A quaternary Diophantine inequality with prime numbers of a special form (II)

Let $N$ be a sufficiently large real number. In this paper, it is proved that, for $1<c<\frac{5179}{3909}$ and for any arbitrary large number $E>0$, the Diophantine inequality \begin{equation*} \big|p_1^c+p_2^c+p_3^c+p_4^c-N\big|<(\log N)^{-E} \end{equation*} is solvable in prime variables $p_1,p_2,p_3,p_4$ such that each of the numbers $p_i+2\,(i=1,2,3,4)$ has, counted according to multiplicity, at most $[\frac{31250}{5179-3909c}]$ prime factors, which constitutes a large improvement upon the previous results of Dimitrov [13] and Li-Xue-Zhang [22], respectively.