Exceptional Sets for Sums of Prime Cubes in Short Interval

Let E N , X denote the number of even integers n , with N − X ≤ n ≤ N , such that n cannot be written as n = p 1 3 + ⋯ + p 8 3 . We prove that if X > N 1 / 36 + ɛ , then E N , X = o X .