Slim Exceptional Sets for Sums of Cubes

http://dx.doi.org/10.4153/CJM-2002-014-4
Canad. J. Math. 54(2002), 417-448

  Published:2002-04-01
 Printed: Apr 2002

We investigate exceptional sets associated with various additive problems involving sums of cubes. By developing a method wherein an exponential sum over the set of exceptions is employed explicitly within the Hardy-Littlewood method, we are better able to exploit excess variables. By way of illustration, we show that the number of odd integers not divisible by $9$, and not exceeding $X$, that fail to have a representation as the sum of $7$ cubes of prime numbers, is $O(X^{23/36+\eps})$. For sums of eight cubes of prime numbers, the corresponding number of exceptional integers is $O(X^{11/36+\eps})$.