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 type over Piatetski-Shapiro sequence. Namely, it is proved that for any given constant A>0 and any sufficiently small epsilon > 0, there holds Sigma(d <= x xi(d,l)=1)| Sigma(A1(x)<= a<A2(x)(a,d)=1) g(a)( Sigma(ap <= xap equivalent to l(modd)ap=[k1/gamma])1-1 phi(d)| Sigma(ap <= xap = [k1/gamma])1)| << x(gamma)(log x)(A), provided that 1 <= A(1)(x) < A(2)(x) <= x(1-epsilon) and g(a)<< tau(s)(r)(a), where l not equal 0 is a fixed integer and xi := xi(gamma) = 2(38)+17/ 38 gamma-2(38)-1/38-epsilon with 1-18/2(38) + 17 < gamma < 1. Moreover, for gamma satisfying 1-0.03208/2(38)+17 < gamma < 1, w e prove that there exist infinitely many primes p such that p+2 = P-2 with P-2 being Piatetski-Shapiro almost-primes of type gamma, and there exist infinitely many Piatetski-Shapiro primes p of type gamma such that p+2 = P-2. These results generalize the result of Pan and Ding [37] and constitutes an improvement upon a series of previous results of [29,31,39,47].