\documentstyle{SibMatJ}
%\TestXML
\topmatter

  \Author           Dimitrov
    \Initial        S.
    \Initial        I.
    \Email          sdimitrov\@tu-sofia.bg
    \Email          xyzstoyan\@gmail.com
    \AffilRef       1
    \AffilRef       2
    \Corresponding  %\?
  \endAuthor

  \Author           Lazarova
    \Initial        M.
    \Initial        D.
    \Email          meglena.laz\@tu-sofia.bg
    \Email          meglena-lazarova\@uniburgas.bg
    \AffilRef       1
    \AffilRef       3
  \endAuthor

  \Affil 1
    \Division       Faculty of Applied Mathematics and Informatics
    \Organization   Technical University of Sofia
    \City           Sofia
    \Country        Bulgaria
  \endAffil

  \Affil 2
    \Division       Department of Bioinformatics and Mathematical Modelling
    \Organization   Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences
    \City           Sofia
    \Country        Bulgaria
  \endAffil

  \Affil 3
    \Division       Department of Mathematics, Infrormatics and Physics
    \Organization   Burgas State University Prof. Dr. Assen Zlatarov
    \City           Sofia
    \Country        Bulgaria
  \endAffil

  \datesubmitted   August 20, 2025\enddatesubmitted
  \dateaccepted    December 11, 2025\enddateaccepted

  \UDclass
    ??? % 11J25  11J71  11L07  11L20
  \endUDclass

  \dedication
    This article is dedicated to Professor Themistocles M. Rassias on the occasion of his 75th anniversary.
  \enddedication

  \title
    On the Distribution of $\alpha p^3$ Modulo One over Special Primes
  \endtitle

  \abstract
    Let $[x]$ denote the integer part of $x\in \Bbb{R}$, and let $\|x\|$ denote the distance from $x$ to the nearest integer.
    In this paper we prove that,  whenever $\alpha$ is irrational number and $\beta$ is any real number,
    then for any fixed $\frac{47}{48}<\gamma<1$, there exist infinitely many prime numbers $p$ satisfying the inequality
    $\|\alpha p^3+\beta\|<p^{\frac{47-48\gamma}{96}+\varepsilon}$ and such that $p=[n^{1/\gamma}]$.
  \endabstract

  \keywords
    distribution modulo one,
    Piatetski-Shapiro primes
  \endkeywords

\endtopmatter

\head
  1. Introduction and Statement of the Result
\endhead

Let $\alpha$ be an irrational real number, let $\beta $ be a real number, and let $\|x\|=\min\nolimits_{n\in \Bbb{Z}}|x-n|$.
It was shown by Vinogradov [1] that if $\theta=\frac{1}{5}-\varepsilon$, then there are infinitely many primes $p$ such that
$$
\|\alpha p+\beta\|<p^{-\theta }.
\tag1
$$
Subsequently, inequality \Tag(1) was sharpened several times, with the best known result to date due to Matom\"{a}ki [2], who obtained $\theta=\frac{1}{3}-\varepsilon$ in the case $\beta=0$.
An analogous problem has also been studied for polynomial expressions involving higher powers of primes.
Ghosh [3] was the first to establish that the inequality
$$
\|\alpha p^2+\beta\|<p^{-\theta}
\tag2
$$
has infinitely many prime solutions $p$, for $\theta=\frac{1}{8}-\varepsilon$.
The result of Ghosh was later improved by Baker and Harman [4] with $\theta=\frac{3}{20}-\varepsilon$ and by Harman [5] with $\theta=\frac{2}{13}-\varepsilon$.
A further natural extension is to consider polynomial expressions of even higher degree.
Harman [6] is credited with the inequality
$$
\|\alpha p^3+\beta\|<p^{-\theta}
\tag3
$$
which holds for infinitely many primes $p$, with $\theta=\frac{1}{19}-\varepsilon$.
Subsequently, Harman's result was sharpened by  Harman [7] with $\theta=\frac{3}{49}-\varepsilon$, by Baker and Harman [4] with $\theta=\frac{1}{12}-\varepsilon$
and by Wong [8] with $\theta=\frac{5}{56}-\varepsilon$.
The study of Diophantine inequalities involving prime numbers naturally leads to questions about the distribution of primes in sparse sequences.
One notable class of such sequences arises from noninteger powers of integers.
In 1953, Piatetski-Shapiro [9] established that for any fixed $\frac{11}{12}<\gamma<1$, there are infinitely many primes of the form $p=[n^{1/\gamma}]$.
These primes are known as Piatetski-Shapiro primes of type $\gamma$.
Later, the range for $\gamma$ was refined by several authors, with the best result to date provided by Rivat and Wu [10].
Specifically, they demonstrated that for any $\frac{205}{243}<\gamma<1$, we have the following lower bound
$$
\sum\Sb p\leq X\\ p=[n^{1/\gamma}]\endSb 1\gg\frac{X^\gamma}{\log X}.
\tag4
$$
The primes of the form $[n^{1/\gamma}]$ are very much in focus nowadays and many problems are solved using them.
We mention, for example, the papers [11] and [12].
Recently, Dimitrov [13] considered a hybrid problem, restricting the set of primes $p$ in \Tag(1) to Piatetski-Shapiro primes.
More precisely, he proved that, for any fixed $\frac{11}{12}<\gamma<1$, there exist infinitely many Piatetski-Shapiro primes $p$ of type $\gamma$
satisfying inequality \Tag(1) for $\theta=\frac{12\gamma-11}{26}-\varepsilon$.
Subsequently, X. Li, J. Li, and  M. Zhang [14] %\?убрать инициалы
generalized the result of Dimitrov [13] by solving \Tag(1) with primes $p=[n_1^{1/\gamma_1}]=[n_2^{1/\gamma_2}]$,
where $\frac{23}{12}<\gamma_1+\gamma_2<2$, and with $\theta=\frac{12(\gamma_1+\gamma_2)-23}{38}-\varepsilon$.
Finally, Baier and Rahaman [15] managed to improve Dimitrov's result by solving \Tag(1) with primes $p=[n^{1/\gamma}]$, where $\frac{8}{9}<\gamma<1$,
and with $\theta=\frac{9\gamma-8}{10}-\varepsilon$.
Very recently, the authors [16] showed that, for any fixed $\frac{13}{14}<\gamma<1$, there are infinitely many primes $p=[n^{1/\gamma}]$
satisfying inequality \Tag(2), for $\theta=\frac{14\gamma-13}{29}+\varepsilon$.
Building on these results, we address inequality \Tag(3) involving Piatetski-Shapiro primes.

\proclaim{Theorem 1}
Let $\gamma$ be fixed with $\frac{47}{48}<\gamma<1$, and assume that  $\alpha$ is irrational and $\beta$ is real.
Then there exist infinitely many Piatetski-Shapiro primes $p$ of type $\gamma$ such that
$$
\|\alpha p^3+\beta\|<p^{\frac{47-48\gamma}{96}+\varepsilon}.
$$
\endproclaim

We remark that \Par*{Theorem 1} is hardly best possible.
It is likely that more sophisticated exponential sum estimates and/or sieve techniques would have allowed us to extend the range of $\gamma$. Nevertheless, we chose not to pursue such directions, and our main message is that inequality \Tag(3) can be solved with infinitely many Piatetski-Shapiro primes.

\head
2. Notations
\endhead

Let $C$ be a sufficiently large positive constant. The letter $p$ will always denote a prime number.
We use $\varepsilon$ to denote an arbitrarily small positive number, which may vary in different occurrences.
The notation $m \sim M$ means that $m$ ranges over the interval $(M, 2M]$.
We denote by $\tau _k(n)$ the number of solutions of the equation $m_1m_2\dots m_k$ $=n$ in natural numbers $m_1,\dots,m_k$.
As usual $\Lambda(n)$ is von Mangoldt's function.
We write $[x]$ for the integer part of $x$, $\{x\}$ for the fractional part, and $\|x\|$ for the distance from $x$ to the nearest integer.
Moreover $e(x)=e^{2\pi i x}$ and $\psi(t)=\{t\}-1/2$.
Let $\gamma$ be a real constant such that $\frac{47}{48}<\gamma<1$.
Since $\alpha$ is irrational, there are infinitely many different convergents $a/q$ to its continued fraction, with
$$
\Bigl|\alpha- \frac{a}{q}\Bigr|<\frac{1}{q^2},\quad (a, q) = 1,\quad a\neq0
\tag5
$$
and $q$ is arbitrary large. Define
\iftex
$$
\align
&N=q^\frac{4}{49-48\gamma};
\tag6
\\
&\Delta=CN^{\frac{47-48\gamma}{96}+\varepsilon};
\tag7
\\
&H=[q^\frac{1}{3}];
\tag8
\\
&M=N^{\frac{49-48\gamma}{96}};
\tag9
\\
&\vartheta=N^{\frac{23}{48}};
\tag10
\\
&\goth{S}(u)=\sum\limits_{1\leq h\leq u}\biggl|\sum_{n\le N}e(\alpha hn^3)\biggr|;
\tag11
\\
&\Sigma=\sum\limits_{p\leq N}(\psi(-(p+1)^\gamma)-\psi(-p^\gamma))e(\alpha h p^3)\log p.
\tag12
\endalign
$$
\else
$$
N=q^\frac{4}{49-48\gamma};
\tag6
$$
$$
\Delta=CN^{\frac{47-48\gamma}{96}+\varepsilon};
\tag7
$$
$$
H=[q^\frac{1}{3}];
\tag8
$$
$$
M=N^{\frac{49-48\gamma}{96}};
\tag9
$$
$$
\vartheta=N^{\frac{23}{48}};
\tag10
$$
$$
\goth{S}(u)=\sum\limits_{1\leq h\leq u}\biggl|\sum_{n\le N}e(\alpha hn^3)\biggr|;
\tag11
$$
$$
\Sigma=\sum\limits_{p\leq N}(\psi(-(p+1)^\gamma)-\psi(-p^\gamma))e(\alpha h p^3)\log p.
\tag12
$$
\fi

\head
3. Auxiliary Lemmas
\endhead

\proclaim{Lemma 1}
For any $M\geq2$, we have
$$
\psi(t)=-\sum\limits_{1\leq |m|\leq M}\frac{e(mt)}{2\pi i m}+\Cal{O}\biggl(\min\Bigl(1, \frac{1}{M\|t\|}\Bigr)\biggr),
$$
where
$$
\min\Bigl(1,\frac{1}{M\|t\|}\Bigr)=\sum\limits_{h=-\infty}^{\infty}b_he(h t)
$$
and
$$
b_h\ll\min\Bigl(\frac{\log 2M}{M}, \frac{1}{|h|}, \frac{M}{|h|^2}\Bigr).
$$
\endproclaim

\demo{Proof}
See [17, p.~245].
\qed\enddemo

\proclaim{Lemma 2}
Let $Q$, $J$, and $L$ be three positive integers,
which %\?
satisfy $1 \leq Q \leq N \log^{-1}N$, $1 \leq J \leq N \log^{-1}N$, and $1 \leq L \leq N \log^{-1}N$.
Then for the sum \Tag(11), we have
$$
\goth{S}^8(u)\ll u^8N^8\Bigl(\frac{1}{Q^4}+\frac{1}{J^2}+\frac{1}{L}\Bigr)+\frac{u^7N^8}{QJL}\sum\limits_{q=1}^Q\sum\limits_{j=1}^J\sum\limits_{l=1}^L
\biggl|\sum\limits_{1\leq h\leq u}e(6\alpha qjlh)\biggr|.
$$
\endproclaim

\demo{Proof}
This lemma is very similar to a result of Li and Zhang [18]. By inspecting the arguments presented in [18, pp. 203--206],
the reader will easily see that the proof of \Par*{Lemma 2} can be obtained in the same way.
\qed\enddemo

\proclaim{Lemma 3}
For any real number $\alpha$ and $X\geq1$, we have
$$
\biggl|\sum\limits_{n\leq X}e(\alpha n)\biggr|\leq\min\Bigl(X,\frac{1}{2\|\alpha\|}\Bigr).
$$
\endproclaim

\demo{Proof}
See [19, Chapter 6, Lemma 4].
\qed\enddemo

\proclaim{Lemma 4}
Suppose that $X, Y\geq1$,  $|\alpha-\frac{a}{q}|<\frac{1}{q^2}$, $(a, q)=1$.
Then
$$
\sum_{n\le X} \, \min \Bigl( Y, \, \frac{1}{ \|\alpha n\| } \Bigr)\ll\frac{XY}{q}+(X+q)\log 2q.
$$
\endproclaim

\demo{Proof}
See [20, Lemma 1].
\qed\enddemo

\proclaim{Lemma 5}
Let $k \geq 3$ be an integer, and suppose that $f(x):[N, N_1]\rightarrow \Bbb{R}$ has continuous derivatives of order up to $k$ on $[N, N_1]$, where $1\leq N < N_1 \leq 2N$.
Suppose further that
$$
0<\lambda_k\leq|f^{(k)}(x)|\leq A\lambda_k, \quad x\in[N, N_1].
$$
Then
$$
\sum_{N<n\le N_1}e(f(n))\ll_{A, k, \varepsilon} N^{1+\varepsilon}(\lambda_k^{\frac{1}{k(k-1)}}+N^{-\frac{1}{k(k-1)}}+N^{-\frac{2}{k(k-1)}}\lambda_k^{-\frac{2}{k^2(k-1)}}).
$$
\endproclaim

\demo{Proof}
See [21, Theorem 1].
\qed\enddemo

\proclaim{Lemma 6}
For any complex numbers $a(n)$ we have
$$
\biggl|\sum_{a<n\le b}a(n)\biggr|^2\leq\Bigl(1+\frac{b-a}{Q}\Bigr)\sum_{|q|\leq Q}\Bigl(1-\frac{|q|}{Q}\Bigr)\sum_{a<n,\, n+q\leq b}a(n+q)\overline{a(n)},
$$
where $Q$ is any positive integer.
\endproclaim

\demo{Proof}
See [22, Lemma 8.17].
\qed\enddemo

\proclaim{Lemma 7}
Suppose that $\alpha \in \Bbb{R}$, $a \in \Bbb{Z}$, $q\in \Bbb{N}$,
$|\alpha-\frac{a}{q}|\leq\frac{1}{q^2}$, $(a, q)=1$. Then
$$
\sum\limits_{p\le N}e(\alpha p^3)\log p\ll N^{1+\varepsilon}\Bigl(\frac{1}{q}+\frac{1}{N^\frac{1}{2}}+\frac{q}{N^3}\Bigr)^\frac{1}{16}.
$$
\endproclaim

\demo{Proof}
See [23, Theorem 1].
\qed\enddemo

\head
4. Proof of the Theorem
\endhead

\specialhead
4.1. Initial steps
\endspecialhead

Our method goes back to Vaughan [20]. We take a periodic with period 1 function such that
$$
F_\Delta(\theta)=
\cases
0&\text{if }-\frac{1}{2}\leq \theta< -\Delta,
\\
1&\text{if }-\Delta\leq \theta< \Delta,
\\
0&\text{if }\quad\, \Delta\leq \theta<\frac{1}{2},
\endcases
$$
where $\Delta$ is defined by \Tag(7). Every nontrivial lower bound for the sum
$$
\sum\Sb p\leq N\\ p=[n^{1/\gamma}]\endSb F_\Delta(\alpha p^3+\beta)\log p
$$
leads directly to \Par*{Theorem 1}. In order to achieve this, we define the sum
$$
\Gamma=\sum\Sb p\leq N\\p=[n^{1/\gamma}]\endSb(F_\Delta(\alpha p^3+\beta)-2\Delta)\log p.
\tag13
$$

\specialhead
4.2. Estimation of ${\Gamma}$
\endspecialhead

\proclaim{Lemma 8}
Suppose that $H, N\geq1$,  $|\alpha-\frac{a}{q}|<\frac{1}{q^2}$, $(a, q)=1$.
Then
$$
\sum_{n\le N} \min \Bigl(1,\, \frac{1}{H \|\alpha n^3+\beta\pm\Delta\|} \Bigr) \ll (NHq)^\varepsilon\bigl(Nq^{-\frac{1}{8}}+N^\frac{7}{8}+NH^{-\frac{1}{8}}+N^\frac{5}{8}H^{-\frac{1}{8}}q^\frac{1}{8}\bigr).
$$
\endproclaim

\demo{Proof}
Using \Par*{Lemma 1}, we write
$$
\aligned
&\sum_{n\le N}\min \Bigl(1,\, \frac{1}{H \|\alpha n^3+\beta\pm\Delta\|} \Bigr)=
\sum_{n\le N}\sum\limits_{h=-\infty}^{\infty}b_he(h(\alpha n^3+\beta\pm\Delta))
\\
&=\sum_{n\le N}\biggl(b_0+\sum\limits_{1\leq|h|\leq H^2}b_he(h(\alpha n^3+\beta\pm\Delta))+\sum\limits_{|h|>H^2}b_he(h(\alpha n^3+\beta\pm\Delta)) \biggr)
\\
&\ll(\log H)\sum\limits_{1\leq|h|\leq H^2}\min\Bigl(\frac{1}{H}, \frac{1}{|h|}\Bigr)\biggl|\sum_{n\le N}e(\alpha hn^3)\biggr|+NH\sum\limits_{|h|>H^2}\frac{1}{h^2}+\frac{N\log H}{H}
\\
&\ll(\log H)\Psi(H)+NH^{-1}\log H,
\endaligned
\tag14
$$
where
$$
\Psi(H)=\sum\limits_{1\leq h\leq H^2}\min\Bigl(\frac{1}{H}, \frac{1}{h}\Bigr)\biggl|\sum_{n\le N}e(\alpha hn^3)\biggr|.
\tag15
$$
By \Tag(11), \Tag(15), and Abel's summation formula, we obtain
$$
\align
\Psi(H)=\frac{\goth{S}(H^2)}{H^2}+\int\limits_{H}^{H^2}\frac{\goth{S}(u)}{u^2}\,du\ll(\log H)\max_{H\leq u\leq H^2}\frac{\goth{S}(u)}{u}.
\tag16
\endalign
$$
It remains to estimate $\goth{S}(u)$. In \Par*{Lemma 2}, we set
$$
Q=J=L=\Bigl[\frac{N}{\log N}\Bigr], \quad  k=6qjl.
\tag17
$$
Now \Tag(11), \Tag(17),
\Par*{Lemma 2},  \Par*{Lemma 3}, and \Par*{Lemma 4} %\?and \Par{Lemma 2}{Lemmas 2}--\Par{Lemma 4}{4}
imply
$$
\aligned
\goth{S}^8(u)&\ll u^8N^{7+\varepsilon}+u^7N^{5+\varepsilon}\sum\limits_{q=1}^Q\sum\limits_{j=1}^J\sum\limits_{l=1}^L \, \min\Bigl(u,\,\frac{1}{\|6\alpha qjl\|}\Bigr)
\\
&\ll u^8N^{7+\varepsilon}+u^7N^{5+\varepsilon}\sum\limits_{k=1}^{6QJL}\sum\limits_{q=1}^Q\sum\limits_{j=1}^J
\sum^L\Sb l=1\\ 6qjl=k\endSb
\min\Bigl(u,\,\frac{1}{\|\alpha k\|}\Bigr)\\
&\ll u^8N^{7+\varepsilon}+u^7N^{5+\varepsilon}\sum\limits_{k=1}^{6QJL}\tau_3(k)\min\Bigl(u,\,\frac{1}{\|\alpha k\|}\Bigr)\\
&\ll u^8N^{7+\varepsilon}+u^7N^{5+\varepsilon}\sum\limits_{k=1}^{6QJL}\min\Bigl(u,\,\frac{1}{\|\alpha k\|}\Bigr)\\
&\ll u^8N^{7+\varepsilon}+u^7N^{5+\varepsilon}\Bigl(\frac{uQJL}{q}+QJL\log q+q\log q\Bigr)\\
&\ll (Nq)^\varepsilon\bigl(u^8N^7+u^7N^8+u^8N^8q^{-1}+u^7N^5q\bigr).
\endaligned
$$
Hence it follows that
$$
\goth{S}(u)\ll (Nq)^\varepsilon(uN^\frac{7}{8}+u^\frac{7}{8}N+uNq^{-\frac{1}{8}}+u^\frac{7}{8}N^\frac{5}{8}q^\frac{1}{8}).
\tag18
$$
Bearing in mind \Tag(14), \Tag(16),  and \Tag(18), we establish the statement in the lemma.
\qed\enddemo

\proclaim{Lemma 9}
Let $\frac{47}{48}<\gamma<1$. For the sum denoted by \Tag(12), the following estimate holds
$$
\Sigma\ll N^{\frac{48\gamma+47}{96}+\varepsilon}.
$$
\endproclaim

\demo{Proof}
From \Tag(9), \Tag(12), \Par*{Lemma 1}, and the simplest splitting up argument, we get
$$
\Sigma\ll(\Sigma_1+\Sigma_2)\log^2N+N^{1/2},
\tag19
$$
where
\iftex
$$
\align
&\Sigma_1=\sum\limits_{m\sim M_1}\frac{1}{m}\biggl|\sum\limits_{n\sim N_1}\Lambda(n)e(\alpha h n^3)\bigl(e(-mn^\gamma)-e(-m(n+1)^\gamma)\bigr) \biggr|,
\tag20
\\
&\Sigma_2=\sum\limits_{n\sim N_1}\min\Bigl(1, \frac{1}{M\|n^\gamma\|}\Bigr),
\tag21
\\
&M_1\leq \frac{M}{2},\quad N_1\leq \frac{N}{2}.
\tag22
\endalign
$$
\else
$$
\Sigma_1=\sum\limits_{m\sim M_1}\frac{1}{m}\biggl|\sum\limits_{n\sim N_1}\Lambda(n)e(\alpha h n^3)
\bigl(e(-mn^\gamma)-e(-m(n+1)^\gamma)\bigr) \biggr|,
\tag20
$$
$$
\Sigma_2=\sum\limits_{n\sim N_1}\min\Bigl(1, \frac{1}{M\|n^\gamma\|}\Bigr),
\tag21
$$
$$
M_1\leq \frac{M}{2},\quad N_1\leq \frac{N}{2}.
\tag22
$$
\fi
Working as in [18, p. 214], and using \Tag(21) and \Tag(22), we deduce
$$
\Sigma_2\ll \bigl(NM^{-1}+N^\frac{\gamma}{2} M^\frac{1}{2}+N^{1-\gamma}\bigr)\log M.
\tag23
$$
In view of \Tag(9) and \Tag(23), we derive
$$
\Sigma_2\ll  N^{\frac{48\gamma+47}{96}+\varepsilon}.
\tag24
$$
Next, we estimate $\Sigma_1$. By \Tag(20) and Abel's summation formula, we have
$$
\Sigma_1\ll N^{\gamma-1}_1\sum\limits_{m\sim M_1}\max_{N_2\in[N_1,2N_1]}|\goth{F}(N_1,N_2)|,
\tag25
$$
where
$$
\goth{F}(N_1,N_2)=\sum\limits_{N_1<n\leq N_2}\Lambda(n)e(\alpha hn^3-mn^\gamma).
\tag26
$$
Assume that
$$
N_1\leq N^{\frac{48\gamma-1}{48\gamma}}.
\tag27
$$
Based on \Tag(9), \Tag(22), \Tag(25), \Tag(26), and \Tag(27), we conclude that
$$
\Sigma_1\ll N^{\frac{48\gamma+47}{96}+\varepsilon}.
\tag28
$$
Henceforth, we suppose that
$$
N^{\frac{48\gamma-1}{48\gamma}}<N_1\leq \frac{N}{2}.
\tag29
$$
We now estimate the sum \Tag(26). Define
$$
f(d,l)=\alpha hd^3l^3-md^\gamma l^\gamma.
\tag30
$$
By \Tag(26), \Tag(30), and Vaughan's identity (see [24]), we establish
$$
\goth{F}(N_1,N_2)=\Phi_1-\Phi_2-\Phi_3-\Phi_4,
\tag31
$$
where
\iftex
$$
\align
&\Phi_1=\sum_{d\le \vartheta}\mu(d)\sum_{\frac{N_1}{d}<l\le \frac{N_2}{d}}e(f(d,l))\log l,
\tag32
\\
&\Phi_2=\sum_{d\le \vartheta}c(d)\sum_{\frac{N_1}{d}<l\le \frac{N_2}{d}}e(f(d,l)),
\tag33
\\
&\Phi_3=\sum_{\vartheta<d\le \vartheta^2}c(d)\sum_{\frac{N_1}{d}<l\le \frac{N_2}{d}}e(f(d,l)),
\tag34
\\
&\Phi_4= \overset{\displaystyle{\sum\sum}}\to{\Sb N_1<dl\le N_2 \\d>\vartheta,\,l>\vartheta\endSb} %\?
a(d)\Lambda(l) e(f(d,l))
\tag35
\endalign
$$
\else
$$
\Phi_1=\sum_{d\le \vartheta}\mu(d)\sum_{\frac{N_1}{d}<l\le \frac{N_2}{d}}e(f(d,l))\log l,
\tag32
$$
$$
\Phi_2=\sum_{d\le \vartheta}c(d)\sum_{\frac{N_1}{d}<l\le \frac{N_2}{d}}e(f(d,l)),
\tag33
$$
$$
\Phi_3=\sum_{\vartheta<d\le \vartheta^2}c(d)\sum_{\frac{N_1}{d}<l\le \frac{N_2}{d}}e(f(d,l)),
\tag34
$$
$$
\Phi_4= \overset{\displaystyle{\sum\sum}}\to{\Sb N_1<dl\le N_2 \\d>\vartheta,\,l>\vartheta\endSb} %\?
a(d)\Lambda(l) e(f(d,l))
\tag35
$$
\fi
and
$$
|c(d)|\leq\log d,\quad  | a(d)|\leq\tau_2(d),
\tag36
$$
and $\vartheta$ is defined by \Tag(10). Let us first consider the sum $\Phi_2$, as defined in \Tag(33). In view of \Tag(30), we get
$$
\Bigl|\frac{\partial^4f(d,l)}{\partial l^4}\Bigr|\asymp m d^4N_1^{\gamma-4}.
\tag37
$$
Now \Tag(37) and \Par*{Lemma 5} for $k=4$ lead to
$$
\sum_{\frac{N_1}{d}<l\le \frac{N_2}{d}}e(f(d, l))\ll
m^\frac{1}{12}d^{-\frac{2}{3}} N_1^{\frac{\gamma}{12}+\frac{2}{3}+\varepsilon}+N_1^{\frac{11}{12}+\varepsilon}d^{-\frac{11}{12}}+m^{-\frac{1}{24}}d^{-1}N_1^{1-\frac{\gamma}{24}+\varepsilon}.
\tag38
$$
From  \Tag(9), \Tag(10), \Tag(33), \Tag(36), and \Tag(38), we find
$$
\Phi_2\ll \bigl(m^\frac{1}{12}\vartheta^\frac{1}{3} N_1^{\frac{\gamma}{12}+\frac{2}{3}}
+N_1^{\frac{11}{12}}\vartheta^\frac{1}{12}+m^{-\frac{1}{24}}N_1^{1-\frac{\gamma}{24}}\bigr)N^\varepsilon.
\tag39
$$
To estimate $\Phi_1$ defined by \Tag(32), we use Abel's summation formula.
Then, following the same reasoning as in the estimation of $\Phi_2$, we have
$$
\Phi_1\ll \bigl(m^\frac{1}{12}\vartheta^\frac{1}{3} N_1^{\frac{\gamma}{12}+\frac{2}{3}}
+N_1^{\frac{11}{12}}\vartheta^\frac{1}{12}+m^{-\frac{1}{24}}N_1^{1-\frac{\gamma}{24}}\bigr)N^\varepsilon.
\tag40
$$
Our next goal is to obtain estimates for the sums defined by \Tag(34) and \Tag(35).
Proceeding as in [13],  we deduce that it suffices to estimate the sum
$$
\Phi'_4=\sum_{D<d\le 2D}a(d)\sum\Sb L<l\le 2L\\ N_1<dl\le N_2\endSb\Lambda(l)e(f(d,l))
\tag41
$$
 under the given conditions
$$
\frac{N_1}{4}\leq DL\le 2N_1, \quad  \frac{N^\frac{1}{2}_1}{2}\leq D\leq \vartheta^2.
\tag42
$$
Consequently, we obtain
$$
\Phi_4\ll|\Phi'_4|\log N_1
\tag43
$$
and the resulting estimate also holds for $\Phi_3$.
Now \Tag(36), \Tag(41), \Tag(42), Cauchy's inequality, and \Par*{Lemma 6} with $Q\leq \frac{L}{2}$ give us
$$
|\Phi'_4|^2\ll\biggl(\frac{LD}{Q}\sum_{1\leq q\leq Q}\sum_{L<l\le 2L}\biggl|\sum_{D_1<d\le D_2}e(g(d))\biggr|+\frac{ (LD)^2}{Q}\biggr)N^\varepsilon,
\tag44
$$
where
$$
D_1=\max{\Bigl\{D,\frac{N_1}{l},\frac{N_1}{l+q}\Bigr\}},\quad
D_2=\min{\Bigl\{2D,\frac{N_2}{l},\frac{N_2}{l+q}\Bigr\}}
\tag45
$$
and
$$
g(d)=f(d,l+q)-f(d,l).
\tag46
$$
By \Tag(30) and \Tag(46), it follows that
$$
|g^{(4)}(d)|\asymp m D^{\gamma-4} q L^{\gamma-1}.
\tag47
$$
Now \Tag(45), \Tag(47), and \Par*{Lemma 5} for $k=4$ yield
$$
\sum\limits_{D_1<d\leq D_2}e(g(d))
\ll m^\frac{1}{12}q^\frac{1}{12}D^{\frac{\gamma}{12}+\frac{2}{3}+\varepsilon}L^{\frac{\gamma}{12}-\frac{1}{12}}+D^{\frac{11}{12}+\varepsilon}+
m^{-\frac{1}{24}}q^{-\frac{1}{24}}D^{1-\frac{\gamma}{24}+\varepsilon}L^{\frac{1}{24}-\frac{\gamma}{24}}.
\tag48
$$
Set
$$
Q=\min([L/4], [Q_0]),
\tag49
$$
where
$$
Q_0=m^{-1}D^{3-\gamma}L^{1-\gamma}.
\tag50
$$
From \Tag(9), \Tag(22), \Tag(29), \Tag(42), and \Tag(50), we have
$$
Q_0>N^{\frac{4367}{4512}}.
$$
Bearing in mind \Tag(44), \Tag(48), \Tag(49), and \Tag(50), %\?\Tag(48)--\Tag(50),
we derive
$$
\aligned
|\Phi'_4|^2&\ll(D^2L^2Q^{-1}+m^\frac{1}{12}Q^\frac{1}{12}D^{\frac{\gamma}{12}+\frac{5}{3}}L^{\frac{\gamma}{12}+\frac{23}{12}}+D^\frac{23}{12}L^2
+m^{-\frac{1}{24}}Q^{-\frac{1}{24}}D^{2-\frac{\gamma}{24}}L^{\frac{49}{24}-\frac{\gamma}{24}})N^\varepsilon
\\
&\ll (D^2L^2L^{-1}+ D^2L^2Q_0^{-1}+m^\frac{1}{12}Q_0^\frac{1}{12}D^{\frac{\gamma}{12}+\frac{5}{3}}L^{\frac{\gamma}{12}+\frac{23}{12}}+D^\frac{23}{12}L^2
\\
&\qquad+m^{-\frac{1}{24}}D^{2-\frac{\gamma}{24}}L^{\frac{49}{24}-\frac{\gamma}{24}}(L^{-\frac{1}{24}}+Q_0^{-\frac{1}{24}}))N^\varepsilon
\\
&\ll (D^2L+mD^{\gamma-1}L^{\gamma+1}+D^\frac{23}{12}L^2+m^{-\frac{1}{24}}D^{2-\frac{\gamma}{24}}L^{2-\frac{\gamma}{24}}+D^\frac{15}{8}L^2)
N^\varepsilon.
\endaligned
\tag51
$$
Now \Tag(42), \Tag(43), and \Tag(51) imply
$$
\Phi_4\ll \bigl( N_1^\frac{1}{2}\vartheta+M^\frac{1}{2}N_1^\frac{\gamma}{2}+N_1^\frac{47}{48}+m^{-\frac{1}{48}}N_1^{1-\frac{\gamma}{48}}\bigr)N^\varepsilon.
\tag52
$$
Proceeding analogously to the estimation of $\Phi_4$ for the sum \Tag(34), we deduce
$$
\Phi_3\ll  ( N_1^\frac{1}{2}\vartheta+M^\frac{1}{2}N_1^\frac{\gamma}{2}+N_1^\frac{47}{48}+m^{-\frac{1}{48}}N_1^{1-\frac{\gamma}{48}})N^\varepsilon.
\tag53
$$
Combining \Tag(31), \Tag(39), \Tag(40), \Tag(52), and \Tag(53), we get
$$
\goth{F}(N_1,N_2)\ll\bigl( N_1^\frac{1}{2}\vartheta+M^\frac{1}{2}N_1^\frac{\gamma}{2}+N_1^\frac{47}{48}
+m^\frac{1}{12}\vartheta^\frac{1}{3} N_1^{\frac{\gamma}{12}+\frac{2}{3}}+N_1^{\frac{11}{12}}\vartheta^\frac{1}{12}
+m^{-\frac{1}{48}}N_1^{1-\frac{\gamma}{48}}\bigr)N^\varepsilon.
\tag54
$$
Based on \Tag(9), \Tag(10), \Tag(25), \Tag(29), and \Tag(54), we conclude
$$
\Sigma_1\ll N^{\frac{48\gamma+47}{96}+\varepsilon}.
\tag55
$$
Taking into account \Tag(19), \Tag(24), \Tag(28), and \Tag(55), we establish the statement in the lemma.
\qed\enddemo

\proclaim{Lemma 10}
Let $\frac{47}{48}<\gamma<1$. For the sum $\Gamma$ defined by \Tag(13) the estimate
$$
\Gamma\ll N^{\frac{48\gamma+47}{96}+\varepsilon}
$$
holds.
\endproclaim

\demo{Proof}
From \Tag(13), we write
$$
\Gamma=\sum\limits_{p\leq N}([-p^\gamma]-[-(p+1)^\gamma])(F_\Delta(\alpha p^3+\beta)-2\Delta)\log p
=\Gamma_1+\Gamma_2,
\tag56
$$
where
\iftex
$$
\align
&\Gamma_1=\sum\limits_{p\leq N}((p+1)^\gamma-p^\gamma)(F_\Delta(\alpha p^3+\beta)-2\Delta)\log p,
\tag57
\\
&\Gamma_2=\sum\limits_{p\leq N}(\psi(-(p+1)^\gamma)-\psi(-p^\gamma))(F_\Delta(\alpha p^3+\beta)-2\Delta)\log p.
\tag58
\endalign
$$
\else
$$
\Gamma_1=\sum\limits_{p\leq N}((p+1)^\gamma-p^\gamma)(F_\Delta(\alpha p^3+\beta)-2\Delta)\log p,
\tag57
$$
$$
\Gamma_2=\sum\limits_{p\leq N}(\psi(-(p+1)^\gamma)-\psi(-p^\gamma))(F_\Delta(\alpha p^3+\beta)-2\Delta)\log p.
\tag58
$$
\fi

\specialhead
Upper bound for $\Gamma_1$
\endspecialhead

The function $F_\Delta(\theta)-2\Delta$ is well known to have the expansion
$$
\sum\limits_{1\leq|h|\leq H}\frac{\sin2\pi h\Delta}{\pi h}\,e(h\theta)+\Cal{O}\Bigl(\min\Bigl(1, \frac{1}{H\|\theta+\Delta\|}\Bigr)+\min\Bigl(1, \frac{1}{H\|\theta-\Delta\|}\Bigr)\Bigr).
\tag59
$$
Now \Tag(57), \Tag(59), and the formula
$$
(p+1)^\gamma-p^\gamma=\gamma p^{\gamma-1}+\Cal{O}(p^{\gamma-2})
$$
lead to
$$
\Gamma_1=\gamma\sum\limits_{p\leq N}p^{\gamma-1}\log p
\sum\limits_{1\leq|h|\leq H}\frac{\sin2\pi h\Delta}{\pi h}\,e(h(\alpha p^3+\beta))+\Cal{O}(\Omega\log N),
\tag60
$$
where
$$
\Omega=\sum\limits_{n=1}^N\biggl(\min\Bigl(1, \frac{1}{H\|\alpha n^3+\beta+\Delta\|}\Bigr)+\min\Bigl(1, \frac{1}{H\|\alpha n^3+\beta-\Delta\|}\Bigr)\biggr).
\tag61
$$
By \Tag(5), \Tag(6), \Tag(8), \Tag(61), and \Par*{Lemma 8}, we derive
$$
\Omega\ll N^\varepsilon\bigl(Nq^{-\frac{1}{8}}+N^\frac{7}{8}+NH^{-\frac{1}{8}}+N^\frac{5}{8}H^{-\frac{1}{8}}q^\frac{1}{8}\bigr)\ll N^{1+\varepsilon} H^{-\frac{1}{8}}\ll N^{\frac{48\gamma+47}{96}+\varepsilon}.
\tag62
$$
Now \Tag(60) and \Tag(62) yield
$$
\Gamma_1\ll\sum\limits_{h=1}^{H}\min\Bigl(\Delta,\frac{1}{h}\Bigr)\biggl|\sum\limits_{p\leq N}p^{\gamma-1}e(\alpha h p^3)\log p \biggr|+N^{\frac{48\gamma+47}{96}+\varepsilon}.
\tag63
$$
Set
$$
\goth{X}(u)=\sum\limits_{h\leq u}\biggl|\sum\limits_{p\leq N}p^{\gamma-1}e(\alpha h p^3)\log p \biggr|.
\tag64
$$
By applying Abel's summation formula, we obtain
$$
\sum\limits_{h=1}^{H}\min\Bigl(\Delta,\frac{1}{h}\Bigr)\biggl|\sum\limits_{p\leq N}p^{\gamma-1}e(\alpha h p^3)\log p \biggr|
=\frac{\goth{X}(H)}{H}+\int\limits_{\Delta^{-1}}^{H}\frac{\goth{X}(u)}{u^2}\,du
\ll(\log H)\max_{\Delta^{-1}\leq u\leq H}\frac{\goth{X}(u)}{u}.
\tag65
$$
Using Abel's summation once more, we get
$$
\sum\limits_{p\leq N}p^{\gamma-1}e(\alpha h p^3)\log p=N^{\gamma-1}G(N)+(1-\gamma)\int\limits_{2}^{N}G(y)y^{\gamma-2}\,dy,
\tag66
$$
where
$$
G(y)=\sum\limits_{p\leq y}e(\alpha h p^3)\log p.
\tag67
$$
According to Dirichlet's approximation theorem, there exist integers $a_h$ and $q_h$ such that
$$
\Bigl|\alpha h-\frac{a_h}{q_h}\Bigr|\leq\frac{1}{q_hq^2},\quad (a_h,q_h)=1,\quad 1\leq q_h\leq q^2.
\tag68
$$
In view of \Tag(67), \Tag(68), and \Par*{Lemma 7}, we deduce
$$
G(y)\ll y^{1+\varepsilon}\bigl(q_h^{-\frac{1}{16}}+y^{-\frac{1}{32}}+y^{-\frac{3}{16}}q_h^{\frac{1}{16}}\bigr).
\tag69
$$
From \Tag(64), \Tag(66), and \Tag(69), we find
$$
\goth{X}(u)\ll N^{\gamma-1+\varepsilon}\sum\limits_{h\leq u}\bigl(Nq_h^{-\frac{1}{16}}+N^{\frac{31}{32}}+N^{\frac{13}{16}}q_h^{\frac{1}{16}}\bigr).
\tag70
$$
Combining \Tag(5), \Tag(8), \Tag(68) and following the reasoning in [13], we derive
$$
q_h\in\Bigl(\frac{q^\frac{2}{3}}{\log N},q^2\Bigr].
\tag71
$$
Based on \Tag(6), \Tag(70), and \Tag(71), we obtain
$$
\goth{X}(u)\ll uN^{\gamma+\varepsilon}q^{-\frac{1}{24}}\ll uN^{\frac{144\gamma-49}{96}+\varepsilon}.
\tag72
$$
By \Tag(6), \Tag(8), \Tag(63), \Tag(65), and \Tag(72), it follows that
$$
\Gamma_1\ll N^{\frac{48\gamma+47}{96}+\varepsilon}.
\tag73
$$

\specialhead
Upper bound for $\Gamma_2$
\endspecialhead

From \Tag(58), \Tag(59) and proceeding as in $\Gamma_1$, we conclude that
$$
\Gamma_2\ll\sum\limits_{h=1}^{H}\min\Bigl(\Delta,\frac{1}{h}\Bigr)
\biggl|\sum\limits_{p\leq N}\bigl(\psi(-(p+1)^\gamma)-\psi(-p^\gamma)\bigr)e(\alpha h p^3)\log p \biggr|+N^{\frac{48\gamma+47}{96}+\varepsilon}.
\tag74
$$
Define
$$
S(u)=\sum\limits_{h\leq u}\biggl|\sum\limits_{p\leq N}(\psi(-(p+1)^\gamma)-\psi(-p^\gamma))e(\alpha h p^3)\log p \biggr|.
\tag75
$$
By \Tag(75) and Abel's summation formula, we deduce
$$
\sum\limits_{h=1}^{H}\min\Bigl(\Delta,\frac{1}{h}\Bigr)
\biggl|\sum\limits_{p\leq N}(\psi(-(p+1)^\gamma)-\psi(-p^\gamma))e(\alpha h p^3)\log p \biggr|
=\frac{S(H)}{H}+\int\limits_{\Delta^{-1}}^{H}\frac{S(u)}{u^2}\,du.
\tag76
$$
Now \Tag(6), \Tag(8), \Tag(74)--\Tag(76) and \Par*{Lemma 9} give us
$$
\Gamma_2\ll(\log H)\max_{\Delta^{-1}\leq u\leq H}\frac{S(u)}{u}+N^{\frac{48\gamma+47}{96}+\varepsilon}\ll N^{\frac{48\gamma+47}{96}+\varepsilon}.
\tag77
$$
Summarizing \Tag(56), \Tag(73), and \Tag(77), we complete the proof of the lemma.
\qed\enddemo

\specialhead
4.3. The end of the proof
\endspecialhead

Taking into account \Tag(4), \Tag(7), \Tag(13) and \Par*{Lemma 10}, we get
$$
\sum\Sb p\leq N\\p=[n^{1/\gamma}]\endSb
F_\Delta(\alpha p^3+\beta)\log p\gg N^{\frac{48\gamma+47}{96}+\varepsilon}.
$$
This completes the proof of \Par*{Theorem 1}.

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\enddocument