Sums and differences of three kth 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(BθN1/10) ways, providing that N≪B3/13. The significance of this is that we may take θ strictly less than 1. We also prove the estimate O(B10/k) (subject to N≪B) which is better for large k. The results extend to representations by an arbitrary fixed non-singular ternary from. However “non-trivial” must then be suitably defined. Consideration of the singular form xk−1y−zk allows us to establish an asymptotic formula for (k−1)-free values of pk+c, when p runs over primes, answering a problem raised by Hooley.