An Additive Problem over Piatetski-Shapiro Primes and Almost-primes

Let $\mathcal{P}_r$ denote an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this paper, we establish a theorem of Bombieri-Vinogradov type for the Piatetski-Shapiro primes $p=[n^{1/\gamma}]$ with $\frac{85}{86}<\gamma<1$. Moreover, we use this result to prove that, for $0.9989445<\gamma<1$, there exist infinitely many Piatetski-Shapiro primes such that $p+2=\mathcal{P}_3$, which improves the previous results of Lu, Wang and Cai, and Peneva.