On sums of powers of almost equal primes

Let k≥2 and s be positive integers, and let n be a large positive integer subject to certain local conditions. We prove that if s≥k2+k+1 and θ>31/40, then n can be expressed as a sum p1k+…+psk, where p1,…,ps are primes with |pj−(n/s)1/k|≤nθ/k. This improves on earlier work by Wei and Wooley [15] and by Huang [8] who proved similar theorems when θ>19/24.