Sums and differences of three $k$th powers

Let k > 2 be a fixed integer exponent and let θ > 9 / 10. We show that a positive integer N can be represented as a non-trivial sum or difference of 3kth powers, using integers of size at most B, in O (Bθ N1 / 10) ways, providing that N ≪ B3 / 13. The significance of this is that we ma...