Sumsets in primes containing almost all even positive integers

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.

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