# A quantitative bound on Furstenberg-Sárközy patterns with shifted prime
power common differences in primes

Mengdi Wang  Department of Mathematics, KTH Royal Institute of Technology, 100
44 Stockholm, Sweden<EMAIL_ADDRESS>
###### Abstract.

Let $k\geqslant 1$ be a fixed integer, and $\mathcal{P}_{N}$ be the set of
primes no more than $N$. We prove that if a set
$\mathcal{A}\subset\mathcal{P}_{N}$ contains no patterns
$p_{1},p_{1}+(p_{2}-1)^{k}$, where $p_{1},p_{2}$ are prime numbers, then

$\frac{|\mathcal{A}|}{|\mathcal{P}_{N}|}\ll(\log\log
N)^{-\frac{1}{4k^{3}+17k^{2}}}.$

###### Contents

  1. 1 Introduction
  2. 2 Notations and Preliminaries
  3. 3 Exponential sum estimates
  4. 4 Restriction for shifted prime powers
  5. 5 A Local Inverse Theorem
  6. 6 Density Increment
  7. A Proof of Lemma 2.3
  8. B Restriction for prime numbers
  9. C Estimation of the singular series (6.2)

##  1\. Introduction

Lovász conjectured that any integer set without non-zero perfect square
differences must have asymptotic density zero. This conjecture was confirmed
by Furstenberg [8] and Sárközy [23] independently. Furstenberg used ergodic
theory and his result is just a purely qualitative one. However, using the
Fourier-analytic density increment method, Sárközy proved the following
quantitative strengthening.

######  Theorem A (Sárközy [23]).

If $\mathcal{A}$ is a subset of $[N]=\left\\{1,\cdots,N\right\\}$ and lacks
non-trivial patterns $x,x+y^{2}(y\neq 0)$, then

$\frac{|\mathcal{A}|}{N}\ll\Bigl{(}\frac{(\log\log N)^{2}}{\log
N}\Bigr{)}^{1/3}.$

Pintz, Steiger and Szemerédi [19] created a double iteration strategy which
refined Sárközy’s proof and improved the above upper bound to

$\frac{|\mathcal{A}|}{N}\ll(\log N)^{-\frac{1}{12}\log\log\log\log N}.$

Subsequently, this method has been generalized to deal with sets without
differences of the form $n^{k}(k\geqslant 3)$ by Balog, Pelikán, Pintz and
Szemerédi [2]. They proved that for any integer $k\geqslant 3$, if
$\mathcal{A}\subset[N]$ lacks non-trivial patterns $x,x+y^{k}(y\neq 0)$, then

$\frac{|\mathcal{A}|}{N}\ll(\log N)^{-\frac{1}{4}\log\log\log\log N}.$

Very recently, Bloom and Maynard gained a much more efficient density
increment argument by showing that no set $\mathcal{A}$ can have many large
Fourier coefficients which are rationals with small and distinct denominators.
Taking the advantage of this density increment argument they can prove

######  Theorem B (Bloom-Maynard [3]).

If $\mathcal{A}\subset[N]$ lacks non-trivial patterns $x,x+y^{2}$ with $y\neq
0$, then

$\frac{|\mathcal{A}|}{N}\ll(\log N)^{-c\log\log\log N}$

for some absolute constant $c>0$.

In the same series of papers (to study difference sets of subsets of
integers), Sárközy also answered a question posed by Erdős, proving that if a
subset of integers does not contain two elements which differ by one less than
a prime number, then this subset has asymptotic density zero. Assume that
$\mathcal{A}$ is a subset of $[N]$, in the following theorems we’ll use
$\mathcal{A}-\mathcal{A}$ to denote the difference set
$\left\\{a_{1}-a_{2}:a_{1},a_{2}\in\mathcal{A}\right\\}$. More formally,
Sárközy proved that

######  Theorem C (Sárközy [24]).

If $\mathcal{A}\subset[N]$ and $p-1\not\in\mathcal{A}-\mathcal{A}$ for all
primes $p$, then

$\frac{|\mathcal{A}|}{N}\ll\frac{(\log\log\log N)^{3}\log\log\log\log
N}{(\log\log N)^{2}}.$

By exploiting a dichotomy depending on whether the exceptional zero of
Dirichlet $L$-functions occurs or not, Ruzsa and Sanders [22] proved the above
upper density bound can be
$\frac{|\mathcal{A}|}{N}\ll\exp\bigl{(}-O\bigl{(}(\log
N)^{1/4}\bigr{)}\bigr{)}$. Very recently, Green [11] improved this bound to
$N^{-c}$ by studying the van der Corput property for the shifted primes, and
such a bound is out of the strength of the density increment argument.
Subsequently, Thorner and Zaman [25] shows that this constant can be
$c=3.4/10^{18}$.

Li and Pan combined the above two questions posed by Lovász and Erdős
respectively and showed that

######  Theorem D (Li-Pan [16]).

Let $h\in\mathbb{Z}[x]$ be a polynomial with positive leading term and zero
constant term. If $\mathcal{A}\subset[N]$ and
$h(p-1)\not\in\mathcal{A}-\mathcal{A}$ for all primes $p$, then

$\frac{|\mathcal{A}|}{N}\ll\frac{1}{\log\log\log N}.$

Rice improved the above triple logarithmic decay to a single logarithmic
bound. Rice also generalized this kind of consideration to a larger class of
polynomials named $\mathcal{P}$-intersective polynomials (see [20, Definition
1]).

######  Theorem E (Rice [20]).

Let $h\in\mathbb{Z}[x]$ be a $\mathcal{P}$-intersective polynomial of degree
$k\geqslant 2$ with positive leading term. If $\mathcal{A}\subset[N]$ and
$h(p)\not\in\mathcal{A}-\mathcal{A}$ for all primes $p$ with $h(p)>0$, then

$\frac{|\mathcal{A}|}{N}\ll(\log N)^{-c}$

for any $0<c<\frac{1}{2k-2}$.

Similar to Green-Tao theorem [13], which shows that the prime numbers contain
arbitrary long arithmetic progressions, one interesting question is to study
whether the sequences of prime numbers contain the above-described patterns.
As the readers may see from Theorem B, primes trivially contain $x,x+y^{2}$
with $y\neq 0$. By adopting the transference principle [12], Li-Pan and Rice
can also prove the corresponding $\mathcal{P}$-intersective polynomial
difference results of subsets of prime numbers.

######  Theorem F (Li-Pan [16], Rice [20]).

Let $h\in\mathbb{Z}[x]$ be a zero constant term polynomial. If
$\mathcal{A}\subset\mathcal{P}_{N}$ is a subset of primes and $\mathcal{A}$
does not contain patterns $p_{1},p_{1}+h(p_{2}-1)$ for all primes
$p_{1},p_{2}$, then

$|\mathcal{A}|=o(|\mathcal{P}_{N}|).$

In this paper, we are planning to provide Theorem F a quantitative relative
upper density bound when $h\in\mathbb{Z}[x]$ takes the form of the perfect
$k$-th power polynomial, i.e. $h(x)=x^{k}$.

######  Theorem 1.1.

Let $k\geqslant 1$ be a fixed integer, and $\mathcal{P}_{N}$ be the set of
primes which are no more than $N$. If $\mathcal{A}\subset\mathcal{P}_{N}$ and
$(p-1)^{k}\not\in\mathcal{A}-\mathcal{A}$ for all primes $p$, then

$\frac{|\mathcal{A}|}{|\mathcal{P}_{N}|}\ll\bigl{(}\log\log
N\bigr{)}^{-\frac{1}{4k^{3}+17k^{2}}}.$

Besides, we believe that our method can be generalised to deal with all
polynomials with zero constant term, and hopefully all
$\mathcal{P}$-intersective polynomials. This would, however, involve more
complicated exponential sum estimates.

As a proof ingredient of Theorem 1.1, in this paper, we are also concerned
with the restriction estimates of primes and prime powers in arithmetic
progressions. Suppose that $1\leqslant b\leqslant d$ are coprime integers, $k$
is an integer, and $M=\left\lfloor N^{1/k}\right\rfloor$. Suppose that
$P=b+d\cdot[X]\subseteq[N]$ is an arithmetic progression, define

$\displaystyle\Lambda_{b,d}(x)=\begin{cases}\frac{\phi(d)}{d}\Lambda(b+dx)\quad&\text{
if }1\leqslant x\leqslant X;\\\ 0&\text{ otherwise,}\end{cases}$ (1.1)

and for any $\alpha\in\mathbb{T}$ define

$\displaystyle
S_{d}(\alpha)=\sum_{y\in[M]}\frac{\phi(d)}{d}ky^{k-1}\Lambda(dy+1)e(y^{k}\alpha).$
(1.2)

Then:

######  Theorem 1.2.

Suppose that $1\leqslant b\leqslant d\leqslant\exp\bigl{(}c^{\prime}\sqrt{\log
N}\bigr{)}$ are coprime integers with some $0<c^{\prime}<1$,

  1. (1)

If $\nu:\mathbb{Z}\to\mathbb{C}$ is a function supported on $[X]$ and
satisfying $|\nu|\leqslant\Lambda_{b,d}$ pointwise, the for any real $p>2$

$\int_{\mathbb{T}}\Bigl{|}\sum_{x\in[X]}\nu(x)e(x\alpha)\Bigr{|}^{p}\,\,\mathrm{d}\alpha\ll
X^{p-1};$

  2. (2)

Let $p>k(k+1)+2$ be a real number. $S_{d}(\alpha)$ is the exponential sum
defined in (1.2), then we have

$\int_{\mathbb{T}}\bigl{|}S_{d}(\alpha)\bigr{|}^{p}\,\mathrm{d}\alpha\ll_{p}N^{p-1}.$

We actually generalize the restriction estimates to modulus
$d\ll\Bigl{(}\exp\bigl{(}c^{\prime}\sqrt{\log N}\bigr{)}\Bigr{)}$, where in
the literature [12], [16], [6] and so on, the modulus is always small
$d\ll(\log N)^{A}$.

### Outline of the argument

As mentioned above, the traditional approach to finding (low complexity)
arithmetic configurations in primes is the so-called (Fourier-analysis)
transference principle which was essentially developed by Green and Tao [9],
[12]. Roughly speaking, if we can bound the subset of integers without
targeted configurations via density increment argument, one can hope to
construct a pseudorandom majorant of the subset of primes to make this
majorant indistinguishable from subsets of integers in certain statistical
senses. In this paper, instead, we’d like to do the density increment
procedure in the subset of primes directly. And to our knowledge, this is the
first attempt to run the density increment in a sparse set.

As expected, compared to the situation in integers, to make the density
increment argument work successfully in primes many questions need to be
overcome. Firstly, the “translation-invariant” property is broken. Assume that
the goal is to find $x,x+y^{k}\in\mathcal{A}$ in the interval $[N]$ and there
is an arithmetic progression $P=a+q\cdot[X]$ inside $[N]$ such that $A$ has an
increment density in $P$, then more-or-less we can transfer the matter to find
$x,x+q^{k-1}y^{k}\in\mathcal{A}^{\prime}$ in the interval $[X]$, where
$\mathcal{A}^{\prime}=(\mathcal{A}-a)/q$. Even though the targeted
configuration changes a bit, we are always finding configurations in intervals
and such a procedure can be continued as long as the length of the intervals
is not too short. In contrast, in our case, we are finding
$p_{1},p_{1}+(p_{2}-1)^{k}$ in shorter and shorter arithmetic progressions
because we need to keep the feature of elements being primes throughout. Due
to the poor understanding of primes in short arithmetic progressions, our
iteration can not be too long which would yield a poor relative density of
$\mathcal{A}$ in $\mathcal{P}_{N}$.

Meanwhile, in primes, the basic idea “non-uniformity yields sufficient bias”
would also be difficult to express. Thanks to the flexibility of Fourier
analysis, we can finally successfully connect counting the weighted (weighted
by the balanced function) number of $p_{1},p_{1}+(p_{2}-1)^{k}$ in
$\mathcal{A}$ with counting them in short arithmetic progressions in primes.
And then we would calculate the later expression by making use of the Hardy-
Littlewood circle method.

We’d like to end up this section with several remarks. Even though in the
statement of Theorem 1.2 the modulus $q$ can be up to
$O\Bigl{(}\exp\bigl{(}c\sqrt{\log N}\bigr{)}\Bigr{)}$, in the proof of Theorem
1.1, we would only use partial result with $q\ll(\log N)^{A}$. The reason is
that when we look for $p_{1},p_{1}+(p_{2}-1)^{k}$ in arithmetic progressions,
it is necessary to ensure such a progression $P=a+q\cdot[X]$ has many primes.
In view of the possible existence of Siegel zero, such a conclusion is only
known for $q\ll(\log N)^{A}$. And for the same reason, we believe it is
impossible to get a single logarithmic upper bound via density increment
argument.

We organize this paper as follows. In Section 2, we set notations and exhibit
several conclusions of primes in (short) arithmetic progressions. In Section
3, we use the Hardy-Littlewood method to study the Fourier transform of the
majorant of shifted prime powers. We then use Bourgain’s method [4] and the
above Fourier transform information to prove the second result of Theorem 1.2
and give the proof of the first result in Appendix B, as the proof strategies
are similar. In Section 5, we’ll prove our main result, a local inverse
theorem, and then in Section 6 use this inverse result to carry out the
density increment argument and so complete the proof of Theorem 1.1.

### Acknowledgements

The author would like to thank her PhD supervisors Xuancheng Shao and Lilu
Zhao for their helpful discussions.

##  2\. Notations and Preliminaries

###  2.1. Notations

For a real number $X\geqslant 1$, we use $[X]$ to denote the discrete interval
$\left\\{1,2\cdots,\left\lfloor X\right\rfloor\right\\}$. $\mathcal{P}$
denotes the set of prime numbers and $\mathcal{P}_{N}=\mathcal{P}\cap[N]$
denotes the set of primes no more than $N$. For
$\alpha\in\mathbb{T}=\mathbb{R}/\mathbb{Z}$, we write $\left\|\alpha\right\|$
for the distance of $\alpha$ to the nearest integer. For any set of integers
$\mathcal{A}$, use $1_{\mathcal{A}}$ to denote its indicator function.

Following standard arithmetical function conventions, we’ll use $\phi$ to
denote the Euler totient function; $\Lambda$ to denote the von Mangoldt
function; $\mu$ to denote the Möbius function.

We’ll use the counting measure on $\mathbb{Z}$, so for a function
$f:\mathbb{Z}\to\mathbb{C}$ its $L^{p}$-norm is defined to be

$\left\|f\right\|_{p}^{p}=\sum_{x}|f(x)|^{p},$

and $L^{\infty}$-norm is defined to be

$\left\|f\right\|_{\infty}=\sup_{x}|f(x)|.$

Besides, we’ll use Haar probability measure on $\mathbb{T}$, so for a function
$F:\mathbb{T}\to\mathbb{C}$ define its $L^{p}$-norm as

$\left\|F\right\|_{p}^{p}=\int_{\mathbb{T}}|F(\alpha)|^{p}\,\mathrm{d}\alpha.$

If $f:A\rightarrow\mathbb{C}$ is a function and $B$ is a non-empty finite
subset of $A$, we write

$\mathbb{E}_{x\in B}f(x)=\frac{1}{|B|}\sum_{x\in B}f(x)$

as the average of $f$ on $B$. We would also abbreviate $\mathbb{E}_{x\in
A}f(x)$ to $\mathbb{E}(f)$ if the supported set $A$ is finite and no confusion
caused.

We’ll use Fourier analysis on $\mathbb{Z}$ with its dual $\mathbb{T}$. Let
$f:\mathbb{Z}\to\mathbb{C}$ be a function, define its Fourier transform by
setting

$\widehat{f}(\alpha)=\sum_{x}f(x)e(x\alpha),$

where $\alpha\in\mathbb{T}$. For functions $f,g:\mathbb{Z}\to\mathbb{C}$
define the convolution of $f$ and $g$ as

$f*g(x)=\sum_{y}f(y)\overline{g}(x-y).$

Then, basic properties of Fourier analysis, for $f,g:\mathbb{Z}\to\mathbb{C}$

  1. (1)

(Fourier inversion formula )
$f(x)=\int_{\mathbb{T}}\widehat{f}(\alpha)e(-x\alpha)\,\mathrm{d}\alpha$;

  2. (2)

(Parseval’s identity) $\left\|f\right\|_{2}=\|\widehat{f}\|_{2}$;

  3. (3)

$\widehat{f*g}=\widehat{f}\cdot\widehat{g}.$

$\varepsilon>0$ is always an arbitrarily small number, $c>0$ is a small
number, and both $c$ and $\varepsilon$ are allowed to change at different
occurrences. For a function $f$ and positive-valued function $g$, write $f\ll
g$ or $f=O(g)$ if there exists a constant $C>0$ such that $|f(x)|\leqslant
Cg(x)$ for all $x$; and write $f\gg g$ if $f$ is also positive-valued and
there is a constant $C>0$ such that $f(x)\geqslant Cg(x)$ for all $x$.

###  2.2. Preliminaries

Suppose that $x$ is a real number, $a$ and $q$ are positive integers, we write

$\psi(x;q,a)=\sum_{\begin{subarray}{c}n\leqslant x\\\ n\equiv
a\pmod{q}\end{subarray}}\Lambda(n).$

Estimating $\psi(x;q,a)$ is one of the central problems in analytic number
theory, and the following two lemmas are [7, P.123 Eq.(11)] and [14, Theorem
10.3] respectively.

######  Lemma 2.1 (Siegel-Walfisz Theorem).

Provided that $1\leqslant a\leqslant q$ are coprime integers with
$q\leqslant(\log x)^{D}$ for some fixed $D>0$, we have

$\psi(x;q,a)=\frac{x}{\phi(q)}+O\bigl{(}x\exp(-c(\log x)^{1/2})\bigr{)}.$

######  Lemma 2.2 (Primes in short arithmetic progressions).

Suppose that $1\leqslant a\leqslant q\leqslant(\log x)^{A}$ are coprime
integers and $x^{0.55+\varepsilon}\leqslant h\leqslant x$, then

$\frac{0.99h}{\phi(q)}<\sum_{\begin{subarray}{c}x<n\leqslant x+h\\\ n\equiv
a\pmod{q}\end{subarray}}\Lambda(n)<\frac{1.01h}{\phi(q)}.$

We can deal with exponential sums in arithmetic progressions (like (1.2)) with
$q$ beyond the limitation of Lemma 2.1 with the help of the below lemma. This
is observed in [22], and the next lemma is a modification of [22, Proposition
4.7]. We will give a brief proof in Appendix A.

######  Lemma 2.3 (Exceptional pair result).

Let $D_{1}\geqslant D_{0}\geqslant 2$ be some parameters. There is an absolute
constant $c_{0}>0$, a character $\chi$ modulus $q_{0}$ with $q_{0}\geqslant
2$, and a real number $\rho$ satisfying $(1-\rho)^{-1}\ll q_{0}^{1/2}(\log
q_{0})^{2}$ such that the following statement holds.

For all real $x\geqslant 1$ and integers $1\leqslant a\leqslant q$ with
$q\leqslant D_{0}$ and $(a,q)=1$, we have

$\psi(x;q,a)=\frac{\overline{\chi_{0}}(a)x}{\phi(q)}-E(x;q,a)+O\biggl{(}x\exp\Bigl{(}-c_{0}\frac{\log
x}{\sqrt{\log x}+\log D_{1}}\Bigr{)}(\log D_{1})^{2}\biggr{)},$

where

$\displaystyle
E(x;q,a)=\begin{cases}\frac{\overline{\chi_{0}\chi}(a)x^{\rho}}{\phi(q)\rho}\qquad&\text{if
}q_{0}|q;\\\ 0&\text{otherwise,}\end{cases}$ (2.1)

and $\chi_{0}$ denotes the principal character of modulus $q$.

Let $P=b+d\cdot[X]$ be an arithmetic progression inside $[N]$ which satisfies
$1\leqslant b\leqslant d$ and $(b,d)=1$. Recall the majorant function
$\Lambda_{b,d}:\mathbb{Z}\to\mathbb{R}_{\geqslant 0}$ defined in (1.1) as

$\Lambda_{b,d}(x)=\begin{cases}\frac{\phi(d)}{d}\Lambda(b+dx)\quad&\text{ if
}1\leqslant x\leqslant X;\\\ 0&\text{ otherwise.}\end{cases}$

We are going to give a simple consequence of Theorem 1.2 (1) as follows.

######  Lemma 2.4 (Alternative restriction).

Let $0<\delta<1$ be a number, there is an absolute constant $0<c^{\prime}<1$
such that for every relatively prime pair $1\leqslant b\leqslant
d\leqslant\exp(c^{\prime}\sqrt{\log X})$, every function
$\nu:\mathbb{Z}\to\mathbb{C}$ supported on $[X]$ and satisfying
$|\nu|\leqslant\Lambda_{b,d}$ pointwise, and every real number $p>2$, we
always have

$\|\widehat{f}\|_{p}^{p}=\int_{\mathbb{T}}|\widehat{f}(\alpha)|^{p}\,\mathrm{d}\alpha\ll_{p}X^{p-1},$

where $f=\nu-\delta\Lambda_{b,d}$.

###### Proof.

Practically, suppose that $p>2$ is a real number, on recalling
$f=\nu-\delta\Lambda_{b,d}$, the use of mean value inequality yields that

$\int_{\mathbb{T}}|\widehat{f}(\alpha)|^{p}\,\mathrm{d}\alpha=\int_{\mathbb{T}}\Bigl{|}\widehat{\nu}(\alpha)-\delta\widehat{\Lambda}_{b,d}(\alpha)\Bigr{|}^{p}\,\mathrm{d}\alpha\ll_{p}\int_{\mathbb{T}}|\widehat{\nu}(\alpha)|^{p}\,\mathrm{d}\alpha+\delta^{p}\int_{\mathbb{T}}|\widehat{\Lambda}_{b,d}(\alpha)|^{p}\,\mathrm{d}\alpha.$

Then it follows from the application of Theorem 1.2 (1) to functions $\nu$ and
$\Lambda_{b,d}$ separately, together with the assumption $0<\delta<1$.

∎

##  3\. Exponential sum estimates

In this section, we aim to estimate the auxiliary exponential summation (1.2)

$S_{d}(\alpha)=\sum_{y\in[M]}\frac{\phi(d)}{d}ky^{k-1}\Lambda(dy+1)e(y^{k}\alpha),$

where $d,k\geqslant 1$ are positive integers, $N^{\prime}$ is a large number,
$M=\left\lfloor N^{\prime 1/k}\right\rfloor$ and $\alpha\in\mathbb{T}$. Here,
in order to avoid any potential confusion (in Section 5) we have changed the
letter $N$ to $N^{\prime}$.

In the light of the Hardy-Littlewood method we’ll partition frequencies
$\alpha\in\mathbb{T}$ into major arcs and minor arcs, and then calculate the
behavior of $S_{d}(\alpha)$ in major and minor arcs respectively. We now
handle these tasks by turns. Assume that
$d\ll\exp\bigl{(}\frac{c_{0}}{20k}\sqrt{\log N^{\prime}}\bigr{)}$ ($c_{0}$ is
the constant in Lemma 2.3) throughout this section, and let us divide the
interval $\mathbb{T}$ into major arcs $\mathfrak{M}$ and minor arcs
$\mathfrak{m}$ which are defined in the following forms

$\displaystyle\mathfrak{M}$ $\displaystyle=\bigcup_{q\leqslant
Q}\bigl{\\{}\alpha\in\mathbb{T}:\left\|q\alpha\right\|\ll
Q/N^{\prime}\bigr{\\}},$ (3.1) $\displaystyle\mathfrak{m}$
$\displaystyle=\mathbb{T}\backslash\mathfrak{M},$

where

$\displaystyle Q=(\log N^{\prime})^{2^{2k}D},$ (3.2)

and $D>0$ is a large number to be specified later.

We’ll calculate the asymptotic formula of the exponential sum $S_{d}(\alpha)$
in major arcs firstly. But before it, an auxiliary lemma.

######  Lemma 3.1.

Let $k\geqslant 1$ be an integer. Let $a$ and $q$ be relatively prime positive
integers. Then for any coprime pair $b,t\geqslant 1$ we have

$\left|\sum_{\begin{subarray}{c}r(q)\\\
(tr+b,q)=1\end{subarray}}e\bigl{(}ar^{k}/q\bigr{)}\right|\ll
q^{1-\frac{1}{k}+\varepsilon},$

where $\varepsilon>0$ is arbitrarily small.

###### Proof.

We firstly claim that the left-hand side expression is multiplicative. To see
this, let $q=q_{1}q_{2}$ with $(q_{1},q_{2})=1$, and
$r=r_{1}q_{2}+r_{2}q_{1}$, on noting that

$(tr+b,q)=(tr_{1}q_{2}+tr_{2}q_{1}+b,q_{1}q_{2})=(tr_{1}q_{2}+b,q_{1})(tr_{2}q_{1}+b,q_{2}),$

we then have

$\sum_{\begin{subarray}{c}r(q_{1}q_{2})\\\
(tr+b,q_{1}q_{2})=1\end{subarray}}e\biggl{(}\frac{ar^{k}}{q_{1}q_{2}}\biggr{)}=\sum_{\begin{subarray}{c}r_{1}(q_{1})\\\
(tr_{1}q_{2}+b,q_{1})=1\end{subarray}}e\biggl{(}\frac{ar_{1}^{k}q_{2}^{k-1}}{q_{1}}\biggr{)}\sum_{\begin{subarray}{c}r_{2}(q_{2})\\\
(tr_{2}q_{1}+b,q_{2})=1\end{subarray}}e\biggl{(}\frac{ar_{2}^{k}q_{1}^{k-1}}{q_{2}}\biggr{)}.$

Thus, without loss of generality, we may assume that $q=p^{m}$ with
$p\in\mathcal{P}$. If $p\arrowvert t$, then for any $r$ modulus $p^{m}$,
$(tr+b,p)=1$ always holds, just by taking note that $(b,t)=1$. Thus, in this
case it follows from [26, Theorem 4.2] that

$\left|\sum_{\begin{subarray}{c}r(p^{m})\\\
(tr+b,p)=1\end{subarray}}e\biggl{(}\frac{ar^{k}}{p^{m}}\biggr{)}\right|=\Bigl{|}\sum_{r(p^{m})}e\biggl{(}\frac{ar^{k}}{p^{m}}\biggr{)}\Bigr{|}\ll(p^{m})^{1-\frac{1}{k}+\varepsilon}.$

Now we assume that $(p,t)=1$. By taking advantage of the Möbius function, one
has

$\displaystyle\sum_{\begin{subarray}{c}r(p^{m})\\\
(tr+b,p)=1\end{subarray}}e\biggl{(}\frac{ar^{k}}{p^{m}}\biggr{)}$
$\displaystyle=\sum_{r(p^{m})}e\biggl{(}\frac{ar^{k}}{p^{m}}\biggr{)}\sum_{d|(tr+b,p)}\mu(d)=\sum_{d|p}\mu(d)\sum_{\begin{subarray}{c}r(p^{m})\\\
d|(tr+b)\end{subarray}}e\biggl{(}\frac{ar^{k}}{p^{m}}\biggr{)}.$

Just from the definition, above Möbius function $\mu$ will vanish except when
$d=1$ and $d=p$, and, thus, above expression is indeed

$\sum_{r(p^{m})}e\biggl{(}\frac{ar^{k}}{p^{m}}\biggr{)}-\sum_{\begin{subarray}{c}r(p^{m})\\\
tr+b\equiv 0\pmod{p}\end{subarray}}e\biggl{(}\frac{ar^{k}}{p^{m}}\biggr{)}.$

By making use of [26, Theorem 4.2] once again one has,

$\displaystyle\biggl{|}\sum_{r(p^{m})}e\biggl{(}\frac{ar^{k}}{p^{m}}\biggr{)}\biggr{|}\ll(p^{m})^{1-\frac{1}{k}+\varepsilon}.$

As for the second term, taking note that $(t,p)=(t,b)=1$, let $t_{p}$ be the
unique solution to the linear congruence equation $tr+b\equiv 0\pmod{p}$ in
the range $[1,p]$, then for $r\in[p^{m}]$, the solutions are in the form of
$r=t_{p}+sp$ with $s\in[p^{m-1}]$. And therefore,

$\sum_{\begin{subarray}{c}r(p^{m})\\\ tr+b\equiv
0\pmod{p}\end{subarray}}e\biggl{(}\frac{ar^{k}}{p^{m}}\biggr{)}=\sum_{s(p^{m-1})}e\biggl{(}\frac{a(t_{p}+sp)^{k}}{p^{m}}\biggr{)}\\\
=e\Bigl{(}\frac{at_{p}^{k}}{p^{m}}\Bigr{)}\sum_{s(p^{m-1})}e\biggl{(}\frac{akt_{p}^{k-1}s+\cdots+ap^{k-1}s^{k}}{p^{m-1}}\biggr{)}.$

Now let $p^{\tau}\parallel k$, if
$\tau\geqslant\left\lceil\frac{m}{2}\right\rceil$, then
$\sum_{s(p^{m-1})}e\biggl{(}\frac{akt_{p}^{k-1}s+\cdots+ap^{k-1}s^{k}}{p^{m-1}}\biggr{)}$
is trivially bounded by $O(1)$; if
$0<\tau<\left\lceil\frac{m}{2}\right\rceil$, let $0<u\leqslant\tau$ be the
integer such that
$\mathrm{gcd}\Bigl{(}at_{p}^{k-1}k,at_{p}^{k-2}p\tbinom{k}{2},\cdots,ap^{k-1},p^{m-1}\Bigr{)}=p^{u}$,
it can be deduced from [26, Theorem 7.1] that

$\biggl{|}\sum_{s(p^{m-1})}e\biggl{(}\frac{akt_{p}^{k-1}s+\cdots+ap^{k-1}s^{k}}{p^{m-1}}\biggr{)}\biggr{|}\leqslant
k\biggl{|}\sum_{s(p^{m-u-1})}e\biggl{(}\frac{a_{1}^{\prime}s+\cdots+a_{k}^{\prime}s^{k}}{p^{m-u-1}}\biggr{)}\biggr{|}\\\
\ll(p^{m-u-1})^{1-\frac{1}{k}+\varepsilon},$

where $\mathrm{gcd}(a_{1}^{\prime},\cdots,a_{k}^{\prime},p)=1$; if $\tau=0$,
it is directly from [26, Theorem 7.1] that

$\left|\sum_{\begin{subarray}{c}r(p^{m})\\\ tr+b\equiv
0\pmod{p}\end{subarray}}e\biggl{(}\frac{ar^{k}}{p^{m}}\biggr{)}\right|=\biggl{|}\sum_{s(p^{m-1})}e\biggl{(}\frac{akt_{p}^{k-1}s+\cdots+ap^{k-1}s^{k}}{p^{m-1}}\biggr{)}\biggr{|}\ll(p^{m-1})^{1-\frac{1}{k}+\varepsilon}.$

Putting all above cases together we obtain

$\left|\sum_{\begin{subarray}{c}r(p^{m})\\\
(tr+b,p)=1\end{subarray}}e\biggl{(}\frac{ar^{k}}{p^{m}}\biggr{)}\right|\ll(p^{m})^{1-\frac{1}{k}+\varepsilon},$

and the lemma follows. ∎

######  Lemma 3.2 (Major-arc behavior of $S_{d}(\alpha)$).

Suppose that $\alpha\in\mathbb{T}$ can be decomposed as
$\alpha=\frac{a}{q}+\beta$ with coprime pair $1\leqslant a\leqslant q$. There
is an absolute constant $0<20kc^{\prime}<c_{0}$, where $0<c_{0}<1$ is the
constant in Lemma 2.3, such that for all $d,q\ll\exp(c^{\prime}\sqrt{\log
N^{\prime}})$

$S_{d}(\alpha)\ll
q^{-\frac{1}{k}+\varepsilon}N^{\prime}(1+N^{\prime}|\beta|)^{-1}+N^{\prime}(1+N^{\prime}|\beta|)\exp\bigl{(}-c\sqrt{\log
N^{\prime}}\bigr{)}$

holds for every sufficiently small $\varepsilon>0$.

###### Proof.

In the first place, we consider the much easier case, that is $\alpha=a/q$.
Expanding the definition of $S_{d}(\alpha)$ with $\alpha=a/q$, and then making
the change of variables $dy+1\mapsto y$ to get that

$\displaystyle S_{d}(a/q)$
$\displaystyle=\sum_{y\in[M]}k\frac{\phi(d)}{d}y^{k-1}\Lambda(dy+1)e\Bigl{(}\frac{ay^{k}}{q}\Bigr{)}$
$\displaystyle=\sum_{\begin{subarray}{c}y\in[dM+1]\\\ y\equiv
1\pmod{d}\end{subarray}}k\frac{\phi(d)}{d}\Bigl{(}\frac{y-1}{d}\Bigr{)}^{k-1}\Lambda(y)e\Bigl{(}\frac{a}{q}\cdot\Bigl{(}\frac{y-1}{d}\Bigr{)}^{k}\Bigr{)}.$

We now split above summation interval into sub-progressions, according to the
residue classes of $\frac{y-1}{d}$ modulus $q$, to obtain that

$S_{d}(a/q)=\sum_{r(q)}\sum_{\begin{subarray}{c}y\in[dM+1]\\\ y\equiv
1\pmod{d}\\\ \frac{y-1}{d}\equiv
r\pmod{q}\end{subarray}}k\frac{\phi(d)}{d}\Bigl{(}\frac{y-1}{d}\Bigr{)}^{k-1}\Lambda(y)e\Bigl{(}\frac{a}{q}\cdot\Bigl{(}\frac{y-1}{d}\Bigr{)}^{k}\Bigr{)}.$

Then it is quite easy to see that

$\displaystyle
S_{d}(a/q)=\sum_{r(q)}e\bigl{(}\frac{ar^{k}}{q}\bigr{)}\Psi(dM+1;dq,dr+1),$
(3.3)

where

$\Psi(dM+1;dq,dr+1)=\sum_{\begin{subarray}{c}y\in[dM+1]\\\ y\equiv
dr+1\pmod{dq}\end{subarray}}\Lambda(y)k\frac{\phi(d)}{d}\Bigl{(}\frac{y-1}{d}\Bigr{)}^{k-1}.$

Our task now is to compute $\Psi(dM+1;dq,dr+1)$, and the main tools are Lemma
2.3 and Abel’s summation formula. Indeed, an application of summation by parts
yields

$\Psi(dM+1;dq,dr+1)=\psi(dM+1;dq,dr+1)\frac{\phi(d)}{d}kM^{k-1}\\\
-\frac{\phi(d)}{d}\int_{1}^{dM+1}\psi(t;dq,dr+1)\biggl{(}k\Bigl{(}\frac{t-1}{d}\Bigr{)}^{k-1}\biggr{)}^{\prime}\,\mathrm{d}t,$
(3.4)

recalling that $\psi(x;q,a)=\sum_{\begin{subarray}{c}n\in[x]\\\ n\equiv
a\pmod{q}\end{subarray}}\Lambda(n)$.

Let $c^{\prime}=\frac{c_{0}}{20k}$, $D_{0}=\exp\bigl{(}c^{\prime}\sqrt{\log
N^{\prime}}\bigr{)}$ and $D_{1}=\exp\bigl{(}2c^{\prime}\sqrt{\log
N^{\prime}}\bigr{)}$, it then follows from Lemma 2.3 that when
$d,q\leqslant\exp(c^{\prime}\sqrt{\log N^{\prime}})$ the above first term
equals to

$\displaystyle\phi(d)kM^{k-1}\Bigl{(}\frac{\overline{\chi}_{0}(dr+1)M}{\phi(dq)}-\frac{E(dM;dq,dr+1)}{d}\Bigr{)}+O\bigl{(}M^{k}\exp\bigl{(}-c\sqrt{\log
N^{\prime}}\bigr{)}\bigr{)}.$ (3.5)

Meanwhile, it can also be seen from Lemma 2.3 that when $dq\leqslant D_{0}$
the integral in (3.4) is equal to

$-\frac{\phi(d)}{d}\frac{\overline{\chi_{0}}(dr+1)}{\phi(dq)}\int_{1}^{dM+1}t\Bigl{(}k\Bigl{(}\frac{t-1}{d}\Bigr{)}^{k-1}\Bigr{)}^{\prime}\,\mathrm{d}t+\frac{\phi(d)}{d}\int_{1}^{dM+1}E(t;dq,dr+1)\Bigl{(}k\Bigl{(}\frac{t-1}{d}\Bigr{)}^{k-1}\Bigr{)}^{\prime}\,\mathrm{d}t\\\
+O\Bigl{(}d^{1-k}\log
N^{\prime}\int_{1}^{dM+1}t^{k-1}\exp\bigl{(}-c_{0}\frac{\log t}{\sqrt{\log
t}+2c^{\prime}\sqrt{\log N^{\prime}}}\bigr{)}\,\mathrm{d}t\Bigr{)}.$ (3.6)

Recalling that (2.1), when $q_{0}|dq$ the second term of (3.6) equals to
$\frac{\phi(d)}{d}$ times

$\frac{\overline{\chi\chi_{0}}(dr+1)}{\phi(dq)}d^{1-k}\int^{dM+1}_{1}\frac{t^{\rho}}{\rho}k(k-1)(t-1)^{k-2}\,\mathrm{d}t=\frac{\overline{\chi\chi_{0}}(dr+1)(dM)^{\rho}}{\phi(dq)\rho}\frac{k(k-1)}{k-1+\rho}M^{k-1};$

otherwise, this term vanishes. Combining the above two cases, and recalling
(2.1) again, we can conclude that the second term of (3.6) is always equal to

$\frac{\phi(d)}{d}E(dM;dq,dr+1)\frac{k(k-1)}{k-1+\rho}M^{k-1}.$

As for the big O-term (the third term) of (3.6), on noticing that

$\frac{c_{0}\log t}{\sqrt{\log t}+2c^{\prime}\sqrt{\log
N^{\prime}}}\geqslant\frac{\frac{c_{0}^{2}}{2}\log
N^{\prime}}{\sqrt{\log(dM)}+2c^{\prime}\sqrt{\log
N^{\prime}}}\geqslant\frac{c_{0}^{2}}{4}\sqrt{\log N^{\prime}}$

whenever $M^{10kc^{\prime}}=M^{c_{0}/2}\leqslant t\leqslant dM+1$, thus, this
term can be bounded up by

$d^{1-k}\log
N^{\prime}\biggl{\\{}\int_{1}^{M^{10kc^{\prime}}}t^{k-1}\,\mathrm{d}t+\exp\bigl{(}-\frac{c_{0}^{2}}{4}\sqrt{\log
N^{\prime}}\bigr{)}\int_{M^{10kc^{\prime}}}^{dM+1}t^{k-1}\,\mathrm{d}t\biggr{\\}}\ll
M^{k}\exp(-c\sqrt{\log N^{\prime}}),$

as $d\log N^{\prime}\exp\bigl{(}-\frac{c_{0}^{2}}{4}\sqrt{\log
N^{\prime}}\bigr{)}\ll\exp\bigl{(}-c\sqrt{\log N^{\prime}}\bigr{)}$ whenever
$d\ll\exp\bigl{(}\frac{c_{0}}{20k}\sqrt{\log N^{\prime}}\bigr{)}$. Therefore,
(3.6) equals to

$-\frac{\phi(d)\overline{\chi_{0}}(dr+1)}{\phi(dq)}(k-1)M^{k}+\frac{\phi(d)}{d}E(dM;dq,dr+1)\frac{k(k-1)}{k-1+\rho}M^{k-1}+O(M^{k}\exp(-c\sqrt{\log
N^{\prime}})).$

We now substitute the above expression and (3.5) into (3.4), and then
substitute it into (3.3). Besides, recalling the definition of
$E(dM+1;dq,dr+1)$ in (2.1) one more time, i.e.

$E(dM;dq,dr+1)=\begin{cases}\frac{\overline{\chi_{0}\chi}(dr+1)(dM)^{\rho}}{\phi(dq)\rho}\qquad&\text{if
}q_{0}|dq;\\\ 0&\text{otherwise,}\end{cases}$

and letting
$C(q;a,d)=\sum_{r(q)\atop(dr+1,q)=1}e\bigl{(}\frac{ar^{k}}{q}\bigr{)}$, we
then have when $q_{0}|dq$,

$S_{d}(a/q)=\frac{\phi(d)}{\phi(dq)}C(q;a,d)\biggl{\\{}M^{k}-\overline{\chi}(dr+1)\frac{kd^{\rho-1}}{k-1+\rho}M^{k-1+\rho}\biggr{\\}}\\\
+O\bigl{(}M^{k}\exp(-c\sqrt{\log N^{\prime}})\bigr{)};$

and when $q_{0}\nmid dq$,

$S_{d}(a/q)=\frac{\phi(d)}{\phi(dq)}C(q;a,d)M^{k}+O\bigl{(}M^{k}\exp(-c\sqrt{\log
N^{\prime}})\bigr{)}.$

Now let $\alpha=a/q+\beta$, it follows from Abel’s summation formula, as well
as $N^{\prime}\asymp M^{k}$, that

$S_{d}(\alpha)=\sum_{y\in[M]}k\frac{\phi(d)}{d}y^{k-1}\Lambda(dy+1)e\biggl{(}\frac{ay^{k}}{q}\biggr{)}e(\beta
y^{k})\\\ =S_{d}(a/q)e(\beta
M^{k})-\int_{1}^{M}\sum_{y\in[t]}k\frac{\phi(d)}{d}y^{k-1}\Lambda(dy+1)e\biggl{(}\frac{ay^{k}}{q}\biggr{)}\Bigl{(}e(\beta
t^{k})\Bigr{)}^{\prime}\,\mathrm{d}t.$

From some computation which is similar to the previous paragraph, it turns out
that when $q_{0}|dq$,

$S_{d}(\alpha)=\frac{\phi(d)}{\phi(dq)}C(q;a,d)\biggl{\\{}\int_{1}^{M^{k}}e(\beta
t)\,\mathrm{d}t-\overline{\chi}(dr+1)d^{\rho-1}\int_{1}^{M^{k}}t^{\frac{\rho-1}{k}}e(\beta
t)\,\mathrm{d}t\biggr{\\}}\\\
+O\biggl{(}N^{\prime}(1+N^{\prime}|\beta|)\exp\bigl{(}-c\sqrt{\log
N^{\prime}}\bigr{)}\biggr{)};$

and when $q_{0}\nmid dq$,

$S_{d}(\alpha)=\frac{\phi(d)}{\phi(dq)}C(q;a,d)\int_{1}^{M^{k}}e(\beta
t)\,\mathrm{d}t+O\biggl{(}N^{\prime}(1+N^{\prime}|\beta|)\exp\bigl{(}-c\sqrt{\log
N^{\prime}}\bigr{)}\biggr{)}.$

If the case $q_{0}|dq$ arises, as $1/2<\rho<1$ is a real number which
satisfies $1-\rho\gg D_{0}^{-1/2}(\log D_{0})^{-2}\gg\exp(-c\sqrt{\log
N^{\prime}})$, it is clearly that the second term is rather smaller than the
first one. Therefore, we can conclude that

$|S_{d}(\alpha)|\ll\frac{\phi(d)}{\phi(dq)}|C(q;a,d)|N^{\prime}(1+N^{\prime}|\beta|)^{-1}+N^{\prime}(1+N^{\prime}|\beta|)\exp\bigl{(}-c\sqrt{\log
N^{\prime}}\bigr{)},$

on noting that $\int_{1}^{M^{k}}e(\beta
t)\,\mathrm{d}t\ll\min\bigl{\\{}M^{k},|\beta|^{-1}\bigr{\\}}\ll
N^{\prime}(1+N^{\prime}|\beta|)^{-1}$. We’ll estimate the multiple factors in
the first term in turns. From elementary properties of Euler totient function
$\phi$, see [1, Theorem 2.5] as an example, one has,

$\frac{\phi(d)}{\phi(dq)}=\frac{\phi(d)}{\phi(d)\phi(q)}\cdot\frac{\phi(\mathrm{gcd}(d,q))}{\mathrm{gcd}(d,q)}\leqslant\phi(q)^{-1}.$

Whilst, it can be verified by Lemma 3.1 with $(a,q)=(d,1)=1$ that

$C(q;a,d)\ll q^{1-\frac{1}{k}+\varepsilon},$

for arbitrary small $\varepsilon>0$. On combining the above three
representations yields the desired upper bound of $S_{d}(\alpha)$.

∎

It is time to deal with the minor arcs estimates of $S_{d}(\alpha)$. In fact,
such kind of minor-arc analysis for exponential sum subjects to a congruence
equation has been developed in [9], the linear cases, and [6], the higher
degree cases. However, the modulus of the congruence equation in their works
are only in logarithmic power size, therefore there are slight differences in
our approach. Because the minor-arc estimates are long and miscellaneous, the
type I and type II sum estimates in the following lemma are only brief and
rough, and we’d like to refer the interested readers to [17, Chapter 3] for
more details.

######  Lemma 3.3 (Minor arcs estimates).

$\sup_{\alpha\in\mathfrak{m}}|S_{d}(\alpha)|\ll N^{\prime}(\log
N^{\prime})^{-D},$

where $D>1$ is the number in (3.2).

###### Proof.

For one thing, in view of Dirichlet approximate theorem, for any frequency
$\alpha\in\mathbb{T}$, there always exists an integer $q\leqslant
N^{\prime}/Q$ such that $\left\|q\alpha\right\|\ll Q/N^{\prime}$. Thus, when
$\alpha$ belongs to minor arcs $\mathfrak{m}$, we must have

$\displaystyle Q\leqslant q\leqslant N^{\prime}/Q.$ (3.7)

For another, from the proof of Lemma 3.2, it is clear that

$S_{d}(\alpha)=\sum_{\begin{subarray}{c}y\in[dM+1]\\\ y\equiv
1\pmod{d}\end{subarray}}k\frac{\phi(d)}{d}\Bigl{(}\frac{y-1}{d}\Bigr{)}^{k-1}\Lambda(y)e\Bigl{(}\alpha\Bigl{(}\frac{y-1}{d}\Bigr{)}^{k}\Bigr{)}.$

After making use of Abel’s summation formula, we’ll get

$S_{d}(\alpha)=\frac{\phi(d)}{d}kM^{k-1}\sum_{\begin{subarray}{c}y\in[dM+1]\\\
y\equiv
1\pmod{d}\end{subarray}}\Lambda(y)e\Bigl{(}\alpha\Bigl{(}\frac{y-1}{d}\Bigr{)}^{k}\Bigr{)}\\\
-\int_{1}^{dM+1}\biggl{(}\sum_{\begin{subarray}{c}y\in[t]\\\ y\equiv
1\pmod{d}\end{subarray}}\Lambda(y)e\Bigl{(}\alpha\Bigl{(}\frac{y-1}{d}\Bigr{)}^{k}\Bigr{)}\biggr{)}\cdot\biggl{(}\frac{\phi(d)}{d}k\Bigl{(}\frac{t-1}{d}\Bigr{)}^{k-1}\biggr{)}^{\prime}\,\mathrm{d}t.$

From the perspecitive of Brun -Titchmarsh theorem ([18, Theorem 2]) it is an
easy matter to verify that

$\biggl{|}\sum_{\begin{subarray}{c}y\in[t]\\\ y\equiv
1\pmod{d}\end{subarray}}\Lambda(y)e\Bigl{(}\alpha\Bigl{(}\frac{y-1}{d}\Bigr{)}^{k}\Bigr{)}\biggr{|}\leqslant\pi(t;d,1)\log
t\ll\frac{t\log t}{\phi(d)},$

whilst derivation gives the upper bound

$\biggl{(}\frac{\phi(d)}{d}k\Bigl{(}\frac{t-1}{d}\Bigr{)}^{k-1}\biggr{)}^{\prime}\ll\frac{\phi(d)}{d^{k}}t^{k-2}.$

Combining above two inequalities it is not hard to find that the integral over
the interval $[1,dM(\log N^{\prime})^{-2D}]$ is negligible, in practice,

$\left|\int_{1}^{dM(\log
N^{\prime})^{-2D}}\biggl{(}\sum_{\begin{subarray}{c}y\in[t]\\\ y\equiv
1\pmod{d}\end{subarray}}\Lambda(y)e\Bigl{(}\alpha\Bigl{(}\frac{y-1}{d}\Bigr{)}^{k}\Bigr{)}\biggr{)}\cdot\biggl{(}\frac{\phi(d)}{d}k\Bigl{(}\frac{t-1}{d}\Bigr{)}^{k-1}\biggr{)}^{\prime}\,\mathrm{d}t\right|\\\
\ll d^{-k}\int_{1}^{dM(\log N^{\prime})^{-2D}}t^{k-1}\log t\,\mathrm{d}t\ll
N^{\prime}(\log N^{\prime})^{-D},$

on recalling that $N^{\prime}\asymp M^{k}$. Therefore, it can be verified that
the absolute value of $S_{d}(\alpha)$ is bounded by

$\frac{\phi(d)}{d}M^{k-1}\sup_{dM(\log N^{\prime})^{-2D}\leqslant t\leqslant
dM+1}\biggl{|}\sum_{\begin{subarray}{c}y\in[t]\\\ y\equiv
1\pmod{d}\end{subarray}}\Lambda(y)e\Bigl{(}\alpha\Bigl{(}\frac{y-1}{d}\Bigr{)}^{k}\Bigr{)}\biggr{|}+O\bigl{(}N^{\prime}(\log
N^{\prime})^{-D}\bigr{)}.$

For convenience, we’ll partition the above exponential sum into dyadic
intervals to get that

$\biggl{|}\sum_{\begin{subarray}{c}y\in[t]\\\ y\equiv
1\pmod{d}\end{subarray}}\Lambda(y)e\Bigl{(}\alpha\Bigl{(}\frac{y-1}{d}\Bigr{)}^{k}\Bigr{)}\biggr{|}\ll\log
N^{\prime}\biggl{|}\sum_{\begin{subarray}{c}y\sim T\\\ y\equiv
1\pmod{d}\end{subarray}}\Lambda(y)e\Bigl{(}\alpha\Bigl{(}\frac{y-1}{d}\Bigr{)}^{k}\Bigr{)}\biggr{|}$

with $1\leqslant T\leqslant t$ and $dM(\log N^{\prime})^{-2D}\leqslant
t\leqslant dM+1$. Besides, it is a fairly immediate consequence of Brun
-Titchmarsh theorem that when $1\leqslant T\leqslant dM(\log
N^{\prime})^{-2D}$ the above right-hand side expression is bounded by
$O\Bigl{(}\frac{dM}{\phi(d)}(\log N^{\prime})^{-D}\Bigr{)}$, thus the
contribution to $S_{d}(\alpha)$ is $O(N^{\prime}(\log N^{\prime})^{-D})$. So,
now, we may assume that $dM(\log N^{\prime})^{-2D}\leqslant T\leqslant dM+1$,
and it can be deduced from Vaughan’s identity with parameter
$U=\Bigl{(}\frac{T}{d}\Bigr{)}^{\frac{1}{2^{k}}}$ that the inner sum of the
above right-hand side is equal to

$\sum_{\begin{subarray}{c}uv\sim T\\\ uv\equiv 1\pmod{d}\\\ u\leqslant
U\end{subarray}}\mu(u)(\log
v)e\Bigl{(}\alpha\Bigl{(}\frac{uv-1}{d}\Bigr{)}^{k}\Bigr{)}-\sum_{\begin{subarray}{c}s\leqslant
U^{2}\\\ sw\sim T\\\ sw\equiv
1\pmod{d}\end{subarray}}a(s)e\Bigl{(}\alpha\Bigl{(}\frac{sw-1}{d}\Bigr{)}^{k}\Bigr{)}\\\
+\sum_{\begin{subarray}{c}sv\sim T\\\ s,v>U\\\ sv\equiv
1\pmod{d}\end{subarray}}b(s)\Lambda(v)e\Bigl{(}\alpha\Bigl{(}\frac{sv-1}{d}\Bigr{)}^{k}\Bigr{)}:=I_{1}-I_{2}+I_{3},$

where $a(s)=\sum_{\begin{subarray}{c}uv=s\\\ u,v\leqslant
U\end{subarray}}\mu(u)\Lambda(v)$ and $b(s)=\sum_{\begin{subarray}{c}uw=s\\\
u>U\end{subarray}}\mu(u)$ are two arithmetic functions. We now estimate
$I_{1}$, $I_{2}$ and $I_{3}$ in turns. On writing $u^{-1}$ as the inverse of
$u$ modulus $d$, that is $uu^{-1}\equiv 1\pmod{d}$ (so does $s^{-1}$, $v^{-1}$
and so on), and then define $c_{d}$ to be the integer such that
$uu^{-1}-1=dc_{d}$, one then has

$|I_{1}|\leqslant\sum_{\begin{subarray}{c}u\leqslant U\\\
(u,d)=1\end{subarray}}\biggl{|}\sum_{\begin{subarray}{c}v\sim\frac{T}{u}\\\
v\equiv u^{-1}\pmod{d}\end{subarray}}(\log
v)e\Bigl{(}\alpha\Bigl{(}\frac{uv-1}{d}\Bigr{)}^{k}\Bigr{)}\biggr{|}\\\
\leqslant\sum_{\begin{subarray}{c}u\leqslant U\\\ (u,d)=1\end{subarray}}\log
T\cdot\sup_{1\leqslant
z\leqslant\frac{2T}{u}}\biggl{|}\sum_{\begin{subarray}{c}\max\left\\{z,\frac{T}{u}\right\\}<v\leqslant\frac{2T}{u}\\\
v\equiv
u^{-1}\pmod{d}\end{subarray}}e\Bigl{(}\alpha\Bigl{(}\frac{uv-1}{d}\Bigr{)}^{k}\Bigr{)}\biggr{|}.$

Now let $v=u^{-1}+dv^{\prime}$ with
$\max\left\\{\frac{z}{d},\frac{T}{ud}\right\\}<v^{\prime}\leqslant\frac{2T}{ud}$,
it is immediate that

$|I_{1}|\ll\sum_{u\leqslant U}\log T\cdot\sup_{1\leqslant
z\leqslant\frac{2T}{u}}\Bigl{|}\sum_{\max\left\\{\frac{z}{d},\frac{T}{ud}\right\\}<v^{\prime}\leqslant\frac{2T}{ud}}e(\alpha(uv^{\prime}+c_{d})^{k})\Bigr{|}.$

It then follows from [17, formula (3.67)] that

$|I_{1}|\ll\frac{T}{d}(\log
N^{\prime})^{C_{1}}\biggl{(}\frac{1}{q}+\frac{d}{T}+\frac{q}{\bigl{(}\frac{T}{d}\bigr{)}^{k}}\biggr{)}^{\frac{1}{2^{k}}}$

holds for some large but fixed number $C_{1}>0$. Secondly, in a similar
routine , it also can be deduced from [17, formula (3.67)] that

$|I_{2}|\leqslant\left\|a\right\|_{\infty}\sum_{\begin{subarray}{c}s\leqslant
U^{2}\\\
(s,u)=1\end{subarray}}\Bigl{|}\sum_{w\sim\frac{T}{sd}}e(\alpha(sw+c_{d})^{k})\Bigr{|}\ll\frac{T}{d}(\log
N^{\prime})^{C_{1}}\biggl{(}\frac{1}{q}+\frac{d}{T}+\frac{q}{\bigl{(}\frac{T}{d}\bigr{)}^{k}}\biggr{)}^{\frac{1}{2^{k}}},$

here we have abused the number $c_{d}$.

As for $I_{3}$, a dyadic argument gives us that for every $U\leqslant
L\leqslant\frac{2T}{U}$

$|I_{3}|\ll\log T\biggl{|}\sum_{\begin{subarray}{c}s\sim L\\\
(s,d)=1\end{subarray}}b(s)\sum_{\begin{subarray}{c}\max\left\\{\frac{T}{s},U\right\\}<v\leqslant\frac{2T}{s}\\\
v\equiv
s^{-1}\pmod{d}\end{subarray}}\Lambda(v)e\Bigl{(}\alpha\Bigl{(}\frac{sv-1}{d}\Bigr{)}^{k}\Bigr{)}\biggr{|}.$

The application of Hölder’s inequality together with making changing of
variables and then expanding the $2^{k-1}$-th power lead us to

$|I_{3}|^{2^{k-1}}\ll(\log T)^{2^{k-1}}\Bigl{(}\sum_{s\sim
L}b(s)^{\frac{2^{k-1}}{2^{k-1}-1}}\Bigr{)}^{2^{k-1}-1}\sum_{\begin{subarray}{c}s\sim
L\\\
(s,d)=1\end{subarray}}\biggl{|}\sum_{\max\left\\{\frac{T}{sd},\frac{U}{d}\right\\}<v\leqslant\frac{2T}{sd}}\Lambda(s^{-1}+dv)e(\alpha(sv+c_{d})^{k})\biggr{|}^{2^{k-1}}\\\
\ll(\log
T)^{2^{k-1}}\left\|b\right\|_{\infty}^{2^{k-1}}L^{2^{k-1}-1}\sum_{\begin{subarray}{c}s\sim
L\\\
(s,d)=1\end{subarray}}\sum_{\max\left\\{\frac{T}{sd},\frac{U}{d}\right\\}<v,v_{1},\cdots,v_{2^{k-1}-1}\leqslant\frac{2T}{sd}}\\\
\Lambda(s^{-1}+dv)\cdots\Lambda(s^{-1}+dv_{2^{k-1}-1})e\bigl{(}\alpha\left\\{(sv+c_{d})^{k}+\cdots-(sv_{2^{k-1}-1}+c_{d})^{k}\right\\}\bigr{)}.$

We now make the following changing of variable, let
$\underline{\omega}=\left\\{\omega_{1},\dots,\omega_{k-1}\right\\}\in\left\\{0,1\right\\}^{k-1}$,
and then let $v_{i}=v+\omega_{1}l_{1}+\cdots+\omega_{k-1}l_{k-1}$ for all
$0\leqslant i\leqslant 2^{k-1}-1$ satisfying $v_{0}=v$ and $v_{i}\neq v_{j}$
for all distinct $i\neq j$. The careful calculation would yield that

$|I_{3}|^{2^{k-1}}\ll(\log
T)^{2^{k-1}}\left\|b\right\|_{\infty}^{2^{k-1}}L^{2^{k-1}-1}\left\|\Lambda\right\|_{\infty}^{2^{k-1}}\bigl{(}\frac{T}{Ld}\bigr{)}^{2^{k-1}-k}\sum_{l_{1},\cdots,l_{k-1}\in[\frac{2T}{Ld}]}\\\
\sum_{\max\left\\{\frac{T}{Ld},\frac{U}{d}\right\\}<v\leqslant\frac{2T}{Ld}}\biggl{|}\sum_{\begin{subarray}{c}s\sim
L\\\ (s,d)=1\end{subarray}}e(\alpha k!l_{1}\cdots
l_{k-1}vs^{k}+\cdots)\biggr{|}.$

It can be derived from [17, formula (3.66)] that

$|I_{3}|\ll\frac{T}{d}(\log
N^{\prime})^{C_{2}}\biggl{(}\frac{1}{q}+\Bigl{(}\frac{d}{T}\Bigr{)}^{1/2}+\frac{q}{\bigl{(}\frac{T}{d}\bigr{)}^{k}}\biggr{)}^{\frac{1}{2^{2k-1}}}$

holds for some large but fixed number $C_{2}>0$. Now we let

$\displaystyle D=\frac{2k(k+3)}{\varepsilon},$ (3.8)

where $0<\varepsilon<1$, and decreasing $\varepsilon$ if necessary such that
$D>\max\left\\{C_{1},C_{2},2k(k+1)+8\right\\}$. Putting everything together
and absorbing terms which are obviously smaller than others, it allows us to
conclude that

$\displaystyle S_{d}(\alpha)\ll N^{\prime}(\log
N^{\prime})^{-D}+N^{\prime}(\log
N^{\prime})^{D}\biggl{(}\frac{1}{q}+\frac{(\log
N^{\prime})^{D}}{M^{1/2}}+\frac{q}{N^{\prime}}\biggr{)}^{\frac{1}{2^{2k-1}}},$
(3.9)

taking note that $dM(\log N^{\prime})^{-2D}\leqslant T\leqslant dM+1$ and
$N^{\prime}\asymp M^{k}$. The lemma is proven by noticing that when $\alpha$
lies in minor arcs $\mathfrak{m}$ the lower and upper bound of $q$ in (3.7)
and (3.2).

∎

##  4\. Restriction for shifted prime powers

The main purpose of this section is to prove the second conclusion of Theorem
1.2. We now introduce an auxiliary set: for any real number $0<\eta\ll 1$,
define the $\eta$-large spectrum of $S_{d}$ to be

$\mathcal{R}_{\eta}=\bigl{\\{}\alpha\in\mathbb{T}:|S_{d}(\alpha)|\geqslant\eta
N\bigr{\\}},$

which is, in fact, a set that contains those frequencies with large Fourier
coefficients of $S_{d}$. The key observation is that the large spectrum set
can not contain many elements.

######  Lemma 4.1 (Large spectrum set is small).

For every real number $0<\eta\ll 1$,

$\mathrm{meas}\mathcal{R}_{\eta}\ll_{\varepsilon}\eta^{-k(k+1)-2-\varepsilon}N^{-1}$

holds for arbitrarily small $\varepsilon=\varepsilon(\eta)>0$.

We can now prove Theorem 1.2 (2) driving from Lemma 4.1.

_Proof ofTheorem 1.2 (2)._

Suppose that $p>k(k+1)+2$ is a real number, using dyadic argument, we have,
for some absolute constant $C>0$,

$\int_{\mathbb{T}}|S_{d}(\alpha)|^{p}\,\mathrm{d}\alpha\leqslant\sum_{j\geqslant-C}\int_{\bigl{\\{}\alpha\in\mathbb{T}:\frac{N}{2^{j+1}}\leqslant|S_{d}(\alpha)|\leqslant\frac{N}{2^{j}}\bigr{\\}}}|S_{d}(\alpha)|^{p}\,\mathrm{d}\alpha\\\
\leqslant\sum_{j\geqslant-C}2^{-jp}N^{p}\mathrm{meas}\bigl{\\{}\alpha\in\mathbb{T}:|S_{d}(\alpha)|\geqslant
2^{-j-1}N\bigr{\\}}.$

It then follows from Lemma 4.1 that

$\int_{\mathbb{T}}|S_{d}(\alpha)|^{p}\,\mathrm{d}\alpha\leqslant
N^{p-1}2^{k(k+1)+2+\varepsilon}\sum_{j\geqslant-C}2^{j(k(k+1)+2+\varepsilon-p)}.$

And then, on noting the assumption that $p>k(k+1)+2$, it is permissible to
take $\varepsilon=\varepsilon(p)$ sufficiently small so that above geometric
series converges. Therefore, it is immediately that the left-hand side
integral is bounded by $O_{p}(N^{p-1})$.

∎

In the rest of this section we’ll concentrate on proving Lemma 4.1. Firstly,
when $\eta$ is sufficiently small, Lemma 4.1 would follow from a mean value
result.

######  Lemma 4.2 (A variant of Hua’s lemma).

Let $p\geqslant k(k+1)+2$ be a positive number, we have

$\int_{\mathbb{T}}|S_{d}(\alpha)|^{p}\,\mathrm{d}\alpha\ll N^{p-1}(\log
N)^{k(k+1)+2}.$

###### Proof.

Firstly, by expanding the $p$-th powers, we notice that when $p=k(k+1)+2$

$\int_{\mathbb{T}}|S_{d}(\alpha)|^{k(k+1)+2}\,\mathrm{d}\alpha=\int_{\mathbb{T}}\Bigl{|}\sum_{y\in[M]}\frac{\phi(d)}{d}ky^{k-1}\Lambda(dy+1)e(\alpha
y^{k})\Bigr{|}^{k(k+1)+2}\,\mathrm{d}\alpha\\\
\ll\sum_{y_{1},\dots,y_{k(k+1)+2}\in[M]}\prod_{1\leqslant i\leqslant
k(k+1)+2}\bigl{(}y_{i}^{k-1}\log(dy_{i}+1)\bigr{)}\int_{\mathbb{T}}e\bigl{(}\alpha\bigl{(}y_{1}^{k}+\cdots-
y_{k(k+1)+2}^{k}\bigr{)}\bigr{)}\,\mathrm{d}\alpha.$

From the trivial upper bounds $y_{i}^{k-1}\log(dy_{i}+1)\ll M^{k-1}\log N$ for
all $i=1,\dots,k(k+1)+2$, one may find that

$\int_{\mathbb{T}}|S_{d}(\alpha)|^{k(k+1)+2}\,\mathrm{d}\alpha\ll\bigl{(}M^{k-1}\log
N\bigr{)}^{k(k+1)+2}\int_{\mathbb{T}}\Bigl{|}\sum_{y\in[M]}e(y^{k}\alpha)\Bigr{|}^{k(k+1)+2}\,\mathrm{d}\alpha.$

Besides, let
$\underline{\alpha}=(\alpha_{1},\cdots,\alpha_{k})\in\mathbb{T}^{k}$ be a
$k$-tuple of reals, and define an exponential sum

$F(\underline{\alpha};M)=\sum_{y\in[M]}e(y\alpha_{1}+\cdots+y^{k}\alpha_{k}).$

From the observation that
$\int_{\mathbb{T}^{k}}|F(\underline{\alpha};M)|^{k(k+1)+2}\,\mathrm{d}\underline{\alpha}$
counts the number of integral solutions to the system of $k$ equations

$y_{1}^{j}+\cdots
y_{1+\frac{k(k+1)}{2}}^{j}=y_{2+\frac{k(k+1)}{2}}^{j}+\cdots+y_{k(k+1)+2}^{j}\qquad
1\leqslant j\leqslant k$

with variables $y_{i}\in[M]\bigl{(}1\leqslant i\leqslant k(k+1)+2\bigr{)}$,
hence

$\int_{\mathbb{T}}\biggl{|}\sum_{y\in[M]}e(y^{k}\alpha)\biggr{|}^{k(k+1)+2}\,\mathrm{d}\alpha\leqslant\sum_{\begin{subarray}{c}|h_{j}|\leqslant
k(k+1)M^{j}\\\ 1\leqslant j\leqslant
k-1\end{subarray}}\int_{\mathbb{T}^{k}}|F(\underline{\alpha};M)|^{k(k+1)+2}e\bigl{(}-h_{1}\alpha_{1}-\cdots-
h_{k-1}\alpha_{k-1}\bigr{)}\,\mathrm{d}\underline{\alpha}.$

Making use of [5, formula (7)] with $p=k(k+1)+2$ together with the triangle
inequality leads us to

$\int_{\mathbb{T}}\Bigl{|}\sum_{y\in[M]}e(y^{k}\alpha)\Bigr{|}^{k(k+1)+2}\,\mathrm{d}\alpha\ll
M^{k(k+1)+2-k}.$

Next, substitute the above inequality into the upper bound of
$\int_{\mathbb{T}}|S_{d}(\alpha)|^{k(k+1)+2}\,\mathrm{d}\alpha$ to get that

$\int_{\mathbb{T}}|S_{d}(\alpha)|^{k(k+1)+2}\,\mathrm{d}\alpha\ll
N^{k(k+1)+1}(\log N)^{k(k+1)+2},$

just on taking note that $N\geqslant M^{k}$.

On the other hand, it is fairly straightforward to verify that

$\left\|S_{d}\right\|_{\infty}\leqslant\sum_{y\in[M]}k\frac{\phi(d)}{d}y^{k-1}\Lambda(dy+1)\ll
N,$

once again recalling that $N\geqslant M^{k}$. We then can jump to conclude
that

$\int_{\mathbb{T}}|S_{d}(\alpha)|^{p}\,\mathrm{d}\alpha\leqslant\left\|S_{d}\right\|_{\infty}^{p-k(k+1)-2}\int_{\mathbb{T}}|S_{d}(\alpha)|^{k(k+1)+2}\,\mathrm{d}\alpha\ll
N^{p-1}(\log N)^{k(k+1)+2}$

whenever $p\geqslant k(k+1)+2$. ∎

_Proof of Lemma 4.1_.

When $\eta\ll(\log N)^{-\frac{k(k+1)+2}{\varepsilon}}$ Lemma 4.1 is just a
corollary of Lemma 4.2. Indeed, on recalling the definition of the large
spectrum set $\mathcal{R}_{\eta}$, Lemma 4.2 leads us to

$(\eta
N)^{k(k+1)+2}\mathrm{meas}\mathcal{R}_{\eta}\leqslant\int_{\mathbb{T}}|S_{d}(\alpha)|^{k(k+1)+2}\,\mathrm{d}\alpha\ll
N^{k(k+1)+1}(\log N)^{k(k+1)+2},$

which gives

$\mathrm{meas}\mathcal{R}_{\eta}\ll\eta^{-k(k+1)-2}(\log N)^{k(k+1)+2}N^{-1}.$

This allows us to conclude that
$\mathrm{meas}\mathcal{R}_{\eta}\ll\eta^{-k(k+1)-2-\varepsilon}N^{-1}$
whenever $\eta\ll(\log N)^{-\frac{k(k+1)+2}{\varepsilon}}$. Thus, in the
remainder part of the proof we can reduce the consideration to those $\eta$ in
the region

$\displaystyle(\log N)^{-\frac{k(k+1)+2}{\varepsilon}}\ll\eta\ll 1.$ (4.1)

We may pick up a discrete sequence
$\left\\{\alpha_{1},\cdots,\alpha_{R}\right\\}\subset\mathbb{T}$ for which
satisfies

  1. (1)

For any $\alpha_{r}\in\left\\{\alpha_{1},\cdots,\alpha_{R}\right\\}$,
$|S_{d}(\alpha_{r})|\geqslant\eta N$;

  2. (2)

For any pair
$\alpha_{i}\neq\alpha_{j}\in\left\\{\alpha_{1},\cdots,\alpha_{R}\right\\}$,
$|\alpha_{i}-\alpha_{j}|\geqslant N^{-1}$;

  3. (3)

For any $\alpha\in\mathcal{R}_{\eta}$, there is some
$\alpha_{r}\in\left\\{\alpha_{1},\cdots,\alpha_{R}\right\\}$ such that
$|\alpha-\alpha_{r}|\leqslant N^{-1}$;

  4. (4)

$R$ is the largest integer such that above three events holds.

Consequently, it suffices to show that

$\displaystyle R\ll_{\varepsilon}\eta^{-k(k+1)-2-\varepsilon}.$ (4.2)

And, clearly, $\mathrm{meas}\mathcal{R}_{\eta}\leqslant 2R/N$.

For $1\leqslant r\leqslant R$, let
$c_{r}\in\left\\{z\in\mathbb{C}:|z|=1\right\\}$ be a number such that
$c_{r}S_{d}(\alpha_{r})=|S_{d}(\alpha_{r})|$. One may find from the first
assumption of above sequence $\left\\{\alpha_{1},\cdots,\alpha_{R}\right\\}$
that

$R^{2}\eta^{2}N^{2}\leqslant\Bigl{(}\sum_{1\leqslant r\leqslant
R}|S_{d}(\alpha_{r})|\Bigr{)}^{2}.$

Expanding the definition of the exponential sum and using Cauchy-Schwarz
inequality successively to obtain that

$\displaystyle R^{2}\eta^{2}N^{2}$
$\displaystyle\leqslant\biggl{|}\sum_{1\leqslant r\leqslant
R}c_{r}\sum_{y\in[M]}k\frac{\phi(d)}{d}y^{k-1}\Lambda(dy+1)e(\alpha_{r}y^{k})\biggr{|}^{2}$
$\displaystyle\ll
N\sum_{y\in[M]}k\frac{\phi(d)}{d}y^{k-1}\Lambda(dy+1)\Bigl{|}\sum_{1\leqslant
r\leqslant R}c_{r}e(\alpha_{r}y^{k})\Bigr{|}^{2},$

just on noting that $\sum_{y\in[M]}k\frac{\phi(d)}{d}y^{k-1}\Lambda(dy+1)\ll
N$. Expanding the square to double the variables $c_{r}$ we then have

$R^{2}\eta^{2}N\ll\sum_{y\in[M]}\sum_{1\leqslant r,r^{\prime}\leqslant
R}c_{r}\overline{c_{r^{\prime}}}k\frac{\phi(d)}{d}y^{k-1}\Lambda(dy+1)e\bigl{(}(\alpha_{r}-\alpha_{r^{\prime}})y^{k}\bigr{)}.$

In view of $c_{r}$, $c_{r^{\prime}}$ are 1-bounded, we can swap the order of
summations and then take absolute value inside $r$ and $r^{\prime}$, it then
follows from the definition of $S_{d}(\alpha)$ that

$R^{2}\eta^{2}N\ll\sum_{1\leqslant r,r^{\prime}\leqslant
R}|S_{d}(\alpha_{r}-\alpha_{r^{\prime}})|.$

Let $\gamma>k$ be a positive number, utilizing Hölder’s inequality one has

$R^{2}\eta^{2\gamma}N^{\gamma}\ll\sum_{1\leqslant r,r^{\prime}\leqslant
R}|S_{d}(\alpha_{r}-\alpha_{r^{\prime}})|^{\gamma}.$

Taking note that $(\log N)^{-D/2}\ll\eta$, where $D$ is defined in (3.8),
whenever $\eta$ satisfies (4.1), Lemma 3.3 with $N^{\prime}=N$ leads us to
that the contribution of $S_{d}(\alpha_{r}-\alpha_{r^{\prime}})$ is negligible
if $\alpha_{r}-\alpha_{r^{\prime}}\in\mathfrak{m}$. We then can assume further
that $\alpha_{r}-\alpha_{r^{\prime}}\in\mathfrak{M}$. Under such
circumstances, the major-arc approximate recorded in Lemma 3.2 with
$N^{\prime}=N$ leads us to

$S_{d}(\alpha_{r}-\alpha_{r^{\prime}})\ll
q^{-\frac{1}{k}+\varepsilon}N(1+N|\alpha_{r}-\alpha_{r^{\prime}}-a/q|)^{-1}$

for some relatively prime pair $1\leqslant a\leqslant q\leqslant Q$(recalling
the definition of major arcs $\mathfrak{M}$). We also notice that when
$q\geqslant\widetilde{Q}$,

$S_{d}(\alpha_{r}-\alpha_{r^{\prime}})\ll\widetilde{Q}^{-\frac{1}{k}+\varepsilon}N.$

Hence, if we take $\widetilde{Q}=\eta^{-2k+\varepsilon}$ the contribution of
$S_{d}(\alpha_{r}-\alpha_{r^{\prime}})$ with $q>\tilde{Q}$ is also negligible.
Combining what we have so far it is allowed to restrict our considerations to
those $q$ which are no more than $\widetilde{Q}$. And we’ll have

$\eta^{2\gamma}R^{2}\ll\sum_{q\leqslant\widetilde{Q}}\sum_{\begin{subarray}{c}a(q)\\\
(a,q)=1\end{subarray}}\sum_{1\leqslant r,r^{\prime}\leqslant
R}q^{-\frac{\gamma}{k}+\varepsilon}(1+N|\alpha_{r}-\alpha_{r^{\prime}}-a/q|)^{-\gamma}\leqslant\sum_{1\leqslant
r,r^{\prime}\leqslant R}G(\alpha_{r}-\alpha_{r^{\prime}}),$

where

$G(\alpha)=\sum_{q\leqslant\widetilde{Q}}\sum_{a(q)}q^{-\frac{\gamma}{k}+\varepsilon}F(\alpha-a/q)$

and

$F(\beta)=(1+N|\sin\beta|)^{-\gamma}.$

We are now in the position of [4, Eq.(4.16)], just need to replace $N^{2}$ by
$N$, replace $\delta$ by $\eta$. Following from Bourgain’s proof, one may find
that when $\gamma\geqslant k+\varepsilon$

$\eta^{2\gamma}R^{2}N^{-2}\ll
N^{-1}\Bigl{\\{}\widetilde{Q}^{\tau}RN^{-1}+R^{2}N^{-2}\widetilde{Q}\cdot\\#\left\\{|u|\leqslant
N:d(\widetilde{Q};u)\geqslant\widetilde{Q}^{\tau}\right\\}\Bigr{\\}},$

where $\tau>0$ is a number to be chosen later, $d(\widetilde{Q};u)$ counts the
number of the divisors of $u$ which are smaller than $\widetilde{Q}$. It
follows from [4, Lemma 4.28] that

$R\ll\eta^{-2\gamma}\widetilde{Q}^{\tau}+\eta^{-2\gamma}\widetilde{Q}^{1-B}R$

with $\tau<B<+\infty$. By careful choice of $\tau$ and $B$, together with
letting $\gamma=\frac{k(k+1)+2}{2}+\varepsilon$ will lead us to the desired
bound (4.2) of $R$.

∎

##  5\. A Local Inverse Theorem

The main goal of this section is to prove the following local inverse result.
Despite it being in Fourier-analysis language, the basic point is still that
if the sum of a function $f$ is zero, and the counting of configurations
weighted by $f$ is non-zero, then either the summation interval is short, or
there is a reasonable bias of $f$ towards a structured sub-region of the
summation interval.

######  Proposition 5.1 (Local inverse theorem).

Let $0<\delta<1$ be a positive parameter satisfying $\delta\gg(\log N)^{-1}$.
Suppose that $(a^{\prime},q^{\prime})=1$ are coprime integers,
$\Lambda_{a^{\prime},q^{\prime}}$ is the majorant function defined in (1.1)
which is supported on the interval $[N^{\prime}]$ with $N^{\prime}\gg
N^{0.9}$, and $a^{\prime}+q^{\prime}\cdot[N^{\prime}]\subseteq[N]$. Let
$k\geqslant 1$, $1\leqslant d\ll\exp\bigl{(}c^{\prime}\sqrt{\log N}\bigr{)}$
be integers and $M=\left\lfloor N^{\prime 1/k}\right\rfloor$. Assume that
$\nu,\nu_{1}:[N^{\prime}]\to\mathbb{C}$ are functions satisfying
$0\leqslant\nu,\nu_{1}\leqslant\Lambda_{a^{\prime},q^{\prime}}$ pointwise, and
$f=\nu_{1}-\delta\Lambda_{a^{\prime},q^{\prime}}$ satisfies $\mathbb{E}(f)=0$.
If

$\delta^{2}N^{\prime
2}\ll\int_{\mathbb{T}}\bigl{|}\widehat{f}(\alpha)S_{d}(\alpha)\widehat{\nu}(\alpha)\bigr{|}\,\mathrm{d}\alpha,$

where
$S_{d}(\alpha)=\sum_{y\in[M]}\frac{\phi(d)}{d}ky^{k-1}\Lambda(dy+1)e(y^{k}\alpha)$
which is defined in (1.2), then one of the following two alternatives holds:

  1. (1)

($N$ is small) $(\log N)^{A}\ll q^{\prime}$;

  2. (2)

($f$ has local structure) there is an arithmetic progression
$P=a+q^{k}\cdot[X]\subseteq[N^{\prime}]$ with $(a,q)=1$,
$q\ll\delta^{-2k^{3}-6k^{2}}$, and $X\gg q^{1-k}(\log
N)^{-2^{2k}D}N^{\prime}$, where $D$ is the number defined in (3.8), such that

$\sum_{x\in P}f(x)\gg\delta^{4k^{3}+16k^{2}}\sum_{n\in P}\Lambda(n).$

By decomposing the integral interval $\mathbb{T}$ into major and minor arcs
$\mathfrak{M}$ and $\mathfrak{m}$ as in (3.1), it can be seen from the
assumption of Proposition 5.1 that

$\displaystyle\delta^{2}N^{\prime
2}\ll\int_{\mathfrak{M}}\bigl{|}\widehat{f}(\alpha)S_{d}(\alpha)\widehat{\nu}(\alpha)\bigr{|}\,\mathrm{d}\alpha+\int_{\mathfrak{m}}\bigl{|}\widehat{f}(\alpha)S_{d}(\alpha)\widehat{\nu}(\alpha)\bigr{|}\,\mathrm{d}\alpha.$
(5.1)

We’d like to show that the integral over the minor arcs is negligible, i.e.
bounded by $c\delta^{2}N^{2}$ for some tiny enough $c>0$, with the help of the
restriction lemmas established in Section 2 and Section 4. By making use of
Hölder’s inequality, the second term in the right-hand side of (5.1), i.e. the
integral over minor arcs, is bounded by

$\quad\Bigl{(}\int_{\mathfrak{m}}\left|S_{d}(\alpha)\right|^{k(k+1)+4}\,\mathrm{d}\alpha\Bigr{)}^{\frac{1}{k(k+1)+4}}\Bigl{(}\int_{\mathbb{T}}\left|\widehat{f}(\alpha)\right|^{\frac{k(k+1)+4}{k(k+1)+3}}\left|\widehat{\nu}(\alpha)\right|^{\frac{k(k+1)+4}{k(k+1)+3}}\,\mathrm{d}\alpha\Bigr{)}^{\frac{k(k+1)+3}{k(k+1)+4}}\\\
\leqslant\sup_{\alpha\in\mathfrak{m}}\bigl{|}S_{d}(\alpha)\bigr{|}^{\frac{1}{k(k+1)+4}}\Bigl{(}\int_{\mathbb{T}}\left|S_{d}(\alpha)\right|^{k(k+1)+3}\,\mathrm{d}\alpha\Bigr{)}^{\frac{1}{k(k+1)+4}}\\\
\Bigl{(}\int_{\mathbb{T}}\left|\widehat{f}(\alpha)\right|^{\frac{2k(k+1)+8}{k(k+1)+3}}\,\mathrm{d}\alpha\Bigr{)}^{\frac{k(k+1)+3}{2k(k+1)+8}}\Bigl{(}\int_{\mathbb{T}}\left|\widehat{\nu}(\alpha)\right|^{\frac{2k(k+1)+8}{k(k+1)+3}}\,\mathrm{d}\alpha\Bigr{)}^{\frac{k(k+1)+3}{2k(k+1)+8}}.$

The application of Lemma 2.4 with $f$ and $p=\frac{2k(k+1)+8}{k(k+1)+3}$,
Theorem 1.2 with $\Lambda_{b,d}=\Lambda_{a^{\prime},q^{\prime}}$ and
$p=\frac{2k(k+1)+8}{k(k+1)+3}$ and with $S_{d}(\alpha)$ and $p=k(k+1)+3$
yields that

$\int_{\mathfrak{m}}\bigl{|}\widehat{f}(\alpha)S_{d}(\alpha)\widehat{\nu}(\alpha)\bigr{|}\,\mathrm{d}\alpha\ll\sup_{\alpha\in\mathfrak{m}}\bigl{|}S_{d}(\alpha)\bigr{|}^{\frac{1}{k(k+1)+4}}N^{\prime\frac{2k(k+1)+7}{k(k+1)+4}}.$

It then follows from Lemma 3.3 and the assumption $(\log N)^{-1}\ll\delta$
that

$\int_{\mathfrak{m}}\bigl{|}\widehat{f}(\alpha)S_{d}(\alpha)\widehat{\nu}(\alpha)\bigr{|}\,\mathrm{d}\alpha\ll(\log
N)^{-\frac{D}{k(k+1)+4}}N^{\prime 2}\ll\delta^{2}N^{\prime 2}.$

Substituting the above inequality into (5.1), one thus has

$\displaystyle\int_{\mathfrak{M}}\bigl{|}\widehat{f}(\alpha)S_{d}(\alpha)\widehat{\nu}(\alpha)\bigr{|}\,\mathrm{d}\alpha\gg\delta^{2}N^{\prime
2}.$ (5.2)

The next step is to show that we can gain from (5.2) a long arithmetic
progression which possesses much more information of the function $f$. To
carry out this procedure we are now proving that the inequality (5.2) can be
reduced to a subset of major arcs $\mathfrak{M}$, and the set only contains
frequencies which are close to rationals with quite small denominators
(depending only on $\delta$).

Let $q$ be a positive integer, define a set with respect to $q$ as

$\mathfrak{M}(q)=\Bigl{\\{}\alpha\in\mathbb{T}:\exists 1\leqslant a\leqslant
q\text{ with }(a,q)=1\text{ such that
}\bigl{|}\alpha-\frac{a}{q}\bigr{|}\leqslant\frac{Q}{qN^{\prime}}\Bigr{\\}},$

where $Q$ is the number defined in (3.2). We now partition the major arcs
$\mathfrak{M}$ defined as in (3.1) into two disjoint ranges

$\mathfrak{M}_{1}=\bigcup_{1\leqslant
q\ll\delta^{-2k^{3}-6k^{2}}}\mathfrak{M}(q)\qquad\text{and}\qquad\mathfrak{M}_{2}=\bigcup_{\delta^{-2k^{3}-6k^{2}}\ll
q\ll Q}\mathfrak{M}(q).$

It follows from the triangle inequalty and (5.2) that

$\displaystyle\delta^{2}N^{\prime
2}\ll\int_{\mathfrak{M}_{1}}\bigl{|}\widehat{f}(\alpha)S_{d}(\alpha)\widehat{\nu}(\alpha)\bigr{|}\,\mathrm{d}\alpha+\int_{\mathfrak{M}_{2}}\bigl{|}\widehat{f}(\alpha)S_{d}(\alpha)\widehat{\nu}(\alpha)\bigr{|}\,\mathrm{d}\alpha.$
(5.3)

In an analogous way of proving that the integral over minor arcs
$\mathfrak{m}$ is negligible, we can also show that the integral over
$\mathfrak{M}_{2}$ is negligible. Indeed, one can deduce from Hölder’s
inequality and restriction estimates, i.e. Lemma 2.4 and Theorem 1.2, that

$\int_{\mathfrak{M}_{2}}\bigl{|}\widehat{f}(\alpha)S_{d}(\alpha)\widehat{\nu}(\alpha)\bigr{|}\,\mathrm{d}\alpha\ll\sup_{\alpha\in{\mathfrak{M}_{2}}}\bigl{|}S_{d}(\alpha)\bigr{|}^{\frac{1}{k(k+1)+4}}N^{\prime\frac{2k(k+1)+7}{k(k+1)+4}}.$

Recalling that $q\ll Q\ll(\log N)^{2^{2k}D}$, the application of Lemma 3.2
shows that the above integral is bounded by

$\Bigl{(}q^{-\frac{1}{k}+\varepsilon}\min\left\\{N^{\prime},|\beta|^{-1}\right\\}+N^{\prime}(1+N^{\prime}|\beta|)\exp\bigl{(}-c\sqrt{\log
N}\bigr{)}\Bigr{)}^{\frac{1}{k(k+1)+4}}N^{\prime\frac{2k(k+1)+7}{k(k+1)+4}}$

for all $\frac{a}{q}+\beta\in\mathfrak{M}_{2}$ with coprime pair $1\leqslant
a\leqslant q$. The assumption $\delta^{-2k^{3}-6k^{2}}\ll q\ll(\log
N)^{2^{2k}D}$ whenever $\alpha=\frac{a}{q}+\beta\in\mathfrak{M}_{2}$ then
gives us that

$\int_{\mathfrak{M}_{2}}\bigl{|}\widehat{f}(\alpha)S_{d}(\alpha)\widehat{\nu}(\alpha)\bigr{|}\,\mathrm{d}\alpha\ll\delta^{2}N^{\prime
2}.$

Combining the above inequality with (5.3) one thus has

$\int_{\mathfrak{M}_{1}}\bigl{|}\widehat{f}(\alpha)S_{d}(\alpha)\widehat{\nu}(\alpha)\bigr{|}\,\mathrm{d}\alpha\gg\delta^{2}N^{\prime
2}.$

Up to now, we complete all the reduction argument.

_Proof of Proposition 5.1._

It can be deduced from the definition of $\mathfrak{M}_{1}$ together with the
above inequality that

$\delta^{2}N^{\prime
2}\ll\sum_{q\ll\delta^{-2k^{3}-6k^{2}}}\int_{\mathfrak{M}(q)}|\widehat{f}(\alpha)\widehat{\nu}(\alpha)S_{d}(\alpha)|\,\mathrm{d}\alpha.$

Taking note that the asymptotic formula in Lemma 3.2 is valid when
$q\ll\delta^{-2k^{3}-6k^{2}}$ and $\delta\gg(\log N)^{-1}$, in consequence of
the above inequality, Hölder’s inequality and Theorem 1.2 and Lemma 3.2, one
has

$\displaystyle\delta^{2}N^{\prime 2}$
$\displaystyle\ll\sum_{q\ll\delta^{-2k^{3}-6k^{2}}}\left\|\widehat{\nu}\right\|_{3}\Bigl{(}\int_{\mathfrak{M}(q)}|\widehat{f}(\alpha)|^{2}\,\mathrm{d}\alpha\Bigr{)}^{1/2}\Bigl{(}\int_{\mathfrak{M}(q)}|S_{d}(\alpha)|^{6}\,\mathrm{d}\alpha\Bigr{)}^{1/6}$
$\displaystyle\ll\sum_{q\ll\delta^{-2k^{3}-6k^{2}}}q^{-1/k+\varepsilon}N^{\prime
5/3}\Bigl{(}\int_{\mathfrak{M}(q)}|\widehat{f}(\alpha)|^{2}\,\mathrm{d}\alpha\Bigr{)}^{1/2}\Bigl{(}\int_{|\beta|\leqslant\frac{Q}{qN^{\prime}}}(1+N^{\prime}|\beta|)^{-6}\,\mathrm{d}\beta\Bigr{)}^{1/6}$
$\displaystyle\ll\sum_{q\ll\delta^{-2k^{3}-6k^{2}}}q^{-1/k+\varepsilon}N^{\prime
3/2}\Bigl{(}\int_{\mathfrak{M}(q)}|\widehat{f}(\alpha)|^{2}\,\mathrm{d}\alpha\Bigr{)}^{1/2}.$

Clearly, on noting that
$\sum_{q\leq\delta^{-2k^{3}-6k^{2}}}q^{-\frac{1}{k}+\varepsilon}\ll\delta^{-2k^{3}-6k^{2}}$,
it then follows from pigeonhole principle that there is some
$q\leq\delta^{-2k^{3}-6k^{2}}$ such that

$\displaystyle\delta^{4k^{3}+16k^{2}}N^{\prime}\ll\int_{\mathfrak{M}(q)}|\widehat{f}(\alpha)|^{2}\,\mathrm{d}\alpha.$
(5.4)

Now, suppose that $P=q\cdot[X]$ is an arithmetic progression of common
difference $q$, the denominator such that (5.4) holds, and of length
$X=Q^{-1}N^{\prime}/(2\pi)$. We are going to show that, if $N^{\prime}$ (thus
$N$) is not small, for any frequency $\alpha$ in $\mathfrak{M}(q)$, the
Fourier coefficient $|\widehat{1}_{-P}(\alpha)|$ is large. For this purpose,
we now consider the Fourier transform of the indicator function $1_{-P}$.
After changing variables $x$ to $-x$ one has

$|\widehat{1}_{-P}(\alpha)|=\biggl{|}\sum_{x}1_{-P}(x)e(x\alpha)\biggr{|}=\biggl{|}\sum_{x}1_{P}(x)e(-x\alpha)\biggr{|}.$

On omitting the indicator function $1_{P}$,

$\displaystyle|\widehat{1}_{-P}(\alpha)|=\biggl{|}\sum_{x\in[X]}e(-\alpha
a_{0}-qx\alpha)\biggr{|}=\biggl{|}\sum_{x\in[X]}e(-qx\alpha)\biggr{|}.$ (5.5)

With the help of the estimate $|1-e(\alpha)|\leqslant
2\pi\left\|\alpha\right\|$, as well as the triangle inequality, we can ensure
that

$|\widehat{1}_{-P}(\alpha)|\geqslant
X-\sum_{x\in[X]}2\pi\left\|qx\alpha\right\|\geqslant X-\pi
X^{2}\left\|q\alpha\right\|.$

It thus follows immediately from the facts $\left\|q\alpha\right\|\ll QN^{-1}$
$(\alpha\in\mathfrak{M}(q))$ and $X=Q^{-1}N^{\prime}/(2\pi)$ that whenever
$\alpha\in\mathfrak{M}(q)$,

$\displaystyle|\widehat{1}_{-P}(\alpha)|\geqslant X/2.$ (5.6)

In order to prove the local inverse theorem, our strategy is to show that the
convolution $f*1_{-P}(x)$ is large at some point $x$, which means that $f$
correlates with a translation of this progression $P$. On combining (5.4) with
(5.6), we have

$\delta^{4k^{3}+16k^{2}}X^{2}N^{\prime}\ll\int_{\mathfrak{M}(q)}|\widehat{1}_{-P}(\alpha)|^{2}|\widehat{f}(\alpha)|^{2}\,\mathrm{d}\alpha.$

Then by removing the restriction of domain of the integral to get that

$\delta^{4k^{3}+16k^{2}}X^{2}N^{\prime}\ll\int_{\mathbb{T}}|\widehat{1}_{-P}(\alpha)|^{2}|\widehat{f}(\alpha)|^{2}\,\mathrm{d}\alpha.$

It can be deduced from Parseval’s identity that

$\displaystyle\delta^{4k^{3}+16k^{2}}X^{2}N^{\prime}\ll\left\|f*1_{-P}\right\|_{2}^{2}\leqslant\left\|f*1_{-P}\right\|_{\infty}\left\|f*1_{-P}\right\|_{1}.$
(5.7)

On noticing that the assumption
$f\geqslant-\delta\Lambda_{a^{\prime},q^{\prime}}$ pointwise yields that for
each natural number $x$

$f*1_{-P}(x)\geqslant-\delta\frac{\phi(q^{\prime})}{q^{\prime}}\sum_{n\in
x+P}\Lambda(a^{\prime}+q^{\prime}n)=-\delta\frac{\phi(q^{\prime})}{q^{\prime}}\sum_{n\in[X]}\Lambda(a^{\prime}+q^{\prime}x+qq^{\prime}n).$

We may firstly assume that there is some $x$ such that
$|f*1_{-P}(x)|\geqslant\delta^{1/2}\frac{\phi(q^{\prime})qX}{\phi(qq^{\prime})}$.
As $\mathrm{supp}(f)\subseteq[N^{\prime}]$, we can also suppose that
$1\leqslant a^{\prime}+q^{\prime}x\leqslant N-qq^{\prime}X$. Since $X\gg
Q^{-1}N^{\prime}\gg N^{\prime 0.9}$ and $q\ll\delta^{-2k^{3}-6k^{2}}\ll(\log
N)^{A}$ whenever $\delta^{-1}\ll\log N$, it follows from Lemma 2.2 and the
assumption $q^{\prime}\ll(\log N)^{A}$ (i.e. case (1) in the statement of the
proposition would not occur) that for any $(\overline{a},qq^{\prime})=1$

$\sum_{\begin{subarray}{c}N-qq^{\prime}X<n\leqslant N\\\
n\equiv\overline{a}\pmod{qq^{\prime}}\end{subarray}}\Lambda(n)<1.01\frac{qq^{\prime}X}{\phi(qq^{\prime})}.$

One thus has

$f*1_{-P}(x)\geqslant-2\delta\frac{\phi(q^{\prime})}{q^{\prime}}\frac{qq^{\prime}X}{\phi(qq^{\prime})}=-2\delta\frac{\phi(q^{\prime})qX}{\phi(qq^{\prime})}.$

Thus, on combining the above analysis with the assumption
$|f*1_{-P}(x)|\geqslant\delta^{1/2}\frac{\phi(q^{\prime})qX}{\phi(qq^{\prime})}$
we must have, in this case,

$\displaystyle
f*1_{-P}(x)\geqslant\delta^{1/2}\frac{\phi(q^{\prime})qX}{\phi(q^{\prime}q)}\gg\delta\frac{qX}{\phi(q)},$
(5.8)

which gains more than we expected.

Therefore, we can suppose that
$\left\|f*1_{-P}\right\|_{\infty}\leqslant\delta^{1/2}\frac{\phi(q^{\prime})qX}{\phi(qq^{\prime})}$.
In view of (5.7), we have

$\sum_{x}|f*1_{-P}(x)|\gg\delta^{4k^{3}+16k^{2}-1/2}\frac{\phi(qq^{\prime})}{\phi(q^{\prime})q}XN^{\prime}.$

On noting that $\frac{q}{\phi(q)}\ll\delta^{-\varepsilon}$ whenever
$q\ll\delta^{-2k^{3}-6k^{2}}$,

$\displaystyle\sum_{x}|f*1_{-P}(x)|\gg\delta^{4k^{3}+16k^{2}}\frac{q}{\phi(q)}XN^{\prime}.$
(5.9)

However, on the other hand, we also notice from the definition of the
convolution that

$\sum_{x}f*1_{-P}(x)=\sum_{y}f(y)\sum_{x}1_{-P}(x-y).$

For each fixed number $y$, the inner sum is always $X$, the length of the
progression $P$. Thus, as a consequence of $\mathbb{E}(f)=0$,

$\displaystyle\sum_{x}f*1_{-P}(x)=0.$ (5.10)

Hence, on combining (5.9) and (5.10), it follows from the pigeonhole
principle, together with $\mathrm{supp}(f)\subset[N^{\prime}]$, that there is
some $x$ such that

$f*1_{-P}(x)\gg\delta^{4k^{3}+16k^{2}}\frac{q}{\phi(q)}X.$

Expanding the convolution one more time, which would allow us to conclude that

$\delta^{4k^{3}+16k^{2}}\frac{q}{\phi(q)}X\ll\sum_{y\in x+P}f(y).$

The last step is to partition the above arithmetic progression
$x+P=x+q\cdot[X]$ into sub-progressions with common difference $q^{k}$, and as
a consequence, each of the sub-progressions has length
$\gg\left\lfloor\frac{X}{q^{k-1}}\right\rfloor\gg q^{1-k}(\log
N)^{-2^{2k}D}N$. It is not hard to find from the above inequality and the
pigeonhole principle that there is such a sub-progression $P^{\prime}$ such
that

$\delta^{4k^{3}+16k^{2}}\frac{q}{\phi(q)}|P^{\prime}|\ll\sum_{y\in
P^{\prime}}f(y).$

For simplification of writing, let $P^{\prime}=a^{\prime}+q^{k}\cdot[X]$,
obviously, $(a^{\prime},q)=1$ and $1\leqslant a^{\prime}\leqslant N^{\prime}$.
It follows from Lemma 2.2 and the assumptions $X\gg q^{1-k}(\log
N)^{-2^{2k}D}N\gg N^{0.55+\varepsilon}$ and $q^{k}\ll(\log N)^{A}$ that

$\sum_{n\in
P^{\prime}}\Lambda(n)=\sum_{\begin{subarray}{c}a^{\prime}<n\leqslant
a^{\prime}+q^{k}X\\\ n\equiv
a^{\prime}\pmod{q^{k}}\end{subarray}}\Lambda(n)\asymp\frac{q}{\phi(q)}|P^{\prime}|,$

which yields that $\delta^{4k^{3}+16k^{2}}\sum_{n\in
P^{\prime}}\Lambda(n)\ll\sum_{n\in P^{\prime}}f(n)$.

∎

##  6\. Density Increment

We are going to prove Theorem 1.1 in this section. The proof strategy is to
run a density increment argument in arithmetic progressions in primes. Roth
uses the corresponding argument in integers to bound the size of sets lacking
3-term arithmetic progressions, and Sárközy employs it to bound the size of
sets lacking Furstenberg-Sárközy configurations $x,x+y^{2}$. Although the
difficulty in dense and sparse sets varies, the basic idea is always to find a
long progression in which $\mathcal{A}$ has an increasing relative density if
$\mathcal{A}$ lacks the desired configurations. We now record it as a lemma.

######  Lemma 6.1 (Lacking configurations leads to increasing density).

Suppose that $Q=a+q^{k}\cdot[N^{\prime}]\subseteq[N]$ is an arithmetic
progression with $(a,q)=1$, $N^{\prime}\gg N^{0.9}$ and $q\ll(\log N)^{A}$.
$\mathcal{A}\subseteq\mathcal{P}$ is a subset of primes without configurations
$p_{1},p_{1}+(p_{2}-1)^{k}$, where $p_{1},p_{2}$ are primes, and

$\sum_{n\in Q}\Lambda\cdot 1_{\mathcal{A}}(n)=\delta\sum_{n\in Q}\Lambda(n).$

If $\delta\gg(\log N)^{-1}$, then there are integers
$q^{\prime}\ll\delta^{-2k^{3}-6k^{2}}$ and $X\gg q^{\prime 1-k}(\log
N^{\prime})^{-2^{2k}D}N^{\prime}$ such that one of the three events happens

  1. (1)

(the common difference is large) $(qq^{\prime})^{k}\gg(\log N)^{A}$; or

  2. (2)

($X$ is small) $X\ll N^{0.9}$; or

  3. (3)

($\mathcal{A}$ has increasing density in a sub-progression) there is a sub-
progression $Q^{\prime}=a+a^{\prime}q^{k}+(qq^{\prime})^{k}\cdot[X]$ inside
$Q$ with $(a^{\prime},q^{\prime})=1$ such that

$\sum_{n\in Q^{\prime}}\Lambda\cdot
1_{\mathcal{A}}(n)\geqslant(\delta+c\delta^{6k^{3}+16k^{2}})\sum_{n\in
Q^{\prime}}\Lambda(n).$

###### Proof.

The same as before, set
$\Lambda_{a,q^{k}}=\frac{\phi(q)}{q}\Lambda(a+q^{k}\cdot)$ and
$1_{\mathcal{A}_{a,q^{k}}}=1_{\mathcal{A}}(a+q^{k}\cdot)$, and let
$f=\Lambda_{a,q^{k}}\cdot
1_{\mathcal{A}_{a,q^{k}}}-\delta\Lambda_{a,q^{k}}:[N^{\prime}]\to\mathbb{C}$
be the weighted balanced function. Clearly, from the assumption, the average
of $f$ is zero, i.e. $\sum_{n\in[N^{\prime}]}f(n)=0$.

Inspired by the proof of [3, Lemma 5] we consider the following expression

$\int_{\mathbb{T}}\widehat{f}(-\alpha)\widehat{\Lambda_{a,q^{k}}\cdot
1_{\mathcal{A}_{a,q^{k}}}}(\alpha)S_{q}(\alpha)\,\mathrm{d}\alpha,$

where $S_{q}(\alpha)=\sum_{y\leqslant
M}\frac{\phi(q)}{q}ky^{k-1}\Lambda(qy+1)e(y^{k}\alpha)$ and $M=\left\lfloor
N^{\prime 1/k}\right\rfloor$. It can be seen from the definition of function
$f$ that the above expression can be splitting into the sum of two integrals

$\int_{\mathbb{T}}\widehat{\Lambda_{a,q^{k}}\cdot
1_{\mathcal{A}_{a,q^{k}}}}(-\alpha)\widehat{\Lambda_{a,q^{k}}\cdot
1_{\mathcal{A}_{a,q^{k}}}}(\alpha)S_{q}(\alpha)\,\mathrm{d}\alpha$

and

$-\delta\int_{\mathbb{T}}\widehat{\Lambda}_{a,q^{k}}(-\alpha)\widehat{\Lambda_{a,q^{k}}\cdot
1_{\mathcal{A}_{a,q^{k}}}}(\alpha)S_{q}(\alpha)\,\mathrm{d}\alpha.$

It is obvious from the orthogonality principle, as well as the notation
$\Lambda_{a,q^{k}}=\frac{\phi(q)}{q}\Lambda(a+q^{k}\cdot)$, that the first
integral is

$\Bigl{(}\frac{\phi(q)}{q}\Bigr{)}^{3}\sum_{x\leqslant
N^{\prime}}\sum_{y\leqslant M}ky^{k-1}\Lambda(qy+1)\Lambda\cdot
1_{\mathcal{A}}(a+q^{k}x)\Lambda\cdot 1_{\mathcal{A}}(a+q^{k}(x+y^{k})),$

which counts the weighted number of patterns $p_{1},p_{1}+(p_{2}-1)^{k}$ in
$\mathcal{A}\cap Q$ (indeed we can take $p_{1}=a+q^{k}x$, $p_{2}=qy+1$ and
$p_{1}+(p_{2}-1)^{k}=a+q^{k}(x+y^{k})$), thus, from our assumption, it is
zero. Meanwhile, by introducing an auxiliary function
$\delta\Lambda_{a,q^{k}}$, one may find that the second integral is the sum of
the following two representations

$-\delta^{2}\int_{\mathbb{T}}\widehat{\Lambda}_{a,q^{k}}(-\alpha)\widehat{\Lambda}_{a,q^{k}}(\alpha)S_{q}(\alpha)\,\mathrm{d}\alpha$

and

$-\delta\int_{\mathbb{T}}\widehat{\Lambda}_{a,q^{k}}(-\alpha)\bigl{(}\widehat{\Lambda_{a,q^{k}}\cdot
1_{\mathcal{A}_{a,q^{k}}}}-\delta\widehat{\Lambda}_{a,q^{k}}\bigr{)}(\alpha)S_{q}(\alpha)\,\mathrm{d}\alpha=-\delta\int_{\mathbb{T}}\widehat{\Lambda}_{a,q^{k}}(-\alpha)\widehat{f}(\alpha)S_{q}(\alpha)\,\mathrm{d}\alpha.$

on recalling that $f=\Lambda_{a,q^{k}}\cdot
1_{\mathcal{A}_{a,q^{k}}}-\delta\Lambda_{a,q^{k}}$ once again. Therefore,
putting the above analysis together we obtain that

$\int_{\mathbb{T}}\widehat{f}(-\alpha)\widehat{\Lambda_{a,q^{k}}\cdot
1_{\mathcal{A}_{a,q^{k}}}}(\alpha)S_{q}(\alpha)\,\mathrm{d}\alpha+\delta\int_{\mathbb{T}}\widehat{\Lambda}_{a,q^{k}}(-\alpha)\widehat{f}(\alpha)S_{q}(\alpha)\,\mathrm{d}\alpha\\\
=-\delta^{2}\int_{\mathbb{T}}\widehat{\Lambda}_{a,q^{k}}(-\alpha)\widehat{\Lambda}_{a,q^{k}}(\alpha)S_{q}(\alpha)\,\mathrm{d}\alpha.$

Taking absolute values both sides and noting that $0<\delta<1$, one thus has,

$\delta^{2}\Bigl{|}\int_{\mathbb{T}}\widehat{\Lambda}_{a,q^{k}}(-\alpha)\widehat{\Lambda}_{a,q^{k}}(\alpha)S_{q}(\alpha)\,\mathrm{d}\alpha\Bigr{|}\\\
\leqslant\int_{\mathbb{T}}\bigl{|}\widehat{f}(\alpha)\widehat{\Lambda}_{a,q^{k}}(\alpha)S_{q}(\alpha)\bigr{|}\,\mathrm{d}\alpha+\int_{\mathbb{T}}\bigl{|}\widehat{f}(\alpha)\widehat{\Lambda_{a,q^{k}}\cdot
1_{\mathcal{A}_{a,q^{k}}}}(\alpha)S_{q}(\alpha)\bigr{|}\,\mathrm{d}\alpha.$
(6.1)

By truncating the integral region on the left-hand side into major arcs
$\cup_{r\leqslant
Q}\cup_{b(r)\,(b,r)=1}\left\\{\frac{b}{r}+\beta:|\beta|\leqslant\frac{Q}{rN^{\prime}}\right\\}$
which are defined in (3.1), it can be deduced from Lemma 3.2 and Lemma B.1
that when $q\ll(\log N)^{A}$ this integral is at least

$\sum_{r\leqslant
Q}\Bigl{(}\frac{\phi(q)}{\phi(qr)}\Bigr{)}^{3}\sum_{\begin{subarray}{c}1\leqslant
b\leqslant r\\\
(b,r)=1\end{subarray}}\biggl{(}\sum_{\begin{subarray}{c}s(r)\\\
(sq+a,r)=1\end{subarray}}e\bigl{(}\frac{sb}{r}\bigr{)}\biggr{)}^{2}\biggl{(}\sum_{\begin{subarray}{c}s(r)\\\
(sq+1,r)=1\end{subarray}}e\bigl{(}\frac{s^{k}b}{r}\bigr{)}\biggr{)}\int_{|\beta|\leqslant\frac{Q}{rN^{\prime}}}\Bigl{(}\int_{1}^{N^{\prime}}e(t\beta)\,\mathrm{d}t\Bigr{)}^{3}\,\mathrm{d}\beta.$

Standard singular integral calculation suggests that (see [26, $2.4-2.5] as an
example)

$N^{\prime
2}\ll\int_{|\beta|\leqslant\frac{Q}{rN^{\prime}}}\Bigl{(}\int_{1}^{N^{\prime}}e(t\beta)\,\mathrm{d}t\Bigr{)}^{3}\,\mathrm{d}\beta.$

If we claim that when $(a,q)=1$ (we’ll prove this claim in Appendix C)

$\displaystyle\sum_{r\leqslant
Q}\biggl{(}\frac{\phi(q)}{\phi(qr)}\biggr{)}^{3}\sum_{\begin{subarray}{c}1\leqslant
b\leqslant r\\\
(b,r)=1\end{subarray}}\biggl{(}\sum_{\begin{subarray}{c}s(r)\\\
(sq+a,r)=1\end{subarray}}e\bigl{(}\frac{sb}{r}\bigr{)}\biggr{)}^{2}\Bigl{(}\sum_{\begin{subarray}{c}s(r)\\\
(sq+1,r)=1\end{subarray}}e\bigl{(}\frac{s^{k}b}{r}\bigr{)}\Bigr{)}\gg 1,$
(6.2)

in consequence of (6.1), at least one of the following two inequalities holds

$\delta^{2}N^{\prime
2}\ll\int_{\mathbb{T}}\bigl{|}\widehat{f}(\alpha)\widehat{\Lambda}_{a,q^{k}}(\alpha)S_{q}(\alpha)\bigr{|}\,\mathrm{d}\alpha,$

or

$\delta^{2}N^{\prime
2}\ll\int_{\mathbb{T}}\bigl{|}\widehat{f}(\alpha)\widehat{\Lambda_{a,q^{k}}\cdot
1_{\mathcal{A}_{a,q^{k}}}}(\alpha)S_{q}(\alpha)\bigr{|}\,\mathrm{d}\alpha.$

In either case, taking note that $(\log N)^{-1}\ll\delta$, it follows from
Proposition 5.1 with $d=q$, $f=\Lambda_{a,q^{k}}\cdot
1_{\mathcal{A}_{a,q^{k}}}-\delta\Lambda_{a,q^{k}}$, and
$\nu=\Lambda_{a,q^{k}}\cdot 1_{\mathcal{A}_{a,q^{k}}}$ or
$\nu=\Lambda_{a,q^{k}}$ that when $q^{k}\ll(\log N)^{A}$ there is an
arithmetic progression $P^{\prime}=a^{\prime}+q^{\prime k}\cdot[X]$ inside
$[X]$ such that

$\delta^{4k^{3}+16k^{2}}\sum_{n\in P^{\prime}}\Lambda(n)\ll\sum_{x\in
P^{\prime}}f(x),$

where $(a^{\prime},q^{\prime})=1$, $q^{\prime}\ll\delta^{-2k^{3}-6k^{2}}$ and
$X\gg q^{\prime 1-k}(\log N^{\prime})^{-2^{2k}D}N^{\prime}$. In the following,
we assume that $(qq^{\prime})^{k}\leqslant(\log N)^{A}$ and $X\gg N^{0.9}$,
and aim to prove that case (3) of the statement is always true. Recalling that
$f=\frac{\phi(q)}{q}\Lambda\cdot
1_{\mathcal{A}}(a+q^{k}\cdot)-\delta\frac{\phi(q)}{q}\Lambda(a+q^{k}\cdot)$
and setting a progression as
$Q^{\prime}=a+a^{\prime}q^{k}+(qq^{\prime})^{k}\cdot[X]$, the above inequality
is indeed

$\delta^{4k^{3}+16k^{2}}\frac{qq^{\prime}}{\phi(q)\phi(q^{\prime})}X\ll\sum_{n\in
Q^{\prime}}\Lambda\cdot 1_{\mathcal{A}}(n)-\delta\sum_{n\in
Q^{\prime}}\Lambda(n).$

On noting that
$\frac{qq^{\prime}}{\phi(q^{\prime})\phi(q)}\gg\frac{qq^{\prime}}{\phi(qq^{\prime})}$,
and Lemma 2.2 yields that

$\sum_{n\in Q^{\prime}}\Lambda(n)\geqslant
0.99\frac{qq^{\prime}X}{\phi(qq^{\prime})},$

we can therefore conclude that

$\sum_{n\in Q^{\prime}}\Lambda\cdot
1_{\mathcal{A}}(n)\geqslant(\delta+c\delta^{4k^{3}+16k^{2}})\sum_{n\in
Q^{\prime}}\Lambda(n).$

∎

We are now ready to accomplish the proof of Theorem 1.1.

_Proof of Theorem 1.1._

Suppose that $\mathcal{A}\subseteq\mathcal{P}_{N}$ has density
$\delta=\frac{|\mathcal{A}|}{|\mathcal{P}_{N}|}$ and $\mathcal{A}$ lacks
patterns $p_{1},p_{1}+(p_{2}-1)^{k}$ with $p_{1},p_{2}$ are prime numbers.
Taking $\mathcal{A}^{\prime}=\mathcal{A}\cap[\frac{N}{\log^{2}N},N]$ as a
subset set of $\mathcal{A}$, then for every element
$n\in\mathcal{A}^{\prime}$,

$\Lambda(n)=\log n\geqslant\log N-2\log\log N.$

And as a consequence,

$\sum_{n\in\mathcal{A}}\Lambda(n)\geqslant\sum_{n\in\mathcal{A}^{\prime}}\Lambda(n)\geqslant(\log
N-10(\log_{3}N)^{2})|\mathcal{A}^{\prime}|.$

Taking note from the assumption $|\mathcal{A}|\geqslant\delta\frac{N}{\log N}$
that the cardinality of $\mathcal{A}^{\prime}$ is at least

$|\mathcal{A}^{\prime}|\geqslant|\mathcal{A}|-N/\log^{2}N\geqslant\bigl{(}\delta-\frac{1}{\log
N}\bigr{)}\frac{N}{\log N}.$

We thus have

$\sum_{n\in[N]}\Lambda\cdot 1_{\mathcal{A}}(n)=\delta^{\prime}N,$

with

$\displaystyle\delta^{\prime}\geqslant\frac{1}{\log
N}\bigl{(}\delta-\frac{1}{\log N}\bigr{)}\bigl{(}\log N-2\log\log
N\bigr{)}\geqslant\delta-O\bigl{(}\frac{\log\log N}{\log N}\bigr{)}.$ (6.3)

By making use of Lemma 6.1 with progression $P=[N]$ one can find an arithmetic
progression $P_{1}=a_{1}+q_{1}^{k}\cdot[X_{1}]\subseteq[N]$ of common
difference $q_{1}\ll\delta^{\prime-2k^{3}-6k^{2}}$ and length $X_{1}\gg
q_{1}^{1-k}(\log N)^{-2^{2k}D}N$ such that

$\sum_{n\in P_{1}}\Lambda\cdot 1_{\mathcal{A}}(n)=\delta_{1}\sum_{n\in
P_{1}}\Lambda(n),$

where the relative density of $\mathcal{A}$ in progression $P_{1}$ satisfies
$\delta_{1}\geqslant\delta^{\prime}+c\delta^{\prime 4k^{3}+16k^{2}}$. Applying
Lemma 6.1 repeatedly gives a sequence of progressions

$P_{i}=a_{1}+a_{2}q_{1}^{k}+\cdots+(q_{1}\cdots q_{i})^{k}\cdot[X_{i}]$

such that the following events happen simultaneously

  1. (1)

$\sum_{n\in P_{i}}\Lambda\cdot 1_{\mathcal{A}}(n)=\delta_{i}\sum_{n\in
P_{i}}\Lambda(n),$

which satisfies
$\delta_{i}\geqslant\delta_{i-1}+c\delta_{i-1}^{4k^{3}+16k^{2}}$ and
$\delta_{1}=\delta^{\prime}+c\delta^{\prime 4k^{3}+16k^{2}}$;

  2. (2)

$a_{i}$ is coprime with $q_{i}$ and $q_{i}\ll\delta_{i-1}^{-2k^{3}-6k^{2}}$
and $X_{i}\gg(\log N)^{-2^{2k}D}q_{i}^{1-k}X_{i-1}$;

  3. (3)

the set $\mathcal{A}\cap P_{i}$ lacks configurations
$p_{1},p_{1}+(p_{2}-1)^{k}$;

  4. (4)

$(q_{1}\cdots q_{i})^{k}\ll(\log N)^{A}$ ;

  5. (5)

$X_{i}\gg N^{0.9}$.

However, this iteration cannot last too long—it must stop at some step
$m\ll\delta^{\prime-4k^{3}-16k^{2}}$, since otherwise $\sum_{n\in
P_{m}}\Lambda(n)\leqslant\sum_{n\in P_{m}}\Lambda\cdot 1_{\mathcal{A}}(n)$
which means that the relative density of $\mathcal{A}$ in progression $P_{m}$
would surpass $1$. And we’ll end up (the iteration) with either the condition
(4) or (5) failed, otherwise we could perform of Lemma 6.1 one more time.

Firstly, we assume that (4) fails, then, in view of (2), we have

$\Bigl{(}\delta^{\prime-2k^{3}-6k^{2}}\Bigr{)}^{O(\delta^{\prime-4k^{3}-16k^{2}})}\gg(\log
N)^{A},$

Taking logarithms of both sides yields that

$\delta^{\prime}\ll(\log_{2}N)^{-\frac{1}{4k^{3}+17k^{2}}}.$

Taking note that when
$\delta^{\prime}\ll(\log_{2}N)^{-\frac{1}{4k^{3}+17k^{2}}}$ we always have

$N^{0.9}\ll N\Bigl{(}(\log
N)^{-2^{2k}D}(\delta^{\prime-2k^{3}-6k^{2}})^{1-k}\Bigr{)}^{O(\delta^{\prime-4k^{3}-16k^{2}})}\ll
X_{m},$

which means that (5) is always true for such a choice of $\delta^{\prime}$.
Combining the above analysis with (6.3), we’ll get the desired upper bound

$\delta\ll(\log_{2}N)^{-\frac{1}{4k^{3}+17k^{2}}}.$

This completes the proof. ∎

##  Appendix A Proof of Lemma 2.3

We firstly record the following three theorems, all of them are in [22,
Chapter 4], also in the book [7].

######  Theorem A.1 ([22, Proposition 4.4]).

Suppose that $D\geqslant 2$, $\chi_{D}$ is a primitive character of modulus
$q_{0}\leqslant D$, $\rho\geqslant 1-\frac{c}{\log 3D}$ is a real such that
$L(\rho,\chi_{D})=0$, then we call $\chi_{D}$ the _exceptional Dirichlet
character at level $D$_, and we have

$(1-\rho)^{-1}\ll q_{0}^{1/2}(\log q_{0})^{2}.$

######  Theorem A.2 ([22, Theorem 4.5]).

There is an absolute constant $c_{1}>0$ such that if $D\geqslant 2$ and $\chi$
is a non-principal character of modulus $q\leqslant D$, then

  1. (1)

if the exceptional Dirichlet character $\chi_{D}$ exists,
$L(\rho,\chi_{D})=0$, and $\chi$ is induced by $\chi_{D}$, then for any real
$x\geqslant 1$ we have

$\psi(x,\chi):=\sum_{n\leqslant
x}\Lambda(n)\chi(n))=-\frac{x^{\rho}}{\rho}+O\Bigl{(}x\exp\bigl{(}-\frac{c_{1}\log
x}{\sqrt{\log x}+\log D}\bigr{)}(\log D)^{2}\Bigr{)},$

where $\rho$ is the exceptional zero;

  2. (2)

if the exceptional Dirichlet character $\chi_{D}$ does not exist, or $\chi$ is
not induced by $\chi_{D}$, then for any real $x\geqslant 1$ we have

$\psi(x,\chi)\ll x\exp\bigl{(}-\frac{c_{1}\log x}{\sqrt{\log x}+\log
D}\bigr{)}(\log D)^{2}.$

######  Theorem A.3 ([22, Theorem 4.6]).

There is an absolute constant $c_{2}>0$ such that if $\chi_{0}$ is the
principal character of modulus $q$, then for all real $x\geqslant 1$ we have

$\psi(x,\chi_{0})=x+O\Bigl{(}x\exp(-c_{2}\sqrt{\log x})+\log q\log x\Bigr{)}.$

_Proof of Lemma 2.3._

By making use of the orthogonality principle, one has

$\psi(x;q,a)=\frac{1}{\phi(q)}\sum_{\chi}\overline{\chi}(a)\sum_{n\leqslant
x}\Lambda(n)\chi(n)=\frac{1}{\phi(q)}\sum_{\chi}\overline{\chi}(a)\psi(x;\chi),$

where the summation is over all Dirichlet characters $\chi$ of modulus $q$.
Suppose that $\chi_{0}\pmod{q}$ is the principal character, one then has,

$\psi(x;q,a)=\frac{\overline{\chi_{0}}(a)}{\phi(q)}\psi(x,\chi_{0})+\frac{1}{\phi(q)}\sum_{\chi\neq\chi_{0}}\overline{\chi}(a)\psi(x;\chi).$

Let $c_{0}=\min\left\\{c_{1},c_{2}\right\\}$, the application of Theorem A.3
gives us that

$\frac{\overline{\chi_{0}}(a)}{\phi(q)}\psi(x,\chi_{0})=\frac{\overline{\chi_{0}}(a)x}{\phi(q)}+O\Bigl{(}x\exp(-c_{0}\sqrt{\log
x})+\log q\log x\Bigr{)}.$

If the exceptional Dirichlet character at level $D_{0}$ exists, suppose it is
$\chi_{D_{0}}\pmod{q_{0}}$ and has zero $\rho$, then Theorem A.1 implies that
$q_{0}\leqslant D_{0}$ and $(1-\rho)^{-1}\ll q_{0}^{1/2}(\log q_{0})^{2}$;
otherwise we’ll set $q_{0}=D_{0}+1$ and $\chi_{D_{0}}$ is a primitive
character of modulus $q_{0}$. When $\chi_{D_{0}}\pmod{q_{0}}$ is exceptional
and $q_{0}|q$, there would be one and only one non-principal character
$\chi\pmod{q}$ such that $\chi$ is induced by $\chi_{D_{0}}$, i.e.
$\chi=\chi_{0}\chi_{D_{0}}$. The first result of Theorem A.2 then shows that
for this character $\chi_{0}\chi_{D_{0}}$ of modulus $q\leqslant
D_{0}\leqslant D_{1}$

$\psi(x,\chi_{0}\chi_{D_{0}})=-\frac{x^{\rho}}{\rho}+O\Bigl{(}x\exp\bigl{(}-\frac{c_{0}\log
x}{\sqrt{\log x}+\log D_{1}}\bigr{)}(\log D_{1})^{2}\Bigr{)};$

otherwise, it follows from the second result of Theorem A.2 that for all non-
principal characters $\chi\pmod{q}$

$\psi(x,\chi)\ll x\exp\bigl{(}-\frac{c_{0}\log x}{\sqrt{\log x}+\log
D_{1}}\bigr{)}(\log D_{1})^{2}.$

The combination of the above four expressions gives Lemma 2.3.

∎

##  Appendix B Restriction for prime numbers

The main task of this appendix is to prove Theorem 1.2 (1). The same as Lemma
4.1, we’ll define the large spectrum set. For any real number $0<\eta\ll 1$,
the $\eta$-large spectrum for the Fourier transform of $\nu$ is defined to be
the set

$\mathcal{R}_{\eta}=\bigl{\\{}\alpha\in\mathbb{T}:|\widehat{\nu}(\alpha)|\geqslant\eta
X\bigr{\\}}.$

As expected, if we can show that

$\displaystyle\mathrm{meas}\mathcal{R}_{\eta}\ll_{\varepsilon}\eta^{-2-\varepsilon}X^{-1},$
(B.1)

then by making use of dyadic argument we can prove Theorem 1.2 (1). Indeed,
assume that $p>2$ is a real number, and let $\varepsilon=\varepsilon(p)$ be
sufficiently small, one has, for some absolute constant $C>0$,

$\int_{\mathbb{T}}|\widehat{\nu}(\alpha)|^{p}\,\mathrm{d}\alpha\leqslant\sum_{j\geqslant-C}\int_{\bigl{\\{}\alpha\in\mathbb{T}:\frac{X}{2^{j+1}}\leqslant|\widehat{\nu}(\alpha)|\leqslant\frac{X}{2^{j}}\bigr{\\}}}|\widehat{\nu}(\alpha)|^{p}\,\mathrm{d}\alpha\ll_{\varepsilon,p}X^{p-1}\sum_{j\geqslant-C}2^{j(p-2+\varepsilon)}\ll_{\varepsilon,p}X^{p-1}.$

And this is Theorem 1.2 (1).

Noting that Parseval’s identity provides a crude bound, that if $\eta\ll(\log
N)^{-1/\varepsilon}$ (B.1) follows. Indeed, since
$|\nu|\leqslant\Lambda_{b,d}$ pointwise, applying Parseval’s identity and
pigeonhole principle successively, one has

$\left\|\widehat{\nu}\right\|_{2}^{2}=\left\|\nu\right\|_{2}^{2}=\sum_{x\in[X]}|\nu(x)|^{2}\leqslant\left\|\Lambda_{b,d}\right\|_{\infty}\sum_{x\in[X]}\Lambda_{b,d}(x)\ll
X\log X.$

For another thing, it can be seen from the definition of $\eta$-large spectrum
set that

$(\eta
X)^{2}\mathrm{meas}\mathcal{R}_{\eta}\leqslant\int_{\mathcal{R}}|\widehat{\nu}(\alpha)|^{2}\,\mathrm{d}\alpha\leqslant\int_{\mathbb{T}}|\widehat{\nu}(\alpha)|^{2}\,\mathrm{d}\alpha.$

On combining the above two inequalities to get that

$\mathrm{meas}\mathcal{R}_{\eta}\leqslant\eta^{-2}(\log X)X^{-1}.$

Thus, we may assume that

$\displaystyle(\log X)^{-1/\varepsilon}\ll\eta\ll 1.$ (B.2)

Let $\alpha_{1},\cdots,\alpha_{R}$ be $X^{-1}$-spaced points in
$\mathcal{R}_{\eta}$, in order to prove (B.1) it suffices to show that

$\displaystyle R\ll_{\varepsilon}\eta^{-2-\varepsilon}.$ (B.3)

In the next step, we would transfer the Fourier analysis from the uncertain
function $\nu$ to $\Lambda_{b,d}$. For this reason, we are now going to
partition $\mathbb{T}$ into major and minor arcs, and show that the Fourier
transform of the majorant function $\widehat{\Lambda}_{b,d}$ possesses
logarithmic savings on the minor arcs. Let $\alpha\in\mathbb{T}$ be a
frequency, we see from Fourier transform and the definition of $\Lambda_{b,d}$
that

$\widehat{\Lambda}_{b,d}(\alpha)=\sum_{x\in[X]}\Lambda_{b,d}(x)e(\alpha
x)=\sum_{x\in[X]}\frac{\phi(d)}{d}\Lambda(b+dx)e(\alpha x).$

Let

$\displaystyle\mathfrak{M}^{*}$ $\displaystyle=\bigcup_{q\leqslant
Q}\bigl{\\{}\alpha\in\mathbb{T}:\left\|q\alpha\right\|\ll Q/X\bigr{\\}},$

and the minor arcs $\mathfrak{m}^{*}$ are the complement of major arcs, i.e.
$\mathfrak{m}^{*}=\mathbb{T}\backslash\mathfrak{M}^{*}$, where $Q=(\log
X)^{5C}$ for some large number $C>0$.

######  Lemma B.1 (Major-arc asymptotic for $\widehat{\Lambda}_{b,d}$).

Suppose that $\alpha=\frac{a}{q}+\beta\in\mathfrak{M}^{*}$ and $1\leqslant
b\leqslant d\leqslant\exp\bigl{(}c^{\prime}\sqrt{\log X}\bigr{)}$ for some
$0<c^{\prime}<1$, then

$|\widehat{\Lambda}_{b,d}(\alpha)|\ll
q^{-1+\varepsilon}X(1+X|\beta|)^{-1}+X\exp\bigl{(}-c\sqrt{\log
X}\bigr{)}(1+X|\beta|).$

###### Proof.

Dues to the proof of this lemma is conceptually the same as Lemma 3.2, we’d
like to omit some details. If $\beta=0$ and $\alpha=\frac{a}{q}$,

$\widehat{\Lambda}_{b,d}(a/q)=\frac{\phi(d)}{d}\sum_{\begin{subarray}{c}r(q)\\\
(b+rd,q)=1\end{subarray}}e\bigl{(}ar/q\bigr{)}\Bigl{\\{}\psi(b+dX;dq,b+rd)-\psi(b+d;dq,b+rd)\Bigr{\\}}.$

On noticing that the discrete interval $[b+d]$ contains at most one element
which is congruent to $b+rd$ modulus $dq$, thereby,

$\widehat{\Lambda}_{b,d}(a/q)=\frac{\phi(d)}{d}\sum_{\begin{subarray}{c}r(q)\\\
(b+rd,q)=1\end{subarray}}e\bigl{(}ar/q\bigr{)}\psi(b+dX;dq,b+rd)+O(q).$

Setting $c^{\prime}=c_{0}/100$, $D_{0}=\exp\bigl{(}c^{\prime}\sqrt{\log
X}\bigr{)}$ and $D_{1}=\exp\bigl{(}2c^{\prime}\sqrt{\log X}\bigr{)}$ in Lemma
2.3, it then follows from summation by parts and Lemma 2.3 that for
$\alpha=\frac{a}{q}+\beta$, when $q_{0}|q$,

$\widehat{\Lambda}_{b,d}(\alpha)=\frac{\phi(d)}{\phi(dq)}\sum_{\begin{subarray}{c}r(q)\\\
(b+rd,q)=1\end{subarray}}e\bigl{(}ar/q\bigr{)}\biggl{\\{}\int_{1}^{X}e(\beta
t)\,\mathrm{d}t-\overline{\chi}(b+dr)\int_{1}^{X}(b+dt)^{\rho-1}e(\beta
t)\,\mathrm{d}t\biggr{\\}}\\\ +O\Bigl{(}X\exp\bigl{(}-c\sqrt{\log
X}\bigr{)}(1+X|\beta|)\Bigr{)};$

and when $q_{0}\nmid q$,

$\widehat{\Lambda}_{b,d}(\alpha)=\frac{\phi(d)}{\phi(dq)}\sum_{\begin{subarray}{c}r(q)\\\
(b+rd,q)=1\end{subarray}}e\bigl{(}ar/q\bigr{)}\int_{1}^{X}e(\beta
t)\,\mathrm{d}t+O\Bigl{(}X\exp\bigl{(}-c\sqrt{\log
X}\bigr{)}(1+X|\beta|)\Bigr{)}.$

We’ll finish the proof by making use of Lemma 2.3 and Lemma 3.1, which is in a
similar path of Lemma 3.2.

∎

The minor arcs analysis is conceptually the same as Lemma 3.3. Again, we
outline the main steps.

######  Lemma B.2 (Minor arcs estimates).

$\sup_{\alpha\in\mathfrak{m}^{*}}\bigl{|}\widehat{\Lambda}_{b,d}(\alpha)\bigr{|}\ll
X(\log X)^{-C}.$

###### Proof.

It is immediate from Fourier transform as well as the definition of
$\Lambda_{b,d}$ that

$\widehat{\Lambda}_{b,d}(\alpha)=\frac{\phi(d)}{d}\sum_{x\in[X]}\Lambda(b+dx)e(x\alpha).$

After changing variables $b+dx\mapsto x$, we have

$\widehat{\Lambda}_{b,d}(\alpha)=\frac{\phi(d)}{d}e\Bigl{(}-\frac{b\alpha}{d}\Bigr{)}\sum_{\begin{subarray}{c}x\in[b,b+dX]\\\
x\equiv b\pmod{d}\end{subarray}}\Lambda(x)e\Bigl{(}\frac{\alpha
x}{d}\Bigr{)}.$

On noting that $b$ is quite small compared with $X$, to prove this lemma it
suffices to show that

$\displaystyle\biggl{|}\sum_{\begin{subarray}{c}x\in[b+dX]\\\ x\equiv
b\pmod{d}\end{subarray}}\Lambda(x)e\Bigl{(}\frac{\alpha
x}{d}\Bigr{)}\biggr{|}\ll X(\log X)^{-C}$ (B.4)

for all frequencies $\alpha\in\mathfrak{m}^{*}$. And the proof is a
modification of Lemma 3.3 and [9, Lemma 4.9]. To begin with, it follows from
[10, formula (3)] that

$\sum_{\begin{subarray}{c}x\in[b+dX]\\\ x\equiv
b\pmod{d}\end{subarray}}\Lambda(x)e\Bigl{(}\frac{\alpha
x}{d}\Bigr{)}=\sum_{\begin{subarray}{c}x\leqslant U\\\ x\equiv
b\pmod{d}\end{subarray}}\Lambda(x)e\Bigl{(}\frac{\alpha
x}{d}\Bigr{)}+\sum_{\begin{subarray}{c}xy\in[b+dX]\\\ x\leqslant U\\\ xy\equiv
b\pmod{d}\end{subarray}}\mu(x)(\log y)e\Bigl{(}\frac{\alpha xy}{d}\Bigr{)}\\\
+\sum_{\begin{subarray}{c}xy\in[b+dX]\\\ x\leqslant U^{2}\\\ xy\equiv
b\pmod{d}\end{subarray}}f(x)e\Bigl{(}\frac{\alpha
xy}{d}\Bigr{)}+\sum_{\begin{subarray}{c}xy\in[b+dX]\\\ x,y>U\\\ xy\equiv
b\pmod{d}\end{subarray}}\mu(x)g(y)e\Bigl{(}\frac{\alpha xy}{d}\Bigr{)},$

where $f,g$ are arithmetic functions defined in [10, formula (4-5)] satisfying
$\left\|f\right\|_{\infty},\left\|g\right\|_{\infty}\ll\log X$. Practically,
the above four terms can be re-grouped and departed into the following type I
and type II sums,

$\sum_{\begin{subarray}{c}xy\in[b+dX]\\\ x\sim L,y\geqslant T\\\ xy\equiv
b\pmod{d}\end{subarray}}f(x)e\Bigl{(}\frac{\alpha xy}{d}\Bigr{)}$

and

$\sum_{\begin{subarray}{c}xy\in[b+dX]\\\ x\sim L\\\ xy\equiv
b\pmod{d}\end{subarray}}f(x)g(y)e\Bigl{(}\frac{\alpha xy}{d}\Bigr{)}.$

Next, we turn our attention to the behavior of frequencies $\alpha$ in minor
arcs temporarily. In view of Dirichlet approximate theorem, for any frequency
$\alpha\in\mathbb{T}$, there always exists an integer $q\leqslant X/Q$ such
that $\left\|q\alpha\right\|\ll Q/X$. Thus, when $\alpha$ belongs to minor
arcs $\mathfrak{m}^{*}$, we must have

$\displaystyle Q\leqslant q\leqslant X/Q.$ (B.5)

The proof of [10, Lemma 4] tells us that (B.4) follows from the following two
claims and (B.5).

### Claim 1

Suppose that $\bigl{|}\alpha-\frac{a}{q}\bigr{|}\leqslant\frac{1}{q^{2}}$ with
$(a,q)=1$ and $1\leqslant a\leqslant q$. Suppose also that $X\geqslant L$,
then we have

$\sum_{\begin{subarray}{c}xy\in[b+dX]\\\ x\sim L,y\geqslant T\\\ xy\equiv
b\pmod{d}\end{subarray}}f(x)e\Bigl{(}\frac{\alpha xy}{d}\Bigr{)}\ll(\log
X)(\log q)\left\|f\right\|_{\infty}\bigl{(}\frac{X}{q}+L+q\bigr{)}.$

### Claim 2

Suppose that $\bigl{|}\alpha-\frac{a}{q}\bigr{|}\leqslant\frac{1}{q^{2}}$ with
$(a,q)=1$ and $1\leqslant a\leqslant q$. Suppose also that $X\geqslant L$,
then we have

$\sum_{\begin{subarray}{c}xy\in[b+dX]\\\ x\sim L\\\ xy\equiv
b\pmod{d}\end{subarray}}f(x)g(y)e\Bigl{(}\frac{\alpha
xy}{d}\Bigr{)}\ll\left\|f_{\infty}\right\|\left\|g\right\|_{\infty}X(\log
q)\Bigl{(}\frac{1}{q}+\frac{L}{X}+\frac{q}{X}\Bigr{)}^{1/4}.$

To prove Claim 1, we notice that

$\Biggl{|}\sum_{\begin{subarray}{c}xy\in[b+dX]\\\ x\sim L,y\geqslant T\\\
xy\equiv b\pmod{d}\end{subarray}}f(x)e\Bigl{(}\frac{\alpha
xy}{d}\Bigr{)}\Biggr{|}\leqslant\sum_{\begin{subarray}{c}x\sim L\\\
(x,d)=1\end{subarray}}f(x)\biggl{|}\sum_{\begin{subarray}{c}T\leqslant
y\leqslant\frac{b+dX}{x}\\\ y\equiv
bx^{-1}\pmod{d}\end{subarray}}e\Bigl{(}\frac{\alpha xy}{d}\Bigr{)}\biggr{|}\\\
\ll\left\|f\right\|_{\infty}\sum_{x\sim L}\Bigl{|}\sum_{\frac{T}{d}\leqslant
y\leqslant\frac{X}{x}}e(xy\alpha)\Bigr{|}\ll\left\|f\right\|_{\infty}\sum_{x\sim
L}\min\bigl{\\{}\frac{X}{x},\left\|x\alpha\right\|^{-1}\bigr{\\}}.$

Thus, Claim 1 follows from [10, Lemma 2]. It remains to prove Claim 2. One may
obtain from an application of Cauchy-Schwarz inequality that

$\Biggl{|}\sum_{\begin{subarray}{c}xy\in[b+dX]\\\ x\sim L\\\ xy\equiv
b\pmod{d}\end{subarray}}f(x)g(y)e\Bigl{(}\frac{\alpha
xy}{d}\Bigr{)}\Biggr{|}^{2}\leqslant\sum_{x\sim
L}f(x)^{2}\sum_{\begin{subarray}{c}s\sim L\\\
(x,d)=1\end{subarray}}\sum_{\begin{subarray}{c}y_{1},y_{2}\leqslant\frac{b+dX}{x}\\\
y_{1},y_{2}\equiv
bx^{-1}\pmod{d}\end{subarray}}g(y_{1})g(y_{2})e\Bigl{(}\frac{\alpha
x(y_{1}-y_{2})}{d}\Bigr{)}.$

Changing variables and exchanging the order of summation yields that above
expression is bounded by

$L\left\|f\right\|_{\infty}^{2}\left\|g\right\|_{\infty}^{2}\sum_{y_{1},y_{2}\leqslant\frac{X}{L}}\Bigl{|}\sum_{\begin{subarray}{c}x\sim
L\\\ (x,d)=1\end{subarray}}e(x(y_{1}-y_{2})\alpha)\Bigr{|}.$

After square (both sides) and another application of Cauchy-Schwarz inequality
gives

$\Biggl{|}\sum_{\begin{subarray}{c}xy\in[b+dX]\\\ x\sim L\\\ xy\equiv
b\pmod{d}\end{subarray}}f(x)g(y)e\Bigl{(}\frac{\alpha
xy}{d}\Bigr{)}\Biggr{|}^{4}\ll\left\|f\right\|_{\infty}^{4}\left\|g\right\|_{\infty}^{4}X^{2}\sum_{y_{1},y_{2}\leqslant\frac{X}{L}}\sum_{\begin{subarray}{c}x\sim
L\\\ (x,d)=1\end{subarray}}\sum_{t\leqslant 2L}e(t(y_{1}-y_{2})\alpha)\\\
\ll\left\|f\right\|_{\infty}^{4}\left\|g\right\|_{\infty}^{4}X^{3}\sum_{j\leqslant\frac{X}{L}}\min\bigl{\\{}L,\left\|j\alpha\right\|^{-1}\bigr{\\}}.$

One then deduces from [10, Lemma 1] that

$\Biggl{|}\sum_{\begin{subarray}{c}xy\in[b+dX]\\\ x\sim L\\\ xy\equiv
b\pmod{d}\end{subarray}}f(x)g(y)e\Bigl{(}\frac{\alpha
xy}{d}\Bigr{)}\Biggr{|}^{4}\ll\left\|f\right\|_{\infty}^{4}\left\|g\right\|_{\infty}^{4}X^{4}(\log
q)\Bigl{(}\frac{1}{q}+\frac{L}{X}+\frac{q}{X}\Bigr{)}.$

∎

We are now left to verify (B.3). For $1\leqslant r\leqslant R$, let
$c_{r}\in\left\\{z\in\mathbb{C}:|z|=1\right\\}$ be a number such that
$c_{r}\widehat{\nu}(\alpha_{r})=|\widehat{\nu}(\alpha_{r})|$. And let
$l_{x}(1\leqslant x\leqslant X)\in\mathbb{C}$ be a sequence such that
$|l_{x}|\leqslant 1$ and $l_{x}\Lambda_{b,d}(x)=\nu(x)$. Routinely, as in
Section 5, one has

$R^{2}\eta^{2}X^{2}\leqslant\biggl{(}\sum_{r\in[R]}|\widehat{\nu}(\alpha_{r})|\biggr{)}^{2}\ll\biggl{|}\sum_{r\in[R]}c_{r}\sum_{x\in[X]}l_{x}\Lambda_{b,d}(x)e(x\alpha_{r})\biggr{|}^{2}\\\
\ll
X\sum_{x\in[X]}\biggl{|}\sum_{r\in[R]}c_{r}\Lambda_{b,d}(x)e(x\alpha_{r})\biggr{|}^{2}\ll
X\sum_{r,r^{\prime}\in[R]}\Bigl{|}\sum_{x\in[X]}\Lambda_{b,d}(x)e\biggl{(}(\alpha_{r}-\alpha_{r^{\prime}})x\biggr{)}\Bigr{|},$

just on noticing that $\sum_{x\in[X]}l_{x}^{2}\Lambda_{b,d}(x)\ll X$. It then
follows from Fourier transform that

$R^{2}\eta^{2}X\ll\sum_{r,r^{\prime}\in[R]}\bigl{|}\widehat{\Lambda}_{b,d}(\alpha_{r}-\alpha_{r^{\prime}})\bigr{|}.$

Put $\gamma=1+\varepsilon$, utilizing Hölder’s inequality one then has

$R^{2}\eta^{2\gamma}X^{\gamma}\ll\sum_{r,r^{\prime}\in[R]}\bigl{|}\widehat{\Lambda}_{b,d}(\alpha_{r}-\alpha_{r^{\prime}})\bigr{|}^{\gamma}.$

Joint Lemma B.1, Lemma B.2, (B.2) and putting
$\widetilde{Q}=\eta^{-1-\varepsilon}$ gives

$\eta^{2\gamma}R^{2}\ll\sum_{q\leqslant\widetilde{Q}}\sum_{\begin{subarray}{c}a(q)\\\
(a,q)=1\end{subarray}}\sum_{1\leqslant r,r^{\prime}\leqslant
R}q^{-\gamma+\varepsilon}(1+X\left\|\alpha_{r}-\alpha_{r^{\prime}}-a/q\right\|)^{-\gamma}\ll\sum_{1\leqslant
r,r^{\prime}\leqslant R}G(\alpha_{r}-\alpha_{r^{\prime}}),$

where

$G(\alpha)=\sum_{q\leqslant\widetilde{Q}}\sum_{a(q)}q^{-\gamma+\varepsilon}F(\alpha-a/q)$

and

$F(\beta)=(1+X|\sin\beta|)^{-\gamma}.$

We are now in the position of [4, Eq.(4.16)], just need to replace $N^{2}$ by
$X$, replace $\delta$ by $\eta$. Following from Bourgain’s proof, on noting
$\gamma=1+\varepsilon$, one then has
$R\ll_{\varepsilon}\eta^{-2-\varepsilon}$.

##  Appendix C Estimation of the singular series (6.2)

Since the idea is rather similar to the book [15, Section 8], the proof here
would be a bit rough. On writing

$A(r)=\biggl{(}\frac{\phi(q)}{\phi(qr)}\biggr{)}^{3}\sum_{\begin{subarray}{c}1\leqslant
b\leqslant r\\\
(b,r)=1\end{subarray}}\biggl{(}\sum_{\begin{subarray}{c}s(r)\\\
(sq+a,r)=1\end{subarray}}e\bigl{(}\frac{sb}{r}\bigr{)}\biggr{)}^{2}\Bigl{(}\sum_{\begin{subarray}{c}s(r)\\\
(sq+1,r)=1\end{subarray}}e\bigl{(}\frac{s^{k}b}{r}\bigr{)}\Bigr{)},$

clearly, it is multiplicative. Suppose that $(a,q)=1$ and $Q=(\log N)^{A}$ for
some large $A>1$, the goal is to show that

$\sum_{r\leqslant Q}A(r)\gg 1.$

Assume for a moment that $r=p^{t}$, then if $p|q$, $(sq+a,p)=(a,p)=1$ and
$(sq+1,p)=1$ holds for all choices of $s\pmod{p^{t}}$ whenever $(a,q)=1$, and
thus, in this case,

$A(p^{t})=\sum_{\begin{subarray}{c}1\leqslant b\leqslant p^{t}\\\
(b,p)=1\end{subarray}}\biggl{(}\sum_{s(p^{t})}e\bigl{(}\frac{sb}{p^{t}}\bigr{)}\biggr{)}^{2}\Bigl{(}\sum_{s(p^{t})}e\bigl{(}\frac{s^{k}b}{p^{t}}\bigr{)}\Bigr{)}=0.$

If $p\nmid q$ and degree $t\geqslant 2$, we may write $s\pmod{p^{t}}$ as the
sum $s=s_{1}p^{t-1}+s_{2}$, and when $(b,p)=1$ we have

$\sum_{\begin{subarray}{c}s(p^{t})\\\
(sq+a,p)=1\end{subarray}}e\bigl{(}\frac{sb}{p^{t}}\bigr{)}=\sum_{s_{1}(p)}\sum_{\begin{subarray}{c}s_{2}(p^{t-1})\\\
(s_{2}q+a,p)=1\end{subarray}}e\Bigl{(}\frac{s_{1}b}{p}+\frac{s_{2}b}{p^{t}}\Bigr{)}=0,$

which means that $A(p^{t})\equiv 0$ whenever degree $t\geqslant 2$. Putting
the above analysis together one thus has

$\displaystyle\sum_{r\leqslant Q}A(r)=\prod_{p\leqslant Q\atop p\nmid
q}\bigl{(}1+A(p)\bigr{)}+O\biggl{(}\sum_{\begin{subarray}{c}r>Q\\\
r:\Box-\mathrm{free}\\\ p|r\Rightarrow p\leqslant Q,p\nmid
q\end{subarray}}|A(r)|\biggr{)},$ (C.1)

where

$A(p)=\phi(p)^{-3}\sum_{\begin{subarray}{c}1\leqslant b\leqslant p\\\
(b,p)=1\end{subarray}}\biggl{(}\sum_{\begin{subarray}{c}s(p)\\\
(sq+a,p)=1\end{subarray}}e\bigl{(}\frac{sb}{p}\bigr{)}\biggr{)}^{2}\Bigl{(}\sum_{\begin{subarray}{c}s(p)\\\
(sq+1,p)=1\end{subarray}}e\bigl{(}\frac{s^{k}b}{p}\bigr{)}\Bigr{)}\qquad(p\nmid
q).$

We shall deal with the big O-term and the “main term” successively, and in the
following, we always suppose that primes $p$ satisfy $p\nmid q$. Firstly, a
modification of the proof of [15, Lemma 8.5] shows that when $(b,p)=1$,

$\biggl{|}\sum_{\begin{subarray}{c}s(p)\\\
(sq+1,p)=1\end{subarray}}e\bigl{(}\frac{s^{k}b}{p}\bigr{)}\biggr{|}\ll
p^{1/2+\varepsilon};$

besides, under the same condition we also have

$\biggl{|}\sum_{\begin{subarray}{c}s(p)\\\
(sq+a,p)=1\end{subarray}}e\bigl{(}\frac{sb}{p}\bigr{)}\biggr{|}\leqslant\biggl{|}\sum_{\begin{subarray}{c}s(p)\\\
(s,p)=1\end{subarray}}e\bigl{(}\frac{sb}{p}\bigr{)}\biggr{|}+\biggl{|}\sum_{\begin{subarray}{c}s(p)\\\
p|(sq+a)\end{subarray}}e\bigl{(}\frac{sb}{p}\bigr{)}\biggr{|}\leqslant 2.$

As a consequence,

$\displaystyle|A(p)|\ll p^{-3/2+\varepsilon}.$ (C.2)

Therefore, the big O-term in (C.1) can be bounded up by $o(1)$, in practice
since $A(r)$ is multiplicative,

$\sum_{\begin{subarray}{c}r>Q\\\ r:\Box-\mathrm{free}\\\ p|r\Rightarrow
p\leqslant Q,p\nmid q\end{subarray}}|A(r)|\ll\sum_{\begin{subarray}{c}r>Q\\\
r:\Box-\mathrm{free}\end{subarray}}\prod_{p|r\atop p\nmid
q}|A(p)|\ll\sum_{r>Q}r^{-3/2+\varepsilon}\ll Q^{-1/2+\varepsilon}.$

By substituting the above inequality into (C.1), we may find that it is
sufficient to show that for some large constant $C>1$

$\displaystyle\prod_{p\leqslant Q\atop p\nmid
q}\bigl{(}1+A(p)\bigr{)}=\prod_{p\leqslant C\atop p\nmid
q}\bigl{(}1+A(p)\bigr{)}\prod_{C\leqslant p\leqslant Q\atop p\nmid
q}\bigl{(}1+A(p)\bigr{)}\gg 1.$ (C.3)

In order to deal with this, let $M(p)$ count the number of solutions of the
following equation

$x_{1}+x_{2}+x_{3}^{k}\equiv 0\pmod{p}$

with variables $x_{1},x_{2},x_{3}\pmod{p}$ and
$p\nmid(qx_{1}+a)(qx_{2}+a)(qx_{3}+1)$. Using orthogonality, one has

$\displaystyle pM(p)$
$\displaystyle=\sum_{h(p)}\biggl{(}\sum_{\begin{subarray}{c}x(p)\\\
(xq+a,p)=1\end{subarray}}e\bigl{(}\frac{xh}{p}\bigr{)}\biggr{)}^{2}\Bigl{(}\sum_{\begin{subarray}{c}x(p)\\\
(xq+1,p)=1\end{subarray}}e\bigl{(}\frac{x^{k}h}{p}\bigr{)}\Bigr{)}$
$\displaystyle=\Bigl{\\{}\sum_{\begin{subarray}{c}h(p)\\\
(h,p)=1\end{subarray}}+\sum_{\begin{subarray}{c}h(p)\\\
(h,p)=p\end{subarray}}\Bigr{\\}}\biggl{(}\sum_{\begin{subarray}{c}x(p)\\\
(xq+a,p)=1\end{subarray}}e\bigl{(}\frac{xh}{p}\bigr{)}\biggr{)}^{2}\Bigl{(}\sum_{\begin{subarray}{c}x(p)\\\
(xq+1,p)=1\end{subarray}}e\bigl{(}\frac{x^{k}h}{p}\bigr{)}\Bigr{)}$
$\displaystyle=\phi(p)^{3}(A(p)+1),$

just on recalling the definition of $A(p)$. Thus, to prove (C.3) it suffices
to establish that

$\prod_{p\leqslant C\atop p\nmid q}\frac{p}{\phi(p)^{3}}M(p)\prod_{C\leqslant
p\leqslant Q\atop p\nmid q}\bigl{(}1+A(p)\bigr{)}\gg 1.$

The next step is to show that $M(p)\geqslant 1$ for primes $p\nmid q$. In
practice, when $p\geqslant 3$, on noting that
$\left\\{x\pmod{p},p\nmid(qx+a)\right\\}$ runs over $\phi(p)$ residue classes
modulus $p$, the using of [15, Lemma 8.7] yields that the sequence
$\left\\{x_{1}+x_{2}\pmod{p},p\nmid(qx_{1}+a)(qx_{2}+a)\right\\}$ runs over at
least $\min\left\\{2\phi(p)-1,p\right\\}\geqslant p$ residue classes modulus
$p$. Therefore, there must be at least one choice of
$x_{1},x_{2},x_{3}\pmod{p}$ such that

$\begin{cases}x_{1}+x_{2}+x_{3}^{k}\equiv 0\pmod{p}\\\
p\nmid(qx_{1}+a)(qx_{2}+a)(qx_{3}+1).\end{cases}$

When $p=2$, it is clearly that $1,1,0\pmod{2}$ is a solution to the system

$\begin{cases}x_{1}+x_{2}+x_{3}^{k}\equiv 0\pmod{2}\\\
2\nmid(qx_{1}+a)(qx_{2}+a)(qx_{3}+1).\end{cases}$

The conclusion $M(p)\geqslant 1$ when $p\nmid q$ then yields that

$\prod_{p\leqslant C\atop p\nmid q}\frac{p}{\phi(p)^{3}}M(p)\prod_{C\leqslant
p\leqslant Q\atop p\nmid q}\bigl{(}1+A(p)\bigr{)}\gg\prod_{p\leqslant C\atop
p\nmid q}p^{-2+\varepsilon}\prod_{C\leqslant p\leqslant Q\atop p\nmid
q}\bigl{(}1+A(p)\bigr{)}.$

Taking note from (C.2) that $1+A(p)\geqslant 1-Cp^{-3/2+\varepsilon}$ and that
$C$ is a constant, thus,

$\prod_{p\leqslant C\atop p\nmid q}p^{-2+\varepsilon}\prod_{C\leqslant
p\leqslant Q\atop p\nmid q}\bigl{(}1+A(p)\bigr{)}\gg\prod_{p\geqslant
C}\bigl{(}1-O\bigl{(}p^{-3/2+\varepsilon}\bigr{)}\bigr{)}\gg 1.$

## References

  * [1] T. Apostol. Introduction to analytic number theory, Springer Science & Business Media (2013).
  * [2] A. Balog, J. Pelikán, J. Pintz, E. Szemerédi. “Difference sets without $k$-th powers.” Acta Mathematica Hungarica, 65(2), 165–187 (1994).
  * [3] T. Bloom, J. Maynard. “A new upper bound for sets with no square differences.” arXiv preprint arXiv:2011.13266. (2020).
  * Bou [89] J. Bourgain. “ On $\Lambda(p)$-subsets of squares.” Israel Journal of Mathematics, 67(3), 291–311 (1989).
  * [5] J. Bourgain, C. Demeter, L. Guth. “Proof of the main conjecture in Vinogradov’s mean value theorem for degrees higher than three.” Annals of Mathematics, 633–682 (2016).
  * [6] S. Chow. “Roth-Waring-Goldbach.” International Mathematics Research Notices, (8) 2341–2374 (2018).
  * [7] H. Davenport. Multiplicative number theory (Vol. 74) Springer Science & Business Media (2013).
  * [8] H. Furstenberg. “Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions.” Journal d’Analyse Mathématique, 31(1), 204–256 (1977).
  * [9] B. Green. “Roth’s theorem in the primes.” Annals of mathematics, 1609–1636 (2005).
  * [10] B. Green. “Some minor arcs estimates relevant to the paper ‘Roth’s theorem in the primes’ ”.
  * Gr [22] B. Green. “On Sárközy’s theorem for shifted primes.” arXiv preprint arXiv:2206.08001.
  * GT [06] B. Green, T. Tao. “Restriction theory of the Selberg sieve, with applications.” Journal de théorie des nombres de Bordeaux, 1(18) 147–182 (2006).
  * GT [08] B. Green, T. Tao. “The primes contain arbitrarily long arithmetic progressions.” Annals of Mathematics, 481–547 (2008).
  * [14] G. Harman. Prime-Detecting Sieves (LMS-33)[M], Princeton university press (2012).
  * [15] L. Hua. Additive theory of prime numbers, American Mathematical Soc. Vol. 13 (2009).
  * [16] H. Li, H. Pan. “Difference sets and polynomials of prime variables. ” Acta Arithmetica, 138, 25–52 (2009).
  * [17] J. Liu, T. Zhan. New Developments in the Additive Theory of Prime Numbers, World Scientific (2011).
  * [18] H. Montgomery, R. Vaughan. “The large sieve.” Mathematika, 20(2), 119–134 (1973).
  * [19] J. Pintz, W. L. Steiger, E. Szemerédi. “On sets of natural numbers whose difference set contains no squares.” Journal of the London Mathematical Society, 2(2), 219–231 (1988).
  * [20] A. Rice. “Sárközy’s theorem for $\mathcal{P}$-intersective polynomials.” arXiv:1111.6559v5 [math.CA]
  * [21] K. Roth. “On certain sets of integers.” Journal of the London Mathematical Society, 1(1) 104–109 (1953).
  * [22] I. Ruzsa, T. Sanders. “ Difference sets and the primes.” Acta Arithmetica, 131, 281–301 (2008).
  * S [781] A. Sárközy. “On difference sets of sequences of integers. I.” Acta Mathematica Academiae Scientiarum Hungarica, 31(1-2), 125–149 (1978).
  * S [783] A. Sárközy. “ On difference sets of sequences of integers. III.” Acta Mathematica Academiae Scientiarum Hungarica, 31(3-4), 355–386 (1978).
  * [25] J. Thorner, A. Zaman. “An explicit version of Bombieri’s log-free density estimate.” arXiv:2208.11123 (2022).
  * [26] R. Vaughan. the Hardy-Littlewood method, Cambreidge (2003).