\documentstyle{SibMatJ}
% \TestXML
\topmatter

  \Author           Song
    \Initial        Y.
    \Gender         he
    \Sign           Yanbo Song
    \Email          syb888202106\@163.com
    \AffilRef       1
  \endAuthor

  \Affil 1
    \Division       School of Information Engineering
    \Organization   Xi'an University
    \City           Xi'an
    \Country        China
  \endAffil

  \datesubmitted    September 25, 2025\enddatesubmitted
  \daterevised      February 16, 2026\enddaterevised
  \dateaccepted     February 17, 2026\enddateaccepted

  \UDclass
    511.32 %èç ïðåæíåé ñòàòüè \?11N05, 11N35, 11L07
  \endUDclass

  \title
    On Numbers of the Form $\theta\lfloor n^c\rfloor+\varphi$ with Smooth Indices
  \endtitle

  \abstract
    In this paper, we study two problems about numbers of the form $\theta\lfloor n^c\rfloor+\varphi$ with smooth indices.
    First, let $1<c<3/2$ be a fixed real number. Let $\theta$ be an irrational number and let $\varphi$ be a real number. 
    Let $T > 5$ be a real number. Then there are infinitely many $y$-smooth numbers $n$ such that
    $$
    \|\theta \lfloor n^c\rfloor + \varphi\| < n^{-\frac{(3-2c)(1-1/T)}{27-1/T}+100\varepsilon},
    $$
 where $y=(\log n)^{T}$.
 Second, let $1<c<2$ be a fixed real number. Let $\theta$ be an irrational number and let $\varphi$ be a~real number. Then for all sufficiently large real $x$ and arbitrary real $R\geq\max\{\frac{8+\tau}{c-\varepsilon},\frac{9}{2-c-\varepsilon}\}+0.124820$ we have
$$
\bigl|\{n\in S(x,y)|\lfloor\theta\lfloor n^c\rfloor+\varphi\rfloor  \text{ is an $R$-almost prime}\}\bigr|\gg \frac{x}{\log^2x},
$$
where $x^{\varepsilon}<y<x$ and $S(x,y)$ is the set of $y$-smooth numbers smaller than $x$.
  \endabstract

  \keywords
    exponential sums,
    smooth numbers,
    Diophantine approximation,
    almost primes
  \endkeywords

\endtopmatter

\head
1. Introduction
\endhead

Recall that a positive integer $n$ is called $y$-smooth, %\?\it íåò
if its largest prime factor~$P^+(n)$ is at most~$y$. Smooth numbers are widely recognized as playing a pivotal role in analytic number theory, and the study of their distribution constitutes a core focus in relevant research. For illustrative examples pertaining to both smooth numbers and their distribution, interested readers are directed to
references [1--4]. %\?the
For $2 \leq y \leq x$, recall the notation 
$$ 
S(x, y) := \{ n\leq x : P^+(n)\leq y\}, \quad \Psi(x, y) := |S(x, y)|. 
$$
The size of the parameter~$y$ with respect to~$x$ is vital in the study of smooth numbers. 
The smaller the value of $y$, the sparser the set $S(x,y)$ becomes; 
consequently, the corresponding problem grows increasingly difficult to tackle. 
If we write $u=\frac{\log x}{\log y}$
then for fixed %\?a
$u$ we have $\Psi(x,y) \sim \rho(u) x$ where $\rho(u)$ denotes
the Dickman function defined by $\rho(u) =1$ for $0\le u\le 1$ and
for $u\ge 1$ is defined as the unique continuous solution to
the differential-difference equation $u{\rho}^{\prime}(u)=-\rho(u-1)$.
This asymptotic formula was first given by Dickman [5] for
fixed values of $u$ and as $x\to \infty$. When $y=(\log x)^\kappa$ for some fixed~$\kappa>1$, by (1.14) in the paper [3] we have
$$
x^{1-1/\kappa-\varepsilon}\leq\Psi(x, (\log x)^\kappa)\leq x^{1-1/\kappa+\varepsilon} \quad (\text{as } x\to\infty).
\tag1
$$
Because of this sparsity, many results about smooth numbers from this %\? 2 this
example above were until recently only known conditional on assumptions such as the GRH. %\?1 ðàç

On the other hand, the following Diophantine inequality
$$
\| \theta n + \varphi \|<n^{-\alpha}
\tag2
$$
have been studied by many scholars.
Vinogradov [6] first considered this Diophantine approximation in prime variables with $\alpha=1/5-\varepsilon$.
Later Harman [7] and Matomaki [8] improved the exponent to $\alpha=7/22$ and $\alpha=1/3-\varepsilon$ respectively.
Instead of considering primes, Baker [9] and Zhu [10] considered \Tag(2) with smooth numbers.
Dimitrov [11] also considered the case for Piatetski-Shapiro primes. The following Diophantine inequality
$$
\| \alpha p^2 + \beta \|<p^{-\theta}
\tag3
$$
is also studied by many scholars.
The first result in this direction is due to Ghosh [12], who obtained $\theta=1/8-\varepsilon$, while the best result to date is due to Harman [7], with $\theta=2/13-\varepsilon$. Dimitrov and Lazarova [13] also considered \Tag(3) with Piatetski-Shapiro primes.
Inspired by these, we consider the problem involving Piatetski-Shapiro sequences with smooth indices, i.e. the sequence of the form $\{\lfloor n^c\rfloor\}_{n=1}^{\infty}$ with $n$ a smooth number.

\proclaim{Theorem 1}
Let $1<c<3/2$ be a fixed real number, let $\theta$ be an irrational number, let $\varphi$ be a real number,  %\?
and let $T > 5$ be a real number. %\?è ýòî real number
There are infinitely many $n\in \Bbb{N}$ free of prime factors greater than $(\log n)^T$ such that
$$
\|\theta \lfloor n^c\rfloor + \varphi\| < n^{-\frac{(3-2c)(1-1/T)}{27-1/T}+100\varepsilon}.
\tag4
$$
\endproclaim

The Piatetski-Shapiro prime is a fundamental object in analytic number theory, named after the Soviet mathematician Israel Moiseevich Piatetski-Shapiro. It refers to primes of the form $\lfloor n^c\rfloor$, where $c>1$ is a real. In 1953 Piatetski-Shapiro [14] showed that there are infinitely many Piatetski-Shapiro primes for $c\in(1,12/11)$. It is worth emphasizing that the range $c\in(1,12/11)$ was improved by Kolesnik [15], Heath-Brown [16], Kolesnik [17] and Liu and Rivat [18]. The best known result is due to  Rivat and Sargos [19], which improved the range for $c$ to $(2426/2817,1)$. Inspired by the above research, we study the almost prime of the form $\lfloor\theta\lfloor n^c\rfloor+\varphi\rfloor$ with smooth indices. Whether there exist primes of the form $\lfloor\theta\lfloor n^c\rfloor+\varphi\rfloor$ with smooth indices is still an open problem.

Before stating \Par*{Theorem 2}, we give the definitions of the type of an irrational number and the $R$-almost primes.

\specialhead
Type of an irrational number
\endspecialhead

For an irrational number $\theta$, its type $\tau$ is defined as follows:
$$
\tau=\sup\bigl\{\rho\in\Bbb{R}: \liminf\Sb n\rightarrow\infty \\ n\in\Bbb{Z}^+\endSb n^{\rho}\|n\alpha\|=0\bigr\},
\tag5
$$
where $\|u\|$ denotes the distance of $u$ from the nearest integer. By Dirichlet's approximation theorem, we have $\tau\geq1$.
If $\theta$ is of finite type $\tau$, then $\theta^{-1}$ and $k\theta$ are also of finite type $\tau$.

\specialhead
$R$-Almost primes
\endspecialhead

For every integer $R\geq1$, we say that a natural number is an $R$-almost prime if it has at most $R$ prime factors, counted with multiplicity.

\proclaim{Theorem 2}
Let $1<c<2$ be a fixed real number. Let $\theta$ be an irrational number of finite type $\tau$ and let $\varphi$ be a real number.
Then for all sufficiently large real $x$ and arbitrary real $R\geq\max\{\frac{8+\tau}{c-\varepsilon},\frac{9}{2-c-\varepsilon}\}+0.124820$ we have
$$
\bigl|\{n\in S(x,y)|\lfloor\theta\lfloor n^c\rfloor+\varphi\rfloor  \text{ is an $R$-almost prime}\}\bigr|\gg \frac{x}{\log^2x},
\tag6
$$
where $x^{\varepsilon}<y<x$.
\endproclaim

\head
2. Some Notations
\endhead

Throughout this paper, we fix a real number $c$ with $1<c<3/2$. Let $S(x,y)$ denote the set of $y$-smooth numbers which is less than $x$. And let $\Psi(x,y)$ denote the cardinal of the set $S(x,y)$. Let $S_a(x,y)$ denote the set of integers in $[x,ax)$ free of prime factors $> y$.
We define $P^{+}(n)$ to be the largest prime factor of $n$.
We will frequently use $\varepsilon$ with an abuse of notation %\?
to mean a small positive number, possibly a different one each time.
Given a real number $x$, we write $e(x)=e^{2\pi ix}$. We use $\lfloor t\rfloor,\{t\}$ and $\|t\|$ to denote the integral part of $t$, the fractional part of $t$ and the distance from $t$ to the nearest integer, respectively. The notation $n\sim N$ means $N<n\leqslant 2N$ and  $f(x)\ll g(x)$ means  $f(x)=O(g(x))$.

\head
3. Some Lemmas
\endhead

Before proving two theorems, we will list some lemmas which are necessary for establishing two theorems.

\proclaim{Lemma 1}
Let $x_1, \dots, x_N$ be a real sequence with $\|x_n\| \ge M^{-1}$ $(1 \le n \le N)$ where $M$ is a positive integer. Then
$$
\sum_{h=1}^M \Biggl| \sum_{n=1}^N e(hx_n)\Biggr|
 \ge N/6.
$$
\endproclaim

\demo{Proof}
For the proof of this lemma, see [20].
\qed\enddemo

To treat the floor functions in the exponential sums, we need the following so-called `Partition of Unity.' %\?

\proclaim{Lemma 2}
For $Z\in\Bbb{N}$ and $x\in\Bbb{R}$ there exist periodic functions $g_{z}(t)$ with period $1$ $(0\leq z\leq 2Z-1)$, such that
$$
\align
&0<g_{z}(t)\leq1, \quad \text{ if }\ \Bigl|t-\frac{z}{2Z}\Bigr|<\frac{1}{2Z},\\
&g_{z}(t)=0, \quad \text{ if }\ \frac{1}{2Z}\leq|t-\frac{z}{2Z}|\leq\frac{1}{2},\\
&g_{z}(t)=\sum\limits_{|n|\leq Z(\log x)^{4}}\beta_{n}^{(z)}e(nt)+O(x^{-\log\log x}),
\endalign
$$
where
$$
|\beta_{n}^{(z)}|\leq\frac{1}{2Z}.
$$
We also have
$$
\sum\limits_{z=0}^{2Z-1}g_{z}(t)=1.
$$
\endproclaim

\demo{Proof}
This is Lemma 2 in [21].
\qed\enddemo

\proclaim{Lemma 3}
Let $a(l)$ be a sequence of complex numbers, and let $Q$ and $L$ be  positive reals. Then
$$
\biggl|\sum\limits_{L<l\leq2L}a(l)\biggr|^2\leq\Bigl(1+\frac{L}{Q}\Bigr)\sum\limits_{0\leq q\leq Q}\Bigl(1-\frac{q}{Q}\Bigr)Re\biggl(\sum\limits_{L\leq l\leq 2L-q}a(l+q)\overline{a(l)}\biggr).
$$
\endproclaim

\demo{Proof}
This is Lemma 5 in [16].
\qed\enddemo

\proclaim{Lemma 4}
Let $M$ be a positive real. Suppose that $M<M'<2M$. Let $f$ be a function with $f^{\prime\prime}(x)$ is %\?
continuous in $[M,M']$. If $\lambda\ll|f^{\prime\prime}(x)|\ll\lambda$ for $x\in[M,M']$. Then
$$
\sum\limits_{M<n<M'}e(f(n))\ll M\lambda^{1/2}+\lambda^{-1/2}.
$$
\endproclaim

\demo{Proof}
For the proof of this lemma, see [22].
\qed\enddemo

We need the following approximation due to of %\?
Vaaler.

\proclaim{Lemma 5}
Let $x$ be a real, and let $\psi(x)=x-\lfloor x\rfloor-1/2$. Then for any $H\geq1$ there exist reals $a_{h}$ and $b_{h}$ such that
$$
\biggl|\psi(x)-\sum\limits_{0<|h|\leq H}a_{h}e(xh)\biggr|\leq\sum\limits_{|h|\leq H}b_{h}e(xh),
$$
with
$$
a_{h}\ll\frac{1}{|h|},\quad b_{h}\ll\frac{1}{H}.
$$
\endproclaim

\demo{Proof}
For the proof of this lemma, see [23].
\qed\enddemo

In order to state next lemma, we introduce some notation. We say that an $N$-element set of integers $A$ has a level of 
distribution $D$ %\?âñþäó
if for a given multiplicative function $f(d)$ we have
$$
\sum_{d\leq D}\max_{\gcd(s,d)=1}\Bigl|\bigl|\{a\in A, a\equiv s\bmod d\}\bigr|-\frac{f(d)}{d}N\Bigr|\leq\frac{N}{\log^2N}.
$$
We define
$$
\delta_R=0.124820,\quad  R\geq5.
$$
We have the following lemma.

\proclaim{Lemma 6}
Suppose that $A$ is an $N$-element set of positive integers with a level of distribution $D$ and degree $\beta$ in the sense that $a<D^{\beta}$ for any $a\in A$ hold %\?
with some real number $\beta<R-\delta_R$. Then
$$
\bigl|\{a\in A: \text{ $a$ is an $R$-almost prime}\}\bigr|\gg\frac{N}{\log^2N}.
$$
\endproclaim

\demo{Proof}
For the proof of this lemma, see [24, Chapter 5, Proposition 1].
\qed\enddemo

We also need the following lemma to detect the condition $d\,| %\?
\lfloor\theta\lfloor n^{c}\rfloor+\phi\rfloor$, which we call the division trick.

\proclaim{Lemma 7}
Let $d$ and $n$ be two nature numbers, let $\theta>0$, $c>0$, and $\phi$ be reals. Then $d|\lfloor\theta\lfloor n^{c}\rfloor+\phi\rfloor$ if and only if $E_{\theta,c,\phi,d}(n):=\lfloor\frac{\theta\lfloor n^{c}\rfloor+\phi}{d}\rfloor
-\lfloor\frac{\theta\lfloor n^{c}\rfloor+\phi-1}{d}\rfloor=1$.
\endproclaim

\demo{Proof}
The proof of this lemma is easy, we left it to readers.
\qed\enddemo

\proclaim{Lemma 8}
Suppose that $2\leq y\leq R\leq n\leq x$, with $n\in S(x,y)$. Then there is a unique triple $(p, u, v)$ satisfying

\Item (i)   $n=puv$,
\Item (ii)  $p\leq y$,
\Item (iii) $R/p<v\leq R$ with $P^{-}(v)\geq p$ and $P^{+}(v)\leq y$,
\Item (iv)  $u\leq x/pv$ with $P^{+}(u)\leq p$.
\endproclaim

\demo{Proof}
This is Lemma 10.1 in [25].
\qed\enddemo

\proclaim{Lemma 9}
Let $I$ be an interval. If $f$ is continuously differentiable on $I$, $f^{\prime}$ is monotonic on $I$, and $\|f^{\prime}\|\geq\lambda>0$ on $I$ then
$$
\sum_{n\in I}e(f(n))\ll \lambda^{-1}.
$$
\endproclaim

\demo{Proof}
This is Theorem 2.1 in [22].
\qed\enddemo

\proclaim{Lemma 10}
If $Z$ is large enough positive real %\?
then
$$
\int\limits_{-1/2}^{1/2}\biggl|\sum_{r\in I}e(\alpha r)\biggr|d\alpha\ll\log Z,
$$
where $I$ is an interval with $I\subset[Z,2Z]$.
\endproclaim

\demo{Proof}
We have
$$
\int\limits_{-1/2}^{1/2}\biggl|\sum_{r\in I}e(\alpha r)\biggr|d\alpha
=\int\limits_{-1/Z}^{1/Z}\biggl|\sum_{r\in I}e(\alpha r)\biggr|d\alpha+\int\limits_{1/Z}^{1/2}
\biggl|\sum_{r\in I}e(\alpha r)\biggr|d\alpha+\int\limits_{-1/2}^{-1/Z}\biggl|\sum_{r\in I}e(\alpha r)\biggr|d\alpha
=:I_1+I_2+I_3.
$$
By trivial estimation we have
$$
I_1\leq\int\limits_{-1/Z}^{1/Z}Zd\alpha\ll1.
\tag7
$$
By \Par*{Lemma 9} we have
$$
I_2\leq\int\limits_{1/Z}^{1/2}\alpha^{-1}d\alpha\ll\log Z.
\tag8
$$
Similarly
$$
I_3\ll\log Z.
\tag9
$$
Combining \Tag(7)--\Tag(9) we have
$$
\int\limits_{-1/2}^{1/2}\biggl|\sum_{r\in I}e(\alpha r)\biggr|d\alpha\ll\log Z.
$$
This completes the proof of this lemma. %\?2 this
\qed\enddemo

\head
4. Proof of \Par*{Theorem 1}
\endhead

By Dirichlet's approximation theorem, for any irrational number $\theta$, there exists an integer $q$ such that
 $$
\Bigl|\theta - \frac aq\Bigr| < \frac 1{q^2}, \quad (a,q)= 1.
\tag10
 $$
We may suppose that $\varepsilon$ is sufficiently small, so that
 $$
 \gamma  : = \frac{(3-2c)(1-1/T)}{27-1/T}-100\varepsilon
 $$
is positive.

Define $x$ by
 $$
x^{\frac{c+\gamma }2} = q.
\tag11
 $$
Suppose that $n \in \Bbb{N}$ satisfies
 $$
\Bigl\|\frac{a[n^c]}q + \varphi\Bigr\| \le x^{-\gamma }, \quad n \in S(x,y), \quad  y = (\log x)^T.
\tag12
 $$
This implies
 $$
\|\theta [n^c] + \varphi\|
\le x^{-\gamma } + \Bigl|\theta - \frac aq\Bigr| \cdot x^c
\le x^{-\gamma } + \frac{x}{x^{1+\gamma }} < n^{-\frac{(3-2c)(1-1/T)}{27-1/T}+100\varepsilon}
$$
from \Tag(10)--\Tag(12). Hence it will suffice to show that \Tag(12) has a solution $n$.

Let $S = S_2(x^{\frac{1+\gamma }2}, y)$ and $J = S_2(x^{\frac{1-\gamma }2},y)$. By (1.1), %\?íåò òàêîãî, ìîæåò, \Tag(11)
\iftex
 $$
 \align
x^{\frac{1+\gamma }2\, (1-\frac 1T) - \frac\varepsilon{16}} &\ll |S| \ll x^{\frac{1+\gamma }2 (1 - \frac 1T)+\frac \varepsilon{16}},
\tag13
\\
x^{\frac{1-\gamma }2\, (1-\frac 1T) - \frac\varepsilon{16}} &\ll |J| \ll x^{\frac{1-\gamma }2 (1 - \frac 1T)+\frac \varepsilon{16}}.
\tag14
\endalign
 $$
 \else
$$
x^{\frac{1+\gamma }2\, (1-\frac 1T) - \frac\varepsilon{16}} \ll |S| \ll x^{\frac{1+\gamma }2 (1 - \frac 1T)+\frac \varepsilon{16}},
\tag13
$$
$$
x^{\frac{1-\gamma }2\, (1-\frac 1T) - \frac\varepsilon{16}} \ll |J| \ll x^{\frac{1-\gamma }2 (1 - \frac 1T)+\frac \varepsilon{16}}.
\tag14
$$
 \fi
We will show that there are solutions of \Tag(12) with
 $$
 n = uv, \quad u \in S, \quad v \in J.
 $$

Suppose there are no such solutions of \Tag(12). Let
 $$
 S_h = \sum_{u\in S} \ \sum_{v \in J}
 e\Bigl(\frac{ha[u^cv^c]}q\Bigr).
 $$
Let $H = \lfloor x^\gamma \rfloor + 1$. We deduce from \Par*{Lemma 1} and \Tag(13), \Tag(14) that
 $$
\sum_{h=1}^H |S_h| \gg g x^{1 - \frac 1T - \frac \varepsilon 8}.
\tag15
 $$
Now we estimate the upper bounds of $\sum_{h=1}^H |S_h|$. First we have
$$
\sum_{h=1}^H |S_h| \ll x^{\gamma}\max_{h\leq H}|S_h|.
\tag16
$$
By \Par*{Lemma 2} with $Z=\lfloor x^{\gamma+\varepsilon}\rfloor+1$, we have
$$
\align
S_h
&=\sum_{u\leq x^{\frac{1+\gamma}{2}(1-1/T)}}a(u)\sum_{v\leq x^{\frac{1-\gamma}{2}(1-1/T)}}b(v)e(ha[u^cv^c]/q)\sum_{z=0}^{2Z-1}g_z(u^cv^c)\\
&=\sum_{z=0}^{2Z-1}\sum_{u\leq x^{\frac{1+\gamma}{2}(1-1/T)}}a(u)\sum_{v\leq x^{\frac{1-\gamma}{2}(1-1/T)}}b(v)e(ha[u^cv^c]/q)g_z(u^cv^c)\\
&=\sum_{z=0}^{2Z-1}\sum_{u\leq x^{\frac{1+\gamma}{2}(1-1/T)}}a(u)\sum_{v\leq x^{\frac{1-\gamma}{2}(1-1/T)}}b(v)(e(ha(u^cv^c-z/2Z)/q)\\
&\qquad+O(H/Z))g_z(u^cv^c)\\
&=\sum_{z=0}^{2Z-1}\sum_{u\leq x^{\frac{1+\gamma}{2}(1-1/T)}}a(u)\sum_{v\leq x^{\frac{1-\gamma}{2}(1-1/T)}}b(v)e(ha(u^cv^c-z/2Z)/q)\\
&\qquad\times g_z(u^cv^c)+O(x^{1-1/T-\varepsilon}),
\endalign
$$
where $a(u)$ and $b(v)$ are the indicator function %\?functions
of smooth number.
By \Par*{Lemma 2}, we have
$$
g_z(u^cv^c)=\sum_{|l|\leq Z\log^4 x}\beta_z^{(l)}e(lu^cv^c)+O(x^{-\log\log x}).
$$
This implies
$$
\align
S_h
&=\sum_{z=0}^{2Z-1}\sum_{|l|\leq Z\log^4 x}\beta_z^l\sum_{u\leq x^{\frac{1+\gamma}{2}(1-1/T)}}a(u)\sum_{v\leq x^{\frac{1-\gamma}{2}(1-1/T)}}b(v)e((ha/q+l)u^cv^c)\\
&\qquad +O(x^{1-1/T-\varepsilon})\\
&\ll x^{\gamma+\varepsilon}|\sum_{u\leq x^{\frac{1+\gamma}{2}(1-1/T)}}a(u)\sum_{v\leq x^{\frac{1-\gamma}{2}(1-1/T)}}b(v)e((ha/q+l)u^cv^c)|\\
&\qquad+O(x^{1-1/T-\varepsilon})\\
&=:x^{\gamma+\varepsilon}S^{'}+O(x^{1-1/T-\varepsilon}).
\endalign
$$
By \Par*{Lemma 3} with $Q=\lfloor x^{4\gamma+8\varepsilon}\rfloor+1$ (note that $Q\ll x^{\frac{1+\gamma}{2}(1-1/T)}-\varepsilon$), we have
$$
S^{'2}\ll \frac{x^{2(1-1/T)}}{Q}+\frac{{x^{1-1/T}}}{Q}\sum_{1\leq s\leq Q}\,\sum_{u\sim x{\frac{(1+\gamma)(1-1/T)}{2}}}
%\times %\?
\,\biggl|\sum_{v\sim x^{\frac{(1-\gamma)(1-1/T)}{2}}}e((ha/q+l)su^{c-1}v^c)\biggr|.
$$
Note that $|ha/q+l|\geq1/q$, and apply \Par*{Lemma 4} to the innermost sum, we see the innermost sum is
$$
\ll x^{\frac{1+\gamma}{2}(1-\frac{1}{T})}x^{\frac{\gamma+\varepsilon}{2}}Q^{1/2}u^{\frac{c-1}{2}}v^{\frac{c-2}{2}}+q^{1/2}u^{\frac{1-c}{2}}v^{\frac{2-c}{2}}.
$$
It thus follows that
$$
S^{'}\ll x^{1-1/t-3\gamma-\varepsilon}.
$$
Upon collecting the estimates above, we find that
$$
\sum_{n=1}^HS_h\ll x^{1-1/t-\varepsilon}.
$$
This leads to a contradiction. This completes the proof of the theorem.

\head
5. Proof of \Par*{Theorem 2}
\endhead

Let $A=\{\lfloor\theta\lfloor n^c\rfloor+\phi\rfloor|\text{$n$ is a $y$-smooth number and $n\leq x$}\}$. First let $D$ be the level of the distribution of $A$. By \Par*{Lemma 5} with $H=D\log^3x$ and \Par*{Lemma 7} we have
$$
\align
\sum_{d\leq D}\max_{\gcd(s,d)=1}\Bigl|\bigl|
&\{a\in A,\,a\equiv s\bmod d\}\bigr|-{\frac{\Psi(x,y)}{d}}\Bigr|
\\
&=\sum_{d\leq D}\,\biggl|\sum\Sb n\leq x\\P^+(n)\leq y\endSb\Bigl(\Bigl\lfloor\frac{\theta\lfloor n^{c}\rfloor+\phi}{d}\Bigr\rfloor
-\Bigl\lfloor\frac{\theta\lfloor n^{c}\rfloor+\phi-1}{d}\Bigr\rfloor\Bigr)-\frac{\Psi(x,y)}{d}\biggr|
\\
&=\sum_{d\leq D}\,\biggl|\sum\Sb n\leq x\\P^+(n)\leq y\endSb\frac{1}{d}
+O\biggl(\sum\Sb n\leq x\\P^+(n)\leq y\endSb\psi\Bigl(\frac{\theta \lfloor n^c\rfloor+\phi}{d}\Bigr)\biggr)
\\
&\qquad+
O\biggl(\sum\Sb n\leq x\\P^+(n)\leq y\endSb\psi\Bigl(\frac{\theta \lfloor n^c\rfloor+\phi-1}{d}\Bigr)\biggr)-\frac{\Psi(x,y)}{d}\biggr|\\
&\ll\sum_{d\leq D}\biggl(\biggl|\sum\Sb n\leq x\\P^+(n)\leq y\endSb\psi\Bigl(\frac{\theta \lfloor n^c\rfloor+\phi}{d}\Bigr)\biggr|+\biggl|\sum\Sb n\leq x\\P^+(n)\leq y\endSb\psi\Bigl(\frac{\theta \lfloor n^c\rfloor+\phi-1}{d}\Bigr)\biggr|\biggr)\\
&\ll\sum_{d\leq D}\sum_{1\leq h\leq H}\frac{1}{h}\,\biggl|\sum\Sb n\leq x\\P^+(n)\leq y\endSb e\Bigl(\frac{\theta h\lfloor n^c\rfloor}{d}\Bigr)\biggr|+O\Bigl(\frac{\Psi(x,y)}{\log^3x}\Bigr)\\
&\ll\max_{1\leq h\leq H}\log x\sum_{d\leq D}\,\biggl|\sum\Sb n\leq x\\P^+(n)\leq y\endSb e\Bigl(\frac{\theta h\lfloor n^c\rfloor}{d}\Bigr)\biggr|+O\Bigl(\frac{\Psi(x,y)}{\log^3x}\Bigr).
\endalign
$$
By \Par*{Lemma 2} with $Z=D\log^5x$ we have
$$
\align
&\sum_{d\leq D}\max_{\gcd(s,d)=1}\Bigl|\bigl|\{a\in A,\,a\equiv s\bmod d\}\bigr|-\frac{\Psi(x,y)}{d}\Bigr|\\
&\ll\max_{1\leq h\leq H}\log x\sum_{z=0}^{2Z-1}\sum_{d\leq D}\,\biggl|\sum\Sb n\leq x\\P^+(n)\leq y\endSb e\Bigl(\frac{\theta h\lfloor n^c\rfloor}{d}\Bigr)\biggr|g_z(n^c)+O\Bigl(\frac{\Psi(x,y)}{\log^3x}\Bigr)
\\
&\ll\max_{1\leq h\leq H}\log x\sum_{z=0}^{2Z-1}\sum_{d\leq D}\,\biggl|\sum\Sb n\leq x\\P^+(n)\leq y\endSb
e\Bigl(\frac{\theta hn^c}{d}-\frac{\theta z}{2dZ}\Bigr)+O\Bigl(\frac{h\theta}{dZ}\Bigr)\biggr|g_z(n^c)+O\Bigl(\frac{\Psi(x,y)}{\log^3x}\Bigr)\\
&\ll\max_{1\leq h\leq H}\log x\sum_{z=0}^{2Z-1}\sum_{d\leq D}\,\biggl|\sum\Sb n\leq x\\P^+(n)\leq y\endSb e\Bigl(\frac{\theta hn^c}{d}\Bigr)\biggr|g_z(n^c)\\
&\qquad+O\Biggl(\log x\sum_{z=0}^{2Z-1}g_z(n^c)\sum_{d\leq D}\sum\Sb n\leq x\\P^+(n)\leq y\endSb\frac{h\theta}{dZ}\Biggr)
+O\Bigl(\frac{\Psi(x,y)}{\log^3x}\Bigr)\\
&\ll\max_{1\leq h\leq H}\log x\sum_{z=0}^{2Z-1}\sum_{d\leq D}\,\biggl|\sum\Sb n\leq x\\P^+(n)\leq y\endSb e\Bigl(\frac{\theta hn^c}{d}\Bigr)\biggr|g_z(n^c)+\frac{\Psi(x,y)}{\log^3x}\\
&\ll\max_{1\leq h\leq H}\log x\sum_{z=0}^{2Z-1}\sum_{d\leq D}\,\biggl|\sum\Sb n\leq x\\P^+(n)\leq y\endSb e\Bigl(\frac{\theta hn^c}{d}\Bigr)g_z(n^c)\biggr|+\frac{\Psi(x,y)}{\log^3x}\\
&\ll\max_{1\leq h\leq H}\log x\sum_{z=0}^{2Z-1}\sum_{d\leq D}\,\biggl|\sum\Sb n\leq x\\P^+(n)\leq y\endSb
e\Bigl(\frac{\theta hn^c}{d}\Bigr)\sum_{|r|\leq Z\log^4x}\beta_r^{(z)}e(rn^c)\biggr|+\frac{\Psi(x,y)}{\log^3x}\\
&\ll\max_{1\leq h\leq H}\log x\sum_{z=0}^{2Z-1}\sum_{d\leq D}\sum_{|r|\leq Z\log^4x}\beta_r^{(z)}\biggl|\sum\Sb n\leq x\\P^+(n)\leq y\endSb
e\Bigl(\Bigl(\frac{\theta h}{d}+r\Bigr)n^c\Bigr)\biggr|+\frac{\Psi(x,y)}{\log^3x}\\
&\ll\max_{1\leq h\leq H}ZD\log^5x\max_{d\leq D}\biggl|\sum\Sb n\leq x\\P^+(n)\leq y\endSb e\Bigl(\Bigl(\frac{\theta h}{d}+r\Bigr)n^c\Bigr)\biggr|+\frac{\Psi(x,y)}{\log^3x}.
\endalign
$$
Now we estimate the innermost sum in the last expression. Let $s=\frac{\theta h}{d}+r$, recall that the type of $\theta$ is~$\tau$, the type of the irrational $\theta/d$ is also $\tau$, then by the definition of the type of the irrational number, we have $\|h\theta/d\|>Ch^{-\tau-\varepsilon}$, where $C$ is a constant.
By \Par*{Lemma 8} we have
$$
\sum\Sb n \sim M \\ P^{+}(m) \leq y\endSb e(sn^c)
=\sum\Sb p \leq y\endSb \sum\Sb M_1 / p<u \leq M_1 \\ P^{-}(u) \geq p \\ P^{+}(u) \leq y\endSb
\sum\Sb  v\sim M/(up)\\P^{+}(v) \leq p\endSb e(s(puv)^c),
$$
where $1\leq M\leq x$ and $M_1<M$ to be determine later.

Next, we divide the ranges of $p$, $v$, and $u$ into $\log^3N$ dyadic intervals to get
$$
\align
\sum_{p \leq y}\sum\Sb M_1 / p<u \leq M_1 \\ P^{-}(u) \geq p \\ P^{+}(u) \leq y\endSb
\sum\Sb v\sim M/(up)\\ P^{+}(v) \leq p\endSb e(s(puv)^{c})
&\ll\sum_{p \leq y} \sum\Sb M_1 / p<u \leq M_1 \\ P^{-}(u) \geq p \\ P^{+}(u) \leq y\endSb
\biggl|\sum\Sb v\sim M/(up)\\ P^{+}(v) \leq p\endSb e(s(puv)^{c})\biggr|\\
&\ll\sum_{p \leq y} \sum_{M_1 / p<u \leq M_1}\biggl|\sum\Sb v\sim M/(up)\\ P^{+}(v) \leq p\endSb e(s(puv)^{c})\biggr|\\
&=\sum_{p \leq y} \sum_{M_1 / p<u \leq M_1}\sum\Sb v\sim M/(up)\\ P^{+}(v) \leq p\endSb
b(u,p)e(s(puv)^{c})
\\
&\leq\sum\Sb 2 \leqslant P \leqslant y \\ P=2^k\endSb \sum\Sb M_1 / 2 P \leqslant L \leqslant M_1 \\
L=2^{r}\endSb\sum\Sb M/4PL\leq K\leq2M/PL \\ K=2^j\endSb|S^{\prime}(P, K, L)|,
\endalign
$$
where
$$
S':=S^{\prime}(P, K, L)=\sum_{v\sim K}\sum\Sb p\sim P \\P^{+}(v)\leqslant p\endSb\sum\Sb u\sim L\\ uvp\sim M\endSb b(u,p)e(sp^{c}u^{c}v^{c}),
\tag17
$$
and $b(u,p)\ll 1$ be a complex number. %\?

Here we note that
$$
\align
S'
&=\int\limits_{-1/2}^{1/2}\sum_{v\sim K}\sum\Sb p\sim P \\P^{+}(v)\leq p\endSb\sum_{u\sim L}e(\alpha up)b(u,p)e(sp^cu^cv^c)\sum_{f\sim M/v}e(-\alpha f)d\alpha\\
&=\int\limits_{-1/2}^{1/2}\int\limits_{-1/2}^{1/2}\sum_{v\sim K}\sum_{f\sim M/v}e(-\alpha f)\sum_{p\sim P}\sum_{u\sim L}e(\alpha up-\gamma p)b(u,p)e(sp^cu^cv^c)\sum\Sb w\sim P\\ w\geq P^+(v)\endSb e(\gamma w)d\alpha d\gamma\\
&=\int\limits_{-1/2}^{1/2}\int\limits_{-1/2}^{1/2}\sum_{v\sim K}\sum_{f\sim M/v}e(-\alpha f)\sum\Sb w\sim P\\ w\geq P^+(v)\endSb
e(\gamma w)\sum_{p\sim P}\sum_{u\sim L}e(\alpha up-\gamma p)b(u,p)e(sp^cu^cv^c)d\alpha d\gamma.
\endalign
$$
So we get
$$
|S'|=\Biggl|\ \int\limits_{-1/2}^{1/2}\int\limits_{-1/2}^{1/2}\sum_{v\sim K}a(v;\alpha,\gamma)\sum_{u\sim L}\sum_{p\sim P}b(u,p;\alpha,\gamma)e(sp^cu^cv^c)d\alpha d\gamma\Biggr|,
$$
where $a(v;\alpha,\gamma)=\sum_{f\sim M/v}e(-\alpha f)\sum\Sb w\sim P\\ w\geq P^+(v)\endSb
e(\gamma w)$ and $b(u,p;\alpha,\gamma)=e(\alpha up-\gamma p)b(u,p)$.

Let $r=up$. Then we have
$$
S'\ll\Biggl|\int\limits_{-1/2}^{1/2}\int\limits_{-1/2}^{1/2}\sum_{v\sim K}a(v;\alpha,\gamma)\sum_{r\sim PL}c(r;\alpha,\gamma)e(sr^cv^c)d\alpha d\gamma\Biggr|,
$$
where $c(r)=\tau(r)b(u,p;\alpha,\gamma)$.

If $(\alpha,\gamma)\in\Omega$, $\max_{v\sim K}|a(v;\alpha,\gamma)|\neq0$, and $\max_{r\sim PL}|c(r;\alpha,\gamma)|\neq0$, let
$$
T_0(\alpha,\gamma)=\sum_{v\sim K}\frac{a(v;\alpha,\gamma)}{\max\limits_{v\sim K}|a(v;\alpha,\gamma)|}\sum_{r\sim PL}\frac{c(r;\alpha,\gamma)}{\max\limits_{r\sim PL}|c(r;\alpha,\gamma)|}e(sr^cv^c).
$$
So we have
$$
\aligned
S'&\ll\biggl|\int\int\limits_{(\alpha,\gamma)\in\Omega}\max_{v\sim K}|a(v;\alpha,\gamma)|\max_{r\sim PL}|c(r;\alpha,\gamma)|\,| %\?
T_0(\alpha,\gamma)|d\alpha d\gamma\biggr|\\
&=\int\int\limits_{(\alpha,\gamma)\in\Omega}\max_{v\sim K}|a(v;\alpha,\gamma)|\max_{r\sim PL}|c(r;\alpha,\gamma)|f(h)T_0(\alpha,\gamma)d\alpha d\gamma,
\endaligned
\tag18
$$
where $|f(h)|=1$.

Now we estimate $T_0(\alpha,\gamma)$. By \Par*{Lemma 3} with $Q=D^4x^{2\varepsilon}$ we have
$$
T_0(\alpha,\gamma)^2\ll \frac{M^2}{Q}+\frac{M}{Q}\biggl(\sum_{q\leq Q}\,\sum_{v\sim K}\biggl|\sum_{r\sim PL}e(sr^c((v+q)^c-v^c))\biggr|\biggr).
$$
Choosing $K=D^4x^{3\varepsilon}$ and $M_1=\frac{M}{D^4x^{3\varepsilon}}$. Utilizing \Par*{Lemma 4} to the innermost sum in the last expression we get
$$
T_0(\alpha,\gamma)\ll D^{-2}x^{1-\varepsilon}+D^{-2}x^{1/2-\varepsilon}(D^{4+\tau/2}x^{1-c/2+\varepsilon}+D^{9/2}x^{c/2})^{1/2}.
$$
By formula \Tag(18) and \Par*{Lemma 10} we have
$$
S'\ll D^{-2}x^{1-\varepsilon}+D^{-2}x^{1/2-\varepsilon}(D^{4+\tau/2}x^{1-c/2+\varepsilon}+D^{9/2}x^{c/2})^{1/2}.
$$
So we have
$$
\sum\Sb n \sim M \\ P^{+}(m) \leq y\endSb e(sn^c)\ll D^{-2}x^{1-\varepsilon}+D^{-2}x^{1/2-\varepsilon}(D^{4+\tau/2}x^{1-c/2+\varepsilon}+D^{9/2}x^{c/2})^{1/2}.
$$
Since
$$
\align
\sum\Sb n\leq x\\P^+(n)\leq y\endSb e\Bigl(\Bigl(\frac{\theta h}{d}+r\Bigr)n^c\Bigr)
&\ll\log x\max_{1\leq M\leq x}\sum\Sb n \sim M \\ P^{+}(m) \leq y\endSb e(sn^c)\\
&\ll D^{-2}x^{1-\varepsilon}+D^{-2}x^{1/2-\varepsilon}(D^{4+\tau/2}x^{1-c/2+\varepsilon}+D^{9/2}x^{c/2})^{1/2}.
\endalign
$$
So we have
$$
\align
\sum_{d\leq D}\max_{\gcd(s,d)=1}
&\Bigl|\bigl|\{a\in A, a\equiv s\bmod d\}\bigr|-\frac{x}{d}\Bigr|\\
&\ll x^{1-\varepsilon}+x^{1/2-\varepsilon}(D^{4+\tau/2}x^{1-c/2+\varepsilon}+D^{9/2}x^{c/2})^{1/2}+\frac{\Psi(x,y)}{\log^3x}\\
&\ll D^{2+\tau/4}x^{1-c/4}+D^{9/4}x^{1/2+c/4}+\frac{\Psi(x,y)}{\log^3x}.
\endalign
$$
So by the definition of the level of the distribution of the set $A$ we have
$$
D\leq \min\{x^{\frac{c-\varepsilon}{8+\tau}},x^{\frac{2-c-\varepsilon}{9}}\}.
$$
So by \Par*{Lemma 6} when
$$
R>\max\Bigl\{\frac{8+\tau}{c-\varepsilon},\frac{9}{2-c-\varepsilon}\Bigr\}+\delta_R=\max\Bigl\{\frac{8+\tau}{c-\varepsilon},\frac{9}{2-c-\varepsilon}\Bigr\}+0.124820.
$$
we have
$$
\bigl|\{n\in S(x,y)|\lfloor\theta\lfloor n^c\rfloor+\varphi\rfloor  \text{ is an $R$-almost prime}\}\bigr|\gg \frac{x}{\log^2x},
$$
where $x^{\varepsilon}<y<x$.

This completes the proof of this theorem. %\?2 this

\acknowledgments
I %\?
would like to thank the anonymous referee for their insightful and constructive comments, which have greatly improved the quality of this paper.

\Refs

\ref\no 1
\by                      Akbal Y.
\paper                   Friable values of Piatetski-Shapiro sequences
\jour                    Proc. Amer. Math. Soc.
\yr                      2017
\vol                     145
\issue                   10
\pages                   4255--4268
\endref

\ref\no 2
\by                      Baker R.C.
\paper                   Smooth numbers in Beatty sequences
\jour                    Acta Arith.
\yr                      2021
\vol                     200
\issue                   4
\pages                   429--438
\endref

\ref\no 3
\by                      Granville A.
\paper                   Smooth numbers: computational number theory and beyond
\inbook                  Algorithmic Number Theory: Lattices, Number Fields, Curves and Cryptography
\publaddr                Cambridge
\publ                    Cambridge University
\yr                      2008
\pages                   267--323
\finalinfo               Math. Sci. Res. Inst. Publ., 44
\endref

\ref\no 4
\by                      Moree P.
\paper                   Integers without large prime factors: from Ramanujan to de Bruijn
\jour                    Integers
\yr                      2014
\vol                     14A
\num                     A5
\size                    13
\endref

\ref\no 5
\by                      Dickman K.
\paper                   On the frequency of numbers containing prime factors of a certain relative magnitude
\jour                    Ark. Mat. Astr. Fys.
\yr                      1930
\vol                     22
%\issue                  -
\pages                   1-14
\endref

\ref\no 6
\by                Vinogradov~I.M.
      \book        The Method of Trigonometric Sums in the Theory of Numbers
      \publ        Wiley-Intersci.
      \publaddr    London and New York
      \yr          1954
\endref

\ref\no 7
\by                Harman~G.
\paper             On the distribution of $\alpha p$ modulo one.~II
\jour              Proc. London Math. Soc. (3)
\yr                1996
\vol               72
\issue             2
\pages             241--260
\endref

\ref\no 8
\by                Matom\"aki K.
\paper             The distribution of $\alpha p$ modulo one
\jour              Math. Proc. Cambridge Philos. Soc.
\yr                2009
\vol               147
\issue             2
\pages             267--283
\endref

\ref\no 9
\by               Baker R.C.
\paper            Diophantine approximation with smooth numbers
\jour             Ramanujan~J.
\yr               2023
\vol              61
\issue            1
\pages            49--54
\endref

\ref\no 10
\by               Zhu W.B.
\paper            Some problems about friable numbers in Piatetski-Shapiro sequence
\jour             Ramanujan~J.
\yr               2024
\vol              63
\issue            3
\pages            603--656
\endref

\ref\no 11
\by                  Dimitrov~S.I.
\paper               On the distribution of $\alpha p$ modulo one over Piatetski-Shapiro primes
\jour                Indian J. Pure Appl. Math.
\yr                  2023
\vol                 54
\issue               3
\pages               858--867
\endref

\ref\no 12
\by                Ghosh~A.
\paper             The distribution of $\alpha p^2$ modulo~1
\jour              Proc. London Math. Soc. (3)
\yr                1981
\vol               42
\issue             2
\pages             252--269
\endref

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

\ref\no 14
\by                  Piatetski-Shapiro I.I.
\paper               On the distribution of prime numbers in sequences of the form $[f(n)]$ %\?¢ ®à¨£¨­ «¥ â ª¨¥ ᪮¡ª¨
\jour                Mat. Sb.
\yr                  1953
\vol                 33 %/75
\issue               3
\pages               559--566
\endref

\ref\no 15
\by                  Kolesnik~G.A.
\paper               The distribution of primes in sequences of the form $\lfloor n^c\rfloor$ %\?¢ ®à¨£¨­ «¥ $[n^c]$
\jour                Math. Notes
\yr                  1967
\vol                 2
\issue               2
\pages               553--560 %117--128
\endref

\ref\no 16
\by                  Heath-Brown~D.R.
\paper               The Pjateckii-\v Sapiro prime number theorem
\jour                J. Number Theory
\yr                  1983
\vol                 16
\issue               2
\pages               242--266
\endref

\ref\no 17
\by                  Kolesnik~G.
\paper               Primes of the form $\lfloor n^c\rfloor$
\jour                Pacific J. Math.
\yr                  1985
\vol                 118
\issue               2
\pages               437--447
\endref

\ref\no 18
\by                  Liu~H.Q. and Rivat~J.
\paper               On the Pjateckii-\v Sapiro prime number theorem
\jour                Bull. Lond. Math. Soc.
\yr                  1992
\vol                 24
\issue               2
\pages               143--147
\endref

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

\ref\no 20
\by                  Baker R.C. and Harman G.
\paper               Small fractional parts of quadratic forms
\jour                Proc. Edinburgh Math. Soc. (2)
\yr                  1982
\vol                 25
\issue               3
\pages               269--277
\endref

\ref\no 21
\by                  Song Y.
\paper               A note on primes of the form $\lfloor\alpha p+\beta\rfloor$
\jour                J.~Number Theory
\yr                  2021
\vol                 225
\issue               1
\pages               1--17
\endref

\ref\no 22
 \by                 Graham~S.W. and Kolesnik~G.
 \book               Van der Corput's Method of Exponential Sums
 \publ               Cambridge University
 \publaddr           Cambridge
 \yr                 1991
 \finalinfo          London Math. Soc. Lecture Note Ser., 126
\endref

\ref\no 23
\by                  Vaaler~J.D.
\paper               Some extremal functions in Fourier analysis
\jour                Bull. Amer. Math. Soc.  (N.S.)
\yr                  1985
\vol                 12
\issue               2
\pages               183--216
\endref

\ref\no 24
\by                  Greaves G.
\book                Sieves in Number Theory
\publaddr            Berlin
\publ                Springer
\yr                  2001
\finalinfo           Ergeb. Math. Grenzgeb. (3), 43 %(Results in Mathematics and Related Areas~(3))
\endref

\ref\no 25
\by                Vaughan R.C.
\paper             A new iterative method in Waring's problem
\jour              Acta Math.
\yr                1989
\vol               162
\iftex
\issue             1--2
\else
\issue             1
\fi
\pages             1--71
\endref
\endRefs

 \enddocument