An additive problem over intersection of two Piatetski–Shapiro prime sets and almost-primes

Let \(\mathcal {P}_{\mathfrak {r}}\) ${\mathcal{P}}_{\mathfrak{r}}$ denote an almost-prime with at most \({\mathfrak {r}}\) $\mathfrak{r}$ prime factors, counted according to multiplicity. In this paper, we establish a mean value theorem of Bombieri–Vinogradov’s type for the intersection of two Piatetski–Shapiro prime sets with \(23/12<\gamma _1+\gamma _2<2\) $23/12<{\gamma }_1+{\gamma }_2<2$ . Moreover, we use this result to prove that, for \(1.992911<\gamma _1+\gamma _2<2\) $1.992911<{\gamma }_1+{\gamma }_2<2$ , there exist infinitely many primes of the form \(p=[n^{1/\gamma _1}]=[n^{1/\gamma _2}]\) $p=[n^{1/{\gamma }_1}]=[n^{1/{\gamma }_2}]$ such that \(p+2=\mathcal {P}_6\) $p+2={\mathcal{P}}_6$ .