On the ternary Goldbach problem with primes in independent arithmetic progressions

We show that for every fixed A > 0 and theta > 0 there is a theta = theta (A, theta) > 0 Nvith the following property. Let n be odd and sufficiently large, and let Q(1) = Q(2) := n(1/2) (logn)(-theta) and Q(3) := (log n)(theta). Then for all q(3) <= Q(3), all reduced residues a(3) mod q(3), almost all q(2) <= Q(2), all admissible residues a(2) mod q(2), almost all q(1) <= Q(1) and all admissible residues a(1) mod q(1), there exists a representation n = p(1) + p(2) + p(3) with primes p(i) equivalent to a(i) (q(i)), i = 1, 2, 3.