The asymptotic formula in Waring's problem (Q2882367)

Let \(R_{s,k}(n)\) denote the number of representations of the natural number \(n\) as the sum of \(s\) \(k\)-th powers of positive integers. A heuristic application of the Hardy-Littlewood circle method suggests the if \(k\geq 3\) and \(s\geq k+1\), then an asymptotic formula of the shape NEWLINE\[NEWLINE R_{s,k}(n)=C(s,k)n^{s/k-1}+o\left(n^{s/k-1}\right) NEWLINE\]NEWLINE holds for \(n\rightarrow \infty\), where \(C(s,k)\) is the product of a singular series and a singular integral. Let \(\tilde{G}(k)\) denote the least integer \(t\) with the property that, for all \(s\geq t\) and all sufficiently large natural numbers \(n\), this asymptotic formula holds. \textit{G. H. Hardy} and \textit{J. E. Littlewood} [Math. Zeitschr. 12, 161--188 (1922; JFM 48.0146.01)] established the bound \(\tilde{G}(k)\leq (k-2)2^{k-2}+5\). Since then, there has been a long record of research, carried out in particular by the author. In the paper under review, he establishes the bound NEWLINE\[NEWLINE \tilde{G}(k)\leq 2k^2+1-\max\limits_{{1\leq j\leq k-1,\;2^j\leq k(2k+1)}} \left\lceil \frac{2kj-2^j}{k+1-j}\right\rceil NEWLINE\]NEWLINE when \(k\geq 2\). In particular, it follows that NEWLINE\[NEWLINE \tilde{G}(k)\leq 2k^2-2\left\lfloor \frac{\log k}{\log 2} \right\rfloor NEWLINE\]NEWLINE for all \(k\geq 2\), and \(\tilde{G}(k)\leq 2k^2-11\) for all \(k\geq 6\). This improves upon the author's recent result \(\tilde{G}(k)\leq 2k^2+2k-3\) achieved via efficient congruencing [the author, Ann. Math. (2) 175, No. 3, 1575--1627 (2012; Zbl 1267.11105)]. To obtain this improvement, he establishes a new mean value theorem for the minor arcs. The strategy of its proof is to adapt the argument of the aforementioned paper to the minor arcs only. In addition, the author proves a new theorem on slim exceptional sets for the asymptotic formula in Waring's problem.