Density versions of the binary Goldbach problem

Let > 1/2. We prove that if A is a subset of the primes such that the relative density of A in every reduced residue class is at least, then almost all even integers can be written as the sum of two primes in A. The constant 1/2 in the statement is best possible. Moreover we give an example to show that for any > 0 there exists a subset of the primes with relative density at least 1 - such that A+A misses a positive proportion of even integers.