  In this paper, we study a density version of the Waring-Goldbach problem.
Suppose that A is a subset of the primes, and the lower density of A in the
primes is larger than 1-1/2k. We prove that every sufficiently large natural
number n satisfying the necessary congruence condition can be expressed as a
sum of s terms of the k-th powers of primes from set A, where s is a positive
integer dependent on k.
