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.
Categories: math.NT