On Waring's problem: Three cubes and a sixth power

We establish that almost all natural numbers not congruent to 5 modulo 9 ar e the sum of three cubes and a sixth power of natural numbers, and show, mo reover, that the number of such representations is almost always of the exp ected order of magnitude. As a corollary, the number of representations of a large integer as the sum of six cubes and two sixth powers has the expect ed order of magnitude. Our results depend on a certain seventh moment of cu bic Weyl sums restricted to minor arcs, the latest developments in the theo ry of exponential sums over smooth numbers, and recent technology for contr olling the major arcs in the Hardy-Littlewood method, together with the use of a novel quasi-smooth set of integers.