\documentstyle{SibMatJ}
% \TestXML
\topmatter

  \Author           Dimitrov
    \Initial        S.
    \Initial        I.
    \Gender         he
    \Email          sdimitrov\@tu-sofia.bg
    \Email          xyzstoyan\@gmail.com
    \AffilRef       1
    \AffilRef       2
  \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

  \datesubmitted    December 17, 2025\enddatesubmitted
  \dateaccepted     December 28, 2025\enddateaccepted

  \UDclass
    ??? %11D75  11P32
  \endUDclass

  \title
    Diophantine Approximation with Mixed Powers Of Piatetski-Shapiro Primes
  \endtitle

  \abstract
    Let $[\,\cdot\,]$ denote the floor function. In this paper, we show that whenever $\eta$ is real and the constants $\lambda _i$ satisfy some necessary conditions,
    then for any fixed $\frac{63}{64}<\gamma<1$ and $\theta>0$, there exist  infinitely many prime triples $p_1,\, p_2,\, p_3$ satisfying the inequality
    $$ %\?
    |\lambda _1p_1 + \lambda _2p_2 + \lambda _3p^2_3+\eta|<(\max \{p_1, p_2, p^2_3\})^{\frac{63-64\gamma}{52}+\theta}
    $$
    and such that $p_i=[n_i^{1/\gamma}]$, $i=1,2,3$.
  \endabstract

  \keywords
    Diophantine approximation,
    Piatetski-Shapiro primes
  \endkeywords

\endtopmatter

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

The study of Diophantine inequalities involving prime numbers constitutes a rapidly evolving field within analytic number theory.
In 1967, A. Baker %\?просто Baker
[1] proved that if  $\lambda_1,\lambda_2,\lambda_3$ are nonzero real numbers, not all of the same sign, $\lambda_1/\lambda_2$ is irrational, $\eta$ is real and $A>0$,
then there exist infinitely many prime triples $p_1,p_2,p_3$ such that
$$
|\lambda_1p_1+\lambda_2p_2+\lambda_3p_3+\eta|<\varepsilon_1,
\tag1
$$
where $\varepsilon_1=(\log \max p_j)^{-A}$.
Subsequently, the right-hand side of \Tag(1) was improved by Ramachandra [2], Vaughan [3], Lau and Liu [4],
Baker and Harman [5], and Harman [6].
The best result to date is due to Matom\"{a}ki [7], with $\varepsilon_1=( \max p_j)^{-\frac{2}{9}+\delta}$ and $\delta>0$.
In 2018, Gambini, Languasco, and Zaccagnini [8] proved the existence of infinitely many triples of
primes $p_1$, $p_2$, and $p_3$ such that
$$
|\lambda_1p_1+\lambda_2p_2+\lambda_3p^2_3+\eta|<\varepsilon_2,
\tag2
$$
where $\varepsilon_2=(\max \{p_1, p_2, p^2_3\})^{-\frac{1}{12}+\delta}$ and $\delta>0$.
Weaker results were previously obtained in [9] and [10].
Another interesting question is the study of Diophantine inequalities involving special prime numbers.
Let $P_l$ is a number with at most $l$ prime factors.
Very recently  Todorova and Georgieva [11] solved inequality \Tag(2) with prime numbers $p_1$, $p_2$, and $p_3$ such that $p_i+2=P_{l_i}$,\;$i=1,\,2,\,3$.
In 1953, Piatetski-Shapiro [12] showed that for any fixed $\frac{11}{12}<\gamma<1$, there exist infinitely many prime numbers
of the form $p = [n^{1/\gamma}]$.
Such primes are called Piatetski-Shapiro primes %\?{\it Piatetski-Shapiro primes\/}
of type $\gamma$.
Subsequently, the interval for $\gamma$ was sharpened many times and the best result to date has been supplied by Rivat and Wu [13] with $\frac{205}{243}<\gamma<1$.
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 [14--16].
In 2022, the author [17] %\?
proved that for any fixed $\frac{37}{38}<\gamma<1$,  inequality \Tag(1) is solvable with infinitely many Piatetski-Shapiro prime triples $p_1$, $p_2$, $p_3$ of type $\gamma$.
As a continuation of these studies, we solve \Tag(2) with Piatetski-Shapiro primes.

\proclaim{Theorem 1}
Suppose that $\lambda_1,\lambda_2,\lambda_3$ are nonzero real numbers, not all of the same sign, that $\lambda_1/\lambda_2$ is irrational, and that $\eta$ is real.
Let $\theta>0$ and $\gamma$ be fixed with $\frac{63}{64}<\gamma<1$.
Then there exist infinitely many ordered triples of Piatetski-Shapiro primes $p_1$, $p_2$, and $p_3$ of type $\gamma$ such that
$$
|\lambda_1p_1+\lambda_2p_2+\lambda_3p^2_3+\eta|<(\max \{p_1, p_2, p^2_3\})^{\frac{63-64\gamma}{52}+\theta}.
$$
\endproclaim

\head
2. Notations
\endhead

The letter $p$ will always denote a prime number. By $\delta$ we denote an arbitrarily small positive number, not the same in all appearances.
As usual, $[t]$ and $\{t\}$ denote the integer part and the fractional part of~$t$, respectively.
Moreover $\psi(t)=\{t\}-\frac{1}{2}$. We write  $e(t)=e^{2\pi it}$.
Let $\gamma$, $\theta$, and $\lambda_0$ be a %\?
real constants such that $\frac{63}{64}<\gamma<1$, $\theta>0$, and $0<\lambda_0<1$.
Since $\lambda_1/\lambda_2$ is irrational, there are infinitely many different convergents
$a_0/q_0$ to its continued fraction, with $|\frac{\lambda_1}{\lambda_2} - \frac{a_0}{q_0}|<\frac{1}{q_0^2}$, $(a_0, q_0) = 1$, $a_0\neq0$ and $q_0$ is arbitrary large.
Denote  %\?внести авторскую правку в формулы!!!
\iftex
$$
\align
&X=q_0^\frac{13}{6};
\tag3
\\
&\Delta
=X^{-\frac{12}{13}}\log X;
\tag4
\\
&\varepsilon
=X^{\frac{63-64\gamma}{52}+\theta};
\tag5
\\
&H=\frac{\log^2X}{\varepsilon};
\tag6
\\
&S_k(t)=\sum\limits\Sb\lambda_0X<p^k\leq X\\ p=[n^{1/\gamma}]\endSb
p^{1-\gamma}e(t p^k)\log p, \quad k=1, 2;
\tag7
\\
&\Sigma(t)=\sum\limits_{\lambda_0X<p^2\leq X}e(t p^2)\log p;
\tag8
\\
&U(t)=\sum\limits_{\lambda_0X<n^2\leq X}e(t n^2);
\tag9
\\
&\Omega(t)=\sum\limits_{\lambda_0X<p^2\leq X}p^{1-\gamma}(\psi(-(p+1)^\gamma)-\psi(-p^\gamma))e(t p^2)\log p;
\tag10
\\
&I_k(t)=\int\limits_{(\lambda_0X)^\frac{1}{k}}^{X^\frac{1}{k}}e(t y^k)\,dy, \quad k=1, 2.
\tag11
\endalign
$$
\else
$$
X=q_0^\frac{13}{6};
\tag3
$$
$$
\Delta
=X^{-\frac{12}{13}}\log X;
\tag4
$$
$$
\varepsilon
=X^{\frac{63-64\gamma}{52}+\theta};
\tag5
$$
$$
H=\frac{\log^2X}{\varepsilon};
\tag6
$$
$$
S_k(t)=\sum\limits\Sb\lambda_0X<p^k\leq X\\ p=[n^{1/\gamma}]\endSb
p^{1-\gamma}e(t p^k)\log p, \quad k=1, 2;
\tag7
$$
$$
\Sigma(t)=\sum\limits_{\lambda_0X<p^2\leq X}e(t p^2)\log p;
\tag8
$$
$$
U(t)=\sum\limits_{\lambda_0X<n^2\leq X}e(t n^2);
\tag9
$$
$$
\Omega(t)=\sum\limits_{\lambda_0X<p^2\leq X}p^{1-\gamma}(\psi(-(p+1)^\gamma)-\psi(-p^\gamma))e(t p^2)\log p;
\tag10
$$
$$
I_k(t)=\int\limits_{(\lambda_0X)^\frac{1}{k}}^{X^\frac{1}{k}}e(t y^k)\,dy, \quad k=1, 2.
\tag11
$$
\fi

\head
3. Preliminary Lemmas
\endhead

\proclaim{Lemma 1}
Let $\varepsilon>0$ and $k\in \Bbb{N}$.
There exists a function $\theta(y)$ which is $k$ times continuously differentiable and such that
\iftex
$$
\alignat2
&\theta(y)=1&\quad&\text{for }\ |y|\leq 3\varepsilon/4;
\\
&0<\theta(y)<1&\quad&\text{for }\ 3\varepsilon/4 <|y|< \varepsilon;
\\
&\theta(y)=0&\quad&\text{for }\ |y|\geq \varepsilon,
\endalignat
$$
\else
$$
\align
\theta(y)=1\quad
&\text{for }\ |y|\leq 3\varepsilon/4;
\\
0<\theta(y)<1\quad
&\text{for }\ 3\varepsilon/4 <|y|< \varepsilon;
\\
\theta(y)=0\quad
&\text{for }\ |y|\geq \varepsilon,
\endalign
$$
\fi
and its Fourier transform
$$
\Theta(x)=\int\limits_{-\infty}^{\infty}\theta(y)e(-xy)dy
$$
satisfies the inequality
$$
|\Theta(x)|\leq\min\Bigl(\frac{7\varepsilon}{4},\frac{1}{\pi|x|},\frac{1}{\pi |x|}
\Bigl(\frac{k}{2\pi |x|\varepsilon/8}\Bigr)^k\Bigr).
$$
\endproclaim

\demo{Proof}
See [18].
\qed\enddemo

\proclaim{Lemma 2}
For any fixed $\frac{2426}{2817}<\gamma<1$, we have
$$
\sum\limits\Sb p\leq X\\ p=[n^{1/\gamma}]\endSb
1\sim \frac{X^\gamma}{\log X}.
$$
\endproclaim

\demo{Proof}
See [19, Theorem 1].
\qed\enddemo

\proclaim{Lemma 3}\Label{L3}
We have

\Item (i)
$\int\nolimits_{-\Delta}^\Delta|S_1(t)|^2\,dt\ll X\log^3X$ and $\int\nolimits_{0}^1|S_1(t)|^2\,dt\ll X^{2-\gamma}\log X$,

\Item (ii)
$\int\nolimits_{-\Delta}^\Delta|I_1( t)|^2\,dt\ll X$ and $\int\nolimits_{-\Delta}^\Delta|I_2( t)|^2\,dt\ll 1$.
\endproclaim

\demo{Proof}
For \Par{L3}{(i)} see [17, Lemma 6].
For \Par{L3}{(ii)} see [11, Lemma 15].
\qed\enddemo

\proclaim{Lemma 4}
Let $|t|\leq\Delta$. Then the asymptotic formula
$$
S_1(t)=\gamma  I_1(t)+ \Cal{O}\Bigl(\frac{X}{e^{(\log X)^{1/5}}}\Bigr)
$$
holds.
\endproclaim

\demo{Proof}
See [17, Lemma 5].
\qed\enddemo

\proclaim{Lemma 5}
Let $\frac{13}{14}<\gamma<1$. Then
$$
\Omega(t)\ll X^{\frac{21-7\gamma}{29}+\delta}\,.
$$
\endproclaim

\demo{Proof}
See [20, Lemma 6].
\qed\enddemo

\proclaim{Lemma 6}
Let $k\geq1$ and $1/2X\leq Y\leq 1/2X^{1-\frac{5}{6k}+\delta}$.
Then there exists a positive constant $c_1(\delta)$, which does not depend on $k$, such that
$$
\int\limits_{-Y}^Y|\Sigma(t)-U(t)|^2\,dt\ll\frac{X^{\frac{2}{k}-2}\log^2X}{Y}+Y^2X+X^{\frac{2}{k}-1}
\exp\Bigl(-c_1\Bigl(\frac{\log X}{\log\log X}\Bigr)^{1/3}\Bigr).
$$
\endproclaim

\demo{Proof}
See [8, Lemmas 1 and 2].
\qed\enddemo

\proclaim{Lemma 7}
Let $\frac{11}{12}<\gamma<1$ and $\Delta\leq|t|\leq H$.
Then there exists a sequence of real numbers $X_1,X_2,\dots %\?
\to \infty $ such that
$$
\min\bigl\{|S_1(\lambda_{1}t)|,|S_1(\lambda_2 t)|\bigr\}\ll X_j^{\frac{37-12\gamma}{26}}\log^5X_j,
\quad j=1,2,\dots.
$$
\endproclaim

\demo{Proof}
See [17, Lemma 7].
\qed\enddemo

\proclaim{Lemma 8}
We have
$$
\int\limits_{0}^1|S_2(t)|^4\,dt\ll X^{2-\gamma+\delta}\,.
$$
\endproclaim

\demo{Proof}
See [21, (12)].
\qed\enddemo

\head
4. Beginning of the Proof
\endhead

Consider the sum
$$
\Gamma(X)=\sum\limits\Sb \lambda_0X<p_1,p_2,p^2_3\leq X\\ p_i=[n^{1/\gamma}_i],i=1,2,3\endSb
\theta(\lambda_1p_1+\lambda_2p_2+\lambda_3p^2_3+\eta)p^{1-\gamma}_1p^{1-\gamma}_2p^{1-\gamma}_3\log p_1\log p_2\log p_3.
\tag12
$$
Using the inverse Fourier transform for the function $\theta(x)$, we obtain
$$
\align
\Gamma(X)
&=\sum\limits_{\lambda_0X<p_1,p_2,p^2_3\leq X\atop{p_i=[n^{1/\gamma}_i],\, i=1,2,3}}p^{1-\gamma}_1p^{1-\gamma}_2p^{1-\gamma}_3\log p_1\log p_2\log p_3
\\
&\qquad\times\int\limits_{-\infty}^{\infty}\Theta(t)e((\lambda_1p_1+\lambda_2p_2+\lambda_3p^2_3+\eta)t)\,dt
\\
&=\int\limits_{-\infty}^{\infty}\Theta(t)S_1(\lambda_1t)S_1(\lambda_2t)S_2(\lambda_3t)e(\eta t)\,dt\,.
\endalign
$$
We decompose $\Gamma(X)$ as follows:
$$
\Gamma(X)=\Gamma_1(X)+\Gamma_2(X)+\Gamma_3(X),
\tag13
$$
where
\iftex
$$
\align
&\Gamma_1(X)=\int\limits_{|t|<\Delta}\Theta(t)S_1(\lambda_1t)S_1(\lambda_2t)S_2(\lambda_3t)e(\eta t)\,dt,
\tag14
\\
&\Gamma_2(X)=\int\limits_{\Delta\leq|t|\leq H}\Theta(t)S_1(\lambda_1t)S_1(\lambda_2t)S_2(\lambda_3t)e(\eta t)\,dt,
\tag15
\\
&\Gamma_3(X)=\int\limits_{|t|>H}\Theta(t)S_1(\lambda_1t)S_1(\lambda_1t)S_2(\lambda_3t)e(\eta t)\,dt.
\tag16
\endalign
$$
\else
$$
\Gamma_1(X)=\int\limits_{|t|<\Delta}\Theta(t)S_1(\lambda_1t)S_1(\lambda_2t)S_2(\lambda_3t)e(\eta t)\,dt,
\tag14
$$
$$
\Gamma_2(X)=\int\limits_{\Delta\leq|t|\leq H}\Theta(t)S_1(\lambda_1t)S_1(\lambda_2t)S_2(\lambda_3t)e(\eta t)\,dt,
\tag15
$$
$$
\Gamma_3(X)=\int\limits_{|t|>H}\Theta(t)S_1(\lambda_1t)S_1(\lambda_1t)S_2(\lambda_3t)e(\eta t)\,dt.
\tag16
$$
\fi
We will estimate $\Gamma_1(X)$, $\Gamma_2(X)$, and $\Gamma_3(X)$, respectively,
in \Sec{Section 5}{Sections 5}--\Sec{Section 7}{7}.
In \Sec*{Section 8} we will complete the proof of \Par*{Theorem 1}.

\head
5. Lower Bound  of $\Gamma_1(X)$
\endhead

\proclaim{Lemma 9}
Let $\frac{13}{14}<\gamma<1$. Then
$$
S_2(t)=\gamma\Sigma(t)+\Cal{O}(X^{\frac{21-7\gamma}{29}+\delta}).
$$
\endproclaim

\demo{Proof}
From \Tag(7), \Tag(8), \Tag(10), and the well-known asymptotic formula
$$
(p+1)^\gamma-p^\gamma=\gamma p^{\gamma-1}+\Cal{O}(p^{\gamma-2})
$$
we write
$$
\aligned
S_2(t)
&=\sum\limits_{\lambda_0X<p^2\leq X}p^{1-\gamma}([-p^\gamma]-[-(p+1)^\gamma])e(t p^2)\log p
\\
&=\sum\limits_{\lambda_0X<p^2\leq X}p^{1-\gamma}((p+1)^\gamma-p^\gamma)e(t p^2)\log p
\\
&\qquad+\sum\limits_{\lambda_0X<p^2\leq X}p^{1-\gamma}(\psi(-(p+1)^\gamma)-\psi(-p^\gamma))e(t p^2)\log p
\\
&=\gamma\Sigma(t)+\Omega(t)+\Cal{O}(1).
\endaligned
\tag17
$$
Bearing in mind \Tag(17) and \Par*{Lemma 5}, we establish the statement in the lemma.
\qed\enddemo

Put
$$
J(X)=\gamma^3\int\limits_{|t|<\Delta}\Theta(t)I_1(\lambda_1t,X)I_1(\lambda_2t,X)I_2(\lambda_3t,X)e(\eta t)\,dt.
\tag18
$$
Now \Tag(4), \Tag(7), \Tag(8), \Tag(9), \Tag(11), \Tag(14), \Tag(18),
Cauchy's inequality, \Par{Lemma 1}{Lemmas 1}, \Par{Lemma 3}{3}, \Par{Lemma 4}{4}, and \Par{Lemma 9}{9} imply
$$
\align
\Gamma_1(X)-J(X)
&=\gamma^2\int\limits_{|t|<\Delta}\Theta(t)(S_1(\lambda_1t)-\gamma I_1(\lambda_1t))I_1(\lambda_2t)I_2(\lambda_3t)e(\eta t)\,dt
\\
&\qquad
+\gamma\int\limits_{|t|<\Delta}\Theta(t)S_1(\lambda_1t)(S_1(\lambda_2t)-\gamma I_1(\lambda_2t))I_2(\lambda_3t)e(\eta t)\,dt
\\
&\qquad
+\int\limits_{|t|<\Delta}\Theta(t)S_1(\lambda_1t)S_1(\lambda_2t)(S_2(\lambda_3t)-\gamma I_2(\lambda_3t))e(\eta t)\,dt
\\
&\ll\varepsilon \frac{X}{e^{(\log X)^{1/5}}}\Biggl[\Biggl(\ \int\limits_{|t|<\Delta}|I_1(\lambda_2t)|^2\,dt\Biggr)^\frac{1}{2}
\Biggl(\ \int\limits_{|t|<\Delta}|I_2(\lambda_3t)|^2\,dt\Biggr)^\frac{1}{2}
\\
&\qquad+\Biggl(\ \int\limits_{|t|<\Delta}|S_1(\lambda_1t)|^2\,dt\Biggr)^\frac{1}{2}
\Biggl(\ \int\limits_{|t|<\Delta}|I_2(\lambda_3t)|^2\,dt\Biggr)^\frac{1}{2}\Biggr]
\\
&\qquad
+\varepsilon \int\limits_{|t|<\Delta}|S_1(\lambda_1t)|\,|S_1(\lambda_2t)|\bigl|\gamma\Sigma(\lambda_3t)-\gamma I_2(\lambda_3t)+\Cal{O}(X^{\frac{21-7\gamma}{29}+\delta})\bigr|\,dt
\endalign
$$
$$
\aligned
&\ll\varepsilon\frac{X^\frac{3}{2}}{e^{(\log X)^{1/6}}}+\varepsilon \int\limits_{|t|<\Delta}|S_1(\lambda_1t)|\,|S_1(\lambda_2t)|\,|\Sigma(\lambda_3t)-U(\lambda_3t)|\,dt
\\
&\qquad+\varepsilon \int\limits_{|t|<\Delta}|S_1(\lambda_1t)|\,|S_1(\lambda_2t)|\,|U(\lambda_3t)-I_2(\lambda_3t)|\,dt
\\
&=\varepsilon\Bigl(\frac{X^\frac{3}{2}}{e^{(\log X)^{1/6}}}+J_1+J_2\Bigr),
\endaligned
\tag19  %\?к чему 19 относится?
$$
say. Using  Cauchy's inequality and \Par{Lemma 2}{Lemmas 2}, \Par{Lemma 3}{3}, and \Par{Lemma 6}{6}, we get
$$
J_1\ll X\Biggl(\ \int\limits_{-\Delta}^\Delta|S_1(\lambda_1t)|^2\,dt\Biggr)^\frac{1}{2}
\Biggl(\ \int\limits_{-\Delta}^\Delta|\Sigma(\lambda_3t)-U(\lambda_3t)|^2\,dt\Biggr)^\frac{1}{2}\ll\frac{X^\frac{3}{2}}{e^{(\log X)^{1/6}}}.
\tag20
$$
By Euler's summation formula, we have
$$
I_2(t)-U(t)\ll1+|t|X.
\tag21
$$
From \Tag(21), Cauchy's inequality, and \Par*{Lemma 3}, we derive
$$
J_2\ll (1+\Delta X)\Biggl(\ \int\limits_{-\Delta}^\Delta|S_1(\lambda_1t)|^2\,dt\Biggr)^\frac{1}{2}\Biggl(\ \int\limits_{-\Delta}^\Delta
|S_1(\lambda_2t)|^2\,dt\Biggr)^\frac{1}{2}\ll\Delta X^2\log^3X.
\tag22
$$
On the other hand for the integral defined by \Tag(18), we write
$$
J(X)=B(X)+\Phi,
\tag23
$$
where
$$
B(X)=\gamma^3\int\limits_{-\infty}^{\infty}\Theta(t)I_1(\lambda_1t)I_1(\lambda_2t)I_2(\lambda_3t)e(\eta t)\,dt.
$$
and
$$
\Phi\ll\int\limits_{\Delta}^{\infty }|\Theta(t)||I_1(\lambda_1t)I_1(\lambda_2t)I_2(\lambda_3t)|\,dt.
\tag24
$$
Arguing as in [22, Lemma 4], we deduce that if
$$
\lambda_0<\min\Bigl(\frac{\lambda_1}{4|\lambda_3|}\,,\frac{\lambda_2}{4|\lambda_3|}\,,\frac{1}{16}\Bigr)
$$
then
$$
B(X)\gg\varepsilon X^\frac{3}{2}.
\tag25
$$
By \Tag(11) and [23, Lemma 4.2], we get
$$
I_k(t)\ll X^{\frac{1}{k}-1}\min(X,\, |t|^{-1}).
\tag26
$$
Using \Tag(24), \Tag(26), and \Par*{Lemma 1}, we obtain
$$
\Phi\ll\frac{\varepsilon X^\frac{1}{2}}{\Delta}.
\tag27
$$
Bearing in mind \Tag(4), \Tag(19), \Tag(20), \Tag(22), \Tag(23), \Tag(25), and  \Tag(27), we establish
$$
\Gamma_1(X)\gg\varepsilon X^\frac{3}{2}.
\tag28
$$

\head
6. Upper Bound of $\Gamma_2(X)$
\endhead

Put
$$
\goth{S} %\?было \mathfrak{S}
(t,X)=\min\bigl\{|S_1(\lambda_{1}t)|,|S_1(\lambda_2 t)|\bigr\}.
\tag29
$$
Taking into account \Tag(15), \Tag(29), \Par*{Lemma 1}, and \Par*{Lemma 7},
%\?\Par{Lemma 1}{Lemmas 1}, and \Par{Lemma 7}{7},
we deduce
$$
\aligned
\Gamma_2(X_j)&\ll\varepsilon\int\limits_{\Delta\leq|t|\leq H}\goth{S}(t, X_j)^\frac{1}{2}|S_1(\lambda_1 t)|^\frac{1}{2}|S_1(\lambda_2 t)|\,|S_2(\lambda_3 t)|\,dt
\\
&\qquad
+\varepsilon\int\limits_{\Delta\leq|t|\leq H}\goth{S}(t, X_j)^\frac{1}{2}|S_1(\lambda_1 t)|\,|S_1(\lambda_2 t)|^\frac{1}{2}|S_2(\lambda_3 t)|\,dt
\\
&\ll\varepsilon X_j^{\frac{37-12\gamma}{52}+\delta}(\Psi_1+\Psi_2),
\endaligned
\tag30
$$
where
\iftex
$$
\align
&\Psi_1=\int\limits_{\Delta}^H|S_1(\lambda_1 t)|^\frac{1}{2}|S_1(\lambda_2 t)|\,|S_2(\lambda_3 t)|\,dt,
\tag31
\\
&\Psi_2=\int\limits_{\Delta}^H|S_1(\lambda_1 t)|\,|S_1(\lambda_2 t)|^\frac{1}{2}|S_2(\lambda_3 t)|\,dt.
\tag32
\endalign
$$
\else
$$
\Psi_1=\int\limits_{\Delta}^H|S_1(\lambda_1 t)|^\frac{1}{2}|S_1(\lambda_2 t)|\,|S_2(\lambda_3 t)|\,dt,
\tag31
$$
$$
\Psi_2=\int\limits_{\Delta}^H|S_1(\lambda_1 t)|\,|S_1(\lambda_2 t)|^\frac{1}{2}|S_2(\lambda_3 t)|\,dt.
\tag32
$$
\fi
We estimate only $\Psi_1$ and the estimation of $\Psi_2$ proceeds in the same way. From \Tag(31) and Cauchy's inequality, we derive
$$
\Psi_1\ll\Biggl(\ \int\limits_{\Delta}^H|S_1(\lambda_2t)|^2\,dt\Biggr)^\frac{1}{2}
\Biggl(\ \int\limits_{\Delta}^H|S_1(\lambda_1t)|^2\,dt\Biggr)^\frac{1}{4}
\Biggl(\ \int\limits_{\Delta}^H|S_2(\lambda_3 t)|^4\,dt\Biggr)^\frac{1}{4}.
\tag33
$$
Using \Par*{Lemma 3}\itm(i), we obtain
$$
\int\limits_{\Delta}^H|S_1(\lambda_kt)|^2\,dt\ll HX_j^{2-\gamma}\log X_j, \quad k=1, 2.
\tag34
$$
By \Par*{Lemma 8}, we find
$$
\int\limits_{\Delta}^H|S_2(\lambda_3t)|^4\,dt\ll HX_j^{2-\gamma+\delta}.
\tag35
$$
Now \Tag(33)--\Tag(35) yield
$$
\Psi_1\ll H X_j^{2-\gamma+\delta}.
\tag36
$$
Combining \Tag(5), \Tag(6), \Tag(30), and \Tag(36), we get
$$
\Gamma_2(X_j)\ll X_j^{\frac{37-12\gamma}{52}+\delta}X_j^{2-\gamma+\delta}=X_j^{\frac{141-64\gamma}{52}+\delta}\ll\frac{\varepsilon X_j^\frac{3}{2}}{\log X_j}.
\tag37
$$

\head
7. Upper Bound of $\Gamma_3(X)$
\endhead

By \Tag(7), \Tag(16), and \Par{Lemma 1}{Lemmas 1}  and \Par{Lemma 2}{2}, it follows
$$
\Gamma_3(X)\ll X^3\int\limits_{H}^{\infty}\frac{1}{t}\Bigl(\frac{k}{2\pi t\varepsilon/8}\Bigr)^k \,dt=\frac{X^3}{k}\Bigl(\frac{4k}{\pi\varepsilon H}\Bigr)^k.
\tag38
$$
Choosing $k=[\log X]$ from \Tag(6) and \Tag(38), we deduce
$$
\Gamma_3(X)\ll1.
\tag39
$$

\head
8. Proof of the Theorem
\endhead

Summarizing  \Tag(5), \Tag(13), \Tag(28), \Tag(37), and \Tag(39), we derive
$$
\Gamma(X_j)\gg\varepsilon X_j^\frac{3}{2}=X_j^{\frac{141-64\gamma}{52}+\theta}\,.
$$
The last estimation implies
$$
\Gamma(X_j) \rightarrow\infty \quad\text{as }  X_j\rightarrow\infty.
\tag40
$$
Bearing in mind  \Tag(12) and \Tag(40) we establish \Par*{Theorem 1}.

\Refs

\ref\no 1
\by	          Baker~A.
\paper        On some Diophantine inequalities involving primes
\jour         J. Reine Angew. Math.
\yr           1967
\vol          228
%\issue       -
\pages        166--181
\endref

\ref\no 2
\by                  Ramachandra K.
\paper               On the sums $\sum\lambda_jf_j(p_j)$
\jour                J.~Reine Angew. Math.
\yr                  1973
\iftex
\vol                 262, 263
\else
\vol                 262
\fi
%\issue              -
\pages               158--165
\endref

\ref\no 3
\by                  Vaughan R.C.
\paper               Diophantine approximation by prime numbers.~I
\jour                Proc. London Math. Soc. (3)
\yr                  1974
\vol                 28
%\issue              -
\pages               373--384
\endref

\ref\no 4
\by                  Lau K.W. and Liu M.C.
\paper               Linear approximation by primes
\jour                Bull. Austral. Math. Soc.
\yr                  1978
\vol                 19
\issue               3
\pages               457--466
\endref

\ref\no 5
\by                  Baker R.C. and Harman~G.
\paper               Diophantine approximation by prime numbers
\jour                J.~London Math. Soc. (2)
\yr                  1982
\vol                 25
\issue               2
\pages               201--215
\endref

\ref\no 6
\by                  Harman~G.
\paper               Diophantine approximation by prime numbers
\jour                J.~London Math. Soc. (2)
\yr                  1991
\vol                 44
\issue               2
\pages               218--226
\endref

\ref\no 7
\by                  Matom\"{a}ki K.
\paper               Diophantine approximation by primes
\jour                Glasg. Math.~J.
\yr                  2010
\vol                 52
\issue               1
\pages               87--106
\endref

\ref\no 8
\by                  Gambini  A.,  Languasco A., and Zaccagnini~A.
\paper               A~Diophantine approximation problem with two primes and one $k$th power of a prime
\jour                J.~Number Theory
\yr                  2018
\vol                 188
%\issue              -
\pages               210--228
\endref

\ref\no 9
\by                  Li W. and Wang T.
\paper               Diophantine approximation with two primes and one square of prime
\jour                Chinese Quart. J. Math.
\yr                  2012
\vol                 27
\issue               3
\pages               417--423
\endref

\ref\no 10
\by                  Mu Q.W. and Qu Y.Y.
\paper               A Diophantine inequality with prime variables and mixed power
\jour                Acta Math. Sinica (Chin. Ser.)
\yr                  2015
\vol                 58
\issue               3
\pages               491--500
\endref

\ref\no 11
\by                  Todorova T.L. and  Georgieva A.
\paper               A Diophantine inequality involving mixed powers of primes with a~specific type
\jour                Mathematics
\yr                  2025
\vol                 13
\num                 3065
\size                21
\endref

\ref\no 12
\by                  Piatetski-Shapiro I.I.
\paper               On the distribution of prime numbers in sequences of the form $[f(n)]$
\jour                Mat. Sb.
\yr                  1953
\vol                 33 %/75
\issue               3
\pages               559--566
\endref

\ref\no 13
\by                  Rivat J. and Wu J.
\paper               Prime numbers of the form $[n^c]$
\jour                Glasg. Math. J.
\yr                  2001
\vol                 43
\issue               2
\pages               237--254
\endref

\ref\no 14
\by                  Maier H. and Rassias M.Th.
\paper               The ternary Goldbach problem with a~missing digit and other primes of special types
\inbook              Analysis at Large: Dedicated to the Life and Work of Jean Bourgain
\publaddr            Springer
\publ                Cham
\yr                  2022
\pages               333--362
\endref

\ref\no 15
\by                  Maier H. and  Rassias M.Th.
\paper               The ternary Goldbach problem with two Piatetski-Shapiro primes and a prime with a missing digit
\jour                Commun. Contemp. Math.
\yr                  2023
\vol                 25
\issue               2
\num                 2150101
\size                45
\endref

\ref\no 16
\by                  Maier H. and  Rassias M.Th.
\paper               Prime avoidance property of $k$th powers of Piatetski-Shapiro primes
\jour                J.~Th\'eor. Nombres Bordeaux
\yr                  2025
\vol                 37
\issue               2
\pages               715--725
\endref

\ref\no 17
\by                  Dimitrov~S.I.
\paper               Diophantine approximation by Piatetski-Shapiro primes
\jour                Indian J. Pure Appl. Math.
\yr                  2022
\vol                 53
\issue               4
\pages               875--883
\endref

\ref\no 18
\by                  Piatetski-Shapiro I.I.
\paper               On a variant of the Waring--Goldbach problem
\jour                Mat. Sb.
\yr                  1952
\vol                 30 %/72
%\issue               3
\pages               105--120
\endref

\ref\no 19
\by                  Rivat~J. and Sargos~P.
\paper               Nombres premiers de la forme $\lfloor n^c\rfloor$ %Primes of the form $\lfloor n^c\rfloor$
                     %Nombres premiers de la forme $[n^c]$
\jour                Canad. J. Math.
\yr                  2001
\vol                 53
\issue               2
\pages               414--433
\endref

\ref\no 20
\by                  Dimitrov~S.I. and  Lazarova M.D.
\paper               On the distribution of $\alpha p^2$ modulo one over primes of the form $[n^c]$
\jour                Ramanujan~J.
\yr                  2025
\vol                 68
\issue               3
\num                 79
\size                13
\endref

\ref\no 21
\by                  Zhai W.G.
\paper               The Waring--Goldbach problem in thin sets of primes
\jour                Acta Math. Sinica (Chin. Ser.)
\yr                  1998
\vol                 41
\issue               3
\pages               595--608
\endref

\ref\no 22
\by                  Dimitrov~S.I. and Todorova T.L.
\paper               Diophantine approximation by prime numbers of a~special form
\jour                Ann. Univ. Sofia, Fac. Math. Inform. %God. Sofii. Univ. ``Sv. Kliment Okhridski." Fac. Mat. Inform.
\yr                  2015
\vol                 102
%\issue               -
\pages               71--90
\endref

\ref\no 23
 \by                   Titchmarsh E.C.
 \book                 The Theory of the Riemann Zeta-Function
 \bookinfo             Second edition
 \publ                 Clarendon and  Oxford University
 \publaddr             New York
 \yr                   1986
\endref
\endRefs

\enddocument