On sums of powers of almost equal primes

Let k >= 2 ands be positive integers, and let n be a large positive integer subject to certain local conditions. We prove that if s >= k(2) + k + 1 and 0 > 31/40, then n can be expressed as a sum p(1)(k) +...+p(s)(k) where p(1,)...,p(s) are primes with broken vertical bar p(j) - (n/s)(1/k)broken vertical bar <= n(theta/k). This improves on earlier work by Wei and Wooley [15] and by Huang [8] who proved similar theorems when theta > 19/24. (C) 2017 Elsevier Inc. All rights reserved.