The ternary Goldbach problem with primes in positive density sets

Abstract

Let $\mathcal{P}$ denote the set of all primes. $P_1,P_2,P_3$ are three subsets of $\mathcal{P}$. Let $\underline{\delta }(P_i)$ $(i=1,2,3)$ denote the lower density of $P_i$ in $\mathcal{P}$, respectively. It is proved that if $\underline{\delta }(P_1)>5/8$, $\underline{\delta }(P_2)\ge 5/8$, and $\underline{\delta }(P_3)\ge 5/8$, then for every sufficiently large odd integer n, there exist $p_i\in P_i$ such that $n=p_1+p_2+p_3$. The condition is the best possible.

13 pages