The author pursues the question of representing a large integer \(n\) as a sum of seven cubes \(x_ i^ 3\) of almost-primes \(P_ r\) (having at most \(r\) prime factors). J. Brüdern [Acta Arith. 72, No. 3, 211–227 (1995: Zbl 0839.11046)] showed that this could be done with \(x_ 1\) prime, \(x_ 7\) a \(P_ {69}\), and the other \(x_ i\) \(P_ 5\) numbers. The author refines and modifies the methods previously used in a search for results containing more satisfying collections of the suffices \(r\). In particular, he can require that all the numbers \(x_ i\) are \(P_ 4\) numbers, and indeed that all but one of these are \(P_ 3\), of which one is a \(P_ 2\). In a related way the author can require, following Brüdern, that \(x_ 1\) is a prime, in which case he needs \(x_ 2\) to be a \(P_ 6\) and the other \(x_ i\) to be \(P_ 3\). It is also possible to represent \(n\) as a sum of three cubes of primes and four cubes of natural numbers. One other variant on this theme is offered, as well as a result on the rather different question of representing a number as a sum of seven cubes of “smooth” numbers.
The methods used develop those used by Brüdern, and involve some improvements in the machinery of the circle method, one of them within reach of the methods of R. C. Vaughan [Acta Math. 162, No. 1/2, 1–71 (1989; Zbl 0665.10033)] and the other involving a more advanced version [J. Reine Angew.Math. 365, 122–170 (1986; Zbl 0574.10046)] of Vaughan’s iterative method restricted to minor arcs. These are used to enlarge the level of distribution accessible in an appeal to the linear sieve. Further, the appeal by Brüdern to the use of weights in the linear sieve is replaced by a use of the Chen–Iwaniec switching principle.