On the representation of large even integers as the sum of eight primes from positive density sets

Let $P$ denote the set of all primes. We have proved that if A is a subset of $P$, and the lower density of A in $P$ is larger than 1/2, then every sufficiently large even integer n can be expressed in the form $n=p_1+\cdots +p_8$, where $p_1,\dots ,p_8\in A$. The constant 1/2 in this statement is the best possible.