A density version of quaternary Goldbach problem

Abstract

Let $\mathcal{P}$ denote the set of all primes, and let $\underline{\hat{I}}´(P)$ 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 $\underline{\hat{I}}´(P_1)+\underline{\hat{I}}´(P_2)>1$ and $\underline{\hat{I}}´(P_3)+\underline{\hat{I}}´(P_4)>1.$ 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.