Weyl sums, mean value estimates, and Waring's problem with friable numbers (Q2833617)

Waring's problem is the question of whether or not, given a positive integer \(k\), there exists positive integers \(s\) and \(N_0\) such that every integer \(N > N_0\) can be written as a sum of \(s\) \(k\)th powers: NEWLINE\[NEWLINEN = n_1^{k} + \cdots + n_s^{k}.NEWLINE\]NEWLINE Denote by \(G(k)\) the least such number \(s\). \textit{T. D. Wooley} [J. Lond. Math. Soc., II. Ser. 51, No. 1, 1--13 (1995; Zbl 0833.11041)] proves that NEWLINE\[NEWLINEG(k) \leq k(\log{k} + \log\log{k} + 2 + O(\log\log{k}/\log{k})).NEWLINE\]NEWLINE To obtain an asymptotic formula for the number of solutions to the above mentioned equation, we need more variables than the bound given in the above inequality. The current best result follows from the \textit{T. D. Wooley}'s work [Proc. Lond. Math. Soc. (3) 111, No. 3, 519--560 (2015; Zbl 1328.11087)], which gives such an asymptotic formula when \(s \geq Ck^2 + O(k)\) for \(c=1.542749\ldots\). Recently, \textit{J. Bourgain} et al. [Ann. Math. (2) 184, No. 2, 633--682 (2016; Zbl 1408.11083)] proved the Vinogradov's main conjecture, which allows \(C=1.\) The authors, using Friable integers concept, obtain an asymptotic formula for Weyl sums in major arcs, non-trivial upper bounds for them in minor arcs, and moreover a mean value estimate for friable Weyl sums with exponent essentially the same as in the classical case. As an application, they study Waring's problem with friable numbers, with the number of summands essentially the same as in the classical case.