We establish that almost all natural numbers are the sum of four cubes of positive integers, one of which is no larger than . The proof makes use of an estimate for a certain eighth moment of cubic exponential sums, restricted to minor arcs only, of independent interest.
"On Waring’s problem: Three cubes and a minicube." Nagoya Math. J. 200 59 - 91, December 2010. https://doi.org/10.1215/00277630-2010-012