On Waring's problem: Three cubes and a sixth power (Q2764667)

It is shown that, for `almost all' positive integers \(n\not \equiv 5\pmod 9\), one has \(r(n)\gg n^{1/6}\), where \(r(n)\) is the number of representations \(n=x^3_1+x^3_2 +x^3_3+x^6_4\) with \(x_i\geq 0\). As a corollary one has \(R(n)\gg n^{4/3}\) for all sufficiently large \(n\), where \(R(n)\) is the number of representations \(n=x^3_1+ \cdots+ x^3_6+x^6_7 +x^6_8\) with \(x_i\geq 0\). The proof depends on an estimate for a minor arc integral involving sums of 7 cubes. In addition to exponential sums over smooth numbers, certain `quasi-smooth' integers are also used.