A density version of Waring-Goldbach problem

Abstract

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 1/2. Let k be a positive integer other than 1, 2, 4, 8, and 9. 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