Title: Diophantine approximation by special primes,
Abstract: We show that whenever $\delta>0$, $\eta$ is real and constants $\lambda_i$
satisfy some necessary conditions, there are infinitely many prime triples
$p_1,\, p_2,\, p_3$ satisfying the inequality $|\lambda_1p_1 + \lambda_2p_2 +
\lambda_3p_3+\eta|<(\max p_j)^{-1/12+\delta}$ and such that, for each
$i\in\{1,2,3\}$, $p_i+2$ has at most $28$ prime factors.