The ternary Goldbach problem with primes in positive density sets

Let PP denote the set of all primes. P1,P2,P3P1,P2,P3 are three subsets of PP. Let δ_(Pi)(i=1,2,3)(i=1,2,3) denote the lower density of PiPi in PP, respectively. It is proved that if δ_(P1)>5/8, δ_(P2)≥5/8, and δ_(P3)≥5/8, then for every sufficiently large odd integer n  , there exist pi∈Pipi∈Pi such that n=p1+p2+p3n=p1+p2+p3. The condition is the best possible.