A density version for the sum of nine cubes of primes

Source abstract

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