Waring's problem for sixteenth powers (Q1919037)

Let \(G(k)\) be the least number \(s\) such that every sufficiently large integer can be written as a sum of \(s\) \(k\)-th powers of natural numbers. The author proves that \(G(16)\leq 111\), improving the bound \(G(16)\leq 112\) due to \textit{T. D. Wooley} [Ann. Math. (2) 135, 131-164 (1992; Zbl 0754.11026)]. Namely the author shows that every sufficiently large integer can be written as a sum of 111 sixteenth powers. This improvement is achieved by a careful refinement of the process of obtaining mean value estimates for smooth Weyl sums in the aforementioned work of Wooley. In particular, the author added the idea of \textit{D. R. Heath-Brown} [J. Lond. Math. Soc. (2) 38, 216-230 (1988; Zbl 0619.10046)] to the process.