DIOPHANTINE APPROXIMATION BY PRIME NUMBERS OF A SPECIAL FORM

We show that for $B > 1$ and for some constants $\lambda_i, i = 1, 2, 3$ subject to certain assumptions, there are infinitely many prime triples $p_1, p_2, p_3$ satisfying the inequality $\mid\lambda_1p_1 + \lambda_2p_2 + \lambda_3p_3 + \eta\mid < [log(max\,p_j )]^{-B}$ and such that $p_1 + 2, \, p_2 + 2\, and \, p_3 + 2$ have no more than 8 prime factors. The proof uses Davenport - Heilbronn adaption of the circle method together with a vector sieve method.