Waring–Goldbach problem for fourth powers with almost equal variables

We consider the expression of a positive integer n by sums of fourth powers of almost equal primes, that is, n = p 1 4 + p 2 4 + ⋯ + p s 4 with p i − N / s 1 / 4 ⩽ N / s 1 / 4 − θ s . We establish that for every sufficiently large integer N satisfying necessary local conditions, this equation holds with s = 17 and θ 17 = 1/196 − ε. Moreover, we prove that when 9 ⩽ s ⩽ 16, almost all integers n can be expressed this way with θ s = (2s − 15)/(12(s + 16)) − ε for s = 9, 10 and (8s − 69)/(4(88s − 717)) − ε for 11 ⩽ s ⩽ 16.