On a Piatetski-Shapiro analog problem over almost-primes

Let N be a sufficiently large number, \(\mathfrak{A}\) and \(\mathfrak{B}\) be subsets of \(\{N+1, \ldots , 2N\}\). We prove that if \(1<c<\frac{6}{5}\), \(|\mathfrak{A}|\, |\mathfrak{B}|\gg N^{2-2\delta}\) and \(\delta>0\) is sufficiently small, then the equation

$$ab=\lfloor n^c\rfloor,\quad a\in\mathfrak{A},\ b\in\mathfrak{B} $$

is solvable, which improves the result of Rivat and Sárközy [14]. We also investigate the solvability of the equation

$$ab=\lfloor P_k^c\rfloor,\quad a\in\mathfrak{A},\ b\in\mathfrak{B},\ 1<c<c_0, $$

where P_k denotes an almost-prime with at most k prime factors and c₀ is a fixed real number depends on k.