Quantitative relations between short intervals and exceptional sets of cubic Waring-Goldbach problem

In this paper, we are able to prove that almost all integers n satisfying some necessary congruence conditions are the sum of j almost equal prime cubes with j = 7, 8, i. e., N = p(1)(3) +... + p(j)(3) with vertical bar p(i) -(N/j)(1/3)vertical bar <= N1/3-delta+epsilon (1 <= i <= j), for some 0 < delta <= 1/90. Furthermore, we give the quantitative relations between the length of short intervals and the size of exceptional sets.