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) + ai-aEuroe+ p (s) (4) with . We establish that for every sufficiently large integer N satisfying necessary local conditions, this equation holds with s = 17 and theta(17) = 1/196 - epsilon. Moreover, we prove that when 9 a (c) 1/2 s a (c) 1/2 16, almost all integers n can be expressed this way with theta(s) = (2s - 15)/(12(s + 16)) - epsilon for s = 9, 10 and (8s - 69)/(4(88s - 717)) - epsilon for 11 a (c) 1/2 s a (c) 1/2 <= 16.