On Chen’s theorem over Piatetski–Shapiro type primes and almost–primes

In this paper, we establish a new mean value theorem of Bombieri–Vinogradov’s type over Piatetski–Shapiro sequence. Namely, it is proved that for any given constant \(A>0\) $A>0$ and any sufficiently small \(\varepsilon >0\) $\epsilon >0$ , there holds \(\begin{aligned} \sum _{\begin{array}{c} d\leqslant x^\xi \\ (d,l)=1 \end{array}}\Bigg |\sum _{\begin{array}{c} A_1(x)\leqslant a<A_2(x)\\ (a,d)=1 \end{array}}g(a) \Bigg (\sum _{\begin{array}{c} ap\leqslant x\\ ap\equiv l\!\!\!\!\!\pmod d\\ ap=[k^{1/\gamma }] \end{array}}1-\frac{1}{\varphi (d)}\sum _{\begin{array}{c} ap\leqslant x\\ ap=[k^{1/\gamma }] \end{array}} 1\Bigg )\Bigg |\ll \frac{x^\gamma }{(\log x)^A}, \end{aligned}\) $$\begin{array}{c}\hfill \sum _{\begin{array}{c}d⩽x^{\xi }\\ (d,l)=1\end{array}}|\sum _{\begin{array}{c}{A}_1(x)⩽a<A_2(x)\\ (a,d)=1\end{array}}g(a)(\sum _{\begin{array}{c}ap⩽x\\ ap\equiv l(modd)\\ ap=[k^{1/\gamma }]\end{array}}1-\frac{1}{\phi (d)}\sum _{\begin{array}{c}ap⩽x\\ ap=[k^{1/\gamma }]\end{array}}1)|\ll \frac{x^{\gamma }}{{(logx)}^A},\end{array}$$ provided that \(1\leqslant A_1(x)<A_2(x)\leqslant x^{1-\varepsilon }\) $1⩽A_1(x)<A_2(x)⩽x^{1-\epsilon }$ and \(g(a)\ll \tau _r^s(a)\) $g(a)\ll {\tau }_r^s(a)$ , where \(l\not =0\) $l\ne 0$ is a fixed integer and \(\begin{aligned} \xi :=\xi (\gamma )=\frac{2^{38}+17}{38}\gamma -\frac{2^{38}-1}{38}-\varepsilon \end{aligned}\) $$\begin{array}{c}\hfill \xi :=\xi (\gamma )=\frac{2^{38}+17}{38}\gamma -\frac{2^{38}-1}{38}-\epsilon \end{array}$$ with \(\begin{aligned} 1-\frac{18}{2^{38}+17}<\gamma <1. \end{aligned}\) $$\begin{array}{c}\hfill 1-\frac{18}{2^{38}+17}<\gamma <1.\end{array}$$ Moreover, for \(\gamma \) $\gamma$ satisfying \(\begin{aligned} 1-\frac{0.03208}{2^{38}+17}<\gamma <1, \end{aligned}\) $$\begin{array}{c}\hfill 1-\frac{0.03208}{2^{38}+17}<\gamma <1,\end{array}$$ we prove that there exist infinitely many primes p such that \(p+2=\mathcal {P}_2\) $p+2={\mathcal{P}}_2$ with \(\mathcal {P}_2\) ${\mathcal{P}}_2$ being Piatetski–Shapiro almost–primes of type \(\gamma \) $\gamma$ , and there exist infinitely many Piatetski–Shapiro primes p of type \(\gamma \) $\gamma$ such that \(p+2=\mathcal {P}_2\) $p+2={\mathcal{P}}_2$ . These results generalize the result of Pan and Ding [37] and constitute an improvement upon a series of previous results of [29, 31, 39, 47].