#  Möbius Disjointness for product flows of rigid dynamical systems and affine
linear flows

Fei Wei  Yau Mathematical Sciences Center, Tsinghua University, Beijing
100084, China<EMAIL_ADDRESS>
###### Abstract.

We obtain that Sarnak’s Möbius Disjointness Conjecture holds for product flows
between affine linear flows on compact abelian groups of zero topological
entropy and a class of rigid dynamical systems. To prove this, we show an
estimate for the average value of the product of the Möbius function and any
polynomial phase over short intervals and arithmetic progressions
simultaneously. In addition, we prove that the logarithmically averaged Möbius
Disjointness Conjecture holds for the product flow between any affine linear
flow on a compact abelian group of zero entropy and any rigid dynamical
system. As an application, we show that the logarithmically averaged Möbius
Disjointness Conjecture holds for every Lipschitz continuous skew product
dynamical system on $\mathbb{T}^{2}$ over a rotation of the circle.
MSC2020. 37A44, 11N37
Keywords. Möbius Disjointness Conjecture; rigid dynamical system; affine
linear flow

##  1\. Introduction

Let $\mu(n)$ be the Möbius function, that is, $\mu(n)$ is $0$ when $n$ is not
square-free (i.e., divisible by a nontrivial square), and $(-1)^{r}$ when $n$
is the product of $r$ distinct primes. The Möbius function plays an important
role in number theory. For example, the Prime Number Theorem is known to be
equivalent to $\sum_{n\leq N}\mu(n)=o(N)$. The Riemann hypothesis holds if and
only if $\sum_{n\leq N}\mu(n)=o(N^{\frac{1}{2}+\epsilon})$ for every
$\epsilon>0$. It is widely believed that certain randomness exists in the
values of the Möbius function, and this randomness predicts significant
cancellations in the summation of $\mu(n)\xi(n)$ for any “reasonable” sequence
$\xi(n)$. This rather vague principle is known as an instance of the “Möbius
randomness principle” (see e.g., [23, Section 13.1]). In [36], Sarnak made
this principle precise by identifying the notion of a “reasonable” sequence as
an arithmetic function realized in a topological dynamical system111Let
$(X,T)$ be a topological dynamical system (or a flow), that is $X$ is a
compact metric space and $T:X\rightarrow X$ a continuous map. For $f\in C(X)$
and $x_{0}\in X$, we say $n\mapsto f(T^{n}x_{0})$ _an arithmetic function
realized in $(X,T)$_. with zero (topological) entropy. Precisely,

######  Conjecture 1.1 (Möbius Disjointness Conjecture).

Let $X$ be a compact Hausdorff space and $T$ a continuous map on $X$ with zero
topological entropy, then

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^{N}\mu(n)f(T^{n}x_{0})=0$
(1.1)

for any $x_{0}\in X$ and $f\in C(X)$.

This conjecture has been verified for a variety of dynamical systems. See [2,
3, 7, 9, 12, 18, 19, 20, 21, 25, 29, 30, 34, 38, 40, 41, 42], to list a few.
One such system closely related to this paper is the rigid (measure-
preserving) dynamical system. The notion of (measure theoretical) rigidity was
introduced by Furstenberg and Weiss in [13]. A measure-preserving dynamical
system222Meaning that $(X,\mathcal{B},\nu)$ is a probability space and $T$ is
a measurable, measure-preserving transformation on $X$.
$(X,\mathcal{B},\nu,T)$ is called _rigid_ if there is a sequence
$\\{n_{j}\\}_{j=1}^{\infty}$ of positive integers such that for any $f(x)\in
L^{2}(X,\nu)$,

$\lim_{j\rightarrow\infty}\|f\circ T^{n_{j}}-f\|_{L^{2}(\nu)}^{2}=0.$

Rigid dynamical systems contain dynamical systems with discrete spectrum (see
e.g., [35]), almost all interval exchange transformations [4, Corollary 3] and
a large class of skew products on the torus over a rotation of the circle
[28].

The main motivation of this research on product flows is as follows. The
Möbius disjointness of product flows between zero entropy dynamical systems
and rigid dynamical systems (which is implied by Conjecture 1.1) is of great
interest in number theory. It is closely related to the distribution of the
average value of the product of the Möbius function and any deterministic
sequence (a sequence realized in zero entropy dynamical systems) over short
arithmetic progressions. The former implies that for any fixed modulo and for
almost all short arithmetic progressions, the average value of the product
between the Möbius function and any deterministic sequence is small. Such an
arithmetic property is important and has been extensively studied in number
theory, see e.g., [31] for the Möbius function, [11, 17] for the product of
the Möbius function and nilsequence. Based on this indication, to solve the
Möbius disjointness of product flows of a zero entropy dynamical system and
rigid dynamical systems, we need to explore more properties of the
distribution, such as the dependence on the modulo. In this paper, we study
the case where the zero entropy dynamical system is an affine linear flows on
compact abelian groups of zero topological entropy, and investigate properties
of the distribution in this case.

An affine linear transformation of a compact abelian group $X$ means $x\mapsto
Ax+b$, where $A$ is an automorphism of $X$ and $b\in X$. A known example of
arithmetic functions realized from affine transformations on abelian groups is
$e^{2\pi\mathfrak{i}P(n)}$ for any $P(x)\in\mathbb{R}[x]$. Throughout the
paper, for simplicity, we use $(X,\nu,T)$ to denote a measure-preserving
dynamical system. For a topological dynamical system $(X,T)$ and $x_{0}\in X$,
we use $\mathcal{M}(x_{0};X,T)$ to denote the weak* closure of
$\\{\frac{1}{N}\sum_{n=0}^{N-1}\delta_{T^{n}x_{0}}:N=1,2,\ldots\\}$ in the
space of Borel probability measures on $X$. It is known that
$\mathcal{M}(x_{0};X,T)$ is non-empty and for any
$\nu\in\mathcal{M}(x_{0};X,T)$, $\nu$ is a $T$-invariant Borel probability
measure on $X$ (see e.g., [39, Theorem 6.9]). Our first main result is the
following theorem.

######  Theorem 1.2.

Let $(X_{1},T_{1})$ be an affine linear flow on a compact abelian group of
zero topological entropy. Let $(X_{2},T_{2})$ be a flow such that for any
$x_{0}\in X_{2}$ and any $\nu\in\mathcal{M}(x_{0};X_{2},T_{2})$,
$(X_{2},\nu,T_{2})$ is rigid with the property that there are sequences
$\\{h_{j}\\}_{j=1}^{\infty}$ and $\\{s_{j}\\}_{j=1}^{\infty}$ (may depend on
$\nu$) of positive integers satisfying

$\lim_{j\rightarrow\infty}\frac{\log\log h_{j}}{\log
h_{j}}\frac{s_{j}}{\varphi(s_{j})}=0$ (1.2)

and for any $g(x)\in L^{2}(X_{2},\nu)$,

$\lim_{j\rightarrow\infty}\frac{1}{h_{j}}\sum_{l=1}^{h_{j}}\|g\circ
T_{2}^{ls_{j}}-g\|_{L^{2}(\nu)}^{2}=0,$ (1.3)

where $\varphi(s)$ is the Euler totient function. Then the Möbius Disjointness
Conjecture holds for the product flow $(X_{1}\times X_{2},T_{1}\times T_{2})$.

It is easy to see from condition (1.3) that there is a sequence
$\\{n_{j}\\}_{j=1}^{\infty}$ of positive integers such that for any $g(x)\in
L^{2}(X_{2},\nu)$, $\lim_{j\rightarrow\infty}\|g\circ
T_{2}^{n_{j}}-g\|_{L^{2}(\nu)}^{2}=0$. Conditions (1.2) and (1.3) are
restrictions on the speed of rigidity for $(X_{2},\nu,T_{2})$. There are many
rigid dynamical systems satisfying these conditions, such as measure-
preserving dynamical systems with discrete spectrum [41, Proposition 5.4], and
measure-preserving dynamical systems with BPV rigidity and PR rigidity333It
was explained in [41, Remark 8.1] that both BPV rigidity and PR rigidity are
included in the scenario described by conditions (1.2), (1.3). Also it was
explained in [41, Remark 8.4] that there is a measure-preserving dynamical
system with discrete spectrum, but not satisfying BPV rigidity and PR
rigidity., which was defined and studied in [25]. In Lemma 4.2, we shall prove
a more general version of Theorem 1.2.

Our second main result is about the estimate on the average value of the
product of the Möbius function and any polynomial phase over short intervals
and arithmetic progressions simultaneously. It is the main ingredient of
proving Theorem 1.2. Below, we adopt the standard notation
$e(\alpha):=e^{2\pi\mathfrak{i}\alpha}$ for $\alpha\in\mathbb{R}$.

######  Theorem 1.3.

Let $s\geq 1$ and $h\geq 3$ be integers. Let $d\geq 0$ be an integer. Suppose
$P(x)\in\mathbb{R}[x]$ is of degree $d$. Then we have

$\limsup_{N\rightarrow\infty}\frac{1}{Ns}\sum_{a=1}^{s}\sum_{n=1}^{N}\left|\frac{1}{h}\sum_{\begin{subarray}{c}m=n+1\\\
m\equiv
a(mod~{}s)\end{subarray}}^{n+hs}\mu(m)e(P(m))\right|\ll\frac{s}{\varphi(s)}\frac{\log\log
h}{\log h},$ (1.4)

where the implied constant at most depends on $d$.

We list some results related to formula (1.4). For $s=1$, (1.4) becomes

$\limsup_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^{N}\left|\frac{1}{h}\sum_{m=n+1}^{n+h}\mu(m)e(P(m))\right|\ll\frac{\log\log
h}{\log h}.$ (1.5)

As a consequence of the results of Matomäki-Radziwiłł [31] and Matomäki-
Radziwiłł-Tao [33], the above estimate holds for $P(x)=e(\alpha x)$,
$\alpha\in\mathbb{R}$. In [17], He-Wang further extended the result to any
$P(x)\in\mathbb{R}[x]$, where the implied constant at most depends on the
degree of $P(x)$. Moreover, they proved that $e(P(n))$ can be replaced by any
nilsequence $n\mapsto F(g^{n}x_{0})$. Here $x_{0}\in G/\Gamma$ (a
nilmanifold), $g\in G$ and $F:G/\Gamma\rightarrow\mathbb{C}$ is a Lipschitz
function. See also [8], [10] for earlier partial results in this direction.

For general $s$, Kanigowski, Lemańczyk and Radziwiłł [25] showed that when
$P(x)$ is a constant (i.e., $d=0$), then for any $\varepsilon>0$ and $h,s$
large enough (at most depending on $\epsilon$), the left-hand side of formula
(1.4)$\leq\epsilon$ whenever $\sum_{p|s}1/p\leq(1-\epsilon)\sum_{p\leq h}1/p$.
In Theorem 1.3, we showed that a similar estimate holds for all polynomials
$P(x)$. Note that $\frac{s}{\varphi(s)}=\prod_{p|s}(1-1/p)^{-1}$ and the
estimate $\sum_{p\leq h}1/p=\log\log h+O(1)$. So the above mentioned result in
[25] is contained in Theorem 1.3. Moreover, we study a general version of
Theorem 1.3 replacing $\mu$ by any bounded multiplicative function. This will
be given in Theorem 3.1.

We below explain the basic idea of proving Theorem 1.3. The proof is inspired
by Green-Tao’s work on the average of the Möbius function and nilsequences
over long intervals [15, 16], and He-Wang’s work on the average of the Möbius
function and nilsequences over short intervals [17]. The main technique they
used is different versions of quantitative factorization of finite polynomials
in nilmanifolds. For our situation, we should estimate the average of the
Möbius function and polynomial phases over short arithmetic progressions.
Firstly, we give an appropriate quantitative factorization of finite
polynomials in abelian nilmanifolds. Secondly, we use the nilpotent “major and
minor arcs” method, introduced in [16]. In the major arc part, we use the
result [26, Theorem 1.6] on sums of multiplicative functions over almost all
short arithmetic progressions. In the minor arc case we use a variant of the
arguments of He-Wang [17, Section 6] to obtain the cancellation we need.

It is still open whether the Möbius Disjointness Conjecture holds for all
rigid dynamical systems or the product flows between them and affine linear
flows on compact abelian groups of zero topological entropy. More precisely,

######  Problem 1.

Let $(X_{1},T_{1})$ be an affine linear flow on a compact abelian group of
zero topological entropy. Let $(X_{2},T_{2})$ be a flow such that for any
$\nu$, a $T_{2}$-invariant Borel probability measure on $X_{2}$,
$(X_{2},\nu,T_{2})$ is rigid. Let $T=T_{1}\times T_{2}$. Is it true that for
any $F\in C(X_{1}\times X_{2})$ and $x\in X_{1}\times X_{2}$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^{N}\mu(n)F(T^{n}x)=0.$ (1.6)

Problem 1 is closely related to the coefficient of $\frac{\log\log h}{\log h}$
in the right-hand side of (1.4). We explain the connection as follows. Usually
the estimate of the left-hand side of (1.4) equals $o_{s}(h)$. We hope
$o_{s}(h)=o(h)$, that is, the little “o” term is independent of $s\geq 1$.
This is related to the Chowla conjecture [5] (Remark 5.2) and gives a
sufficient condition to Problem 1. So we need to make explicitly the
dependence of $o_{s}$ on $s$. From Theorem 1.3, we know that there is a
function $\phi$ such that $o_{s}(h)=\phi(s)\frac{\log\log h}{\log h}$. So if
we restrict the decay rate of the rigidity

$\|g\circ T_{2}^{s_{j}}-g\|_{L^{2}(\nu)}^{2}$

of $(X_{2},\nu,T_{2})$ to
$O\Big{(}\exp\big{(}-(\phi(s_{j}))^{A}\big{)}\Big{)}$ for a sequence
$\\{s_{j}\\}_{j=1}^{\infty}$ with
$\lim_{j\rightarrow\infty}\phi(s_{j})=\infty$ and a given $A>1$, then we have
the Möbius disjointness for $(X_{1}\times X_{2},T_{1}\times T_{2})$.

We have a positive answer to Problem 1 under the logarithmic average, namely
we restrict formula (1.6) to logarithmic average $\frac{1}{\log
N}\sum_{n=1}^{N}$ rather than the standard Cesàro average
$\frac{1}{N}\sum_{n=1}^{N}$.

######  Theorem 1.4.

Make the same assumptions as Problem 1. Then the logarithmically averaged
Möbius Disjointness Conjecture holds for the product flow $(X_{1}\times
X_{2},T_{1}\times T_{2})$. That is, for any $F\in C(X_{1}\times X_{2})$ and
$x\in X_{1}\times X_{2}$,

$\lim_{N\rightarrow\infty}\frac{1}{\log
N}\sum_{n=1}^{N}\frac{\mu(n)F(T^{n}x)}{n}=0.$ (1.7)

where $T=T_{1}\times T_{2}$.

The major ingredients of the proof of Theorem 1.4 are the method used to prove
Theorem 1.2, Tao’s result [37], and Frantzikinakis-Host’s result [12] on
logarithmic average of the self-correlations of the Möbius function twisted by
polynomial phases (see equation (5.10)). We also refer to [22] for other
results on logarithmically averaged Möbius Disjointness Conjecture.

It was considered in [6, 20, 25, 27, 30, 40] that the Möbius disjointness for
the skew product dynamical systems $(\mathbb{T}^{2},T_{\alpha,\psi})$, where
$\mathbb{T}^{1}=\mathbb{R}/\mathbb{Z}$,
$\mathbb{T}^{2}=\mathbb{T}^{1}\times\mathbb{T}^{1}$, $\alpha\in[0,1)$ and

$T_{\alpha,\psi}(x,y)=(x+\alpha,y+\psi(x))$ (1.8)

with $\psi$ a continuous map from $\mathbb{T}^{1}$ to itself. So far, it is
known that the Möbius Disjointness Conjecture holds for
$(\mathbb{T}^{2},T_{\alpha,\psi})$ with $\psi$ of class $C^{1+\epsilon}$. That
is for any $\alpha\in[0,1)$ and $\psi$ of class $C^{1+\epsilon}$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^{N}\mu(n)f(T^{n}_{\alpha,\psi}(x_{0},y_{0}))=0,~{}~{}~{}\forall(x_{0},y_{0})\in\mathbb{T}^{2},~{}~{}~{}\forall
f\in C(\mathbb{T}^{2}).$

It is not known whether the Möbius disjointness of
$(\mathbb{T}^{2},T_{\alpha,\psi})$ holds for any $\psi$ of Lipschitz
continuous. As a corollary of Theorem 1.4, we obtain that the logarithmically
averaged Möbius Disjointness Conjecture holds for this class of skew products.

######  Corollary 1.5.

Suppose that $\alpha\in[0,1)$, $\psi:\mathbb{T}^{1}\rightarrow\mathbb{T}^{1}$
is Lipschitz continuous and $T_{\alpha,\psi}$ is in (1.8). Then the
logarithmically averaged Möbius Disjointness Conjecture holds for product
flows between $(\mathbb{T}^{2},T_{\alpha,\psi})$ and affine linear flows on
compact abelian groups of zero topological entropy. In particular, for any
$P(x)\in\mathbb{R}[x]$, $(x_{0},y_{0})\in\mathbb{T}^{2}$ and $f\in
C(\mathbb{T}^{2})$,

$\lim_{N\rightarrow\infty}\frac{1}{\log
N}\sum_{n=1}^{N}\frac{\mu(n)e(P(n))f(T^{n}_{\alpha,\psi}(x_{0},y_{0}))}{n}=0.$
(1.9)

Notation. We call $e(f(n))$ a polynomial phase when $f(x)\in\mathbb{R}[x]$ is
a polynomial. We use $1_{S}$ to denote the indicator of a predicate $S$, that
is $1_{S}=1$ when $S$ is true and $1_{S}=0$ when $S$ is false. We also denote
$1_{A}(n)=1_{n\in A}$ for any subset $A$ of $\mathbb{N}$. For any finite set
$C$, $|C|$ denotes the cardinality of $C$.

We use $\\{x\\}$ and $\lfloor x\rfloor$ to denote the fractional part and the
integer part of a real number $x$, respectively. We use
$\|x\|_{\mathbb{R}/\mathbb{Z}}$ to denote the distance between $x$ and the set
$\mathbb{Z}$, i.e., $\|x\|_{\mathbb{R}/\mathbb{Z}}=\min(\\{x\\},1-\\{x\\})$.
For ease of notation, we sometimes drop the subscript and write simply
$\|x\|$. For $y\in\mathbb{R}^{m}$, $|y|$ is the $l^{\infty}$-norm of $y$. For
integer $N\geq 1$, we use $[N]$ to denote the set $\\{1,\ldots,N\\}$.

For two arithmetic functions $f(n)$ and $g(n)$, $f=o(g)$ means
$\lim_{n\rightarrow\infty}f(n)/g(n)=0$; Let $g(n)$ be a positive functions of
$n$, $f\ll g$ means that there is an absolute constant $c$ such that $|f|\leq
cg$; $f=h+O(g)$ means $f-h\ll g$.

Acknowledgments. I would like to thank Kaisa Matomäki for valuable discussions
about the coefficient $(\log N)^{o(1)}$ in Theorem 3.1. I am very grateful for
JinXin Xue for helpful suggestions on the manuscript. This paper is supported
by the fellowship of China Postdoctoral Science Foundation 2020M670273.

##  2\. A quantitative factorization theorem

In this section, we shall deduce a quantitative factorization for certain
polynomials in two parameters. It is a variation of [15, Theorems 1.19 and
10.2] or [17, Theorem 3.6] that is adapted to our situation. The difference of
these three versions lies in the intervals the parameters belong to. In [15],
all parameters are in long intervals. In [17], for polynomials $p(n,h)$, the
first parameter $n$ varies in a long interval $[1,N]$, while $h$ varies in a
short interval $[1,H]$ in the sense that $H$ can grow arbitrarily slower than
$N$. In our case, $n$ is also allowed to vary in a long interval, while $h$
lies in a short arithmetic progression $\\{h\in[1,Hs]:n+h\equiv
a(mod~{}s)\\}$. Here, $H,s$ are arbitrary positive integers, which can grow
arbitrarily slower than $N$. Since the parameter $s$ can be much larger than
$H$, it is not enough for our purpose if we just apply [17] to $h=ls+a$. Our
factorization theorem is as follows.

######  Theorem 2.1.

Let $m\geq 1$, $d\geq 0$ be integers. Let
$f:\mathbb{Z}\rightarrow\mathbb{R}^{m}$ be a polynomial of degree $d$ and
$g(n,h)=f(n+h)$. Let $R>2$. Let $s\geq 1$ be an integer. Then for any $B\geq
1,N,H\in\mathbb{N}$ such that $N>(Hs)^{O(1)}$ and $H>R^{O(1)}$, there is an
integer $W\in[R,R^{O_{d}(B^{m})}]$, a connected subgroup $G^{\prime}$ of
$\mathbb{R}^{m}$, a set $\mathcal{N}\subseteq[N]$ with
$|\mathcal{N}|\geq(1-W^{-B/2})N$, and a polynomial decomposition

$g(n,h)=\mathcal{E}(n,h)+g^{\prime}(n,h)+\gamma(n,h)$

with $\mathcal{E},g^{\prime},\gamma:\mathbb{Z}^{2}\rightarrow\mathbb{R}^{m}$
satisfying

(i). $|\mathcal{E}(n,h+1)-\mathcal{E}(n,h)|\ll_{d}\frac{1}{sH/W}$ for
$h\in[Hs]$.

(ii). $g^{\prime}(n,h)$ takes values in $G^{\prime}$, and for any
$n\in\mathcal{N}$, there are at least $(1-W^{-B/2})s$ integers of $a\in[s]$
such that $\\{g^{\prime}(n,h)(mod~{}\Gamma^{\prime})\\}_{h\in\mathcal{A}_{a}}$
is totally $W^{-B}$-equidistributed in $G^{\prime}/\Gamma^{\prime}$, where
$\mathcal{A}_{a}=\\{1\leq h\leq Hs:h\equiv a(mod~{}s)\\}$ and
$\Gamma^{\prime}=G^{\prime}\cap\mathbb{Z}^{m}$.

(iii). There is an integer $q$ with $1\leq q\leq W$ such that
$\\{\gamma(n,h)(mod~{}\mathbb{Z}^{m})\\}_{(n,h)\in\mathbb{Z}^{2}}$ is
$qs$-periodic both in $n$ and $h$.

Theorem 2.1 is fundamental in the proof of Theorem 3.1. Before proving it, we
need some preparations and show some intermediate results. We first recall
some definitions.

######  Definition 2.2.

$($[15, Definition 2.7]$)$ Let $d\geq 0$ and
$f:\mathbb{Z}\rightarrow\mathbb{R}/\mathbb{Z}$ be a polynomial of degree $d$.
Write $f(n)=\sum_{i=0}^{d}\alpha_{i}\binom{n}{i}$ with each
$\alpha_{i}\in\mathbb{R}/\mathbb{Z}$. For any integer $N\geq 1$, the
smoothness norm of $f$ is defined by

$\|f\|_{C^{\infty}[N]}:=\max_{1\leq i\leq
d}N^{i}\|\alpha_{i}\|_{\mathbb{R}/\mathbb{Z}}.$

######  Definition 2.3.

$($[15, Definition 1.2]$)$ Let $G/\Gamma$ be a nilmanifold endowed with the
unique normalized Haar measure. Let $\delta>0$ and $\mathcal{A}$ a finite
arithmetic progression in $\mathbb{Z}$. A finite sequence
$\\{x_{n}\\}_{n\in\mathcal{A}}$ in $G/\Gamma$ is said to be
$\delta$-equidistributed in $G/\Gamma$ if

$\Big{|}\mathbb{E}_{n\in\mathcal{A}}F(x_{n})-\int_{G/\Gamma}F\Big{|}\leq\delta\|F\|_{Lip}$

for all Lipschitz function $F:G/\Gamma\rightarrow\mathbb{C}$, where

$\mathbb{E}_{n\in\mathcal{A}}F(x_{n})=\frac{1}{|\mathcal{A}|}\sum_{n\in\mathcal{A}}F(x_{n})$

and

$\|F\|_{Lip}:=\sup_{x\in G/\Gamma}|F(x)|+\sup_{\begin{subarray}{c}x,y\in
G/\Gamma\\\ x\neq y\end{subarray}}\frac{|F(x)-F(y)|}{d_{G/\Gamma}(x,y)}.$
(2.1)

A finite sequence $\\{x_{n}\\}_{n\in\mathcal{A}}$ in $G/\Gamma$ is said to be
totally $\delta$-equidistributed in $G/\Gamma$ if the sequence
$\\{x_{n}\\}_{n\in\mathcal{A}^{\prime}}$ is $\delta$-equidistributed in
$G/\Gamma$ for all arithmetic progressions
$\mathcal{A}^{\prime}\subseteq\mathcal{A}$ of length at least $\delta N$.

The next result gives a simple connection between the coefficients of a
polynomial and its smoothness norm.

######  Lemma 2.4.

$($[16, Lemma 3.2]$)$ Suppose that
$f:\mathbb{Z}\rightarrow\mathbb{R}/\mathbb{Z}$ is a polynomial of the form
$\sum_{i=0}^{d}\beta_{i}n^{i}$. Then there is a positive integer $D=O_{d}(1)$
such that
$\|D\beta_{i}\|_{\mathbb{R}/\mathbb{Z}}\ll_{d}N^{-i}\|f\|_{C^{\infty}[N]}$ for
all $i=1,\ldots,d$.

The following is a “strong recurrence” result for polynomials
$f:\mathbb{Z}\rightarrow\mathbb{R}$.

######  Lemma 2.5.

$($[15, Lemma 4.5]$)$ Suppose that $f:\mathbb{Z}\rightarrow\mathbb{R}$ is a
polynomial of degree $d$ with $d\geq 0$. Suppose that $N\geq 1$,
$\delta\in(0,1/2)$ and $\epsilon\in(0,\delta/2]$. If $f(n)(mod~{}\mathbb{Z})$
belongs to an interval $I\subseteq\mathbb{R}/\mathbb{Z}$ of length $\epsilon$
for at least $\delta N$ integers of $n\in[N]$. Then there is a
$D\in\mathbb{Z}$ with $0<|D|\ll_{d}\delta^{-O_{d}(1)}$ such that
$\|Df(mod~{}\mathbb{Z})\|_{C^{\infty}[N]}\ll_{d}\epsilon\delta^{-O_{d}(1)}$.

The next one is a special case of the quantitative equidistribution result on
polynomials $f:\mathbb{Z}\rightarrow\mathbb{R}^{m}$ ([15, Theorem 2.9] and
[17, Corollary 3.2]).

######  Lemma 2.6.

Let $m\geq 1$, $d\geq 0$ and $N\geq 1$ be integers. Suppose that
$f:\mathbb{Z}\rightarrow\mathbb{R}^{m}$ is a polynomial of degree $d$. Let
$R>2$. If $\\{f(n)(mod~{}\mathbb{Z}^{m})\\}_{n\in[N]}$ is not totally
$R^{-1}$-equidistributed in $\mathbb{R}^{m}/\mathbb{Z}^{m}$, then there is a
$D\in\mathbb{Z}^{m}$ with $0<|D|\leq R^{O_{m,d}(1)}$ such that $\|D\cdot
f(mod~{}\mathbb{Z})\|_{C^{\infty}[N]}\leq R^{O_{m,d}(1)}$.

Now, we prove a variant version of the above result for the equidistribution
in arithmetic progressions that is applicable to our case. In the following,
constants in the asymptotic notation $\ll$ and $O(\cdot)$ are allowed to
depend on $m$ and $d$. These constants are not necessarily the same in each
occurrence.

######  Proposition 2.7.

Let $m\geq 1$, $d\geq 0$, and let
$g(n,h):\mathbb{Z}^{2}\rightarrow\mathbb{R}^{m}$ be a polynomial. Suppose that

$g(n,h)=\sum_{\begin{subarray}{c}j,k\geq 0\\\ j+k\leq
d\end{subarray}}\beta_{j,k}n^{j}h^{k}$

with each $\beta_{j,k}\in\mathbb{R}^{m}$. Let $\widetilde{R}>2$ and
$N,H\in\mathbb{N}$ with $N,H\geq\widetilde{R}^{O(1)}$. Let $a,s$ be integers
with $s\geq 1$ and $1\leq a\leq s$. Write $\mathcal{A}_{a}=\\{h:1\leq h\leq
Hs,h\equiv a(mod~{}s)\\}$. Then at least one of the following holds:

(i). $\\{g(n,h)(mod~{}\mathbb{Z}^{m})\\}_{h\in\mathcal{A}_{a}}$ is totally
$\widetilde{R}^{-1}$-equidistributed in $\mathbb{R}^{m}/\mathbb{Z}^{m}$ for
all but at most $\widetilde{R}^{-1}Ns$ values of $(n,a)\in[N]\times[s]$.

(ii). there is a $Q\in\mathbb{Z}^{m}$ with $0<|Q|\ll\widetilde{R}^{O(1)}$ such
that for any $i=1,\ldots,d$ and $j=0,\ldots,d$ with $i+j\leq d$ and for any
$k$ with $i\leq k\leq d-j$,
$\|s^{i}Q\cdot\beta_{j,k}\|_{\mathbb{R}/\mathbb{Z}}\ll\widetilde{R}^{O(1)}N^{-j}H^{-i}s^{i-k}.$

###### Proof.

Suppose that

$g(n,ls+a)=\sum_{\begin{subarray}{c}i,j\geq 0\\\ i+j\leq
d\end{subarray}}w_{i,j}l^{i}n^{j},$ (2.2)

where

$w_{i,j}=s^{i}\sum_{k=i}^{d-j}a^{k-i}\binom{k}{i}\beta_{j,k}.$ (2.3)

Assume claim $(i)$ does not hold. Denote by $\mathcal{A}$ the set of all
$(n,a)$ such that $\\{g(n,h)(mod~{}\mathbb{Z}^{m})\\}_{h\in\mathcal{A}_{a}}$
is not totally $\widetilde{R}^{-1}$-equidistributed in
$\mathbb{R}^{m}/\mathbb{Z}^{m}$. Then $|\mathcal{A}|\geq\widetilde{R}^{-1}Ns$.
For each $(n,a)\in\mathcal{A}$, by Lemma 2.6, there is a $D\in\mathbb{Z}^{m}$
with $0<|D|\leq\widetilde{R}^{O(1)}$ such that

$\|D\cdot g(n,\cdot
s+a)(mod~{}\mathbb{Z})\|_{C^{\infty}[H]}\ll\widetilde{R}^{O(1)}.$ (2.4)

By the pigeonhole principle, there is a common $D$ independent of $(n,a)$ with
$0<|D|\leq\widetilde{R}^{O(1)}$ such that formula (2.4) holds for at least
$\widetilde{R}^{-O(1)}Ns$ values of
$(n,a)\in\mathcal{A}\subseteq[N]\times[s]$. We still denote the set of such
$(n,a)$ by $\mathcal{A}$. By Lemma 2.4, there is an integer $D_{1}=O(1)$ such
that for any $i=1,\ldots,d$ and $(n,a)\in\mathcal{A}$,

$\|D_{1}\sum_{j=0}^{d-i}D\cdot
w_{i,j}n^{j}\|_{\mathbb{R}/\mathbb{Z}}\ll\widetilde{R}^{O(1)}H^{-i}.$ (2.5)

By enlarging the constant implied in the constant $O(1)$ in the estimate of
$|\mathcal{A}|\geq\widetilde{R}^{-O(1)}Ns$, there are at least
$\widetilde{R}^{-O(1)}s$ choices of $a\in[s]$ such that for each $a$, there
are at least $\widetilde{R}^{-O(1)}N$ choices of $n\in[N]$ with
$(n,a)\in\mathcal{A}$. Denote the set of such $a$ by $\mathcal{B}$. Then
$|\mathcal{B}|\geq\widetilde{R}^{-O(1)}s$. Treat $D_{1}\sum_{j=0}^{d-i}D\cdot
w_{i,j}n^{j}$ as the polynomial of $n$. Then by Lemmas 2.5 and 2.6 again, for
each $a\in\mathcal{B}$, there is an integer $D_{2}$ with
$0<|D_{2}|\ll\widetilde{R}^{O(1)}$, such that $\|D_{2}D_{1}D\cdot
w_{i,j}\|_{\mathbb{R}/\mathbb{Z}}\ll\widetilde{R}^{O(1)}H^{-i}N^{-j}$ for
$i=1,\ldots,d$ and $j=1,\ldots,d$ with $i+j\leq d$. We may assume that $D_{2}$
is independent of $a$, since this can be obtained after pigeonholding in the
$\widetilde{R}^{O(1)}$ possible choices of $D_{2}$ and enlarging the constant
$O(1)$ in the estimate of $|\mathcal{B}|$. This leads to

$\|D_{2}D_{1}\sum_{j=1}^{d-i}D\cdot
w_{i,j}n^{j}\|_{\mathbb{R}/\mathbb{Z}}\ll\widetilde{R}^{O(1)}H^{-i}$ (2.6)

for $i=1,\ldots,d$. By (2.5), (2.6) and the triangle inequality, we obtain the
diophantine information about $w_{i,0}$. Summarize the above analysis,
together with (2.3), we have that

$\|D_{2}D_{1}s^{i}\sum_{k=i}^{d-j}a^{k-i}\binom{k}{i}D\cdot\beta_{j,k}\|_{\mathbb{R}/\mathbb{Z}}\ll\widetilde{R}^{O(1)}N^{-j}H^{-i}$
(2.7)

holds for any $a\in\mathcal{B}$, $i=1,\ldots,d$ and $j=0,\ldots,d$ with
$i+j\leq d$. For fixed $(i,j)\in\\{1,\ldots,d\\}\times\\{0,\ldots,d\\}$, we
treat $D_{2}D_{1}s^{i}\sum_{k=i}^{d-j}a^{k-i}\binom{k}{i}D\cdot\beta_{j,k}$ as
a polynomial of $a$. By an argument similar to that of (2.7), there is an
integer $D_{3}$ with $0<|D_{3}|\ll\widetilde{R}^{O(1)}$ such that for any
$i=1,\ldots,d$ and $j=0,\ldots,d$ with $i+j\leq d$ and for any $i\leq k\leq
d-j$,

$\|s^{i}D_{3}D_{2}D_{1}D\cdot\beta_{j,k}\|_{\mathbb{R}/\mathbb{Z}}\ll\widetilde{R}^{O(1)}N^{-j}H^{-i}s^{i-k}.$

Let $Q=D_{3}D_{2}D_{1}D$. We obtain claim (ii). ∎

######  Corollary 2.8.

Let $m\geq 1$, $d\geq 0$, and let $f(n)=\sum_{i=0}^{d}\alpha_{i}n^{i}$ with
each $\alpha_{i}\in\mathbb{R}^{m}$. Suppose that $g(n,h)=f(n+h)$. Let
$\widetilde{R}>2$ and $N,H\in\mathbb{N}$ with $N,H\geq\widetilde{R}^{O(1)}$.
Let integers $s\geq 1$ and $1\leq a\leq s$. Write $\mathcal{A}_{a}=\\{h:1\leq
h\leq Hs,h\equiv a(mod~{}s)\\}$. Then at least one of the following holds:

(i). $\\{g(n,h)(mod~{}\mathbb{Z}^{m})\\}_{h\in\mathcal{A}_{a}}$ is totally
$\widetilde{R}^{-1}$-equidistributed in $\mathbb{R}^{m}/\mathbb{Z}^{m}$ for
all but at most $\widetilde{R}^{-1}Ns$ values of $(n,a)\in[N]\times[s]$.

(ii). there is a $Q\in\mathbb{Z}^{m}$ with $0<|Q|\ll\widetilde{R}^{O(1)}$ such
that
$\|sQ\cdot\alpha_{l}\|_{\mathbb{R}/\mathbb{Z}}\leq\widetilde{R}^{O(1)}H^{-1}N^{-(l-1)}$
for $l=1,\ldots,d$.

###### Proof.

Suppose that

$g(n,h)=\sum_{\begin{subarray}{c}j,k\geq 0\\\ j+k\leq
d\end{subarray}}\beta_{j,k}n^{j}h^{k},$

where $\beta_{j,k}\in\mathbb{R}^{m}$. By the assumption,
$\beta_{j,k}=\alpha_{j+k}\binom{j+k}{j}$. Assume that $(i)$ does not hold. By
Proposition 2.7, there is a $Q_{1}\in\mathbb{Z}^{m}$ with
$0<|Q_{1}|\ll\widetilde{R}^{O(1)}$ such that for any $i=1,\ldots,d$ and
$j=0,\ldots,d$ with $i+j\leq d$ and for any $k$ with $i\leq k\leq d-j$,
$\|s^{i}Q_{1}\cdot\alpha_{j+k}\binom{j+k}{j}\|_{\mathbb{R}/\mathbb{Z}}\ll\widetilde{R}^{O(1)}N^{-j}H^{-i}s^{i-k}.$
Let $i=k=1$, $j=l-1$. Let $Q=d!Q_{1}$. We obtain claim (ii). ∎

Now using Corollary 2.8, we are ready to prove Theorem 2.1.

###### Proof of Theorem 2.1.

Suppose that $f(n)=\sum_{i=0}^{d}\alpha_{i}n^{i}$ with each
$\alpha_{i}\in\mathbb{R}^{m}$. Let $\widetilde{R}=R^{B}$. If
$\\{g(n,h)(mod~{}\mathbb{Z}^{m})\\}_{h\in\mathcal{A}_{a}}$ is totally
$\widetilde{R}^{-1}$-equidistributed in $\mathbb{R}^{m}/\mathbb{Z}^{m}$ for
all but at most $\widetilde{R}^{-1}Ns$ values of $(n,a)\in[N]\times[s]$, then
choose $\mathcal{E}(n,h)=\gamma(n,h)=0,g^{\prime}(n,h)=g(n,h)$ and
$G^{\prime}=\mathbb{R}^{m}$, $W=R$. Otherwise, applying Corollary 2.8 to
$g(n,h)$, then there is a $Q\in\mathbb{Z}^{m}$ with
$0<|Q|\ll\widetilde{R}^{O(1)}$ such that for $j=1,\ldots,d$,

$\|sQ\cdot\alpha_{j}\|_{\mathbb{R}/\mathbb{Z}}\leq\widetilde{R}^{O(1)}H^{-1}N^{-(j-1)}.$

We may choose $\beta_{j}\in\mathbb{R}^{m}$ with
$Q\cdot\beta_{j}\in\mathbb{Z}/s$ and
$|\alpha_{j}-\beta_{j}|<\widetilde{R}^{O(1)}H^{-1}N^{-(j-1)}/s$. We further
choose $\tau_{j}\in\frac{\mathbb{Z}^{m}}{qs}$ for some integer $q$ with with
$0<|q|<\widetilde{R}^{O(1)}$ and $Q\cdot\beta_{j}=Q\cdot\tau_{j}$. Let
$\mathcal{E}_{1}(n,h)=\sum_{j=1}^{d}(\alpha_{j}-\beta_{j})(n+h)^{j}+\alpha_{0}$
and $\gamma_{1}(n,h)=\sum_{j=1}^{d}\tau_{j}(n+h)^{j}$. Define

$g_{1}(n,h)=g(n,h)-\mathcal{E}_{1}(n,h)-\gamma_{1}(n,h).$

Then $Q\cdot g_{1}(n,h)=0$. So $g_{1}(n,h)$ takes values in $G_{1}$, a
connected proper subgroup $G_{1}$ of $G$, which is isomorphic (after a natural
projection of coordinates, denoted by $\pi_{1}$) to $\mathbb{R}^{m-1}$.
Additionally, using $N>(Hs)^{O(1)}$, one can verify that
$|\mathcal{E}_{1}(n,h+1)-\mathcal{E}_{1}(n,h)|\ll_{d}\frac{\widetilde{R}^{O(1)}}{Hs}$
for any $h\in[Hs]$. Consider the lattice
$\Gamma_{1}:=G_{1}\cap\mathbb{Z}^{m}$. Then it is not hard to check that
$\Gamma_{1}$ is isomorphic to $A_{1}\mathbb{Z}^{m-1}$, where $A_{1}\in
GL_{m-1}(\mathbb{R})$ with integer entries of height $\widetilde{R}^{O(1)}$.
Let $R_{1}=\widetilde{R}\cdot\widetilde{R}^{O(1)}=R^{O(B)}$ and
$\widetilde{R}_{1}=R_{1}^{B}$. We now check weather
$\\{g_{1}(n,h)(mod~{}\Gamma_{1})\\}_{h\in\mathcal{A}_{a}}$ is totally
$\widetilde{R_{1}}^{-1}$-equidistributed in $G_{1}/\Gamma_{1}$ for all but at
most $\widetilde{R}_{1}^{-1}Ns$ values of $(n,a)\in[N]\times[s]$. If it is so,
we terminate the process and then choose
$\mathcal{E}(n,h)=\mathcal{E}_{1}(n,h)$,
$\gamma(n,h)=\gamma_{1}(n,h),g^{\prime}(n,h)=g_{1}(n,h)$ and
$G^{\prime}=G_{1}$, $W=R_{1}$.

If it is not the case, we continue the process. Write
$f_{1}(n)=\sum_{i=1}^{d}\alpha_{i,1}n^{i}$ such that $g_{1}(n,h)=f_{1}(n+h)$.
Then $A_{1}^{-1}\circ\pi_{1}(f_{1}(n))$, $A_{1}^{-1}\circ\pi_{1}(g_{1}(n,h))$
are polynomials taking values in $\mathbb{R}^{m-1}$. We then again apply
Corollary 2.8 to $A_{1}^{-1}\circ\pi_{1}(g_{1}(n,h))$ with $m$ replaced by
$m-1$ and $\widetilde{R}$ replaced by $\widetilde{R_{1}}$. Then by an argument
similar to that of $g(n,h)$, we can find the corresponding
$\widetilde{\mathcal{E}_{2}}(n,h)$, $\widetilde{\gamma_{2}}(n,h)$ and so
define $\mathcal{E}_{2}(n,h)$, $\mathcal{\gamma}_{2}(n,h)$ by
$\pi_{1}^{-1}\circ A_{1}(\widetilde{\mathcal{E}_{2}}(n,h))$ and by
$\pi_{1}^{-1}\circ A_{1}(\widetilde{\mathcal{\gamma}_{2}}(n,h))$,
respectively. Define $g_{2}(n,h)$ by
$g_{1}(n,h)-\mathcal{E}_{2}(n,h)-\gamma_{2}(n,h)$. Assume that in the $l$-th
step, we apply Corollary 2.8 to $g_{l-1}(n,h)$ with $\widetilde{R}$ replaced
by $\widetilde{R}_{l-1}$, where $\widetilde{R}_{l-1}=(R_{l-1})^{B}$ and
$R_{l-1}=\widetilde{R}_{l-2}\widetilde{R}_{l-2}^{O(1)}$. Then we obtain

$\bullet$ $\mathcal{E}_{l}(n,h)$ satisfies
$|\mathcal{E}_{l}(n,h+1)-\mathcal{E}_{l}(n,h)|\ll_{d}(\widetilde{R}_{l-1})^{O(1)}/(Hs)$
for $h\in[Hs]$;

$\bullet$
$\\{\gamma_{l}(n,h)(mod~{}\mathbb{Z}^{m})\\}_{(n,h)\in\mathbb{Z}^{2}}$ is
$q_{l}s$-periodic for some positive integer
$q_{l}\leq(\widetilde{R}_{l-1})^{O(1)}$;

$\bullet$
$g(n,h)=\sum_{i=1}^{l}\mathcal{E}_{i}(n,h)+g_{l}(n,h)+\sum_{i=1}^{l}\gamma_{i}(n,h)$,
where $g_{l}(n,h)$ is a polynomial taking values in $G_{l}$, and
$\\{\sum_{i=1}^{l-1}\gamma_{i}(n,h)(mod~{}\mathbb{Z}^{m})\\}_{(n,h)\in\mathbb{Z}^{2}}$
is $(\prod_{i=1}^{l-1}q_{i})s$-periodic with
$\prod_{i=1}^{l-1}q_{i}\leq\widetilde{R}_{l-1}$.

Since $m$ is a given non-negative integer, the total number of iterations is
bounded by $m$. This implies that for some positive integer $l$,
$\\{g_{l}(n,h)(mod~{}\Gamma_{l})\\}_{h\in\mathcal{A}_{a}}$ is totally
$(\widetilde{R}_{l})^{-1}$-equidistributed in $G_{l}/\Gamma_{l}$ for all but
$(\widetilde{R}_{l})^{-1}Ns$ values of $(n,a)\in[N]\times[s]$, where $G_{l}$
is a connected proper subgroup of $\mathbb{R}^{m}$. Let $G^{\prime}=G_{l}$ and
$\Gamma^{\prime}=\Gamma_{l}=G^{\prime}\cap\mathbb{Z}^{m}$. Choose
$W=R_{l}=R^{O(B^{m})}$, $\mathcal{E}(n,h)=\sum_{i=1}^{l}\mathcal{E}_{i}(n,h)$,
$g^{\prime}(n,h)=g_{l}(n,h)$ and $\gamma(n,h)=\sum_{i=1}^{l}\gamma_{i}(n,h)$.
So we obtain that
$\\{g^{\prime}(n,h)(mod~{}\Gamma^{\prime})\\}_{h\in\mathcal{A}_{a}}$ is
totally $W^{-B}$-equidistributed in $G^{\prime}/\Gamma^{\prime}$ for all but
at most $W^{-B}Ns$ values of $(n,a)\in[N]\times[s]$. Let $\mathcal{N}$ denote
the set of $n\in[N]$ which satisfies that at least $(1-W^{-B/2})s$ choices of
$a\in[s]$ has the property that
$\\{g^{\prime}(n,h)(mod~{}\Gamma^{\prime})\\}_{h\in\mathcal{A}_{a}}$ is
totally $W^{-B}$-equidistributed in $G^{\prime}/\Gamma^{\prime}$. It is not
hard to check that

$(N-|\mathcal{N}|)W^{-B/2}s\leq W^{-B}Ns.$

Then $|\mathcal{N}|\geq(1-W^{-B/2})N$. We complete the proof. ∎

######  Remark 2.9.

From the above proof, it is not hard to see that $g^{\prime}(n,h)$ given in
Theorem 2.1(ii) is in form of $g^{\prime}(n,h)=f^{\prime}(n+h)$ for some
polynomial sequence $f^{\prime}:\mathbb{N}\rightarrow G^{\prime}$.

##  3\. Proof of Theorem 1.3

Actually, Theorem 1.3 follows from a more general result (Theorem 3.1 below).
Before stating the result, let us recall the following distance function
introduced by Granville and Soundararajan (see e.g., [14]). Given integer
$s\geq 1$, for two multiplicative functions $f(n)$ and $g(n)$ with
$|f(n)|,|g(n)|\leq 1$ for all $n\geq 1$, define

$\mathbb{D}_{s}(f(n),g(n);X):=\Bigg{(}\sum_{\begin{subarray}{c}p\leq X\\\
p\nmid
s\end{subarray}}\frac{1-\text{Re}(f(p)\overline{g(p)})}{p}\Bigg{)}^{\frac{1}{2}}$

This distance function was used in [1] to measure the pretentiousness between
any multiplicative function $f(n)$ and some function for which exceptional
modulus $s$ does exist. Throughout define

$\displaystyle\begin{aligned} M_{s}(f;X,T):=&\inf_{\chi(mod~{}s)}\inf_{|t|\leq
T}\mathbb{D}_{s}(f\chi,n\mapsto n^{it};X)^{2},\\\ M(f;X,T,Y):=&\inf_{s\leq
Y}M_{s}(f;X,T)=\inf_{\begin{subarray}{c}s\leq Y\\\
\chi(mod~{}s)\end{subarray}}\inf_{|t|\leq T}\mathbb{D}_{s}(f\chi,n\mapsto
n^{it};X)^{2}.\end{aligned}$ (3.1)

######  Theorem 3.1.

Let $s\geq 1$, $H\geq 3$ and $d\geq 0$ be integers. Let $\beta(n)$ be a
multiplicative function with $|\beta(n)|\leq 1$ for all $n\geq 1$. Let
$P(x)\in\mathbb{R}[x]$ have degree $d$ and
$F:\mathbb{R}/\mathbb{Z}\rightarrow\mathbb{C}$ be a Lipschitz function. For
$N$ large enough with $s\log H\leq\frac{1}{2}(\log N)^{1/32}$, we have

$\displaystyle\begin{aligned}
&\sum_{a=1}^{s}\sum_{n=1}^{N}\left|\sum_{\begin{subarray}{c}1\leq h\leq Hs\\\
n+h\equiv
a(mod~{}s)\end{subarray}}\beta(n+h)F\big{(}P(n+h)(mod~{}\mathbb{Z})\big{)}\right|\ll
NHs\frac{s}{\varphi(s)}\frac{\log\log H}{\log H}\\\ &+NHs\frac{1}{(\log
N)^{1/200}}+(\log N)^{o(1)}NHs\frac{M\big{(}\beta;N/\log N,2N,(\log
N)^{1/32}\big{)}+1}{\exp\Big{(}\frac{1}{2}M\big{(}\beta;N/\log N,2N,(\log
N)^{1/32}\big{)}\Big{)}},\end{aligned}$ (3.2)

where the implied constant depends on $d$ and $\|F\|_{Lip}$ at most.

We remark that the constant implied in formula (3.2) would be universal for
$F$ from the family consisting of all Lipschitz functions whose Lipschitz norm
is bounded by a given constant.

The factor $(\log N)^{o(1)}$ in (3.2) comes from the restriction that the
summation on $n,h$ is over a certain dense set $\mathcal{S}$ of natural
numbers (see Subsection 3.2). As pointed in the remark below [32, Theorem
9.2], $(\log
N)^{o(1)}M(\beta;\cdot,\cdot,\cdot)\exp(-\frac{1}{2}M(\beta;\cdot,\cdot,\cdot))$
would be $o_{N}(1)$ if $M(\beta;\cdot,\cdot,\cdot)$ tends to infinity faster
than $\log\log N$. In case $M(\beta;\cdot,\cdot,\cdot)$ tends to infinity
slowly, it seems to be hard to show that $(\log
N)^{o(1)}M(\beta;\cdot,\cdot,\cdot)\exp(-\frac{1}{2}M(\beta;\cdot,\cdot,\cdot))$
is small. For this case, we use a slightly different method to estimate the
left-hand side of $(\ref{1/18})$ (see Remark 3.11) to avoid the factor $(\log
N)^{o(1)}$.

######  Remark 3.2.

Note that a simply connected abelian Lie group is isomorphic to
$\mathbb{R}^{m}$, and any discrete subgroup of $\mathbb{R}^{m}$ is a lattice.
So for the abelian nilmanifold, we may assume that $G=\mathbb{R}^{m}$ and
$\Gamma=\mathbb{Z}^{m}$ after a linear transformation. It is not hard to check
that by an argument similar to the proof of Theorem 3.1, we can extend
$\mathbb{R}/\mathbb{Z}$ in Theorem 3.1 to abelian nilmanifolds $G/\Gamma$, and
extend $P(x)$ to polynomial sequences $p:\mathbb{Z}\rightarrow G$ in the form
of $p(n)=a_{1}^{b_{1}(n)}\cdot\cdot\cdot a_{k}^{b_{k}(n)}$ for
$a_{1},\ldots,a_{k}\in G$ and $b_{1}(x),\ldots,b_{k}(x)\in\mathbb{R}[x]$ with
integer coefficients.

Theorem 1.3 follows directly from Theorem 3.1 through the following well-known
result (see, e.g., [33, (1.12)]) about the “non-pretentious” nature of the
Möbius function.

######  Proposition 3.3.

Let $N$ be large enough. Then for $\varepsilon>0$ sufficiently small,

$M(\mu;N/\log N,N,(\log N)^{1/32})\geq(1/3-\epsilon)\log\log N+O(1).$

In the rest part of this section, we shall prove Theorem 3.1. We now outline
the strategy in our proof. Using the quantitative factorization theorem (i.e.,
Theorem 2.1), we split the proof into two cases, the major and minor arc
cases. This point is more clear if the Lipschitz function $F$ has zero mean.
For this situation, the major arc part corresponds to
$g^{\prime}=0~{}(G^{\prime}=\\{0\\})$, while the minor arc part corresponds to
$g^{\prime}\neq 0~{}(G^{\prime}=\mathbb{R})$ in Theorem 2.1. In the major arc
part, we use the result [26, Theorem 1.6] to sums of multiplicative functions
over almost all short arithmetic progressions. In the minor arc case, we use a
variant of the arguments of He-Wang [17, Section 6] to obtain the cancellation
we need.

Recall that we use $[N]$ to denote the set $\\{1,\ldots,N\\}$. In the
following, for integers $n,a,s$, the notation $n\equiv a(s)$ means $n\equiv
a$(mod $s$). In this section, many implicit constants $O(1)=O_{d}(1)\geq 1$
will appear and may not be the same in each occurrence. For convenience, we
use a common constant $C_{0}=O_{d}(1)$ that is large enough for all these
purposes. Let

$B=100C_{0}^{2},~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}B_{2}=\frac{1}{10}C_{0}^{-2}B.$
(3.3)

The basic relation in this section among $s$, $H$, $N$ is as in Theorem 3.1,
i.e.,

$s\log H\leq\frac{1}{2}(\log N)^{1/32}.$

For given $H$ sufficiently large, set

$W=\lfloor\log H\rfloor.$ (3.4)

Suppose $P(n)=\sum_{i=0}^{d}\alpha_{i}n^{i}$ with each
$\alpha_{i}\in\mathbb{R}$ and $g(n,h)=P(n+h)$. Applying Theorem 2.1 to
$g(n,h)$ with $m=1$, there is a group $G^{\prime}=\mathbb{R}$ or $\\{0\\}$, a
set

$\mathcal{N}\subseteq[N]$ (3.5)

with $|\mathcal{N}|\geq(1-W^{-B/2})N$, and a decomposition of

$g(n,h)=\mathcal{E}(n,h)+g^{\prime}(n,h)+\gamma(n,h)$ (3.6)

into polynomials
$\mathcal{E},g^{\prime},\gamma:\mathbb{Z}^{2}\rightarrow\mathbb{R}$ with the
following properties:

(i). $|\mathcal{E}(n,h+1)-\mathcal{E}(n,h)|\ll_{d}\frac{1}{sH/W}$ for
$h\in[Hs]$.

(ii). $g^{\prime}(n,h)$ takes values in $G^{\prime}$, and for any
$n\in\mathcal{N}$, there are at least $(1-W^{-B/2})s$ integers of $a\in[s]$
such that $\\{g^{\prime}(n,h)\Gamma^{\prime}\\}_{h\in\mathcal{A}_{a}}$ is
totally $W^{-B}$-equidistributed in $G^{\prime}/\Gamma^{\prime}$, where
$\mathcal{A}_{a}=\\{1\leq h\leq Hs:h\equiv a(s)\\}$ and
$\Gamma^{\prime}=G^{\prime}\cap\mathbb{Z}$.

(iii). For some integer $q$ with $1\leq q\leq W$,
$\\{\gamma(n,h)(mod~{}\mathbb{Z})\\}_{(n,h)\in\mathbb{Z}^{2}}$ is
$qs$-periodic.

In addition, we assume that $q\in(W/2,W]$, otherwise we consider a multiplier
of $q$. Let $\beta:\mathbb{N}\rightarrow\mathbb{C}$ be a multiplicative
function with $|\beta(n)|\leq 1$ for each $n\in\mathbb{N}$ and let
$F:\mathbb{R}/\mathbb{Z}\rightarrow\mathbb{C}$ be a Lipschitz function with
$\|F\|_{Lip}\leq 1$ (see equation (2.1) for the definition of
$\|\cdot\|_{Lip}$).

###  3.1. Splitting into major and minor arc cases

Given $s\geq 1$, for any $a=1,\ldots,s$ and $n\in\mathbb{N}$, choose
$\theta_{n,a}\in\mathbb{C}$ with $|\theta_{n,a}|=1$ such that

$\Bigg{|}\sum_{\begin{subarray}{c}h\leq Hs\\\ n+h\equiv
a(s)\end{subarray}}\beta(n+h)F\big{(}g(n,h)(mod~{}\mathbb{Z})\big{)}\Bigg{|}=\theta_{n,a}\sum_{\begin{subarray}{c}h\leq
Hs\\\ n+h\equiv
a(s)\end{subarray}}\beta(n+h)F\big{(}g(n,h)(mod~{}\mathbb{Z})\big{)}.$ (3.7)

We first split the interval $[1,Hs]$ into disjoint subintervals
$I_{1},\ldots,I_{W^{2}}$, each has length $W^{-2}Hs$. To apply the periodicity
of $\gamma(n,h)(mod~{}\mathbb{Z})$, we further split each subinterval $I_{t}$
into progressions $n+h\equiv js+a(qs)$ for $j\in\\{0,\ldots,q-1\\}$. For any
given $j\in\\{0,\ldots,q-1\\}$ and $t\in\\{1,\ldots,W^{2}\\}$, define
$\mathcal{E}_{a,n,j,t}=\mathcal{E}(n,h^{\prime})$, where $h^{\prime}$ is the
smallest integer in $I_{t}$ with $n+h^{\prime}\equiv js+a(qs)$. Then by
property (i), for any $h\in I_{t}$,

$|\mathcal{E}_{a,n,j,t}-\mathcal{E}(n,h)|\leq\frac{W}{Hs}W^{-2}Hs=W^{-1}.$
(3.8)

Choose $\gamma_{a,n,j,t}\in[0,1)$ with
$\gamma_{a,n,j,t}(mod~{}\mathbb{Z})=\gamma(n,h)(mod~{}\mathbb{Z})$ for any
$h\in I_{t}$ satisfying $n+h\equiv js+a(qs)$. Note that
$\gamma(n,h)(mod~{}\mathbb{Z})$ is $qs$-periodic, so $\gamma_{a,n,j,t}$ is
independent of $h$ with $n+h\equiv js+a(qs)$.

Define the function444In the general form, we should define
$G_{a,n,j,t}=\gamma_{a,n,j,t}^{-1}G^{\prime}\gamma_{a,n,j,t}$ and
$\Gamma_{a,n,j,t}=G_{a,n,j,t}\cap\Gamma$. In the non-abelian case, for any
given $j,t$, there may be $s^{2}$ distinct $G_{a,n,j,t}$ and
$\Gamma_{a,n,j,t}$. Since we are in the abelian case, $G_{a,n,j,t}=G^{\prime}$
and $\Gamma_{a,n,j,t}=G^{\prime}\cap\Gamma=\Gamma^{\prime}$.
$F_{a,n,j,t}:G^{\prime}/\Gamma^{\prime}\rightarrow\mathbb{C}$ as
$F_{a,n,j,t}(y\Gamma^{\prime})=\theta_{n,a}F\big{(}(\mathcal{E}_{a,n,j,t}+\gamma_{a,n,j,t}+y)(mod~{}\mathbb{Z})\big{)}$
for any $y\in G^{\prime}$. Note that
$G^{\prime}/\Gamma^{\prime}=\mathbb{R}/\mathbb{Z}$ or $\\{0\\}$. It is easy to
see that $F_{a,n,j,t}$ is well-defined. By formulas (3.6) and (3.8),

$|F_{a,n,j,t}(g^{\prime}(n,h)\Gamma^{\prime})-\theta_{n,a}F\big{(}g(n,h)(mod~{}\mathbb{Z})\big{)}|\leq
W^{-1}$ (3.9)

holds for all $h\in I_{t}$ with $n+h\equiv js+a(qs)$. Then by the above
formula and formula (3.7),

$\displaystyle\begin{aligned}
&\sum_{a=1}^{s}\sum_{n=1}^{N}\Bigg{|}\sum_{\begin{subarray}{c}h\leq Hs\\\
n+h\equiv
a(s)\end{subarray}}\beta(n+h)F\big{(}g(n,h)(mod~{}\mathbb{Z})\big{)}\Bigg{|}\\\
=&\displaystyle\sum_{a=1}^{s}\sum_{n=1}^{N}\sum_{t=1}^{W^{2}}\sum_{j=0}^{q-1}\sum_{\begin{subarray}{c}h\in
I_{t}\\\ n+h\equiv
js+a(qs)\end{subarray}}\beta(n+h)F_{a,n,j,t}(g^{\prime}(n,h)\Gamma^{\prime})+O(W^{-1}HNs).\end{aligned}$
(3.10)

We write $F_{a,n,j,t}$ as $E_{a,n,j,t}+\widetilde{F}_{a,n,j,t}$, where
$E_{a,n,j,t}=\int_{G^{\prime}/\Gamma^{\prime}}F_{a,n,j,t}$ is a constant and
$\widetilde{F}_{a,n,j,t}$ has zero average on $G^{\prime}/\Gamma^{\prime}$. In
the following, we shall apply different methods to estimate the major arc part

$\sum_{a=1}^{s}\sum_{n=1}^{N}\sum_{t=1}^{W^{2}}\sum_{j=0}^{q-1}\sum_{\begin{subarray}{c}h\in
I_{t}\\\ n+h\equiv js+a(qs)\end{subarray}}\beta(n+h)E_{a,n,j,t},$ (3.11)

and the minor arc part

$\sum_{a=1}^{s}\sum_{n=1}^{N}\sum_{t=1}^{W^{2}}\sum_{j=0}^{q-1}\sum_{\begin{subarray}{c}h\in
I_{t}\\\ n+h\equiv
js+a(qs)\end{subarray}}\beta(n+h)\widetilde{F}_{a,n,j,t}(g^{\prime}(n,h)\Gamma^{\prime}).$
(3.12)

###  3.2. The estimate of major arc case.

For the major arc estimate, we shall show the following result.

######  Theorem 3.4.

Let $s\geq 1$ and $H\geq 3$ be integers. Let $N$ be large enough with $s\log
H\leq\frac{1}{2}(\log N)^{1/32}$. Let $\beta(n)$ be a multiplicative function
with $|\beta(n)|\leq 1$ for all $n\geq 1$. Then

$\displaystyle(\ref{afternoon2})\ll$ $\displaystyle
NHs\Bigg{(}\frac{s}{\varphi(s)}\frac{\log\log H}{\log H}+\frac{1}{(\log
N)^{1/200}}\Bigg{)}$ $\displaystyle+(\log
N)^{o(1)}NHs\frac{M\big{(}\beta;N/\log N,2N,(\log
N)^{1/32}\big{)}+1}{\exp\Big{(}\frac{1}{2}M\big{(}\beta;N/\log N,2N,(\log
N)^{1/32}\big{)}\Big{)}}.$

We first bound formula (3.11) by

$\displaystyle\begin{aligned}
&(\ref{afternoon2})\ll\sum_{a=1}^{s}\sum_{n=1}^{N}\sum_{t=1}^{W^{2}}\sum_{j=0}^{q-1}\left|\sum_{\begin{subarray}{c}h\in
I_{t}\\\ n+h\equiv js+a(qs)\end{subarray}}\beta(n+h)\right|\\\
&=\sum_{c=1}^{qs}\sum_{n=1}^{N}\sum_{t=1}^{W^{2}}\left|\sum_{\begin{subarray}{c}h\in
I_{t}\\\ n+h\equiv c(qs)\\\
n+h\in\mathcal{S}\end{subarray}}\beta(n+h)\right|+\sum_{c=1}^{qs}\sum_{n=1}^{N}\sum_{t=1}^{W^{2}}\left|\sum_{\begin{subarray}{c}h\in
I_{t}\\\ n+h\equiv c(qs)\\\
n+h\notin\mathcal{S}\end{subarray}}\beta(n+h)\right|,\end{aligned}$ (3.13)

where $\mathcal{S}$ is given as follows. Let

$P_{1}=(\log H)^{6000},Q_{1}=H/(\log H)^{4}.$ (3.14)

Then we define $P_{j}$, $Q_{j}$ inductively in the following way, which is the
same as [31, Section 2],

$P_{j}=\exp(j^{4j}(\log Q_{1})^{j-1}\log P_{1}),~{}~{}Q_{j}=\exp(j^{4j+2}(\log
Q_{1})^{j}).$ (3.15)

Let $N_{0}=\sqrt{N}$. Let $J$ be the largest index such that
$Q_{J}\leq\exp\big{(}\sqrt{\log N_{0}}\big{)}$. For $j=1,\ldots,J$, denote by

$\mathcal{P}_{j}=\\{p:~{}\text{a~{}prime~{}in}~{}[P_{j},Q_{j}]~{}\text{with}~{}p\nmid
s\\}.$

Define

$\mathcal{S}=\\{n\leq
2N:n{\rm~{}has~{}at~{}least~{}one~{}prime~{}factor~{}in}~{}\mathcal{P}_{j}~{}{\rm
for~{}all~{}}j\in\\{1,\ldots,J\\}\\}.$ (3.16)

We now estimate the contribution from $n+h\notin\mathcal{S}$ in formula
(3.13).

######  Lemma 3.5.

Let $\mathcal{S}$ be as the above.

$\sum_{c=1}^{qs}\sum_{n=1}^{N}\sum_{t=1}^{W^{2}}\left|\sum_{\begin{subarray}{c}h\in
I_{t}\\\ n+h\equiv c(qs)\\\
n+h\notin\mathcal{S}\end{subarray}}\beta(n+h)\right|\ll
NHs\frac{s}{\varphi(s)}\frac{\log\log H}{\log H}.$

###### Proof.

By the fundamental sieve, for $j=1,\ldots,J$, we have

$\sum_{\begin{subarray}{c}n\leq X\\\ \big{(}n,~{}\prod_{{P_{j}<p\leq
Q_{j},p\nmid s}}~{}p\big{)}=1\end{subarray}}1\ll X\frac{\log P_{j}}{\log
Q_{j}}\prod_{\begin{subarray}{c}P_{j}<p\leq Q_{j}\\\
p|s\end{subarray}}(1-1/p)^{-1}+O(sQ_{j}^{2}).$ (3.17)

So by the above and choices (3.14), (3.15),

$\displaystyle\sum_{c=1}^{qs}\sum_{n=1}^{N}\sum_{t=1}^{W^{2}}\left|\sum_{\begin{subarray}{c}h\in
I_{t}\\\ n+h\equiv c(qs)\\\
n+h\notin\mathcal{S}\end{subarray}}\beta(n+h)\right|\leq\sum_{n=1}^{N}\sum_{\begin{subarray}{c}m=n\\\
m\notin\mathcal{S}\end{subarray}}^{n+Hs}1=\sum_{\begin{subarray}{c}m=1\\\
m\notin\mathcal{S}\end{subarray}}^{N+Hs}\sum_{n=m-Hs}^{m}1$ $\displaystyle\leq
Hs\sum_{j=1}^{J}\sum_{\begin{subarray}{c}n\leq N+Hs\\\
(n,~{}\prod_{{P_{j}<p\leq Q_{j},p\nmid s}}~{}p)=1\end{subarray}}1\ll
NHs\frac{\log P_{1}}{\log Q_{1}}\frac{s}{\varphi(s)}$ $\displaystyle\ll
NHs\frac{s}{\varphi(s)}\frac{\log\log H}{\log H}$

as claimed. ∎

Hence, to prove Theorem 3.4, we are left to bound

$\sum_{c=1}^{qs}\sum_{n=1}^{N}\sum_{t=1}^{W^{2}}\left|\sum_{\begin{subarray}{c}h\in
I_{t}\\\ n+h\equiv c(qs)\\\
n+h\in\mathcal{S}\end{subarray}}\beta(n+h)\right|.$ (3.18)

Let $\beta_{1}$ denote the completely multiplicative function defined by
$\beta_{1}(p)=\beta(p)$ for all prime numbers $p$. We first bound the above in
terms of $\beta_{1}$ rather than $\beta$ by a standard technique (see, e.g.,
[33, p2177-p2178]). The Dirichlet inverse of $\beta_{1}$ is $\mu\beta_{1}$.
Then we can write $\beta=\beta_{1}*\alpha$, where $*$ is the Dirichlet
convolution and $\alpha=\beta*\mu\beta_{1}$. Observe that $\alpha(n)$ is
multiplicative and for all primes $p$, $\alpha(p)=0$ and $|\alpha(p^{j})|\leq
2$ for $j\geq 2$. So from the Euler product, we see that
$\sum_{n=1}^{\infty}|\alpha(n)|n^{-\frac{3}{4}}=O(1)$. Note that $W<P_{1}$. So
$1_{\mathcal{S}}(ab)=1_{\mathcal{S}}(b)$ when $a\leq W$ and $b\leq(N+Hs)/a$.
Then

$\displaystyle(\ref{11/8evening})=$
$\displaystyle\sum_{c=1}^{qs}\sum_{n=1}^{N}\sum_{t=1}^{W^{2}}\sum_{a\leq
W}|\alpha(a)|\left|\sum_{\begin{subarray}{c}b\in\mathbb{N}\\\ ab\equiv
c(qs)\\\ ab\in
n+I_{t}\end{subarray}}1_{\mathcal{S}}(b)\beta_{1}(b)\right|+\sum_{c=1}^{qs}\sum_{n=1}^{N}\sum_{t=1}^{W^{2}}\sum_{a>W}|\alpha(a)|\left|\sum_{\begin{subarray}{c}b\in\mathbb{N}\\\
ab\equiv c(qs)\\\ ab\in
n+I_{t}\end{subarray}}1_{\mathcal{S}}(ab)\beta_{1}(b)\right|$ $\displaystyle=$
$\displaystyle\sum_{c=1}^{qs}\sum_{n=1}^{N}\sum_{t=1}^{W^{2}}\sum_{a\leq
W}|\alpha(a)|\left|\sum_{\begin{subarray}{c}b\in\mathbb{N},ab\equiv c(qs)\\\
ab\in
n+I_{t}\end{subarray}}1_{\mathcal{S}}(b)\beta_{1}(b)\right|+O\left(\sum_{n=1}^{N}\sum_{t=1}^{W^{2}}\sum_{a>W}|\alpha(a)|\Bigg{(}\sum_{b\in\mathbb{N},ab\in
n+I_{t}}1\Bigg{)}\right)$ $\displaystyle=$
$\displaystyle\sum_{c=1}^{qs}\sum_{n=1}^{N}\sum_{t=1}^{W^{2}}\sum_{a\leq
W}|\alpha(a)|\left|\sum_{\begin{subarray}{c}b\in\mathbb{N},ab\equiv c(qs)\\\
ab\in
n+I_{t}\end{subarray}}1_{\mathcal{S}}(b)\beta_{1}(b)\right|+O\Big{(}W^{-1/4}NHs\Big{)}.$
(3.19)

Now the estimate of formula (3.11) is reduced to estimating

$\sum_{c=1}^{qs}\sum_{n=1}^{N}\sum_{t=1}^{W^{2}}\sum_{a\leq
W}|\alpha(a)|\left|\sum_{\begin{subarray}{c}b\in\mathbb{N},ab\equiv c(qs)\\\
ab\in n+I_{t}\end{subarray}}1_{\mathcal{S}}(b)\beta_{1}(b)\right|.$ (3.20)

We split the summation over $b$ according to gcd$(a,qs)$. Note that
$|\alpha(n)|\leq\tau(n)$ (the divisor function) for all $n\in\mathbb{N}$. Then

$\displaystyle\begin{aligned} (\ref{10/29noonformula1})=&\sum_{t\leq
W^{2}}\sum_{\begin{subarray}{c}e\leq W\\\ e\mid
qs\end{subarray}}\sum_{\begin{subarray}{c}a\leq W/e\\\
(a,qs/e)=1\end{subarray}}|\alpha(ae)|\sum_{{1\leq c\leq qs/e}}\sum_{n\leq
N}\left|\sum_{\begin{subarray}{c}b\in n/ae+I_{t}/ae\\\ b\equiv
c(qs/e)\end{subarray}}1_{\mathcal{S}}(b)\beta_{1}(b)\right|\\\ =&\sum_{t\leq
W^{2}}\sum_{\begin{subarray}{c}e\leq W\\\ e\mid
qs\end{subarray}}\sum_{\begin{subarray}{c}a\leq W/e\\\
(a,qs/e)=1\end{subarray}}|\alpha(ae)|ae\sum_{{1\leq c\leq qs/e}}\sum_{n\leq
N/ae}\left|\sum_{\begin{subarray}{c}b\in n+I_{t}/ae\\\ b\equiv
c(qs/e)\end{subarray}}1_{\mathcal{S}}(b)\beta_{1}(b)\right|+O\Big{(}NsW^{3}q(\log
W)^{2}\Big{)}.\end{aligned}$ (3.21)

To give an upper bound for (3.21), we apply the following estimate on sums of
multiplicative functions in short arithmetic progressions.

######  Lemma 3.6.

Given $X$ with $\sqrt{N}\leq X\leq N/2$ and $1\leq s\leq(\log X)^{1/32}$. Let
$3\leq H\leq X/s$. Let $f(n)$ be a multiplicative function with $|f(n)|\leq 1$
for all $n\geq 1$. Then

$\displaystyle\sum_{\begin{subarray}{c}a=1\\\
(a,s)=1\end{subarray}}^{s}\sum_{x=X}^{2X}\Bigg{|}\sum_{\begin{subarray}{c}n=x\\\
n\equiv a(mod~{}s)\end{subarray}}^{x+Hs}1_{\mathcal{S}}(n)f(n)\Bigg{|}^{2}\ll$
$\displaystyle H^{2}X\varphi(s)\Bigg{(}\frac{(\log
Q_{1})^{1/3}}{P_{1}^{1/6-\eta}}+2^{2J}\frac{M(f;X,2X,s)+1}{\exp\Big{(}M(f;X,2X,s)\Big{)}}\Bigg{)}$
$\displaystyle+H^{2}X\varphi(s)\frac{1}{(\log X)^{1/65}},$ (3.22)

where $\eta=1/150$, $P_{1},Q_{1}$ and $\mathcal{S}$, $J$ are given in (3.14)
and (3.16), respectively.

###### Proof.

The proof is almost identical to the proof of [26, Theorem 1.6] (see also [41,
Proposition 6.1]), in which it gives an upper bound of left-hand side of (3.6)
without restricting $n$ to $\mathcal{S}$. The difference is that when we apply
the Halász-type inequality to a Dirichlet polynomial of the form $\sum_{X\leq
n\leq 2X}\frac{1_{\mathcal{S}}(n)g(n)\chi(n)}{n^{1+it}}$ for some function
$g(n)$, we should rewrite

$\sum_{X\leq n\leq
2X}\frac{1_{\mathcal{S}}(n)g(n)\chi(n)}{n^{1+it}}=\sum_{\mathcal{J}\subseteq\\{1,\ldots,J\\}}(-1)^{|\mathcal{J}|}\sum_{X\leq
n\leq 2X}\frac{1_{C_{\mathcal{J}}}(n)g(n)\chi(n)}{n^{1+it}},$

where $C_{\mathcal{J}}$ is the set
$\\{n\in\mathbb{N}:(n,\prod_{p\in\cup_{j\in\mathcal{J}}\mathcal{P}_{j}}p)=1\\}$
and $1_{C_{\mathcal{J}}}(n)$ is a multiplicative function (see [31, Lemma 5]).
∎

By the above result, we have the following result.

######  Corollary 3.7.

Let $s\geq 1$. Let $N_{1}$ be large enough with $N^{2/3}\leq N_{1}\leq N$ and
$s\leq\frac{3}{4}(\log N_{1})^{1/32}$. Let $\beta_{1}(n)$ be a completely
multiplicative function with $|\beta_{1}(n)|\leq 1$ for all $n\geq 1$. Then
for any $H$ with $3\leq H\leq\frac{N_{1}}{s(\log N_{1})^{1/3}}$,

$\displaystyle\begin{aligned}
\sum_{a=1}^{s}\sum_{x=1}^{N_{1}}\Bigg{|}\sum_{\begin{subarray}{c}n=x\\\
n\equiv
a(mod~{}s)\end{subarray}}^{x+Hs}1_{\mathcal{S}}(n)\beta_{1}(n)\Bigg{|}\ll&N_{1}Hs\Big{(}\frac{(\log
Q_{1})^{1/6}}{P_{1}^{1/12-\eta/2}}+\frac{1}{(\log N_{1})^{1/200}}\Big{)}\\\
&+N_{1}Hs2^{J}\Bigg{(}\frac{M(\beta_{1};N_{1}/(\log
N_{1})^{2/3},N_{1},s)+1}{\exp\Big{(}M(\beta_{1};N_{1}/(\log
N_{1})^{2/3},N_{1},s)\Big{)}}\Bigg{)}^{1/2}.\end{aligned}$ (3.23)

###### Proof.

By the Cauchy-Schwarz inequality,

$\sum_{a=1}^{s}\sum_{x=N_{1}/(\log
N_{1})^{1/3}}^{N_{1}}\Bigg{|}\sum_{\begin{subarray}{c}n=x\\\ n\equiv
a(mod~{}s)\end{subarray}}^{x+Hs}1_{\mathcal{S}}(n)\beta_{1}(n)\Bigg{|}\leq(sN_{1})^{\frac{1}{2}}\Bigg{(}\sum_{a=1}^{s}\sum_{x=N_{1}/(\log
N_{1})^{1/3}}^{N_{1}}\Bigg{|}\sum_{\begin{subarray}{c}n=x\\\ n\equiv
a(mod~{}s)\end{subarray}}^{x+Hs}1_{\mathcal{S}}(n)\beta_{1}(n)\Bigg{|}^{2}\Bigg{)}^{\frac{1}{2}}.$
(3.24)

Note that $M(f;X,T,Y)$ defined as in equation (3.1) is increasing in $X$ and
decreasing both in $T$ and in $Y$, and $(x+1)e^{-x}$ is a decreasing function
of $x$ when $x\in[0,\infty)$. Then by Lemma 3.6,

$\displaystyle\sum_{a=1}^{s}\sum_{x=N_{1}/(\log
N_{1})^{1/3}}^{N_{1}}\Bigg{|}\sum_{\begin{subarray}{c}n=x\\\ n\equiv
a(mod~{}s)\end{subarray}}^{x+Hs}1_{\mathcal{S}}(n)\beta_{1}(n)\Bigg{|}^{2}$
$\displaystyle=\sum_{d|s}\sum_{\begin{subarray}{c}a=1\\\
(a,s/d)=1\end{subarray}}^{s/d}\sum_{x=N_{1}/(\log
N_{1})^{1/3}}^{N_{1}}\Bigg{|}\sum_{\begin{subarray}{c}n=x/d\\\ n\equiv
a(mod~{}s/d)\end{subarray}}^{x/d+Hs/d}1_{\mathcal{S}}(dn)\beta_{1}(dn)\Bigg{|}^{2}$
$\displaystyle\leq\sum_{d|s}d\sum_{\begin{subarray}{c}a=1\\\
(a,s/d)=1\end{subarray}}^{s/d}\sum_{x=N_{1}/(d(\log
N_{1})^{1/3})}^{N_{1}/d}\Bigg{|}\sum_{\begin{subarray}{c}n=x\\\ n\equiv
a(mod~{}s/d)\end{subarray}}^{x+Hs/d}1_{\mathcal{S}}(n)\beta_{1}(n)\Bigg{|}^{2}+O(N_{1}s)$
$\displaystyle\ll
H^{2}N_{1}\Big{(}\sum_{d|s}\varphi(\frac{s}{d})\Big{)}\Bigg{(}\frac{(\log
Q_{1})^{1/3}}{P_{1}^{1/6-\eta}}+\frac{1}{(\log
N_{1})^{1/65}}+2^{2J}\frac{M(\beta_{1};N_{1}/(\log
N_{1})^{2/3},N_{1},s)+1}{\exp\Big{(}M(\beta_{1};N_{1}/(\log
N_{1})^{2/3},N_{1},s)\Big{)}}\Bigg{)}$ $\displaystyle\ll
H^{2}N_{1}s\Bigg{(}\frac{(\log Q_{1})^{1/3}}{P_{1}^{1/6-\eta}}+\frac{1}{(\log
N_{1})^{1/65}}+2^{2J}\frac{M(\beta_{1};N_{1}/(\log
N_{1})^{2/3},N_{1},s)+1}{\exp\Big{(}M(\beta_{1};N_{1}/(\log
N_{1})^{2/3},N_{1},s)\Big{)}}\Bigg{)}.$

In the first inequality above, we used the facts that
$\beta_{1}(dn)=\beta_{1}(d)\beta_{1}(n)$ by the complete multiplicity of
$\beta_{1}$ and $1_{\mathcal{S}}(dn)=1_{\mathcal{S}}(n)$ when
$n\leq\frac{2N}{d}$ and $d|s$ by the definition of $\mathcal{S}$. So by
formula (3.24), we obtain formula (3.23). ∎

According to the previous analysis and Corollary 3.7, we now prove Theorem
3.4.

###### Proof of Theorem 3.4.

Note that $q\leq W$ and $W=\lfloor\log H\rfloor$ (i.e., formula (3.4)). So for
$N,H$ sufficiently large, $s\log H\leq\frac{1}{2}(\log N)^{1/32}$ implies that
$qs\leq\frac{3}{4}\big{(}\log(N/W)\big{)}^{1/32}$. Observe that $\frac{(\log
Q_{1})^{1/6}}{P_{1}^{1/12-1/300}}$ in (3.23) is $O\Big{(}\frac{1}{(\log
H)^{400}}\Big{)}$ by the choices (3.14). Applying Corollary 3.7 to formula
(3.21), we have that for $N$ large enough with $s\log H\leq\frac{1}{2}(\log
N)^{1/32}$,

$\displaystyle\sum_{t\leq W^{2}}\sum_{\begin{subarray}{c}e\leq W\\\ e\mid
qs\end{subarray}}\sum_{\begin{subarray}{c}a\leq W/e\\\
(a,qs/e)=1\end{subarray}}|\alpha(ae)|ae\sum_{{1\leq c\leq qs/e}}\sum_{n\leq
N/ae}\Bigg{|}\sum_{\begin{subarray}{c}b\in n+I_{t}/ae\\\ b\equiv
c(qs/e)\end{subarray}}1_{\mathcal{S}}(b)\beta_{1}(b)\Bigg{|}$
$\displaystyle\ll$ $\displaystyle\sum_{t\leq
W^{2}}\sum_{\begin{subarray}{c}e\leq W\\\ e\mid
qs\end{subarray}}\sum_{\begin{subarray}{c}a\leq W/e\\\
(a,qs/e)=1\end{subarray}}|\alpha(ae)|ae(\frac{qs}{e}\frac{N}{ae}\frac{W^{-2}Hs}{aqs})\Bigg{(}\frac{1}{(\log
H)^{400}}+\frac{1}{(\log N)^{1/200}}$ $\displaystyle
2^{J}\Big{(}M(\beta_{1};N/\log
N,2N,qs)+1\Big{)}^{1/2}\exp\Big{(}-\frac{1}{2}M(\beta_{1};N/\log
N,2N,qs)\Big{)}\Bigg{)}$ $\displaystyle\ll$ $\displaystyle
NHs\Bigg{(}\sum_{m\leq
W}\frac{|\alpha(m)|\tau(m)}{m}\Bigg{)}\Bigg{(}\frac{1}{(\log
H)^{400}}+\frac{1}{(\log N)^{1/200}}$
$\displaystyle+2^{J}\frac{M(\beta_{1};N/\log N,2N,(\log
N)^{1/32})+1}{\exp\Big{(}\frac{1}{2}M(\beta_{1};N/\log N,2N,(\log
N)^{1/32})\Big{)}}\Bigg{)}$ $\displaystyle\ll$ $\displaystyle
NHs\Bigg{(}\frac{1}{(\log H)^{400}}+\frac{1}{(\log N)^{1/200}}+(\log
N)^{o(1)}\frac{M(\beta_{1};N/\log N,2N,(\log
N)^{1/32})+1}{\exp\Big{(}\frac{1}{2}M(\beta_{1};N/\log N,2N,(\log
N)^{1/32})\Big{)}}\Bigg{)}.$

In the last inequality we used the estimate $2^{J}\ll(\log N)^{o(1)}$ by the
definition of $J$ after equation (3.15), and
$\sum_{m=1}^{\infty}\frac{|\alpha(m)|\tau(m)}{m}=O(1)$ by the Euler product
and the fact that $M(\beta;X,T,Y)=M(\beta_{1};X,T,Y)$ by the definition of
$\beta_{1}$ after formula (3.18). By the above and Lemma 3.5, together with
formulas (3.13), (3.19), (3.21), we obtain the conclusion stated in this
theorem. ∎

###  3.3. The estimate of minor arc case.

For the minor arc estimate, we shall prove the following result.

######  Theorem 3.8.

Let $s\geq 1$, $H\geq 3$ and $N\geq Hs$ be integers. Let
$\beta:\mathbb{N}\rightarrow\mathbb{C}$ be a multiplicative function with
$|\beta(n)|\leq 1$ for each $n\in\mathbb{N}$. Then we have

$(\ref{afternoon3})=\sum_{a=1}^{s}\sum_{n=1}^{N}\sum_{t=1}^{W^{2}}\sum_{j=0}^{q-1}\sum_{\begin{subarray}{c}h\in
I_{t}\\\ n+h\equiv
js+a(qs)\end{subarray}}\beta(n+h)\widetilde{F}_{a,n,j,t}(g^{\prime}(n,h)\Gamma^{\prime})\ll
NHs\frac{s}{\varphi(s)}\frac{\log\log H}{\log H}.$

Let $P,Q$ be parameters to be determined later with $W<P<Q<H$. Given an
integer $s\geq 1$, define $\mathcal{P}$ as the set of primes $p$ in $(P,Q]$
which are coprime to $s$. Note that here in our setting $(p,s)=1$ in
$\mathcal{P}$, which is a little different from that of [17, Section 6]. Let
$\widetilde{\mathcal{S}}=\\{n:1\leq n\leq N,\exists p\in\mathcal{P},p\mid
n\\}$. Similar to the proof of [17, Lemma 6.2], we have

$\sum_{\begin{subarray}{c}h\leq Hs\\\ n+h\in\widetilde{\mathcal{S}}\\\
n+h\equiv
a(s)\end{subarray}}\Bigg{|}\beta(n+h)-\sum_{p\in\mathcal{P}}\sum_{l\in\mathbb{N}}\frac{1_{pl=n+h}\beta(p)\beta(l)}{1+|\\{q\in\mathcal{P}:q\mid
l\\}|}\Bigg{|}\ll\frac{H}{P}.$ (3.25)

By the above inequality, we conclude the following result.

######  Lemma 3.9.

The expression

$\sum_{a=1}^{s}\sum_{n=1}^{N}\sum_{t=1}^{W^{2}}\sum_{j=0}^{q-1}\sum_{\begin{subarray}{c}h\in
I_{t}\\\ n+h\equiv
js+a(qs)\end{subarray}}1_{\widetilde{\mathcal{S}}}(n+h)\beta(n+h)\widetilde{F}_{a,n,j,t}(g^{\prime}(n,h)\Gamma^{\prime})$
(3.26)

is approximated by

$\sum_{a=1}^{s}\sum_{n\in\mathcal{N}}\sum_{t=1}^{W^{2}}\sum_{j=0}^{q-1}\sum_{p\in\mathcal{P}}\sum_{\begin{subarray}{c}l\in\mathbb{N}\\\
pl\equiv js+a(qs)\end{subarray}}\frac{1_{pl\in
n+I_{t}}\beta(p)\beta(l)}{1+|\\{q\in\mathcal{P}:q\mid
l\\}|}\widetilde{F}_{a,n,j,t}(g^{\prime}(n,pl-n)\Gamma^{\prime})$ (3.27)

up to an error $O(P^{-1}HNs+W^{-B/2}HNs)$, where $\mathcal{N}$ is the set
defined in equation (3.5).

In the following, we show that the estimate of (3.27) is reduced to proving
(3.33). Let

$P=2^{r_{-}},~{}~{}Q=2^{r_{+}},~{}~{}r_{-}=\lfloor 2\log\log
H\rfloor+1,~{}~{}r_{+}=\lfloor 1/10\log H\rfloor+1.$ (3.28)

Then (3.27) becomes

$\sum_{r\in(r_{-},r_{+}]}\sum_{a=1}^{s}\sum_{n\in\mathcal{N}}\sum_{t=1}^{W^{2}}\sum_{j=0}^{q-1}\sum_{\begin{subarray}{c}p\in(2^{r-1},2^{r}]\\\
(p,s)=1\end{subarray}}\sum_{\begin{subarray}{c}l\in\mathbb{N}\\\ pl\equiv
js+a(qs)\end{subarray}}\frac{1_{pl\in
n+I_{t}}\beta(p)\beta(l)}{1+|\\{q\in\mathcal{P}:q\mid
l\\}|}\widetilde{F}_{a,n,j,t}(g^{\prime}(n,pl-n)\Gamma^{\prime}).$ (3.29)

Given an integer $r\in(r_{-},r_{+}]$ and a prime $p\in(2^{r-1},2^{r}]$ with
$(p,s)=1$, by the choice of $P$ and $W$, we see that $(p,q)=1$. Fix an integer
$\overline{p}$ with $\overline{p}p\equiv 1(qs)$. Given $a=1,\ldots,s$ and
$j=0,\ldots,q-1$, let $b_{a}\in[1,s]$ with $\overline{p}b_{a}\equiv a(s)$.
Write $\overline{p}b_{a}=l_{a}s+a$ for some integer $l_{a}$. Define
$J_{p,a,j}$ as the integer in $[0,q-1]$ with

$J_{p,a,j}\equiv p(j-l_{a})(mod~{}q).$ (3.30)

In the following, for simplicity, we use j to denote $J_{p,a,j}$. Now the
expression inside the summation over $r$ in formula (3.29) becomes

$\displaystyle\begin{aligned} &\sum_{\begin{subarray}{c}p\in(2^{r-1},2^{r}]\\\
(p,s)=1\end{subarray}}\sum_{n\in\mathcal{N}}\sum_{t=1}^{W^{2}}\sum_{a=1}^{s}\sum_{j=0}^{q-1}\sum_{\begin{subarray}{c}l\in\mathbb{N}\\\
l\equiv\overline{p}(js+a)(qs)\end{subarray}}\frac{1_{pl\in
n+I_{t}}\beta(p)\beta(l)}{1+|\\{q\in\mathcal{P}:q\mid
l\\}|}\widetilde{F}_{a,n,j,t}(g^{\prime}(n,pl-n)\Gamma^{\prime})\\\
=&\sum_{\begin{subarray}{c}p\in(2^{r-1},2^{r}]\\\
(p,s)=1\end{subarray}}\sum_{n\in\mathcal{N}}\sum_{t=1}^{W^{2}}\sum_{a=1}^{s}\sum_{j=0}^{q-1}\sum_{\begin{subarray}{c}l\in\mathbb{N}\\\
l\equiv js+a(qs)\end{subarray}}\frac{1_{pl\in
n+I_{t}}\beta(p)\beta(l)}{1+|\\{q\in\mathcal{P}:q\mid
l\\}|}\widetilde{F}_{pa,n,\textbf{j},t}(g^{\prime}(n,pl-n)\Gamma^{\prime}).\end{aligned}$
(3.31)

Here we view the index $pa$ appearing in $\widetilde{F}$ as $pa(mod~{}s)$. By
Cauchy-Schwarz inequality, we can bound equation (3.31) by

$\displaystyle\leq\sum_{a=1}^{s}\sum_{j=0}^{q-1}\sum_{t=1}^{W^{2}}\sum_{\begin{subarray}{c}l\leq\frac{N+Hs}{2^{r-1}}\\\
l\equiv js+a(qs)\end{subarray}}\frac{|\beta(l)|}{1+|\\{q\in\mathcal{P}:q\mid
l\\}|}\Bigg{|}\sum_{n\in\mathcal{N}}\sum_{\begin{subarray}{c}p\in(2^{r-1},2^{r}]\\\
pl\in n+I_{t}\\\
(p,s)=1\end{subarray}}\beta(p)\widetilde{F}_{pa,n,\textbf{j},t}(g^{\prime}(n,pl-n)\Gamma^{\prime})\Bigg{|}$
$\displaystyle\leq\left(\frac{N}{2^{r-1}}W^{2}\right)^{1/2}\Bigg{(}\sum_{a=1}^{s}\sum_{j=0}^{q-1}\sum_{t=1}^{W^{2}}\sum_{\begin{subarray}{c}l\leq\frac{N+Hs}{2^{r-1}}\\\
l\equiv
js+a(qs)\end{subarray}}\Bigg{|}\sum_{n\in\mathcal{N}}\sum_{\begin{subarray}{c}p\in(2^{r-1},2^{r}]\\\
pl\in n+I_{t}\\\
(p,s)=1\end{subarray}}\beta(p)\widetilde{F}_{pa,n,\textbf{j},t}(g^{\prime}(n,pl-n)\Gamma^{\prime})\Bigg{|}^{2}\Bigg{)}^{1/2}$
$\displaystyle=\left(\frac{N}{2^{r-1}}W^{2}\right)^{1/2}\cdot S^{1/2},$ (3.32)

where

$S:=\sum_{a=1}^{s}\sum_{j=0}^{q-1}\sum_{t=1}^{W^{2}}\sum_{\begin{subarray}{c}l\leq\frac{N+Hs}{2^{r-1}}\\\
l\equiv
js+a(qs)\end{subarray}}\Bigg{|}\sum_{n\in\mathcal{N}}\sum_{\begin{subarray}{c}p\in(2^{r-1},2^{r}]\\\
pl\in n+I_{t}\\\
(p,s)=1\end{subarray}}\beta(p)\widetilde{F}_{pa,n,\textbf{j},t}(g^{\prime}(n,pl-n)\Gamma^{\prime})\Bigg{|}^{2}.$

Now the proof of Theorem 3.8 is reduced to proving

$S\ll 2^{r}W^{-(B_{2}+2)}H^{2}Ns^{2},$ (3.33)

where the parameter $B_{2}$ is given in formula (3.3).

###### Proof of Theorem 3.8.

(Assume formula (3.33)) By formulas (3.29), (3.31), (3.32) and (3.33), we have

$\sum_{a=1}^{s}\sum_{n\in\mathcal{N}}\sum_{t=1}^{W^{2}}\sum_{j=0}^{q-1}\sum_{p\in\mathcal{P}}\sum_{\begin{subarray}{c}l\in\mathbb{N}\\\
pl\equiv js+a(qs)\end{subarray}}\frac{1_{pl\in
n+I_{t}}\beta(p)\beta(l)}{1+|\\{q\in\mathcal{P}:q\mid
l\\}|}\widetilde{F}_{a,n,j,t}(g^{\prime}(n,pl-n)\Gamma^{\prime})\ll
r_{+}W^{-\frac{B_{2}}{2}}HNs.$

Then by Lemma 3.9,

$\sum_{a=1}^{s}\sum_{n=1}^{N}\sum_{t=1}^{W^{2}}\sum_{j=0}^{q-1}\sum_{\begin{subarray}{c}h\in
I_{t}\\\ n+h\equiv
js+a(qs)\end{subarray}}1_{\widetilde{\mathcal{S}}}(n+h)\beta(n+h)\widetilde{F}_{a,n,j,t}(g^{\prime}(n,h)\Gamma^{\prime})\ll(r_{+}W^{-\frac{B_{2}}{2}}+P^{-1}+W^{-\frac{B}{2}})HNs.$

We know that $B=100C_{0}^{2}$, $B_{2}=\frac{1}{10}C_{0}^{-2}B$ by formula
(3.3), $P=2^{\lfloor 2\log\log H\rfloor+1}$, $Q=2^{\lfloor 1/10\log
H\rfloor+1}$ by (3.28), and $W=\lfloor\log H\rfloor$ by (3.4). Note that
$B\geq 10$. Then

$\sum_{a=1}^{s}\sum_{n=1}^{N}\sum_{t=1}^{W^{2}}\sum_{j=0}^{q-1}\sum_{\begin{subarray}{c}h\in
I_{t}\\\ n+h\equiv
js+a(qs)\end{subarray}}1_{\widetilde{\mathcal{S}}}(n+h)\beta(n+h)\widetilde{F}_{a,n,j,t}(g^{\prime}(n,h)\Gamma^{\prime})\ll(\log
H)^{-1}NHs.$ (3.34)

We are left with estimating the contribution from

$\sum_{a=1}^{s}\sum_{n=1}^{N}\sum_{t=1}^{W^{2}}\sum_{j=0}^{q-1}\sum_{\begin{subarray}{c}h\in
I_{t}\\\ n+h\equiv js+a(qs)\\\
n+h\notin\widetilde{\mathcal{S}}\end{subarray}}\beta(n+h)\widetilde{F}_{a,n,j,t}(g^{\prime}(n,h)\Gamma^{\prime}).$
(3.35)

By the fundamental sieve (see formula (3.17)), we have

$(\ref{evening4})\leq\sum_{n=1}^{N}\sum_{\begin{subarray}{c}m=n\\\
m\notin\widetilde{\mathcal{S}}\end{subarray}}^{n+Hs}1=\sum_{\begin{subarray}{c}m=1\\\
m\notin\widetilde{\mathcal{S}}\end{subarray}}^{N+Hs}\sum_{n=m-Hs}^{m}1\ll
NHs\frac{\log P}{\log Q}\frac{s}{\varphi(s)}\ll
NHs\frac{s}{\varphi(s)}\frac{\log\log H}{\log H}.$

∎

In the end, we focus on showing formula (3.33). Denote by
$\mathcal{A}_{n,p,t}=\\{l\in\mathbb{N}:pl-n\in I_{t}\\}$ and
$\mathcal{A}_{n_{1},n_{2},p_{1},p_{2},t}=\mathcal{A}_{n_{1},p_{1},t}\cap\mathcal{A}_{n_{2},p_{2},t}$.
It is not hard to check that $|\mathcal{A}_{n_{1},n_{2},p_{1},p_{2},t}|\ll
2^{-r}W^{-2}Hs$. For given $t\in[W^{2}]$,
$n_{1}\in\mathcal{N},p_{1},p_{2}\in(2^{r-1},2^{r}]$,

$|\\{n_{2}:n_{2}\in\mathcal{N}:\mathcal{A}_{n_{1},n_{2},p_{1},p_{2},t}\neq\emptyset\\}|\leq
W^{-2}Hs.$ (3.36)

Expanding the square term in $S$, we have that $S$ is bounded by

$\displaystyle\begin{aligned}
&\sum_{a=1}^{s}\sum_{n_{1},n_{2}\in\mathcal{N}}\sum_{\begin{subarray}{c}p_{1},p_{2}\in(2^{r-1},2^{r}]\\\
(p_{1}p_{2},s)=1\end{subarray}}\sum_{j=0}^{q-1}\sum_{t=1}^{W^{2}}\\\
&\Bigg{|}\sum_{\begin{subarray}{c}l\in\mathcal{A}_{n_{1},n_{2},p_{1},p_{2},t}\\\
l\equiv
js+a(qs)\end{subarray}}\widetilde{F}_{p_{1}a,n_{1},\textbf{j}_{1},t}(g^{\prime}(n_{1},p_{1}l-n_{1})\Gamma^{\prime})\overline{\widetilde{F}_{p_{2}a,n_{2},\textbf{j}_{2},t}}(g^{\prime}(n_{2},p_{2}l-n_{2})\Gamma^{\prime})\Bigg{|},\end{aligned}$
(3.37)

where $\textbf{j}_{1}=J_{p_{1},a,j}$ and $\textbf{j}_{2}=J_{p_{2},a,j}$
defined as equation $(\ref{definition of capitalj})$. To prove formula (3.33),
it is enough to prove

$(\ref{10/28formula5})\ll 2^{r}W^{-(B_{2}+2)}H^{2}Ns^{2}.$ (3.38)

From Appendix A, the proof of formula (3.38) can be reduced to proving the
following proposition, which is a variant version of [17, Proposition 6.9] in
arithmetic progressions that is adapted to our situation.

######  Proposition 3.10.

Let $C_{0}$ be a large enough constant only depending on $d$, and let
$B_{2}\geq 10$, $B\geq 10C_{0}^{2}B_{2}$, $H\geq\max(W^{B},2^{10r_{+}})$.
Given $s\geq 1$, $j=0,\ldots,q-1$ and $t=1,\ldots,W^{2}$. Let $r_{-}$, $r_{+}$
be as in formula (3.28). For $r\in(r_{-},r_{+}]$, $n_{1}\in\mathcal{N}$ and a
prime number $p_{1}\in(2^{r-1},2^{r}]$ with $(p_{1},s)=1$, the set
$\Omega_{r,n_{1},p_{1},B_{2}}$ is defined to be the set of all triples:

$(a,n,p)\in[1,s]\times\mathcal{N}\times\\{p:p\in(2^{r-1},2^{r}],(p,s)=1\\}.$

such that

(i) $p$ is prime;

(ii) $|\mathcal{A}_{n_{1},n,p_{1},p,t}|\geq 2^{-r}W^{-(B_{2}+2)}Hs$;

(iii)

$\Bigg{|}\sum_{\begin{subarray}{c}l\in\mathcal{A}_{n_{1},n,p_{1},p,t}\\\
l\equiv
js+a(qs)\end{subarray}}\widetilde{F}_{p_{1}a,n_{1},\textbf{j}_{1},t}(g^{\prime}(n_{1},p_{1}l-n_{1})\Gamma^{\prime})\overline{\widetilde{F}_{pa,n,\textbf{j},t}}(g^{\prime}(n,pl-n)\Gamma^{\prime})\Bigg{|}\geq\frac{W^{-B_{2}}|\mathcal{A}_{n_{1},n,p_{1},p,t}|}{qs}.$

Then

$|\Omega_{r,n_{1},p_{1},B_{2}}|<2^{r}W^{-B_{2}-2}Hs^{2}.$ (3.39)

By the previous analysis and formulas (3.33) and (3.38), to prove Theorem 3.8,
it remains to proving Proposition 3.10. Next, we explain the main issue to
prove Proposition 3.10. In the proof, we mainly apply the method of [17,
Section 7], in which the authors deal with $g^{\prime}(n,pl-n)$, a polynomial
sequence of variables $n,p,l$ in general nilmanifolds. Specially
$g^{\prime}(n,pl-n)$ is a polynomial of variables $p,l$ in
$\mathbb{R}/\mathbb{Z}$ if $g^{\prime}(n,h)=f(n+h)$ for some polynomial
$f:\mathbb{Z}\rightarrow\mathbb{R}$. While for our case, it is not enough if
we just treat $g^{\prime}(n,p(sl+a)-n)$ as a polynomial of variables $p,l$.
Instead, we treat $g^{\prime}(n,p(sl+a)-n)$ as a polynomial of variables
$a,p,l$ because $a$ may vary in a wide range $[1,s]$ compared with $[1,H]$.
Combining with this idea and the proof of [17, Proposition 6.9], under the
assumption that (3.39) does not hold, we shall obtain a conclusion
contradicting the property that the sequence
$\\{g^{\prime}(n,h)\\}_{h\in\mathcal{A}_{a}}$ is totally
$W^{-B/2}$-equidistribution in $\mathbb{R}/\mathbb{Z}$ for almost all $a$ in
$[1,s]$ (see property (ii) below formula (3.6)). For self-containing, we give
the detailed proof of Proposition 3.10 in Appendix A.

Finally, we are ready to prove Theorem 3.1.

###### Proof of Theorem 3.1.

By formulas (3.10), (3.11), (3.12) and Theorems 3.4, 3.8, we obtain the
statement in this theorem. ∎

######  Remark 3.11.

In the estimate of the major arc case (see Subsection 3.2), if we do not split
the summation over $h$ into $n+h\in\mathcal{S}$ and $n+h\notin\mathcal{S}$, we
shall obtain the following estimate under the same assumptions as Theorem 3.1,

$\displaystyle\begin{aligned}
&\sum_{a=1}^{s}\sum_{n=1}^{N}\left|\sum_{\begin{subarray}{c}1\leq h\leq Hs\\\
n+h\equiv
a(mod~{}s)\end{subarray}}\beta(n+h)F\big{(}P(n+h)(mod~{}\mathbb{Z})\big{)}\right|\ll
NHs\Big{(}\frac{s}{\varphi(s)}\frac{\log\log H}{\log H}\Big{)}^{1/2}\\\
&+NHs\Big{(}\frac{\log\log H}{(\log H)^{1/2}}+\frac{1}{(\log
N)^{1/600}}\Big{)}+NHs\frac{M\big{(}\beta;N/\log N,2N,(\log
N)^{1/32}\big{)}+1}{\exp\Big{(}\frac{1}{2}M\big{(}\beta;N/\log N,2N,(\log
N)^{1/32}\big{)}\Big{)}}.\end{aligned}$ (3.40)

Compared with (3.2), it removes $(\log N)^{o(1)}$ before
$M(\beta;\cdot,\cdot,\cdot)\exp(-\frac{1}{2}M(\beta;\cdot,\cdot,\cdot))$.

##  4\. Proof of Theorem 1.2

To prove Theorem 1.2, we need some preparation. The following result is on the
self-correlation of $\mu(n)e(P(n))$ in short arithmetic progressions on
average.

######  Lemma 4.1.

Let $s\geq 1$ and $h\geq 3$ be integers. Suppose that $P(x)\in\mathbb{R}[x]$
is of degree $d\geq 0$. Then

$\limsup_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^{N}\Bigg{|}\frac{1}{h}\sum_{l=1}^{h}\mu(n+ls)e\big{(}P(n+ls)\big{)}\Bigg{|}^{2}\ll\frac{s}{\varphi(s)}\frac{\log\log
h}{\log h},$ (4.1)

where the implied constant depends on $d$ at most.

###### Proof.

Given $s\geq 1$ and $h\geq 3$, for $N$ large enough,

$\displaystyle\sum_{n=1}^{N}\Bigg{|}\sum_{l=1}^{h}\mu(n+ls)e\big{(}P(n+ls)\big{)}\Bigg{|}=\sum_{a=1}^{s}\sum_{\begin{subarray}{c}n=1\\\
n\equiv
a(mod~{}s)\end{subarray}}^{N}\Bigg{|}\sum_{l=1}^{h}\mu(n+ls)e\big{(}P(n+ls)\big{)}\Bigg{|}$
$\displaystyle=\sum_{a=1}^{s}\sum_{m=1}^{N/s}\Bigg{|}\sum_{l=1}^{h}\mu(ms+ls+a)e\big{(}P(ms+ls+a)\big{)}\Bigg{|}+O(hs)$
$\displaystyle=\sum_{a=1}^{s}\sum_{m=1}^{N/s}\Bigg{|}\sum_{\begin{subarray}{c}n=(m+1)s\\\
n\equiv
a(mod~{}s)\end{subarray}}^{(m+h+1)s}\mu(n)e\big{(}P(n)\big{)}\Bigg{|}+O(hs)$
$\displaystyle=\sum_{a=1}^{s}\sum_{\begin{subarray}{c}x=1\\\
s|x\end{subarray}}^{N}\Bigg{|}\sum_{\begin{subarray}{c}n=x\\\ n\equiv
a(mod~{}s)\end{subarray}}^{x+hs}\mu(n)e\big{(}P(n)\big{)}\Bigg{|}+O(hs),$

then

$\sum_{n=1}^{N}\Bigg{|}\sum_{l=1}^{h}\mu(n+ls)e\big{(}P(n+ls)\big{)}\Bigg{|}=\frac{1}{s}\sum_{a=1}^{s}\sum_{x=1}^{N}\Bigg{|}\sum_{\begin{subarray}{c}n=x\\\
n\equiv
a(mod~{}s)\end{subarray}}^{x+hs}\mu(n)e\big{(}P(n)\big{)}\Bigg{|}+O(N).$ (4.2)

By the above and Theorem 1.3,

$\displaystyle\limsup_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^{N}\Bigg{|}\sum_{l=1}^{h}\mu(n+ls)e\big{(}P(n+ls)\big{)}\Bigg{|}^{2}$
$\displaystyle\leq
h\limsup_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^{N}\Bigg{|}\sum_{l=1}^{h}\mu(n+ls)e\big{(}P(n+ls)\big{)}\Bigg{|}$
$\displaystyle\ll h^{2}\frac{s}{\varphi(s)}\frac{\log\log h}{\log h}$

as claimed. ∎

In the following, we prove a more general result than Theorem 1.2.

######  Lemma 4.2.

Let $(X,T)$ be a topological dynamical system. Suppose $x_{0}\in X$ satisfies
the following condition: for any $\nu\in\mathcal{M}(x_{0};X,T)$, there is a
dense set $\mathcal{F}\subseteq C(X)$ such that for each $g(x)\in\mathcal{F}$
we can find sequences $\\{h_{j}\\}_{j=1}^{\infty}$ and
$\\{s_{j}\\}_{j=1}^{\infty}$ (may depend on $\nu$, $g$) of positive integers
with

$\lim_{j\rightarrow\infty}\frac{\log\log h_{j}}{\log
h_{j}}\frac{s_{j}}{\varphi(s_{j})}=0$

and

$\lim_{j\rightarrow\infty}\frac{1}{h_{j}}\sum_{l=1}^{h_{j}}\|g\circ
T^{ls_{j}}-g\|_{L^{2}(\nu)}^{2}=0.$

Then for any $P(x)\in\mathbb{R}[x]$ and $f(x)\in C(X)$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^{N}\mu(n)e(P(n))f(T^{n}x_{0})=0.\
$ (4.3)

###### Proof.

Assume on the contrary, there is a $P(x)\in\mathbb{R}[x]$ and $f(x)\in C(X)$
such that formula (4.3) does not hold, then there is a constant $c_{0}>0$ and
an increasing sequence $\\{N_{m}\\}_{m=1}^{\infty}$ of positive integers such
that

$\frac{1}{N_{m}}\Bigg{|}\sum_{n=0}^{N_{m}-1}\mu(n)e(P(n))f(T^{n}x_{0})\Bigg{|}\geq
2c_{0}.$ (4.4)

Note that $X$ is a compact metric space. Then there is a subsequence of
$\\{N_{m}\\}_{m=1}^{\infty}$ (denoted by $\\{N_{m}\\}_{m=1}^{\infty}$ again
for convenience) and a $T$-invariant measure $\nu$ on $X$ such that
$\nu_{N_{m}}=\frac{1}{N_{m}}\sum_{n=0}^{N_{m}-1}\delta_{T^{n}x_{0}}$ weak*
converges to $\nu$ as $m\rightarrow\infty$, i.e., for any $h(x)\in C(X)$,

$\lim_{m\rightarrow\infty}\int_{X}h(x)d\nu_{N_{m}}=\lim_{m\rightarrow\infty}\frac{1}{N_{m}}\sum_{n=0}^{N_{m}-1}h(T^{n}x_{0})=\int_{X}h(x)d\nu.$

By formula (4.4) and the condition stated in this lemma, there is a $g(x)\in
C(X)$ and sequences $\\{h_{j}\\}_{j=1}^{\infty}$, $\\{s_{j}\\}_{j=1}^{\infty}$
of positive integers satisfying

$\lim_{j\rightarrow\infty}\frac{\log\log h_{j}}{\log
h_{j}}\frac{s_{j}}{\varphi(s_{j})}=0$ (4.5)

and

$\lim_{j\rightarrow\infty}\frac{1}{h_{j}}\sum_{l=1}^{h_{j}}\|g\circ
T^{ls_{j}}-g\|_{L^{2}(\nu)}^{2}=0.$ (4.6)

Moreover, for $m=1,2,\ldots$,

$\frac{1}{N_{m}}\Bigg{|}\sum_{n=0}^{N_{m}-1}\mu(n)e(P(n))g(T^{n}x_{0})\Bigg{|}\geq
c_{0}.$ (4.7)

Let $\widetilde{g}(n)=g(T^{n}x_{0})$. Note that

$\displaystyle\lim_{m\rightarrow\infty}\frac{1}{N_{m}}\sum_{n=0}^{N_{m}-1}|\widetilde{g}(n+ls_{j})-\widetilde{g}(n)|^{2}=$
$\displaystyle\lim_{m\rightarrow\infty}\frac{1}{N_{m}}\sum_{n=0}^{N_{m}-1}|g(T^{ls_{j}+n}x_{0})-g(T^{n}x_{0})|^{2}$
$\displaystyle=$ $\displaystyle\|g\circ T^{ls_{j}}-g\|_{L^{2}(\nu)}^{2}.$

So by equation (4.6),

$\lim_{j\rightarrow\infty}\frac{1}{h_{j}}\sum_{l=1}^{h_{j}}\lim_{m\rightarrow\infty}\frac{1}{N_{m}}\sum_{n=0}^{N_{m}-1}|\widetilde{g}(n+ls_{j})-\widetilde{g}(n)|^{2}=0.$
(4.8)

Let $\delta=\frac{c_{0}}{2(\|\widetilde{g}\|_{l^{\infty}}+1)}$. Then by Lemma
4.1 and equations (4.5), (4.8), there is a positive integer $j_{0}$ and a
positive integer $m_{0}$ such that whenever $m>m_{0}$,

$\frac{1}{N_{m}}\sum_{n=0}^{N_{m}-1}\Bigg{|}\frac{1}{h_{j_{0}}}\sum_{l=1}^{h_{j_{0}}}\mu(n+ls_{j_{0}})e\big{(}P(n+ls_{j_{0}})\big{)}\Bigg{|}^{2}<\delta^{2}/9$

and

$\frac{1}{h_{j_{0}}}\sum_{l=1}^{h_{j_{0}}}\frac{1}{N_{m}}\sum_{n=0}^{N_{m}-1}|\widetilde{g}(n+ls_{j_{0}})-\widetilde{g}(n)|^{2}<\delta^{2}/9.$

Then by the Cauchy-Schwarz inequality,

$\frac{1}{N_{m}}\sum_{n=0}^{N_{m}-1}\Bigg{|}\frac{1}{h_{j_{0}}}\sum_{l=1}^{h_{j_{0}}}\mu(n+ls_{j_{0}})e\big{(}P(n+ls_{j_{0}})\big{)}\Bigg{|}<\delta/3$
(4.9)

and

$\frac{1}{h_{j_{0}}}\sum_{l=1}^{h_{j_{0}}}\frac{1}{N_{m}}\sum_{n=0}^{N_{m}-1}|\widetilde{g}(n+ls_{j_{0}})-\widetilde{g}(n)|<\delta/3.$
(4.10)

Observe that

$\frac{1}{N_{m}}\sum_{n=0}^{N_{m}-1}\mu(n)e\big{(}P(n)\big{)}\widetilde{g}(n)=\frac{1}{N_{m}}\sum_{n=0}^{N_{m}-1}\mu(n+ls_{j_{0}})e\big{(}P(n+ls_{j_{0}})\big{)}\widetilde{g}(n+ls_{j_{0}})+O(\frac{ls_{j_{0}}}{N_{m}}).$

Then there is a positive integer $m_{1}$ such that whenever $m>m_{1}$,

$\Bigg{|}\frac{1}{N_{m}}\sum_{n=0}^{N_{m}-1}\mu(n)e\big{(}P(n)\big{)}\widetilde{g}(n)-\frac{1}{N_{m}}\sum_{n=0}^{N_{m}-1}\frac{1}{h_{j_{0}}}\sum_{l=1}^{h_{j_{0}}}\mu(n+ls_{j_{0}})e\big{(}P(n+ls_{j_{0}})\big{)}\widetilde{g}(n+ls_{j_{0}})\Bigg{|}<\delta/3.$
(4.11)

Let $m_{2}=\max\\{m_{0},m_{1}\\}$. Then by formulas (4.9), (4.10) and (4.11),
for $m>m_{2}$, we have

$\displaystyle\Bigg{|}\frac{1}{N_{m}}\sum_{n=0}^{N_{m}-1}\mu(n)e\big{(}P(n)\big{)}\widetilde{g}(n)\Bigg{|}<$
$\displaystyle\Bigg{|}\frac{1}{N_{m}}\sum_{n=0}^{N_{m}-1}\frac{1}{h_{j_{0}}}\sum_{l=1}^{h_{j_{0}}}\mu(n+ls_{j_{0}})e\big{(}P(n+ls_{j_{0}})\big{)}\widetilde{g}(n+ls_{j_{0}})\Bigg{|}+\delta/3$
$\displaystyle\leq$
$\displaystyle\Bigg{|}\frac{1}{N_{m}}\sum_{n=0}^{N_{m}-1}\frac{1}{h_{j_{0}}}\sum_{l=1}^{h_{j_{0}}}\mu(n+ls_{j_{0}})e\big{(}P(n+ls_{j_{0}})\big{)}\big{(}\widetilde{g}(n+ls_{j_{0}})-\widetilde{g}(n)\big{)}\Bigg{|}$
$\displaystyle+\Bigg{|}\frac{1}{N_{m}}\sum_{n=0}^{N_{m}-1}\frac{1}{h_{j_{0}}}\sum_{l=1}^{h_{j_{0}}}\mu(n+ls_{j_{0}})e\big{(}P(n+ls_{j_{0}})\big{)}\widetilde{g}(n)\Bigg{|}+\delta/3$
$\displaystyle\leq$
$\displaystyle\frac{1}{N_{m}}\sum_{n=0}^{N_{m}-1}\frac{1}{h_{j_{0}}}\sum_{l=1}^{h_{j_{0}}}|\widetilde{g}(n+ls_{j_{0}})-\widetilde{g}(n)|$
$\displaystyle+\frac{\|\widetilde{g}\|_{l^{\infty}}}{N_{m}}\sum_{n=0}^{N_{m}-1}\Bigg{|}\frac{1}{h_{j_{0}}}\sum_{l=1}^{h_{j_{0}}}\mu(n+ls_{j_{0}})e\big{(}P(n+ls_{j_{0}})\big{)}\Bigg{|}+\delta/3$
$\displaystyle<$ $\displaystyle c_{0}/2.$

This contradicts formula (4.7). Hence we obtain the claim in this lemma. ∎

######  Remark 4.3.

With the help of Theorem 3.1, we see that the results listed in Lemmas 4.1 and
4.2 also hold if $\mu$ is replaced by a more general “non-pretentious”
1-bounded multiplicative function (such as the Liouville function and
$\mu(n)\chi(n)$, where $\chi$ is a given Dirichlet character).

Note that functions realized in affine linear flows on compact abelian groups
of zero topological entropy can be approximated by finite linear combinations
of piece-wise polynomial phases (see [30, Subsections 2.1, 2.2]). As an
application of Lemma 4.2, we now prove Theorem 1.2.

###### Proof of Theorem 1.2.

In the following, we prove that for any $F\in C(X_{1}\times X_{2})$ and $x\in
X_{1}\times X_{2}$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^{N}\mu(n)F(T^{n}x)=0,$ (4.12)

where $T=T_{1}\times T_{2}$. Let $\mathcal{S}$ be the *-subalgebra (i.e.,
subalgebra closed under the complex conjugation) of $C(X_{1}\times X_{2})$
generated by $\\{\psi\cdot f:\psi$ is a character of $X_{1}$, $f\in C(X_{2})$
$\\}$. By the Stone-Weierstrass theorem (see, e.g., [KR83, Theorem 3.4.14]),
$\mathcal{S}$ is dense in $C(X_{1}\times X_{2})$. So it suffices to show that
for each $F\in C(X_{1}\times X_{2})$ in form of $\psi\cdot f$, equation (1.7)
holds. Given $x=(x_{1},x_{2})$, there is a positive integer $w$ such that

$\frac{1}{N}\sum_{n\leq
N}\mu(n)F(T^{n}x)=\frac{1}{N}\sum_{l=0}^{w-1}\sum_{\begin{subarray}{c}n\leq
N\\\ n\equiv l(mod~{}w)\end{subarray}}\mu(n)e(\phi(n))f(T_{2}^{n}x_{2}),$
(4.13)

where $\phi(n)$’s are some polynomials in $n$ with coefficients depending on
$w$, $l$ [30, equation (2.10)]. By Remark 4.3, Lemma 4.2 holds with $\mu(n)$
replaced by $\mu(n)1_{n\equiv l(mod~{}w)}$. Hence formula (4.13) converges to
zero as $N\rightarrow\infty$. ∎

##  5\. Proof of Theorem 1.4

In this section, we first show a sufficient condition to Problem 1, which
states the validity of Sarnak’s conjecture for product flows between affine
linear flows on compact abelian groups of zero topological entropy and all
rigid dynamical systems.

######  Proposition 5.1.

Assume that for any given $P(x)\in\mathbb{R}[x]$,

$\limsup_{N\rightarrow\infty}\frac{1}{Ns}\sum_{a=1}^{s}\sum_{n=1}^{N}\left|\sum_{\begin{subarray}{c}m=n+1\\\
m\equiv a(mod~{}s)\end{subarray}}^{n+hs}\mu(m)e(P(m))\right|=o(h)$ (5.1)

where the little “o” term is independent of $s\geq 1$. Then the Möbius
Disjointness Conjecture holds for the product flow $(X_{1}\times
X_{2},T_{1}\times T_{2})$, where $(X_{1},T_{1})$ is an affine linear flow on a
compact abelian group of zero entropy and $(X_{2},T_{2})$ is a flow such that
for any $x_{2}\in X_{2}$ and any $\nu\in\mathcal{M}(x_{2};X_{2},T_{2})$,
$(X_{2},\nu,T_{2})$ is rigid.

###### Proof.

Let $P(x)\in\mathbb{R}[x]$. Let $(X,T)$ be a flow satisfying that for any
$x_{0}\in X$ and any $\nu\in\mathcal{M}(x_{0};X,T)$, $(X,\nu,T)$ is rigid. We
claim that for any $x_{0}\in X$ and $g\in C(X)$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^{N}\mu(n)e(P(n))g(T^{n}x_{0})=0.$
(5.2)

Suppose that the sequence
$\frac{1}{N_{m}}\sum_{n=0}^{N_{m}-1}\delta_{T^{n}x_{0}}$ weak* converges to a
Borel probability measure $\nu$ on $X$. Let $\widetilde{g}(n)=g(T^{n}x_{0})$.
For any given $\epsilon>0$, let
$\delta=\frac{\epsilon}{3(\|\widetilde{g}\|_{l^{\infty}}+1)}$. Assume that
equation (5.1) holds, by equation (4.2), we have that

$\limsup_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^{N}\Bigg{|}\frac{1}{h}\sum_{l=1}^{h}\mu(n+ls)e\big{(}P(n+ls)\big{)}\Bigg{|}$

tends to zero uniformly with respect to $s$ as $h\rightarrow\infty$. By the
above and the rigidity of $(X,\nu,T)$, there is a positive integer $h_{0}$
such that

$\limsup_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^{N}\Bigg{|}\frac{1}{h_{0}}\sum_{l=1}^{h_{0}}\mu(n+ls)e\big{(}P(n+ls)\big{)}\Bigg{|}<\delta$
(5.3)

holds for any $s\geq 1$, and there is a positive integer $s_{j_{0}}$ such that

$\|g\circ
T^{s_{j_{0}}}-g\|_{L^{2}(\nu)}^{2}=\lim_{m\rightarrow\infty}\frac{1}{N_{m}}\sum_{n=1}^{N_{m}}|\widetilde{g}(n+s_{j_{0}})-\widetilde{g}(n)|^{2}<\Big{(}\frac{2\delta}{h_{0}+1}\Big{)}^{2}.$

Then by the Cauchy-Schwarz inequality,

$\limsup_{m\rightarrow\infty}\frac{1}{N_{m}}\sum_{n=1}^{N_{m}}|\widetilde{g}(n+s_{j_{0}})-\widetilde{g}(n)|<\frac{2\delta}{h_{0}+1}.$
(5.4)

By formula (5.4) and the triangle inequality,

$\displaystyle\begin{aligned}
&\limsup_{m\rightarrow\infty}\frac{1}{N_{m}}\sum_{n=1}^{N_{m}}\frac{1}{h_{0}}\sum_{l=1}^{h_{0}}|\widetilde{g}(n+ls_{j_{0}})-\widetilde{g}(n)|\leq\frac{1}{h_{0}}\sum_{l=1}^{h_{0}}\limsup_{m\rightarrow\infty}\frac{1}{N_{m}}\sum_{n=1}^{N_{m}}|\widetilde{g}(n+ls_{j_{0}})-\widetilde{g}(n)|\\\
&\leq\frac{1}{h_{0}}\sum_{l=1}^{h_{0}}\sum_{i=1}^{l}\limsup_{m\rightarrow\infty}\frac{1}{N_{m}}\sum_{n=1}^{N_{m}}|\widetilde{g}(n+is_{j_{0}})-\widetilde{g}(n+(i-1)s_{j_{0}})|\\\
&=\frac{1}{h_{0}}\sum_{l=1}^{h_{0}}\sum_{i=1}^{l}\limsup_{m\rightarrow\infty}\frac{1}{N_{m}}\sum_{n=1}^{N_{m}}|\widetilde{g}(n+s_{j_{0}})-\widetilde{g}(n)|<\delta.\end{aligned}$
(5.5)

We apply an argument similar to the proof of Lemma 4.2 with formulas (4.9) and
(4.10) are replaced by (5.3) and (5.5), respectively, then

$\limsup_{m\rightarrow\infty}\Bigg{|}\frac{1}{N_{m}}\sum_{n=1}^{N_{m}}\mu(n)e\big{(}P(n)\big{)}g(T^{n}x_{0})\Bigg{|}<\epsilon.$

Letting $\epsilon\rightarrow 0$, we have
$\lim_{m\rightarrow\infty}\frac{1}{N_{m}}\sum_{n=1}^{N_{m}}\mu(n)e\big{(}P(n)\big{)}g(T^{n}x_{0})=0$.
Since this equality holds for any sequence $\\{N_{m}\\}_{m}$ such that
$\frac{1}{N_{m}}\sum_{n=0}^{N_{m}-1}\delta_{T^{n}x_{0}}$ weak* converges,
equation (5.2) holds. It is obvious that the indicator function $1_{n\equiv
l(mod~{}w)}$ is periodic for any integer $l$ and any positive integer $w$.
Then $1_{n\equiv l(mod~{}w)}$ is a finite combination of $e(\frac{na}{w})$ for
$a=1,2,\ldots,w$. So equation (5.2) also holds with $\mu(n)$ replaced by
$\mu(n)1_{n\equiv l(mod~{}w)}$. By equations (4.13) and (5.2), we obtain the
claim in this proposition. ∎

######  Remark 5.2.

When $P(x)$ is a constant, condition (5.1) in the above proposition is implied
by Chowla’s conjecture. In fact, under the Chowla conjecture, for integer
$s\geq 1$,

$\limsup_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^{N}\left|\sum_{l=1}^{h}\mu(n+ls)\right|^{2}=6h/\pi^{2}.$

Note that

$\frac{1}{Ns}\sum_{a=1}^{s}\sum_{n=1}^{N}\left|\sum_{\begin{subarray}{c}m=n+1\\\
m\equiv
a(mod~{}s)\end{subarray}}^{n+hs}\mu(m)\right|=\frac{1}{N}\sum_{n=1}^{N}\left|\sum_{l=1}^{h}\mu(n+ls)\right|+O(1).$

So by the Cauchy-Schwarz inequality, the left-hand side of the above equation
$\leq c\sqrt{h}$, where $c$ is an absolute constant. It is obvious that this
bound is independent of $s$.

In the following, we shall prove Theorem 1.4, which gives a positive answer to
Problem 1 under the logarithmic average. Let $X$ be a compact metric space,
$T:X\rightarrow X$ a continuous map and $x_{0}\in X$. Then $C(X)$ is countably
generated as a C*-algebra [24, Remark 3.4.15]. For any integer $N\geq 1$,
consider the bounded linear functional $\frac{1}{\log
N}\sum_{n=1}^{N}\frac{1}{n}\delta_{T^{n}x_{0}}:f\mapsto\frac{1}{\log
N}\sum_{n=1}^{N}\frac{1}{n}f(T^{n}x_{0})$ for any $f\in C(X)$. Note that the
norms of these functionals are uniformly bounded with respect to $N$, by
Alaoglu-Bourbaki theorem, the bounded ball in $C(X)^{\sharp}$, the set of all
bounded linear functionals on $C(X)$, is weak*-compact. Moreover, the bounded
ball is also metrizable since $C(X)$ is countably generated. So there exist
(weak*) limit points of $\\{\frac{1}{\log
N}\sum_{n=1}^{N}\frac{1}{n}\delta_{T^{n}x_{0}}\\}_{N=2}^{\infty}$. For a limit
point along the subsequence $\\{N_{m}\\}_{m=1}^{\infty}$ in $C(X)^{\sharp}$,
denoted by $\rho$, by the Riesz representation theorem and [39, Theorem 6.8],
it is not hard to show that there is a unique $T$-invariant Borel probability
measure $\nu$ on $X$ such that for any $f(x)\in C(X)$,

$\rho(f)=\lim_{m\rightarrow\infty}\frac{1}{\log
N_{m}}\sum_{n=1}^{N_{m}}\frac{f(T^{n}x_{0})}{n}=\int_{X}f(x)d\nu(x).$ (5.6)

The next one is a general version of Theorem 1.4.

######  Proposition 5.3.

Let $(X,T)$ be a topological dynamical system such that for any $x_{0}\in X$
and any $\nu$ in the weak* closure of $\\{\frac{1}{\log
N}\sum_{n=1}^{N}\frac{1}{n}\delta_{T^{n}x_{0}}:N=2,\ldots\\}$ in the space of
Borel measures555Though $\frac{1}{\log
N}\sum_{n=1}^{N}\frac{1}{n}\delta_{T^{n}x_{0}}$ is not a probability measure,
it is not hard to show that $\nu$ is a $T$-invariant Borel probability measure
on X. on $X$, $(X,\nu,T)$ is rigid. Then the logarithmically averaged Möbius
Disjointness Conjecture holds for the product flow between $(X,T)$ and any
affine linear flow on a compact abelian group of zero entropy.

###### Proof.

Let $P(x)\in\mathbb{R}[x]$. We claim that for any $x_{0}\in X$ and $g\in
C(X)$,

$\lim_{N\rightarrow\infty}\frac{1}{N}\sum_{n=1}^{N}\frac{\mu(n)e(P(n))g(T^{n}x_{0})}{n}=0.$
(5.7)

Assume the contrary that formula (5.7) does not hold. Then there is a
$\delta>0$ and a subsequence $\\{N_{m}\\}_{m=1}^{\infty}$ of natural numbers
such that

$\frac{1}{\log
N_{m}}\Bigg{|}\sum_{n=1}^{N_{m}}\frac{\mu(n)e(P(n))g(T^{n}x_{0})}{n}\Bigg{|}>\delta.$
(5.8)

By (5.6), there is a subsequence of $\\{N_{m}\\}_{m=1}^{\infty}$ (denoted by
$\\{N_{m}\\}_{m=1}^{\infty}$ again for convenience) and a $T$-invariant Borel
probability measure $\nu$ on $X$, such that $\nu_{N_{m}}=\frac{1}{\log
N_{m}}\sum_{n=1}^{N_{m}}\frac{1}{n}\delta_{T^{n}x_{0}}$ weak* converges to
$\nu$ as $m\rightarrow\infty$, i.e., for any $f\in C(X)$,

$\lim_{m\rightarrow\infty}\int_{X}f(x)d\nu_{N_{m}}=\lim_{m\rightarrow\infty}\frac{1}{\log
N_{m}}\sum_{n=1}^{N_{m}}\frac{f(T^{n}x_{0})}{n}=\int_{X}f(x)d\nu.$

Let $\widetilde{g}(n)=g(T^{n}x_{0})$. Then by the condition in this
proposition and the above equation, there is a sequence
$\\{n_{j}\\}_{j=1}^{\infty}$ of positive integers such that

$\lim_{j\rightarrow\infty}\lim_{m\rightarrow\infty}\frac{1}{\log
N_{m}}\sum_{n=1}^{N_{m}}\frac{1}{n}|\widetilde{g}(n+n_{j})-\widetilde{g}(n)|^{2}=\lim_{j\rightarrow\infty}\int_{X}|g\circ
T^{n_{j}}(x)-g(x)|^{2}d\nu(x)=0.$ (5.9)

On the other hand, by the remark after [12, Corollary 1.5] and [37, Theorem
2], we have

$\lim_{N\rightarrow\infty}\frac{1}{\log
N}\sum_{n=1}^{N}\frac{\mu(n+h_{1})\mu(n+h_{2})e(P(n))}{n}=0$ (5.10)

for any $h_{1}\neq h_{2}\in\mathbb{N}$. Hence

$\displaystyle\begin{aligned} &\lim_{N\rightarrow\infty}\frac{1}{\log
N}\sum_{n=1}^{N}\frac{1}{n}\Bigg{|}\frac{1}{h}\sum_{l=1}^{h}\mu(n+ls)e\big{(}P(n+ls)\big{)}\Bigg{|}^{2}\\\
&=\frac{1}{h^{2}}\sum_{l_{1}=1}^{h}\sum_{l_{2}=1}^{h}\lim_{N\rightarrow\infty}\frac{1}{\log
N}\sum_{n=1}^{N}\frac{1}{n}\mu(n+l_{1}s)\mu(n+l_{2}s)e\big{(}P(n+l_{1}s)-P(n+l_{2}s)\big{)}\\\
&=\frac{1}{h^{2}}\sum_{l=1}^{h}\lim_{N\rightarrow\infty}\frac{1}{\log
N}\sum_{n=1}^{N}\frac{\mu^{2}(n+ls)}{n}+\frac{1}{h^{2}}\sum_{l_{1}=1}^{h}\sum_{\begin{subarray}{c}l_{2}=1\\\
l_{2}\neq l_{1}\end{subarray}}^{h}\lim_{N\rightarrow\infty}\frac{1}{\log
N}\sum_{n=1}^{N}\frac{\mu(n+l_{1}s)\mu(n+l_{2}s)e(Q(n))}{n}\\\
&=\frac{6}{\pi^{2}}\frac{1}{h},\end{aligned}$ (5.11)

where $Q(n)=P(n+l_{1}s)-P(n+l_{2}s)$ is some polynomial with coefficients
depending on $l_{1},l_{2},s$. So by the Cauchy-Schwarz inequality,

$\limsup_{N\rightarrow\infty}\frac{1}{\log
N}\sum_{n=1}^{N}\frac{1}{n}\Bigg{|}\frac{1}{h}\sum_{l=1}^{h}\mu(n+ls)e\big{(}P(n+ls)\big{)}\Bigg{|}=o_{h}(1),$
(5.12)

where the “$o$” term is independent of $s=1,2,\ldots$. Note that for any
bounded function $\widetilde{f}(n)$,

$\frac{1}{\log N}\sum_{n=1}^{N}\frac{\widetilde{f}(n)}{n}=\frac{1}{\log
N}\sum_{n=1}^{N}\frac{\widetilde{f}(n+m)}{n+m}+O(\frac{m}{\log
N})=\frac{1}{\log N}\sum_{n=1}^{N}\frac{\widetilde{f}(n+m)}{n}+O(\frac{m}{\log
N}).$ (5.13)

Then by equations (5.9), (5.12), (5.13), and an almost identical proof to
Proposition 5.1 with a slight difference by replacing the standard average
form by the logarithmic average, we obtain (5.7) and so the claim in this
proposition. ∎

###### Proof of Theorem 1.4.

In order to apply Proposition 5.3 to prove this theorem, we just need to check
that for any $\nu$ in the weak* closure of $\\{\frac{1}{\log
N}\sum_{n=1}^{N}\frac{1}{n}\delta_{T^{n}x_{0}}:N=1,2,\ldots\\}$, $(X,T,\nu)$
is rigid. This comes from the fact that $\nu$ is a $T$-invariant Borel
probability measure on $X$ and the condition in this theorem. ∎

As an application of Theorem 1.4, we prove Corollary 1.5.

###### Proof of Corollary 1.5.

By lifting $\psi:\mathbb{T}^{1}\rightarrow\mathbb{T}^{1}$ to the real line, we
can write $\psi(x)=cx+\psi_{1}(x)$ for all $x\in\mathbb{T}^{1}$, where
$c\in\mathbb{Z}$ and $\psi_{1}:\mathbb{T}^{1}\rightarrow\mathbb{R}$ is a
continuous 1-periodic function. Note that the set of linear combinations of
characters on $\mathbb{T}^{2}$ are dense in $C(\mathbb{T}^{2})$. Together with
(4.13), it is sufficient to prove that for any fixed $P(x)\in\mathbb{R}[x]$,
$(x_{1},x_{2})\in X$ and $b=(b_{1},b_{2})\in\mathbb{Z}^{2}$,

$\lim_{N\rightarrow\infty}\frac{1}{\log
N}\sum_{n=1}^{N}\frac{\mu(n)e(P(n))e\big{(}\langle
b,T_{\alpha,\psi}^{n}(x_{1},x_{2})\rangle\big{)}}{n}=0.$ (5.14)

Suppose that $T_{\alpha,\psi}^{n}(x_{1},x_{2})=(y_{1}(n),y_{2}(n))$, where
$y_{1}(n)=x_{1}+n\alpha$ and
$y_{2}(n)=c\frac{n(n-1)}{2}\alpha+cnx_{1}+x_{2}+\sum_{j=0}^{n-1}\psi_{1}(x_{1}+j\alpha)$.
It follows that

$\langle
b,T_{\alpha,\psi}^{n}(x_{1},x_{2})\rangle=b_{1}y_{1}(n)+b_{2}y_{2}(n)=Q(n)+b_{1}y_{1}(n)+b_{2}(x_{2}+\sum_{j=0}^{n-1}\psi_{1}(x_{1}+j\alpha)),$

where $Q(n)=b_{2}(c\frac{n(n-1)}{2}\alpha+cnx_{1})$. Let
$\widetilde{P}(n)=P(n)+Q(n)$. We rewrite (5.14) as

$\lim_{N\rightarrow\infty}\frac{1}{\log
N}\sum_{n=1}^{N}\frac{\mu(n)e(\widetilde{P}(n))e\big{(}\langle
b,T_{\alpha,\psi_{1}}^{n}(x_{1},x_{2})\rangle\big{)}}{n}=0.$ (5.15)

For the case $\alpha\in\mathbb{Q}$, an argument similar to that of [30,
Proposition 3.1] leads to (5.15). For
$\alpha\in\mathbb{R}\backslash\mathbb{Q}$, it follows from [6, Section 6.1]
that $(\mathbb{T}^{2},T_{\alpha,\psi_{1}},\nu)$ is rigid for any $\nu$, a
$T_{\alpha,\psi_{1}}$-invariant Borel probability measure on $\mathbb{T}^{2}$.
By Theorem 1.4, we have equation (5.15). ∎

##  Appendix A Proof of Proposition 3.10

In this section, we first show that the proof of formula (3.33) can be reduced
to proving Proposition 3.10. Then we give a proof of Proposition 3.10. These
results were used in Section 3.

######  Proposition 3.10 (restated).

Let $C_{0}$ be a large enough constant only depending on $d$, $B_{2}\geq 10$,
$B\geq 10C_{0}^{2}B_{2}$, $H\geq\max(W^{B},2^{10r_{+}})$. Let $s\geq 1$,
$j=0,\ldots,q-1$ and $t=1,\ldots,W^{2}$. Let $r_{-}$, $r_{+}$ be as in formula
(3.28). For $r\in(r_{-},r_{+}]$, $n_{1}\in\mathcal{N}$ and a prime number
$p_{1}\in(2^{r-1},2^{r}]$ with $(p_{1},s)=1$, the set
$\Omega_{r,n_{1},p_{1},B_{2}}$ is defined to be the set of all triples:

$(a,n,p)\in[1,s]\times\mathcal{N}\times\\{p:p\in(2^{r-1},2^{r}],(p,s)=1\\}.$

such that

(i) $p$ is prime;

(ii) $|\mathcal{A}_{n_{1},n,p_{1},p,t}|\geq 2^{-r}W^{-(B_{2}+2)}Hs$;

(iii)

$\Bigg{|}\sum_{\begin{subarray}{c}l\in\mathcal{A}_{n_{1},n,p_{1},p,t}\\\
l\equiv
js+a(qs)\end{subarray}}\widetilde{F}_{p_{1}a,n_{1},\textbf{j}_{1},t}(g^{\prime}(n_{1},p_{1}l-n_{1})\Gamma^{\prime})\overline{\widetilde{F}_{pa,n,\textbf{j},t}}(g^{\prime}(n,pl-n)\Gamma^{\prime})\Bigg{|}\geq\frac{W^{-B_{2}}|\mathcal{A}_{n_{1},n,p_{1},p,t}|}{qs}.$

Then

$|\Omega_{r,n_{1},p_{1},B_{2}}|<2^{r}W^{-B_{2}-2}Hs^{2}.$

We first show that Proposition 3.10 implies formula (3.33).

###### Proof of inequality (3.33) (Assume Proposition 3.10).

We first decompose (3.37) into three parts $S_{1}+S_{2}+S_{3}$ defined below.
Then we estimate each term.

By formula (3.36),

$\displaystyle S_{1}:=$
$\displaystyle\sum_{j=0}^{q-1}\sum_{t=1}^{W^{2}}\sum_{a=1}^{s}\sum_{\begin{subarray}{c}p_{1},p_{2}\in(2^{r-1},2^{r}]\\\
(p_{1}p_{2},s)=1\end{subarray}}\sum_{\begin{subarray}{c}n_{1},n_{2}\in\mathcal{N}\\\
|\mathcal{A}_{n_{1},n_{2},p_{1},p_{2},t}|<2^{-r}W^{-(B_{2}+2)}Hs\end{subarray}}$
$\displaystyle\Bigg{|}\sum_{\begin{subarray}{c}l\in\mathcal{A}_{n_{1},n_{2},p_{1},p_{2},t}\\\
l\equiv
js+a(qs)\end{subarray}}\widetilde{F}_{p_{1}a,n_{1},\textbf{j}_{1},t}(g^{\prime}(n_{1},p_{1}l-n_{1})\Gamma^{\prime})\overline{\widetilde{F}_{p_{2}a,n_{2},\textbf{j}_{2},t}}(g^{\prime}(n_{2},p_{2}l-n_{2})\Gamma^{\prime})\Bigg{|}$
$\displaystyle\ll
sqW^{2}\sum_{n_{1}\in\mathcal{N}}\sum_{\begin{subarray}{c}p_{1},p_{2}\in(2^{r-1},2^{r}]\\\
(p_{1}p_{2},s)=1\end{subarray}}\sum_{\begin{subarray}{c}n_{2}\in\mathcal{N}\\\
\mathcal{A}_{n_{1},n_{2},p_{1},p_{2},t}\neq\emptyset\end{subarray}}(|\mathcal{A}_{n_{1},n_{2},p_{1},p_{2},t}|/qs)$
$\displaystyle\ll 2^{r}W^{-(B_{2}+2)}H^{2}Ns^{2}.$

By Proposition 3.10,

$\displaystyle S_{2}:=$
$\displaystyle\sum_{j=0}^{q-1}\sum_{t=1}^{W^{2}}\sum_{n_{1}\in\mathcal{N}}\sum_{\begin{subarray}{c}p_{1}\in(2^{r-1},2^{r}]\\\
(p_{1},s)=1\end{subarray}}\sum_{\begin{subarray}{c}a,n_{2},p_{2}\\\
(a,n_{2},p_{2})\in\Omega_{r,n_{1},p_{1},B_{2}}\\\
|\mathcal{A}_{n_{1},n_{2},p_{1},p_{2},t}|\geq
2^{-r}W^{-(B_{2}+2)}Hs\end{subarray}}$
$\displaystyle\Bigg{|}\sum_{\begin{subarray}{c}l\in\mathcal{A}_{n_{1},n_{2},p_{1},p_{2},t}\\\
l\equiv
js+a(qs)\end{subarray}}\widetilde{F}_{p_{1}a,n_{1},\textbf{j}_{1},t}(g^{\prime}(n_{1},p_{1}l-n_{1})\Gamma^{\prime})\overline{\widetilde{F}_{p_{2}a,n_{2},\textbf{j}_{2},t}}(g^{\prime}(n_{2},p_{2}l-n_{2})\Gamma^{\prime})\Bigg{|}$
$\displaystyle\ll$ $\displaystyle
qW^{2}(2^{-r}W^{-2}Hs/qs)N2^{r}(2^{r}W^{-B_{2}-2}Hs^{2})\ll
2^{r}NH^{2}s^{2}W^{-(B_{2}+2)}.$

By the definition of $\Omega_{r,n_{1},p_{1},B_{2}}$ and formula (3.36),

$\displaystyle S_{3}:=$
$\displaystyle\sum_{j=0}^{q-1}\sum_{t=1}^{W^{2}}\sum_{n_{1}\in\mathcal{N}}\sum_{\begin{subarray}{c}p_{1}\in(2^{r-1},2^{r}]\\\
(p_{1},s)=1\end{subarray}}\sum_{\begin{subarray}{c}a,n_{2},p_{2}\\\
(a,n_{2},p_{2})\notin\Omega_{r,n_{1},p_{1},B_{2}}\\\
|\mathcal{A}_{n_{1},n_{2},p_{1},p_{2},t}|\geq
2^{-r}W^{-(B_{2}+2)}Hs\end{subarray}}$
$\displaystyle\Bigg{|}\sum_{\begin{subarray}{c}l\in\mathcal{A}_{n_{1},n_{2},p_{1},p_{2},t}\\\
l\equiv
js+a(qs)\end{subarray}}\widetilde{F}_{p_{1}a,n_{1},\textbf{j}_{1},t}(g^{\prime}(n_{1},p_{1}l-n_{1})\Gamma^{\prime})\overline{\widetilde{F}_{p_{2}a,n_{2},\textbf{j}_{2},t}}(g^{\prime}(n_{2},p_{2}l-n_{2})\Gamma^{\prime})\Bigg{|}$
$\displaystyle\ll$
$\displaystyle\sum_{j=0}^{q-1}\sum_{t=1}^{W^{2}}s\sum_{n_{1}\in\mathcal{N}}\sum_{\begin{subarray}{c}p_{1},p_{2}\in(2^{r-1},2^{r}]\\\
(p_{1}p_{2},s)=1\end{subarray}}\sum_{\begin{subarray}{c}n_{2}\in\mathcal{N}\\\
\mathcal{A}_{n_{1},n_{2},p_{1},p_{2},t}\neq\emptyset\end{subarray}}\frac{W^{-B_{2}}|\mathcal{A}_{n_{1},n,p_{1},p,t}|}{qs}$
$\displaystyle\ll$ $\displaystyle NH^{2}s^{2}2^{r}W^{-B_{2}-2}.$

Then we have

$S\leq(\ref{10/28formula5})=S_{1}+S_{2}+S_{3}\ll
2^{r}NH^{2}s^{2}W^{-(B_{2}+2)},$

which is claimed in formula (3.33). ∎

We now prove Proposition 3.10.

###### Proof of Proposition 3.10.

For any given $j,t,p_{1},n_{1}$, assume the contrary that

$|\Omega_{r,n_{1},p_{1},B_{2}}|\geq 2^{r}W^{-B_{2}-2}Hs^{2}.$

Write $\mathcal{A}_{n_{1},p_{1},t}=[m_{1},m_{2}]$ with $m_{1}<m_{2}$ positive
integers and $|m_{1}-m_{2}|\leq 2\cdot 2^{-r}W^{-2}Hs$. Let
$c_{0}=qs(\lfloor\frac{m_{1}}{qs}\rfloor-2)+js$. Then
$\\{l:l\in\mathcal{A}_{n_{1},p_{1},t},l\equiv
js+a(qs)\\}\subseteq\\{qsl+c_{0}+a:l\in[L]\\}$, where
$L=\lfloor\frac{m_{2}}{qs}\rfloor-\lfloor\frac{m_{1}}{qs}\rfloor+3\leq 4\cdot
2^{-r}W^{-3}H+4$. Let $(a,n,p)\in\Omega_{r,n_{1},p_{1},B_{2}}$. By conditions
$(ii)$ and $(iii)$, we have

$\Bigg{|}\sum_{l\in\mathcal{A}^{\prime}_{n,p}}\widetilde{F}_{p_{1}a,n_{1},\textbf{j}_{1},t}(g^{\prime}(n_{1},p_{1}(qsl+c_{0}+a)-n_{1})\Gamma^{\prime})\overline{\widetilde{F}_{pa,n,\textbf{j},t}}(g^{\prime}(n,p(qsl+c_{0}+a)-n)\Gamma^{\prime})\Bigg{|}\geq
W^{-B_{2}}|\mathcal{A}^{\prime}_{n,p}|,$ (A.1)

where $\mathcal{A}^{\prime}_{n,p}$ is a subinterval of $[L]$ with length
$\lfloor\frac{|\mathcal{A}_{n_{1},n,p_{1},p,t}|}{qs}\rfloor$ or
$\lfloor\frac{|\mathcal{A}_{n_{1},n,p_{1},p,t}|}{qs}\rfloor+1$. In the
following, we apply the method used in [17, Section 7] to our situation with
appropriate modifications.

Denote by $g_{a,n_{1},p_{1}}(u)=g^{\prime}(n_{1},qsu+p_{1}(c_{0}+a)-n_{1})$
and $g_{a,n,p}(u)=g^{\prime}(n,qsu+p(c_{0}+a)-n)$. By formula (A.1), the
sequence
$\\{(g_{a,n_{1},p_{1}}(p_{1}l)\Gamma^{\prime},g_{a,n,p}(pl)\Gamma^{\prime})\\}_{l\in\mathcal{A}^{\prime}_{n,p}}$
is not $2^{-2}W^{-B_{2}}$-equidistributed in $G^{\prime}/\Gamma^{\prime}\times
G^{\prime}/\Gamma^{\prime}$. By [17, Lemma 2.10], we can find a short length
$L^{\prime}_{n,p}\geq 2^{-5}W^{-2B_{2}}L$ such that the sequence
$\\{(g_{a,n_{1},p_{1}}(p_{1}l)\Gamma^{\prime},g_{a,n,p}(pl)\Gamma^{\prime})\\}_{l\in[L^{\prime}_{n,p}]}$
is not $2^{-5}W^{-2B_{2}}$-equidistributed in
$G^{\prime}/\Gamma^{\prime}\times G^{\prime}/\Gamma^{\prime}$. Then by Lemma
2.6, there is a character $\eta_{a,n,p}\in\mathbb{Z}^{2}$ of
$G^{\prime}/\Gamma^{\prime}\times G^{\prime}/\Gamma^{\prime}$ with
$0<|\eta_{a,n,p}|<W^{O(B_{2})}$ such that

$\|\eta_{a,n,p}\circ(g_{a,n_{1},p_{1}}(p_{1}l)\Gamma^{\prime},g_{a,n,p}(pl)\Gamma^{\prime})\|_{C^{\infty}[L^{\prime}_{n,p}]}\ll
W^{O(B_{2})}.$ (A.2)

As $L^{\prime}_{n,p}\gg W^{-2B_{2}}L$, this implies that

$\|\eta_{a,n,p}\circ(g_{a,n_{1},p_{1}}(p_{1}l)\Gamma^{\prime},g_{a,n,p}(pl)\Gamma^{\prime})\|_{C^{\infty}[L]}\ll
W^{O(B_{2})}.$ (A.3)

By the pigeonhole principle, we can find a non-zero additive character
$\eta\in\mathbb{Z}^{2}$ of $G^{\prime}/\Gamma^{\prime}\times
G^{\prime}/\Gamma^{\prime}$ such that for a set
$\Omega_{r,n_{1},p_{1},B_{2}}^{*}\subseteq\Omega_{r,n_{1},p_{1},B_{2}}$ with

$|\Omega_{r,n_{1},p_{1},B_{2}}^{*}|\geq 2^{r}W^{-O(B_{2})}Hs^{2},$ (A.4)

formula (A.3) holds and $\eta_{a,n,p}=\eta$. Let
$\mathcal{P}_{r,n_{1},p_{1}}=\\{p:2^{r-1}<p\leq 2^{r},(p,s)=1$, there are at
least $W^{-O(B_{2})}Hs^{2}$ choices of $(a,n)$ such that
$(a,n,p)\in\Omega_{r,n_{1},p_{1},B_{2}}^{*}\\}$. Since

$\Omega_{r,n_{1},p_{1},B_{2}}^{*}\leq|\mathcal{P}_{r,n_{1},p_{1}}|\cdot
s\cdot|\\{n:n\in\mathcal{N},\mathcal{A}_{n_{1},n,p_{1},p,t}\neq\emptyset\\}|+2^{r-1}W^{-O(B_{2})}Hs^{2},$

we have

$|\mathcal{P}_{r,n_{1},p_{1}}|\gg 2^{r}W^{-O(B_{2})}.$ (A.5)

Furthermore, for any $p\in\mathcal{P}_{r,n_{1},p_{1}}$, there are at least
$W^{-O(B_{2})}s$ numbers of $a\in[s]$ such that
$(a,n,p)\in\Omega_{r,n_{1},p_{1},B_{2}}^{*}$ for some $n$ by formula (3.36).
By Remark 2.9, we may suppose that
$g^{\prime}(n,h)=\sum_{i=0}^{d^{\prime}}\alpha_{i}^{\prime}(n+h)^{i}$ for some
$d^{\prime}$ with $1\leq d^{\prime}\leq d$. Write
$\eta=\eta_{1}\oplus\eta_{2}$. Due to the observation that the space of
Lipschitz functions on $\mathbb{R}/\mathbb{Z}$ is essentially spanned by the
space of pure phase functions $e(m\theta)$ and by a similar argument to the
proof of [15, Proposition 3.1], we have $\eta_{1}\neq 0$ and $\eta_{2}\neq 0$.
Suppose

$\eta_{1}\circ(g_{a,n_{1},p_{1}}(p_{1}l)\Gamma^{\prime})=\sum_{\begin{subarray}{c}i_{1},i_{2}\geq
0\\\ i_{1}+i_{2}\leq
d^{\prime}\end{subarray}}\gamma_{i_{1},i_{2}}a^{i_{1}}l^{i_{2}}.$

and

$\eta_{2}\circ(g_{a,n,p}(u)\Gamma^{\prime})=\sum_{\begin{subarray}{c}l_{1},l_{2},l_{1}^{\prime}\geq
0\\\ l_{1}+l_{2}\leq d^{\prime},l_{1}^{\prime}\leq
l_{1}\end{subarray}}\beta_{l_{1},l_{2},l_{1}^{\prime}}p^{l_{1}}a^{l_{1}^{\prime}}u^{l_{2}}.$
(A.6)

Then

$\eta\circ(g_{a,n_{1},p_{1}}(p_{1}l)\Gamma^{\prime},g_{a,n,p}(pl)\Gamma^{\prime})=\sum_{\begin{subarray}{c}i_{1},i_{2}\geq
0\\\ i_{1}+i_{2}\leq
d^{\prime}\end{subarray}}\gamma_{i_{1},i_{2}}a^{i_{1}}l^{i_{2}}+\sum_{i=0}^{d^{\prime}}\sum_{w=0}^{d^{\prime}-i}\sum_{w^{\prime}=0}^{w}\beta_{w,i,w^{\prime}}p^{i+w}l^{i}a^{w^{\prime}}.$
(A.7)

Now we treat the above as a polynomial in $l$. By formula (A.3) and Lemma 2.4,
for any $(a,n,p)\in\Omega^{*}_{r,n_{1},p_{1},B_{2}}$, there is a positive
integer $Z_{1}\ll O(1)$ such that for all $1\leq i\leq d^{\prime}$,

$\Bigg{\|}Z_{1}\Bigg{(}\sum_{w^{\prime}=0}^{d^{\prime}-i}\gamma_{w^{\prime},i}a^{w^{\prime}}+\sum_{w^{\prime}=0}^{d^{\prime}-i}\Big{(}\sum_{w\geq
w^{\prime}}^{d^{\prime}-i}\beta_{w,i,w^{\prime}}p^{w+i}\Big{)}a^{w^{\prime}}\Bigg{)}\Bigg{\|}_{\mathbb{R}/\mathbb{Z}}\ll
W^{O(B)}L^{-i}\ll 2^{ir}W^{O(B_{2})}H^{-i}.$ (A.8)

Using the pigeonhole principle, one can make $Z_{1}$ independent of $(a,n,p)$
after substituting $\Omega^{*}_{r,n_{1},p_{1},B_{2}}$ with a smaller subset
whose cardinality still satisfies formula (A.4). By formula (A.5), for any
$p\in\mathcal{P}_{r,n_{1},p_{1}}$, there is a set
$\mathcal{C}_{r,n_{1},p_{1},p}\subseteq[s]$ with

$|\mathcal{C}_{r,n_{1},p_{1},p}|\gg W^{-O(B_{2})}s,$ (A.9)

such that for any $a\in\mathcal{C}_{r,n_{1},p_{1},p}$, $(a,p)$ satisfies
formula (A.8). We now view

$Z_{1}\Bigg{(}\sum_{w^{\prime}=0}^{d^{\prime}-i}\gamma_{w^{\prime},i}a^{w^{\prime}}+\sum_{w^{\prime}=0}^{d^{\prime}-i}\Big{(}\sum_{w\geq
w^{\prime}}^{d^{\prime}-i}\beta_{w,i,w^{\prime}}p^{w+i}\Big{)}a^{w^{\prime}}\Bigg{)}$

as a polynomial in $a$. Then applying Lemma 2.5 (with
$\epsilon=2^{ir}W^{O(B_{2})}H^{-i}$ and $\delta=W^{-O(B_{2})}$) to formula
(A.8), there is a positive integer $Z_{2}$ with $Z_{2}\ll W^{O(B_{2})}$ such
that for any $i=1,\ldots,d^{\prime}$,

$\Bigg{\|}Z_{2}Z_{1}\Bigg{(}\sum_{w^{\prime}=0}^{d^{\prime}-i}\gamma_{w^{\prime},i}a^{w^{\prime}}+\sum_{w^{\prime}=0}^{d^{\prime}-i}\Big{(}\sum_{w\geq
w^{\prime}}^{d^{\prime}-i}\beta_{w,i,w^{\prime}}p^{w+i}\Big{)}a^{w^{\prime}}\Bigg{)}(mod~{}\mathbb{Z})\Bigg{\|}_{C^{\infty}[s]}\ll
2^{ir}W^{O(B_{2})}H^{-i}.$

By Lemma 2.4 again, there is a positive integer $Z_{3}$ with $Z_{3}\ll O(1)$
such that for $i=1,\ldots,d^{\prime}$ and $1\leq w^{\prime}\leq d^{\prime}-i$,

$\Bigg{\|}Z_{3}Z_{2}Z_{1}\Big{(}\gamma_{w^{\prime},i}+\sum_{w\geq
w^{\prime}}^{d^{\prime}-i}\beta_{w,i,w^{\prime}}p^{w+i}\Big{)}\Bigg{\|}_{\mathbb{R}/\mathbb{Z}}\ll
2^{ir}W^{O(B_{2})}H^{-i}s^{-w^{\prime}}.$ (A.10)

Again by the pigeonhole principle, one can make $Z_{2}$ and $Z_{3}$ both
independent of $p\in\mathcal{P}_{r,n_{1},p_{1}}$ after substituting
$\mathcal{P}_{r,n_{1},p_{1}}$ with a smaller subset whose cardinality remains
satisfying formula (A.5). By formulas (A.8) and (A.10), and the triangle
inequality, we have that for $i=1,\ldots,d^{\prime}$ and $w^{\prime}=0$,

$\Bigg{\|}Z_{3}Z_{2}Z_{1}\Big{(}\gamma_{0,i}+\sum_{w\geq
0}^{d^{\prime}-i}\beta_{w,i,0}p^{w+i}\Big{)}\Bigg{\|}_{\mathbb{R}/\mathbb{Z}}\ll
2^{ir}W^{O(B_{2})}H^{-i}.$ (A.11)

Applying Lemmas 2.5 and 2.4 again to the following polynomials

$Z_{3}Z_{2}Z_{1}\Big{(}\gamma_{w^{\prime},i}+\sum_{w\geq
w^{\prime}}^{d^{\prime}-i}\beta_{w,i,w^{\prime}}p^{w+i}\Big{)}$

of $p\in[2^{r}]$, then there are positive integers $Z_{4},Z_{5}$ with
$Z_{4}\ll W^{O(B_{2})}$ and $Z_{5}\ll O(1)$, such that for any
$i=1,\ldots,d^{\prime}$, $0\leq w^{\prime}\leq d^{\prime}-i$ and
$w^{\prime}\leq w\leq d^{\prime}-i$,

$\|Z_{5}Z_{4}Z_{3}Z_{2}Z_{1}\beta_{w,i,w^{\prime}}\|_{\mathbb{R}/\mathbb{Z}}\ll
2^{-wr}W^{O(B_{2})}H^{-i}s^{-w^{\prime}}.$ (A.12)

Let $Z=Z_{5}Z_{4}Z_{3}Z_{2}Z_{1}$. Then $Z\ll W^{O(B_{2})}$ and the character
$Z\eta_{2}$ satisfies

$|Z\eta_{2}|\ll|Z||\eta|\ll W^{O(B_{2})}.$ (A.13)

Now we choose a sufficiently large positive number $C_{0}=O(1)$ which serves
as the implicit constant in the exponent $O(B_{2})$ appearing in the above. We
write formula (A.12) as

$\|Z\beta_{l_{1},l_{2},l_{1}^{\prime}}\|_{\mathbb{R}/\mathbb{Z}}\ll
2^{-l_{1}r}W^{C_{0}B_{2}}H^{-l_{2}}s^{-l_{1}^{\prime}},$ (A.14)

which holds for any $l_{1}\geq 0,l_{2}\geq 1$, $l_{1}+l_{2}\leq d^{\prime}$
and $l_{1}^{\prime}\leq l_{1}$.

By formula (3.36), the number of pairs $(n,p)$ with
$(n,p)\in\Omega_{r,n_{1},p_{1},B_{2}}^{*}$ is at most $2^{r}Hs$. Since
$|\Omega_{r,n_{1},p_{1},B_{2}}^{*}|\geq 2^{r}W^{-C_{0}B_{2}}Hs^{2}$, there is
a pair $(n,p)\in\mathcal{N}\times\\{p:p\in(2^{r-1},2^{r}],(p,s)=1\\}$ such
that at least $W^{-C_{0}B_{2}}s$ choices of $a\in[s]$ have the property
$(a,n,p)\in\Omega_{r,n_{1},p_{1},B_{2}}^{*}$. For such $(a,n,p)$, we consider
the set $\mathcal{U}_{a}=\\{u\in\mathbb{Z}:qsu+p(c_{0}+a)-n\in[Hs]\\}$. By
formula (A.1), there is an $l_{0}\in[L]$ such that
$qspl_{0}+p(c_{0}+a)-n\in[Hs]$. Since $L\leq 4\cdot 2^{-r}W^{-3}H+4$,
$-Hs<p(c_{0}+a)-n\leq Hs$. So for any $u\in\mathcal{U}_{a}$, $|u|\leq 2H/q$.
Note that
$\lfloor\frac{H}{q}\rfloor-2\leq|\mathcal{U}_{a}|\leq\lfloor\frac{H}{q}\rfloor+2$.
Fix any subinterval $\mathcal{U}_{a}^{\prime}\subseteq\mathcal{U}_{a}$ with
length $\lfloor\frac{2W^{-2C_{0}B_{2}-3}H}{q}\rfloor$. For any
$u_{1},u_{2}\in\mathcal{U}_{a}^{\prime}$, by formulas (A.6) and (A.14), we
have the following estimate.

$\displaystyle\begin{aligned} &\|Z\eta_{2}\circ
g^{\prime}(n,qsu_{1}+p(c_{0}+a)-n)\Gamma^{\prime}-Z\eta_{2}\circ
g^{\prime}(n,qsu_{2}+p(c_{0}+a)-n)\Gamma^{\prime}\|_{\mathbb{R}/\mathbb{Z}}\\\
=&\|Z\sum_{\begin{subarray}{c}l_{1}\geq 0,l_{2}\geq 1,l_{1}^{\prime}\geq 0\\\
l_{1}+l_{2}\leq d^{\prime},l_{1}^{\prime}\leq
l_{1}\end{subarray}}\beta_{l_{1},l_{2},l_{1}^{\prime}}p^{l_{1}}a^{l_{1}^{\prime}}(u_{1}-u_{2})\sum_{h=0}^{l_{2}-1}u_{1}^{h}u_{2}^{l_{2}-1-h}\|_{\mathbb{R}/\mathbb{Z}}\\\
\ll&\sum_{\begin{subarray}{c}l_{1}\geq 0,l_{2}\geq 1,l_{1}^{\prime}\geq 0\\\
l_{1}+l_{2}\leq d^{\prime},l_{1}^{\prime}\leq
l_{1}\end{subarray}}W^{-C_{0}B_{2}-3}q^{-l_{2}}\leq
C_{0}W^{-C_{0}B_{2}}.\end{aligned}$ (A.15)

Set $F(x)=e(Z\eta_{2}(x))$, which is a Lipschitz function from
$G^{\prime}/\Gamma^{\prime}$ to $\mathbb{C}$ with $\|F\|_{Lip}\leq 2\pi
W^{C_{0}B_{2}}$ by formula (A.13). Let
$\mathcal{B}_{a}^{\prime}=\\{qsu+p(c_{0}+a)-n:u\in\mathcal{U}_{a}^{\prime}\\}$.
Then by formula (A.15),

$|\mathbb{E}_{h\in\mathcal{B}_{a}^{\prime}}F(g^{\prime}(n,h)\Gamma^{\prime})|>1-C_{0}W^{-C_{0}B_{2}}>1/2.$

Note that $\mathcal{B}_{a}^{\prime}$ is an arithmetic progression in
$\mathcal{B}_{a}=\\{h\in[Hs]:n+h\equiv pa(s)\\}$ with length greater than
$W^{-2C_{0}B_{2}-4}H$. By the above, it follows that the sequence
$\\{g^{\prime}(n,h)\Gamma^{\prime}\\}_{h\in\mathcal{B}_{a}}$ is not totally
$\min(W^{-2C_{0}B_{2}-4},\frac{1}{4\pi}W^{-C_{0}B_{2}})$-equidistributed in
$G^{\prime}/\Gamma^{\prime}$. By the assumption in the proposition,
$\min(W^{-2C_{0}B_{2}-4},\frac{1}{4\pi}W^{-C_{0}B_{2}})\geq W^{-C_{0}^{-1}B}$.
So there is an $n\in\mathcal{N}$ such that for at least $W^{-C_{0}B_{2}}s$
choices of $a\in[s]$,
$\\{g^{\prime}(n,h)\Gamma^{\prime}\\}_{h\in\mathcal{B}_{a}}$ is not totally
$W^{-C_{0}^{-1}B}$-equidistributed in $G^{\prime}/\Gamma^{\prime}$. This
contradicts the construction of $\mathcal{N}$ in formula (3.5). We complete
the proof. ∎

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