On a Piatetski-Shapiro analog problem over almost-primes (Q6057439)

The first result proved by the authors has two sets \(\mathfrak{A},\mathfrak{B}\subset\{N+1,\cdots,2N\}\) that are such that \(\vert \mathfrak{A}\vert \vert \mathfrak{B}\vert \gg N^{2-2\delta}\) for some sufficiently small \(\delta\). Then the product set \(\mathfrak{A}\mathfrak{B}\) contains an element of the form \(\lfloor n^c\rfloor\) where \(1<c<6/5\) and \(n\) is an integer. This improves on an earlier result of \textit{J. Rivat} and \textit{A. Sárközy} [Acta Math. Hung. 74, No. 3, 245--260 (1997; Zbl 0923.11038)].\par The second result has more stringent conditions. The sets \(\mathfrak{A}\) and \(\mathfrak{B}\) are now supposed to satisfy \(\vert \mathfrak{A}\vert \gg N\) and \(\vert \mathfrak{B}\vert \gg N\). It is then proved that the product set \(\mathfrak{A}\mathfrak{B}\) contains an element of the form \(\lfloor P_9^c\rfloor\) where \(1<c<91673/90000\) and \(P_9\) is an integer that has at most 9 prime factors.\par After extracting a main term, the error term is studied by the double large sieve inequality and a spacement result due to \textit{O. Robert} and \textit{P. Sargos} [J. Reine Angew. Math. 591, 1--20 (2006; Zbl 1165.11067)]\par In the almost prime case, a similar process is used with a divisibility condition added. The lower bound sieve machinery then applies to ensure the existence of a \(\lfloor P_{48}^c\rfloor\). This result is subsequently refined by employing a weighted sieve. Both steps are in fact carried out simultaneously.