Almost primes of the form \(\lfloor p^c \rfloor\) (Q266153)

Piatetski-Shapiro sequences are those sequences of the form NEWLINE\[NEWLINE {\mathbb N}^{c} = \left( [n^c] \right)_{n \in {\mathbb N}} \;\;(c > 1, \;c \notin {\mathbb N}), NEWLINE\]NEWLINE where \([t]\) denotes the integer part of any real number \(t\). For any \(R \geq 1\), a natural number is called an \(R\)-almost prime if it has at most \(R\) prime factors, counted with multiplicity. The paper under review investigates almost prime values of \([p^c]_{p \in {\mathbb P}}\) where \({\mathbb P} = \{ 2,3,5, \dots \}\), the set of primes in two different regimes. In the first regime, the result is that: If \((R,c_R), \;R = 8,9, \dots , 18, 19\) is any admissible pair from a certain table 1.1, then for any fixed \(c \in (1, c_R]\) there is a real number \(\eta > 0\) such that the lower bound NEWLINE\[NEWLINE |\{\text{prime} \;p \leq x : [p^c] \text{ is an \(R\)-almost prime} \}| \geq \eta \frac {x}{\log^2 x} NEWLINE\]NEWLINE holds for all sufficiently large \(x\).NEWLINENEWLINEIn the second regime, it is established that: For fixed \(c \geq 11/5\) there is a positive integer \(R\) with NEWLINE\[NEWLINER \leq \begin{cases} 16 c^3 + 179 c^2 &\text{if }c \in [11/5, 3),\\ 16c^3 + 88c^2 &\text{if } c \geq 3, \end{cases}NEWLINE\]NEWLINE and a real number \(\eta > 0\) such that the lower bound NEWLINE\[NEWLINE |\{\text{prime } p \leq x : [p^c] \text{ is an \(R\)-almost prime} \}| \geq \eta \frac {x}{\log^2 x} NEWLINE\]NEWLINE holds for all sufficiently large \(x\).NEWLINENEWLINEThe results are based on bounds of bilinear exponential sums, the notion of level of distribution from sieve theory. For related works, the readers are referred to [\textit{A. Balog}, Publ. Math. Orsay 89/01, 3--11 (1989; Zbl 0712.11056); \textit{X. Cao} and \textit{W. Zhai}, Acta Math. Sin., Chin. Ser. 51, No. 6, 1187--1194 (2008; Zbl 1174.11395); \textit{I. I. Piatetski-Shapiro}, Mat. Sb., Nov. Ser. 33(75), 559--566 (1953; Zbl 0053.02702); \textit{J. Rivat} and \textit{P. Sargos}, Can. J. Math. 53, No. 2, 414--433 (2001; Zbl 0970.11035); \textit{J. Rivat} and \textit{J. Wu}, Glasg. Math. J. 43, No. 2, 237--254 (2001; Zbl 0987.11052)].