrees greater than $\deg(p_{1}(n))$
does not change, whereas the number of equivalence classes of elements having
the minimal degree in $\SS_{m}$ decreases by 1 when we pass from $\SS$ to
$\SS_{m}$. Then $\SS_{m}$ precedes $\SS$ and by our PET-induction hypothesis,
the conclusion of Theorem 4.2 holds for the system $\SS_{m}.$

Hence, we can find $y_{m}\in X,$ $n_{m+1}\in\tilde{A}_{n_{1},\dots,n_{m}}\cap
C_{1}$ such that $\rho(T^{q_{t,l}(n_{m+1})}y_{m},y_{m})<\delta_{m}$ for every
$t$, $1\leq t\leq d$, and any $l$, $0\leq l\leq m.$ In light of
$p_{t,l}(n)=q_{t,l}(n),\forall n\in C_{1},$ one has
$\rho(T^{p_{t,l}(n_{m+1})}y_{m},y_{m})<\delta_{m}$, $1\leq t\leq d$, $0\leq
l\leq m.$ Choose $b\in\\{0,1,\dots,N\\}$ such that $\eqref{by T continuity}$
holds for all $y^{\prime}$ from the $\delta_{m}$-neighborhood of $y_{m}$,
denote $x_{m+1}=T^{-p_{1}(n_{m+1})}T^{b}y_{m}.$ Then, for each $t=1,\dots,d,$
we have

$\displaystyle\rho(T^{p_{t}(n_{m+1}+n_{m}+\dots+n_{l+1})}T^{-p_{t}(n_{m}+\dots+n_{l+1})}x_{m+1},x_{m})$
$\displaystyle=$ $\displaystyle\rho(T^{b}T^{p_{t,l}(n_{m+1})}y_{m},x_{m})$
$\displaystyle<$ $\displaystyle\epsilon_{m},\quad l=0,\dots,m-1,$

and

$\displaystyle\rho(T^{p_{t}(n_{m+1})}x_{m+1},x_{m})$ $\displaystyle=$
$\displaystyle\rho(T^{p_{t}(n_{m+1})}T^{-p_{1}(n_{m+1})}T^{b}y_{m},x_{m})$
$\displaystyle<$ $\displaystyle\epsilon_{m}.$

Hence by the choice of $\epsilon_{m},$ one gets

$\rho(T^{p_{t}(n_{m+1}+n_{m}+\dots+n_{l+1})}x_{m+1},x_{l})<\frac{\epsilon}{2},\
\text{for}\ l=0,\dots,m-1$

and

$\rho(T^{p_{t}(n_{m+1})}x_{m+1},x_{m})<\frac{\epsilon}{2}.$

By the above procedure, we have found the sequence of points
$x_{0},x_{1},x_{2},\dots\in X$ and the sequence of integers
$n_{1},n_{2},\dots$ with $n_{1}\in\tilde{A},$
$n_{m+1}\in\tilde{A}_{n_{1},\dots,n_{m}},$ $m\in\mathbb{N}$ satisfying the
inequality (4.2). Since $X$ is compact, there exist $l,m,l<m,$ such that
$\rho(x_{l},x_{m})<\frac{\epsilon}{2}.$ Put
$n=n_{m}+\dots+n_{l+1}\in\tilde{A}\subseteq A,$ then
$\rho(T^{p_{t}(n)}x_{m},x_{l})<\frac{\epsilon}{2},$ for each $t,$ $1\leq t\leq
d,$ and hence $\rho(T^{p_{t}(n)}x_{m},x_{m})<\epsilon.$ ∎

######  Corollary 4.3.

If $(X,T)$ is minimal and $p_{1},\dots,p_{d}\in\widetilde{SGP},$ then for any
IP-set $A$, any non-empty open set $V\subset X$, there exists $n\in A$ such
that

$V\cap T^{-p_{1}(n)}V\cap\dots\cap T^{-p_{d}(n)}V\neq\emptyset.$

##  5\. Proof of Theorem 1.1

In this section, we proceed to use the PET-induction to prove our main result
Theorem 1.1. As in Section 3, we first prove the special case (Theorem 3.4).
Likewise, before carrying out our main result we focus on the elements in
$\widetilde{SGP}$. We start with a brief overview of our idea of this proof by
the PET-induction, precisely, in order to prove the Theorem 1.1 for any system
$P$ in $\widetilde{SGP}$, we will start from $\Phi(P)=(d,0,\dots)$ (this is
true by Theorem 3.3). After that, we assume the result holds for any system
$P^{\prime}$ with $\Phi(P^{\prime})<\Phi(P)$. Then we show the result holds
for $P$, and we complete the proof.

To simplify our exposition, we introduce some notations given by Zhang and
Zhao in [17].

######  Definition 5.1.

Let $a,b\in\mathbb{R}$, $p\in\widetilde{SGP}$ and $N_{0}=1000.$ We denote
$a>>_{N}b$ if $a>b>0$ and $a>Nb$; we denote $a>>_{N_{0}}b$ by $a>>b$; we use
the symbol $a\approx_{N}b$ when $|a|>>_{N}|a-b|$ and $|b|>>_{N}|a-b|$; by
$a\approx b$ we mean $a\approx_{N_{0}}b$; $p=_{C}q$ iff $p(n)=q(n)$ for any
$n\in C$.

Recall that $A(p(n))$ is the sum of the coefficients of the maximal-degree
components of the generalized polynomial $p(n)$ (see Definition 2.21). We have
the following lemmas from [17].

######  Lemma 5.2.

[17, Lemma 4.3] Let $h(n)=\sum_{k=1}^{l}c_{k}\left\lceil
p_{k}(n)\right\rceil\in\widetilde{SGP_{d}}$, where $c_{k}\in\mathbb{Z}$,
$p_{k}(n)=\prod_{i=1}^{t}L(a_{i,1}n^{j_{i,1}},\dots,a_{i,l_{i}}n^{j_{i,l_{i}}})$
with $|a_{i,m}|>>1$, $1\leq i\leq t,1\leq m\leq l_{i}$, $|A(h(n))|>>1$ and
$\deg(h)\geq 2$. Let
$m(h)=\frac{2\sum_{k=1}^{l}|c_{k}\deg(p_{k}(n))A(p_{k}(n))|}{|\sum_{k=1}^{l}c_{k}\deg(p_{k}(n))A(p_{k}(n))|}.$
There are Nil Bohr0-sets $C,$ $C_{1}$ such that for any $m\in C,$ $|m|>m(h)$
we can find $D(h(n),m)\in\widetilde{SGP_{d-1}}$ with
$\deg(D(h(n),m))<\deg(h(n))$ such that

$D(h(n),m)=_{C_{1}}h(n+m)-h(n)-h(m),$
$A(D(h(n),m))\approx_{2N_{0}}\deg(h)mA(h(n)).$

We call $D(h(n),m)$ the derivative of $h(n)$ w.r.t. $m$.

######  Lemma 5.3.

[17, Lemma 4.5] Let $d\in\mathbb{N}$, $p_{i}\in\widetilde{SGP},i=1,\dots,d$,
with $(p_{1},\dots,p_{d})$ non-degenerate, $\deg(p_{i})\geq 2,1\leq i\leq d$
and each $p_{i}$ satisifies conditions in Lemma 5.2 and
$|A(p_{i}-p_{j})|>>1,1\leq i\neq j\leq d.$ Then there exists a Nil Bohr0-set
$C$ and a sequence $\\{r(n)\\}_{n=0}^{\infty}$ of natural numbers, such that
for any $l\in\mathbb{N}$ and $k_{0},k_{1},\dots,k_{l}\in C$ with
$|k_{0}|>r(0)$ and $|k_{i}|>|k_{i-1}|+r(|k_{i-1}|),1\leq i\leq l$, there
exists a Nil Bohr0-set $C_{1}$ and $q_{i,j}\in\widetilde{SGP}$ with

$q_{i,j}(n):=D(p_{i}(n),k_{j})+p_{i}(n)-p_{1}(n)=_{C_{1}}p_{i}(n+k_{j})-p_{i}(k_{j})-p_{1}(n)$

and

(5.1) $|A(q_{i,j}(n))|>>1,|A(q_{i,j}(n)-q_{i^{\prime},j^{\prime}}(n))|>>1$

for all
$(i,j)\neq(i^{\prime},j^{\prime})\in\\{1,\dots,d\\}\times\\{0,1,\dots,l\\}$.

######  Lemma 5.4.

Let $(X,T)$ be a dynamical system, $U,V_{1},\dots,V_{d}$ be non-empty open
subsets of $X$ and $q\in\mathbb{Z}$. We denote

$N=\\{n\in\mathbb{Z}:U\cap T^{-p_{1}(n)}V_{1}\cap\dots\cap
T^{-p_{d}(n)}V_{d}\neq\emptyset\\},$ $N_{1}=\\{n\in\mathbb{Z}:U\cap
T^{-p_{1}(qn)}V_{1}\cap\dots\cap T^{-p_{d}(qn)}V_{d}\neq\emptyset\\}.$

If $N_{1}$ is an IP∗-set, then $N$ is an IP∗-set.

###### Proof.

We notice that for any $n\in N_{1},$ $qn\in N.$ Thus $qN_{1}\subseteq N.$
Since $N_{1}$ is an IP∗-set and by Lemma 2.9, one gets $N$ is an IP∗-set. ∎

######  Remark 5.5.

Notice that for any $0\neq a\in\mathbb{R}$ and $k\in\mathbb{N}$, if we choose

$C=\\{n\in\mathbb{Z}:\\{abn^{k}\\}\in(-\frac{1}{4},\frac{1}{4}),\\{a\left\lceil
bn^{k}\right\rceil\\}\in(-\frac{1}{4},\frac{1}{4}),\\{bn^{k}\\}\in(-\frac{1}{4|a|},-\frac{1}{4|a|})\\},$

then we have $\left\lceil a\left\lceil
bn^{k}\right\rceil\right\rceil=_{C}\left\lceil abn^{k}\right\rceil$. Combining
this fact with Lemma 5.4, from now on we always assume that for any
$p_{1},\dots,p_{d}\in\widetilde{SGP}$ with $(p_{1},\dots,p_{d})$ non-
degenerate, $|A(p_{i})|>>1$ and $|A(p_{i}-p_{j})|>>1$ for any $1\leq i\neq
j\leq d$.

Before showing the proof of Theorem 1.1, we show the case of special integer-
valued generalized polynomials.

######  Theorem 5.6.

Let $(X,T)$ be a topologically mildly mixing minimal system,
$p_{1},\dots,p_{d}\in\widetilde{SGP}$ and $(p_{1},\dots,p_{d})$ be non-
degenerate, $d\in\mathbb{N}$. Then $(X,T)$ is
$\\{p_{1},\dots,p_{d}\\}_{\Delta}$-IP∗-transitive. That is, for all non-empty
open subsets $U,V_{1},\dots,V_{d}$ of $X$,

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

is an IP∗-set.

###### Proof.

Let $d\in\mathbb{N}$ and $P=\\{p_{1},\dots,p_{d}\\}$. We start from the system
whose weight vector is $(d,0,\dots)$. That is, the degree of every element of
$P$ is $1$. This case is Theorem 3.3.

Now let $P\subset\widetilde{SGP}$ be a system whose weight vector
$>(d,0,\dots)$, and we assume that for all systems $P^{\prime}$ preceding $P$,
we have $(X,T)$ is $P^{\prime}_{\Delta}$-IP∗-transitive. Now we show that
$(X,T)$ is $P_{\Delta}$-IP∗-transitive.

Step 1. $(X,T)$ is $P$-IP∗-transitive.

Since the intersection of two IP∗-sets is still an IP∗-set, it is sufficient
to show that for any $p\in P$, and for any given non-empty open subsets $U,V$
of $X$,

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

is an IP∗-set.

If $\deg(p)=1$, by Lemma 3.2, $N(p,U,V)$ is an IP∗-set.

Now we assume that $\deg(p)\geq 2$. Given an IP-set $A.$ By Lemma 2.24, there
exists $\delta>0,$ $Q\subset\widehat{SGP_{b}}$ (for some $b\in\mathbb{N}$) and

$C=C(\delta,Q)=\bigcap_{q(n)\in
Q}\\{n\in\mathbb{Z}:\\{q(n)\\}\in(-\delta,\delta)\\}$

such that for each $m\in C$, $m$ is good w.r.t. $p(n).$ Let $\tilde{C}=C\cap
A$. Since $C$ is an IP∗-set, one has $\tilde{C}$ is an IP-set. We may assume
that $\tilde{C}=\\{n_{\alpha}:\alpha\in\mathcal{F}\\}$ for some sequence
$\\{n_{i}\\}_{i=1}^{\infty}\subset\mathbb{Z}$. As $(X,T)$ is minimal, there is
some $N\in\mathbb{N}$ such that $X=\cup_{i=0}^{N}T^{i}U$. For each
$i=0,1,\dots,N$, by Corollary 4.3, there are some $y_{i}\in T^{i}V$ and
$\alpha_{i}\in\mathcal{F}$ such that $|n_{\alpha_{i}}|>m(p)$ (see Lemma 5.2
for the notation $m(p)$) and

$T^{p(n_{\alpha_{i}})}y_{i}\in T^{i}V,\ i=0,1,\dots,N.$

We may assume that $\alpha_{0}<\alpha_{1}<\dots<\alpha_{N}$ and
$n_{\alpha_{i}}\neq n_{\alpha_{j}}$ where $0\leq i\neq j\leq N.$ Let $V_{i}$
be a neighborhood of $y_{i}$ such that

$T^{p(n_{\alpha_{i}})}V_{i}\subseteq T^{i}V,\ i=0,1,\dots,N.$

By Lemma 5.2, there is a Nil Bohr0-set $C_{1}$ and
$D(p(n),n_{\alpha_{i}})\in\widetilde{SGP_{d-1}}$ with

$\deg(D(p(n),n_{\alpha_{i}}))<\deg(p(n)),i=0,\dots,N$

such that

$D(p(n),n_{\alpha_{i}})=_{C_{1}}p(n_{\alpha_{i}}+n)-p(n_{\alpha_{i}})-p(n).$

Let

$P^{\prime}=\\{D(p(n),n_{\alpha_{i}}):i=0,1,\dots,N\\}.$

Then $P^{\prime}\subset\widetilde{SGP}$. Since for any $i=0,\dots,N$,
$\deg(D(p(n),n_{\alpha_{i}}))<\deg(p(n))$, thus we have
$\Phi(P^{\prime})<\Phi(\\{p\\})$.

For any $i\neq i^{\prime}\in\\{{0,1,\dots,N}\\}$, since $n_{\alpha_{i}}\neq
n_{\alpha_{i^{\prime}}}$, by Lemma 5.2,

$A(D(p(n),n_{\alpha_{i}}))\approx_{2N_{0}}\deg(p)n_{\alpha_{i}}A(p(n)),\
|A(D(p(n),n_{\alpha_{i}}))|>>1,$
$A(D(p(n),n_{\alpha_{i^{\prime}}}))\approx_{2N_{0}}\deg(p)n_{\alpha_{i}^{\prime}}A(p(n)),\
|A(D(p(n),n_{\alpha_{i^{\prime}}}))|>>1,$
$A(D(p(n),n_{\alpha_{i}})-D(p(n),n_{\alpha_{i^{\prime}}}))\approx\deg(p)(n_{\alpha_{i}}-n_{\alpha_{i}^{\prime}})A(p(n)),$

then

$|A(D(p(n),n_{\alpha_{i}})-D(p(n),n_{\alpha_{i^{\prime}}}))|>>1.$

By the inductive hypothesis, there are some $x\in X$ and $\beta\in\mathcal{F}$
with $\beta>\alpha_{N},$ and $n_{\beta}\in
C_{1}\cap\\{n_{\alpha}:\alpha\in\mathcal{F}\\}$ such that

$T^{D(p(n_{\beta}),n_{\alpha_{i}})}x\in V_{i},\ \forall
i\in\\{0,1,\dots,N\\}.$

Then we have

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

Hence

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

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

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

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

$N(p,U,V)\cap A\neq\emptyset.$

Since $A$ is an arbitrary IP-set, which shows that $(X,T)$ is
$P$-IP∗-transitive.

Step 2. $(X,T)$ is $P_{\Delta}$-IP∗-transitive.

We will show that for all non-empty open subsets $U,V_{1},\dots,V_{d}$ of $X$,
any IP-set $A$, there exists $n\in A$ such that

$U\cap T^{-p_{1}(n)}V_{1}\cap\dots\cap T^{-p_{d}(n)}V_{d}\neq\emptyset.$

By permuting the indices, we may assume that
$\deg(p_{1})\leq\deg(p_{2})\leq\dots\leq\deg(p_{d}).$ Presume that
$\deg(p_{w})=1$ and $\deg(p_{w+1})\geq 2$, $1\leq w<d$. If for any $p\in P$,
$\deg p\geq 2$, we put $w=0$. Let $\\{r(n)\\}_{n=0}^{\infty}$ be the sequence
in Lemma 5.3 w.r.t. $(p_{w+1},\dots,p_{d})$.

Since $(X,T)$ is minimal, there is some $N\in\mathbb{N}$ such that
$X=\cup_{i=0}^{N}T^{i}U$. By Lemma 2.24, there exists $\delta>0,$
$Q\subset\widehat{SGP_{f}}$ (for some $f\in\mathbb{N}$) and

$C=C(\delta,Q)=\bigcap_{q(n)\in
Q}\\{n\in\mathbb{Z}:\\{q(n)\\}\in(-\delta,\delta)\\}$

such that for each $m\in C$, $m$ is good w.r.t. $p(n).$ Let $\tilde{C}=C\cap
A$. Since $C$ is an IP∗-set, one has $\tilde{C}$ is an IP-set. Clearly, there
exists an IP-set
$\\{n_{\alpha}:\alpha\in\mathcal{F}\\}\subseteq\tilde{C}\subseteq A.$ By Lemma
2.10, there are $\\{\alpha_{j}\\}_{j=0}^{N}\subset\mathcal{F}$ (where
$\alpha_{0}<\alpha_{1}<\dots<\alpha_{N}$) such that
$|n_{\alpha_{0}}|>r(0),|n_{\alpha_{j}}|>|n_{\alpha_{j-1}}|+r(|n_{\alpha_{j-1}}|)$
for all $j=1,\dots,N$, and $V_{t}^{(N)}\subset V_{t}$ for $t=1,\dots,d$ such
that

(5.2) $T^{p_{t}(n_{\alpha_{j}})}T^{-j}V_{t}^{(N)}\subset V_{t},\ j=0,\dots,N.$

By Lemma 2.25, there is a Nil Bohr0-set $C_{1}$ associated to
$\\{p_{1},\dots,p_{d}\\}$ and $\\{n_{\alpha_{0}},\dots,n_{\alpha_{N}}\\}$.
This means that for any $(t,j)\in\\{1,\dots,d\\}\times\\{0,\dots,N\\}$, there
exists $D(p_{t}(n),n_{\alpha_{j}})\in\widetilde{SGP}$ with
$\deg(D(p_{t}(n),n_{\alpha_{j}}))<\deg(p_{t}(n))$ such that

$D(p_{t}(n),n_{\alpha_{j}})=_{C_{1}}p_{t}(n_{\alpha_{j}}+n)-p_{t}(n_{\alpha_{j}})-p_{t}(n).$

For $1\leq t\leq d$ and $0\leq j\leq N$, let

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

and

$q_{t,j}(n)=D(p_{t}(n),n_{\alpha_{j}})+p_{t}(n)-p_{1}(n),$

then $q_{t,j}(n)\in\widetilde{SGP}$ and

$p_{t,j}(n)=_{C_{1}}q_{t,j}(n).$

For $w\geq 1,1\leq t\leq w$, since $\deg(p_{t})=1$, we have
$D(p_{t}(n),n_{\alpha_{j}})=0$, and then

$q_{t,j}(n)=p_{t}(n)-p_{1}(n),\ j=0,1,\dots,N.$

Let

$P^{\prime}=\\{p_{2}(n)-p_{1}(n),\dots,p_{w}(n)-p_{1}(n)\\}\cup\\{q_{t,j}(n):t=w+1,\dots,d,j=0,1,\dots,N\\}.$

Then $P^{\prime}\subset\widetilde{SGP}$ and when $t\in\\{w+1,\dots,d\\},$
$q_{t,0}(n),\dots,q_{t,N}(n)$ are equivalent to $p_{t}(n).$ So, the number of
equivalence classes having degrees greater than $1$ does not change, whereas
the number of equivalence classes of polynomials having the minimal degree $1$
in $P$ decreases by 1 when we pass from $P$ to $P^{\prime}$, hence
$\Phi(P^{\prime})<\Phi(P).$ Notice that when $2\leq s\neq t\leq w$,

$|A(q_{s,i}-q_{t,j})|=A(|p_{s}-p_{t}|)>>1,i,j\in\\{0,1,\dots,N\\};$

when $2\leq s\leq w$ and $w+1\leq t\leq d$, then

$q_{s,i}-q_{t,j}=p_{s}(n)-D(p_{t}(n),n_{\alpha_{j}})-p_{t}(n),\
i,j\in\\{0,1,\dots,N\\},$

thus

$|A(q_{s,i}-q_{t,j})|=|A(p_{t})|>>1,\ i,j\in\\{0,1,\dots,N\\};$

when $2\leq s,t\leq w,$ since

$|n_{\alpha_{0}}|>r(0),|n_{\alpha_{j}}|>|n_{\alpha_{j-1}}|+r(|n_{\alpha_{j-1}}|),j=1,\dots,N$

by Lemma 5.3, one has

$|A(q_{s,i})|>>1,|A(q_{s,i}-q_{t,j})|>>1,i,j\in\\{0,1,\dots,N\\}.$

For $w=0$, if $p_{t}(n)$ is not equivalent to $p_{1}(n)$, $q_{t,j}\sim p_{t},$
for $(t,j)\in\\{1,\dots,d\\}\times\\{0,1,\dots,N\\}$. If $p_{t}(n)$ is
equivalent to $p_{1}(n)$, then the degrees of these elements decrease:
$\deg(q_{t,j})<\deg(p_{t})=\deg(p_{1}),\ 0\leq j\leq N$. In this case

$P^{\prime}=\\{q_{t,j}(n):(t,j)\in\\{1,\dots,d\\}\times\\{0,1,\dots,N\\}\\}.$

We still have $P^{\prime}\subset\widetilde{SGP}$ and
$\Phi(P^{\prime})<\Phi(P)$. Since
$|n_{\alpha_{0}}|>r(0),|n_{\alpha_{j}}|>|n_{\alpha_{j-1}}|+r(|n_{\alpha_{j-1}}|)$
for all $j=1,\dots,N$, by Lemma 5.3, one has

$|A(q_{t,j})|>>1\ \text{and}\ |A(q_{t,j}-q_{s,i})|>>1$

for all $(t,j)\neq(s,i)\in\\{1,2,\dots,d\\}\times\\{0,1,\dots,N\\}.$

Both of the above conditions satisfy the inductive hypothesis and therefore
there is $x\in V_{1}^{(N)}$ and $\beta\in\mathcal{F}$ such that
$\beta>\alpha_{N}$, $n_{\beta}\in
C_{1}\cap\\{n_{\alpha}:\alpha\in\mathcal{F}\\}$ and

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

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

$\displaystyle T^{p_{t}(n_{\beta}+n_{\alpha_{b}})}z$
$\displaystyle=T^{p_{t}(n_{\beta}+n_{\alpha_{b}})}T^{-b}T^{-p_{1}(n_{\beta})}x$
$\displaystyle=T^{p_{t}(n_{\alpha_{b}})}T^{-b}T^{p_{t}(n_{\beta}+n_{\alpha_{b}})-p_{t}(n_{\alpha_{b}})-p_{1}(n_{\beta})}x$
$\displaystyle=T^{p_{t}(n_{\alpha_{b}})}T^{-b}T^{p_{t,b}(n_{\beta})}x$
$\displaystyle\in T^{p_{t}(n_{\alpha_{b}})}T^{-b}V_{t}^{(N)}\subseteq V_{t}.$

This implies that

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

with $\alpha=\beta\cup\alpha_{b}\in\mathcal{F}$ as
$\beta\cap\alpha_{b}=\emptyset$. Since $n_{\alpha}\in A$, it follows that
$(X,T)$ is $P_{\Delta}$-IP∗-transitive. Thence the proof of the theorem is
complete.

∎

We are ready to prove Theorem 1.1.

###### Proof of Theorem 1.1.

Let $p_{1},\dots,p_{d}\in\mathcal{G}$. Then by Lemma 2.20, there exist
$h_{i}(n)\in\widetilde{SGP}$, $i=1,2,\dots,d$ and
$C=C(\delta,q_{1},\dots,q_{t})$ such that $p_{i}(n)=h_{i}(n),\forall n\in
C,i=1,2,\dots,d$.

Set

$N=\\{n\in\mathbb{Z}:U\cap T^{-h_{1}(n)}V_{1}\cap\dots\cap
T^{-h_{d}(n)}V_{d}\neq\emptyset\\},$

by Theorem 5.6, $N\cap C$ is also an IP∗-set. Since for any $n\in N\cap
C\subseteq C$, $p_{i}(n)=h_{i}(n),i=1,2,\dots,d$, we have

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

that is,

$N\cap C\subseteq\\{n\in\mathbb{Z}:U\cap T^{-p_{1}(n)}V_{1}\cap\dots\cap
T^{-p_{d}(n)}V_{d}\neq\emptyset\\},$

hence

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

is an IP∗-set. ∎

Acknowledgments

Y. Cao is supported by China Postdoctoral Science Foundation (Grant no.
2023M741635). J. Zhao (corresponding author) is supported by NSF of China
(Grant no. 12301226) and NSF of Zhejiang Province (Grant no. LQ23A010006).

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# Multimodal Exponentially Modified Gaussian Oscillators
††thanks: This work is funded by the Hasler Foundation under project number
22027.

Christopher Hahne${}^{\textsuperscript{\href
https://orcid.org/0000-0003-2786-9905}}$ University of Bern, Bern, Switzerland
<EMAIL_ADDRESS>

###### Abstract

Acoustic modeling serves audio processing tasks such as de-noising, data
reconstruction, model-based testing and classification. Previous work dealt
with signal parameterization of wave envelopes either by multiple Gaussian
distributions or a single asymmetric Gaussian curve, which both fall short in
representing super-imposed echoes sufficiently well. This study presents a
three-stage Multimodal Exponentially Modified Gaussian (MEMG) model with an
optional oscillating term that regards captured echoes as a superposition of
univariate probability distributions in the temporal domain. With this,
synthetic ultrasound signals suffering from artifacts can be fully recovered,
which is backed by quantitative assessment. Real data experimentation is
carried out to demonstrate the classification capability of the acquired
features with object reflections being detected at different points in time.
The code is available at {https://github.com/hahnec/multimodal_emg}.

###### Index Terms:

Acoustic, Multimodal, Gaussian, Feature, Classification

##  I Introduction

###  I-A Background, Motivation and Objective

Acoustic signal modeling plays a key role in the technology stack of
tomorrow’s sensor systems, with important use cases in medical analysis, non-
destructive testing, robotics and vehicle navigation. In ultrasound sensing,
real world data is characterized by a heterogeneous superposition of echoes
varying in magnitude and shape as they are reflected from an environment of
complex topologies. This work presents a robust, non-linear regression
framework for acoustic feature extraction to support signal compression,
denoising, data recovery, model-based testing and - most importantly -
classification at low computational cost. An overview of the proposed
processing pipeline is outlined in Fig. 1.

Echo detection in Time-of-Flight (ToF) applications is a well-studied subject
across disciplines, adopting methods including dictionary-based approaches [1,
2], signal clusters [3] just as Recurrent Neural Networks (RNNs) [4].
Parametric sonar echo modeling is a related research subject that typically
supports simulation tasks by reconstructing echoes from probability
distributions. Pioneering work in this field was conducted by Demirli and
Saniie [5, 6], who employed an oscillating Gaussian model to estimate single
echoes by means of an iterative optimization scheme. A later study of the
authors has taken the direction of more realistic echo representation by
introducing additional parameters in an attempt to cover asymmetric envelope
shapes [7, 8]. It is worth noting that these asymmetric Gaussian regression
models find application in other domains such as laser beam modeling [9].
Despite the existence of these fundamental models, echo-based feature
extraction and classification remain under-investigated and have only recently
attracted attention from the field of computational biology [10].
Classification based on ultrasound signals has traditionally been addressed by
means of wavelet coefficients serving as acoustic features [11, 12]. Zhang et
al. conducted a survey on object classification for breast cancer detection
based on two features (sound speed and attenuation) in simulated ultrasound
B-scan images. The authors combined the Conjugate Gradient (CG) with a
Bayesian framework and Neural Network (NN) as classifiers and concluded that
the CG-NN outperforms a single feature Gauss-Newton-based classification [13].
More recently, Sterling et al. use a uni-modal Exponentially Modified Gaussian
(EMG) at several audible frequencies to obtain object features by using an MLE
optimization framework with the overall goal to make predictions on 5
different material classes [14]. While mainly focusing on the damping and
frequency parameters in a synthetic dataset of adequate size, the authors
claim to achieve a reasonable classification accuracy comparable to human
perception.

Figure 1: Overview of the proposed acoustic feature processing pipeline.

###  I-B Statement of Contribution/Methods

While these models serve as solid groundwork for model-based simulations,
their performance falls short in object recognition applications such as
presence detection scenarios. This requires realistic modeling of multiple
components as materials absorb, transmit and reflect transducer signals in
various ways, yielding different energy distributions that expose a multimodal
representation of asymmetrically shaped Gaussian curves.

For this purpose, this paper introduces a Multimodal Exponentially Modified
Gaussian (MEMG) model with an oscillating term to regard multiple echoes as
univariate probability distributions in the time domain. This acoustic feature
model is sub-divided into three-stages of Non-Linear Least-Squares (NLLS)
regression with each being solved by the Levenberg-Marquardt (LM) algorithm to
achieve fast convergence. As the proposed framework offers characteristic
features, it thereby enables echo segmentation and lays the groundwork for
classification tasks.

The experimental section complements existing work by empirically
demonstrating that a single skew term borrowed from stochastic calculus
sufficiently represents asymmetric shapes of recorded chirp reflections. This
supports the reconstruction of overlapped phase signals corrupted by noise and
qualifies MEMG to be used for object classification purposes. In particular,
feeding MEMG features to a Random Forest classifier shows that the proposed
model helps recognize echoes across frames acquired at different points in
time.

##  II Acoustic Feature Model

###  II-A Oscillating Exponentially Modified Gaussian Components

Let an EMG with oscillation term be defined as

$\displaystyle
m(\mathbf{p};\mathbf{x})=\alpha\,\mathcal{N}(\mathbf{x}|\mu,\sigma)\,\Phi(\mathbf{x}|\eta,\mu,\sigma)\,A(\mathbf{x}|\mu,f,\phi)$
(1)

where $\mathbf{x}\in\mathbb{R}^{X}$ denotes the time domain with a total
number of $X$ samples and
$\mathbf{p}~{}=~{}\left[\alpha,\mu,\sigma,\eta,f,\phi\right]^{\intercal}~{}\in~{}\mathbb{R}^{D}$
contains the normalization $\alpha$, mean $\mu$, spread $\sigma$, skew $\eta$,
frequency $f$ and phase $\phi$. With this, the Gaussian function is given by

$\displaystyle\mathcal{N}(\mathbf{x}|\mu,\sigma)=\exp\left(-\frac{\left(\mathbf{x}-\mu\right)^{2}}{2\sigma^{2}}\right)$
(2)

and the term $\Phi(\mathbf{x}|\eta,\mu,\sigma)$ covering an asymmetric shape
with

$\displaystyle\Phi(\mathbf{x}|\eta,\mu,\sigma)=\left(1+\text{erf}\left(\eta\frac{\mathbf{x}-\mu}{\sigma\sqrt{2}}\right)\right)$
(3)

by means of the error function $\text{erf}(\cdot)$. The oscillating function
is obtained by

$\displaystyle A(\mathbf{x}|\mu,f,\phi)=\cos\left(2\pi
f\left(\mathbf{x}-\mu\right)+\phi\right)$ (4)

using the cosine. To consider superpositions of multiple components, we
aggregate EMGs by

$\displaystyle
M\left(\mathbf{\hat{p}};\mathbf{x}\right)=\sum_{k=1}^{K}m\left(\mathbf{p}_{k};\mathbf{x}\right)$
(5)

with
$\mathbf{\hat{p}}=\left[\mathbf{p}_{1}^{\intercal},\mathbf{p}_{2}^{\intercal},\dots,\mathbf{p}_{K}^{\intercal}\right]^{\intercal}~{}\in~{}\mathbb{R}^{DK}$
concatenated parameters and $k=\\{1,2,\dots,K\\}$ components, accordingly.

Prior to the MEMG model estimation, an oscillating signal is routed through a
band-pass filter that eliminates spectral intensities around the dominant
frequency detected by a Fourier transform. A second pre-processing stage
counteracts the power loss over distance, which is caused by anisotropic
radiation. Each incoming frame $y(\mathbf{x})$ is treated with an exponential
fit of $\nicefrac{{a}}{{y(\mathbf{x})^{b}}}$ where $a$ and $b$ are fitted
variables.

###  II-B Three-Stage Non-Linear Least Squares Objective

To assess candidates $\mathbf{\hat{p}}$, a loss function
$L\left(\mathbf{\hat{p}}\right)$ is defined as

$\displaystyle L\left(\mathbf{\hat{p}}\right)=\left\lVert
y_{n}(\mathbf{x})-M\left(\mathbf{\hat{p}};\mathbf{x}\right)\right\rVert_{2}^{2}=\mathbf{f}$
(6)

where $\mathbf{f}\in\mathbb{R}^{X}$ is the residual vector and
$y_{n}(\mathbf{x})$ represents the band-pass filtered measurement data at
frame number $n$. The objective in (6) is minimized using LM steps given as

$\displaystyle\mathbf{\hat{p}}^{(j+1)}=\mathbf{\hat{p}}^{(j)}-\left(\mathbf{J}^{\intercal}\mathbf{J}+\delta\mathbf{D}^{\intercal}\mathbf{D}\right)^{-1}\mathbf{J}^{\intercal}\mathbf{f}$
(7)

with the Jacobian $\mathbf{J}\in\mathbb{R}^{X\times DK}$ w.r.t.
$\mathbf{\hat{p}}^{(j)}$ at each iteration $j$ and diagonal matrix
$\mathbf{D}~{}=~{}\text{diag}\left[\mathbf{J}^{\intercal}\mathbf{J}\right]$
with an adaptive damping term $\delta$. Each Jacobian is computed from
analytical derivatives. The optimal solution vector $\mathbf{\hat{p}}^{\star}$
is obtained by

$\displaystyle\mathbf{\hat{p}}^{\star}=\underset{\mathbf{\hat{p}}^{(j)}}{\operatorname{arg\,min}}\,L\left(\mathbf{\hat{p}}^{(j)}\right)$
(8)

representing the best approximation out of $\mathbf{\hat{p}}^{(j)}$
candidates.

Iterative optimizations are known to be error-prone for start values with
large numerical distance from the solution. For a robust regression, LM
iterations are split into a three-stage process with (a) Hilbert-only EMG
parameter regression of $[\alpha_{k},\mu_{k},\sigma_{k},\eta_{k}]$, which are
used for (b) oscillation parameter optimization $[f_{k},\phi_{k}]$ and (c) a
joint parameter minimization of $\mathbf{p}_{k}$ at a final step. Initial
positional estimates $\mu^{(1)}_{k}$ are obtained by using a threshold $\tau$
for the gradient of the Hilbert-transformed magnitude
$\nabla_{x}\left|\mathcal{H}\left[y_{n}(\mathbf{x})\right]\right|$ as seen in

$\displaystyle\mu^{(1)}_{k}=\\{x_{i}|x_{i}\in\mathbf{x}\wedge\nabla_{x}\left|\mathcal{H}\left[y_{n}(\mathbf{x})\right]\right|>\tau\\}$
(9)

where $\mathcal{H}\left[\cdot\right]$ denotes the Hilbert transform. Other
parameters are initialized for all $k$ with
$\alpha^{(1)}_{k}=y(\mu^{(1)}_{k})$, $\sigma^{(1)}_{k}=\mathbf{1}$,
$\eta^{(1)}_{k}=\mathbf{0}$, $f^{(1)}_{k}=f_{e}$ and
$\phi^{(1)}_{k}=\mathbf{0}$ where $f_{e}$ is the operating frequency given in
$\mathrm{kHz}$ to avoid a large condition number in the Jacobian. Phase
estimates are constrained to be $\phi^{(j)}_{k}\in\left(-\pi,\pi\right]$.

###  II-C Feature Processing

Given the extracted echo feature dimensions, one may infer quantitative
information about their estimation reliability.

####  II-C1 Confidence Measures

For blind MEMG assessment, we define the confidence $C_{n}$ per A-scan by

$C_{n}=\left(\left\lVert\frac{M(\mathbf{\hat{p}}^{\star};\mathbf{x})}{\max\left({y_{n}(\mathbf{x})}\right)}-\frac{y_{n}(\mathbf{x})}{\max\left({y_{n}(\mathbf{x})}\right)}\right\rVert_{2}\right)^{-1}\,,\quad\forall
n$ (10)

with an inverted $\ell_{2}$ norm $(\lVert\cdot\rVert_{2})^{-1}$ so that large
values indicate higher certainty. Similarly, this can be expressed as a per-
component confidence $c_{k}$ which writes

$c^{(n)}_{k}=\left(\left\lVert\hat{M}(\mathbf{p}^{\star}_{k},\mathbf{x})-\hat{y}_{n}(\mathbf{p}^{\star}_{k},\mathbf{x})\right\rVert_{2}\right)^{-1}\,,\quad\forall(k,n)$
(11)

where $\hat{M}(\cdot)$ and $\hat{y}_{n}(\cdot)$ denote the fitted and raw
signals in the range of the detected position $\mu^{\star}_{k}$ at
$5\sigma^{\star}_{k}$, respectively.

####  II-C2 Standardization

Let a solution parameter vector be reshaped to a matrix
$\mathbf{\hat{p}}^{\star}\in\mathbb{R}^{DK}\rightarrow\mathbf{\hat{P}}^{(n)}_{k}\in\mathbb{R}^{D\times
K}$ with $K$ components and $D$ features that belongs to a set
$\\{\mathbf{\hat{P}}^{(1)}_{k},\mathbf{\hat{P}}^{(2)}_{k},\dots,\mathbf{\hat{P}}^{(N)}_{k}\\}$
with $N$ as the total number of A-scan frames. After removal of components
suffering from feature outliers, each $\mathbf{\hat{P}}^{(k)}_{n}$ is
standardized according to

$\mathbf{P}^{(n)}_{k}=\frac{\mathbf{\hat{P}}^{(n)}_{k}-\boldsymbol{\mu}_{s}}{\boldsymbol{\sigma}_{s}}\,,\quad\forall(k,n)$
(12)

with $\boldsymbol{\mu}_{s}\in\mathbb{R}^{D\times 1}$ and
$\boldsymbol{\sigma}_{s}\in\mathbb{R}^{D\times 1}$ for valid feature
evaluation.

####  II-C3 Object Detection Classifier

The classification goal is to distinguish between pronounced echoes and
reverberation clutter based on the acquired feature space. More specifically,
the object detection task is formulated as an identification of unimodal
components across frames whereas the classifier is unaware of the acquisition
times, i.e. to which frame an echo belongs to. This implies that positional
features $\mu_{k}^{\star}$ are left out in the classification to enable
detection of echoes at different points in time independent of the ToF.

Using a set of standardized echo features $\mathbf{\hat{P}}^{(n)}_{k}$ for
training, corresponding components are determined in test frames via the
Random Forest (RF) classifier as an ensemble learning method. The estimation
parameters, e.g. number of decision trees, are selected based on stratified
cross-validation and can be found in the experimental section III-B.

##  III Experimental Work

Validation of the presented framework is conducted by simulation and
experimentation. First, the model’s reconstruction capability is tested by
means of a simulated oscillating MEMG signal suffering from noise with
quantitative analysis from the Peak-Signal-to-Noise-Ratio (PSNR). Second, real
ultrasonic sensor data is used to scrutinize the relevance of the skew
parameter $\eta$ introduced in (1). Finally, a set of features extracted from
several real data frames is fed to proposed feature metrics to demonstrate
object detection.

###  III-A Denoising from Simulation

For denoising evaluation, a simulated Ground-Truth (GT) signal
$g(\mathbf{x})=\sum_{k=1}^{K=4}m\left(\mathbf{p}_{k};\mathbf{x}\right)$ with
8-bit signed integer quantization at a sampling rate
$f_{s}=300~{}$\mathrm{kHz}$$ is used. Quantitative validation is carried out
using a PSNR given by

$\text{PSNR}(s(\mathbf{x}))=20\log_{10}\left((2^{8}-1)/\lVert
g(\mathbf{x})-s(\mathbf{x})\rVert_{2}\right)$ (13)

where $s(\mathbf{x})$ is a measured signal. Equation (2) is used for
$g(\mathbf{x})+\mathcal{N}(\mathbf{x}|0,10)$ to mimic noisy measurements.

Figure 2 depicts the denoising result obtained by gradient sample separation
of 20 and $\tau=0.1$ with a quantitative
$\text{PSNR}(y(\mathbf{x}))=65.06~{}$\mathrm{dB}$$ for the raw data and
$\text{PSNR}(M\left(\mathbf{\hat{p}};\mathbf{x}\right))=104.91~{}$\mathrm{dB}$$
for the oscillating MEMGs achieving a PSNR gain of $\approx
40~{}$\mathrm{dB}$$. It is believed that the presented model will perform
similarly well when recovering sound waves where a portion of samples is
corrupted or went missing.

Figure 2: Denoising simulated chirp echoes in $y_{n}(\mathbf{x})$ with a
$39~{}$\mathrm{dB}$$ PSNR gain. The lower close-up indicates that recovery
with model $M(\mathbf{\hat{p}}^{\star};\mathbf{x})$ is even possible when
echoes vary in $f$ and $\phi$ for the ground-truth signal $g(\mathbf{x})$.

###  III-B Model Performance from Real Data

For experimental analysis, real sensor data is acquired from a single airborne
transducer at $f_{e}=175~{}$\mathrm{kHz}$$ operating frequency, which collects
A-scan frames of $L=126$ samples every 100 $\mathrm{ms}$. A convex-shaped
piece of metal with 4-by-7 $\mathrm{cm}$ size served as a target for a total
number of $N=21$ frames. The heuristic conventions made in the evaluation are
explained hereafter. Samples within the blind zone are omitted. The gradient
is computed by sample distance 1 and $\tau=100$. Frequency and phase are
disregarded as commonly done in NDT. LM iterations are limited to 200 each.
Feature outliers are identified and rejected by $\mu^{\star}_{k}<0$ and
$\sigma^{\star}_{k}<0$ while all remaining components undergo further
assessment.

One objective of the experimental validation is to scrutinize the importance
of the skew parameter $\eta$. Exemplary frame confidences $C_{n}$ and
$c^{(n)}_{k}$ from real acquisitions are provided in Table I. The result
suggests that skew values $\eta\neq 0$ help create a more realistic estimation
when compared to previous work [5, 6]. Moreover, this outcome backs the study
by Sterling et al. [14] who explored ring-down shapes of audible signals to
distinguish between object materials. Figure 3 depicts frames from the test
set for qualitative inspection.

TABLE I: Confidences averaged over $N=21$ frames.

| Multimodal Gaussian [6] |  MEMG [Ours]
---|---|---
| $\frac{1}{N}\sum_{n=1}^{N}C_{n}$
---
1.606 | 1.804
| $\frac{1}{NK}\sum_{n=1}^{N}\sum_{k=1}^{K}c^{(n)}_{k}$
---
27.090 | 84.059

Classification of MEMG features is validated through the detection of a
distinct reflector across frames. Here, $\mu_{k}^{\star}$ is left out to test
how well features are capable of recognizing objects regardless of their
distance. The power loss from anistropic radiation is compensated with
$a=140.18$ and $b=1.16$. Frames are split into train and test sets by
fractions of 0.7 and 0.3, respectively. Only 10 RF trees with maximum depth of
6, minimum of 1 sample per leaf and 2 per split are used.

Figure 3: Object tracking in real data frames of the test set showing the
Ground-Truth (GT) and $\mathbf{p}^{\star}_{s}$ detected by a MEMG-based Random
Forest classifier.

Figure 3 shows qualitative results where the GT from the transducer and
detected echo components $\mathbf{p}^{\star}_{s}$ are highlighted. Prediction
is assessed by the F-score, which yields a $F_{1}=1.0$ when feeding
$\alpha^{\star}_{k}$, $\sigma^{\star}_{k}$, $\eta^{\star}_{k}$, $c_{k}$. The
impact of $\alpha$ is further analyzed by entirely excluding it from the RF
classifier still giving $F_{1}=1.0$. A confusion matrix and feature importance
diagram are shown in Fig. 4 for intuitive interpretability of the class
prediction.

Figure 4: Classification of objects from Fig. 3 with feature importance (left)
and confusion matrix (right) in absence of any $\alpha_{k}$ and $\mu_{k}$.

##  IV Conclusions

This study shows that super-imposed ultrasound echoes are robustly represented
by an accumulation of oscillating EMGs. Quantitative assessments confirm this
hypothesis in a simulation analysis by reconstructing a signal with a
$40~{}$\mathrm{dB}$$ PSNR gain. Real data experimentation is carried out to
demonstrate the relevance of Gaussian skew. The echo classification capability
is supported by an F1-score of 1.0 regardless of the arrival time. This
suggests that the presented framework is a viable acoustic feature extractor
in ToF applications.

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