A density version of quaternary Goldbach problem

Abstract

Let $\mathcal{P}$ denote the set of all primes, and let $\underlineδ(P)$ denote the relative lower density of a subset $P$ in $\mathcal{P}$. Suppose that $P_1, P_2, P_3, P_4$ are four subsets of primes with $\underlineδ(P_1)+\underlineδ(P_2)>1$ and $ \underlineδ(P_3)+\underlineδ(P_4)>1.$ Then for every sufficiently large even integer $n$, there exist primes $p_i \in P_i$ $(i=1,2,3,4)$ such that $n=p_1+p_2+p_3+p_4$. The condition is the best possible.