An additive problem over Piatetski–Shapiro primes and almost-primes

Abstract

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 (Acta Math Sin (Engl Ser) 34(2):255–264, 2018), Wang and Cai (Int J Number Theory 7(5):1359–1378, 2011), and Peneva (Monatsh Math 140(2):119–133, 2003).