A density version for the sum of nine cubes of primes

Abstract

Let $\mathbb{P}$ be the set of primes. Suppose that $P_i$, $i=1,2,\dots ,9$ are nine subsets of $\mathbb{P}$ with relative lower density ${\underline{d}}_{\mathbb{P}}(P_i)={\mu }_i$. When ${\sum }_{i=1}^9{\mu }_i^3>8$, we prove that every sufficiently large odd integer can be written as $n={\sum }_{i=1}^9{p}_i^3$ with $p_i\in P_i$, $i=1,2,\dots ,9$.