A density version of Waring–Goldbach problem

Let [Formula: see text] denote the set of all primes and [Formula: see text] be a positive integer. Suppose that A is a subset of [Formula: see text] with [Formula: see text], where [Formula: see text] is the lower density of A relative to [Formula: see text]. We prove that every sufficiently large natural number n satisfying the necessary congruence condition can be written in the form [Formula: see text], where [Formula: see text] and [Formula: see text] for all [Formula: see text].