On Waring's problem for cubes and smooth Weyl sums (Q2766386)

For a natural number \(n\), write \(P(n)\) for the largest prime factor of \(n\), and denote by \(T(n)\) the number of positive integral solutions of the equation \(m_1^3+\cdots+m_8^3=n\) subject to \(P(m_j)\leq \exp(c_1(\log n\log\log n)^{1/2})\) for \(1\leq j\leq 8\), with some \(c_1>0\). The main theorem of this paper asserts the existence of positive constants \(c_1\) and \(c_2\) such that one has NEWLINE\[NEWLINET(n)>n^{5/3} \exp(-c_2(\log n\log\log n)^{1/2})NEWLINE\]NEWLINE for all sufficiently large \(n\). This theorem refines the corresponding result for the sum of nine cubes obtained recently by \textit{G. Harcos} [Acta Arith. 80, 165-185 (1997; Zbl 0871.11066)]. As a by-product, it is pointed out that almost all natural numbers are the sum of four cubes of `smooth numbers' which means here natural numbers having only small prime factors as above. NEWLINENEWLINENEWLINEThe most important ingredient of the proof is a sharp mean value estimate for the smooth Weyl sum NEWLINE\[NEWLINEf(\alpha;P,R)=\sum_{x\in{\mathcal A(P,R)}} \exp(2\pi i\alpha x^3),NEWLINE\]NEWLINE where \({\mathcal A}(P,R)\) denotes the set of natural numbers \(m\) up to \(P\) such that the largest prime factor of \(m\) does not exceed \(R\). In order to prove the above conclusions via the Hardy-Littlewood method, it is crucial to obtain the estimate of the form NEWLINE\[NEWLINE \int_0^1|f(\alpha;P,R)|^s d\alpha\ll P^{s-3}, NEWLINE\]NEWLINE for some \(s<8\), while this bound is immediate from a theorem of Vaughan when \(s\geq 8\). In fact, this estimate is established here for \(s\geq 7.691\) under reasonable conditions on \(P\) and \(R\), by appealing to the new technique on handling fractional moments of \(f(\alpha;P,R)\), which is refered as \textit{T. Wooley}'s `breaking classical convexity' [Invent. Math. 122, 421-451 (1995; Zbl 0851.11055)].NEWLINENEWLINENEWLINEBesides, the paper includes upper bounds for the above mean value with \(4<s\leq 7.691\) stemming from `breaking classical convexity', and a sophisticated method on treatment of the major arc contribution associated with smooth Weyl sums. These are of independent interest beyond the context of this paper, and may be of use in related research areas. For an example of application, see another paper of the authors [``On Waring's problem: a square, four cubes and a biquadrate'', Math. Proc. Camb. Philos. Soc. 127, 193-200 (1999; Zbl 0944.11029)].