  Let $A$ be a subset of primes up to $x$. If we assume $A$ is well-distributed
(in the Siegel-Walfisz sense) in any arithmetic progressions to moduli
$q\leqslant(\log x)^c$ for any $c>0$, then the sumset $A+A$ has density 1/2 in
the natural numbers as $x$ tends to infinity, which also yields almost all even
positive integers could be represented as the sums of two primes in $A$ as $x$
tends to infinity. This result, improving the previous results in such special
case, could be compared with the classical estimation for the exceptional set
of binary Goldbach problem.
