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On Waring’s problem for smaller exponents. II. (English) Zbl 0601.10037

[For part I, cf. Proc. Lond. Math. Soc., III. Ser. 52, 445-463 (1986; Zbl 0601.10035).]
The asymptotic formula for the number of representations of a (large) integer \(N\) as sums of \(s\) \(k\)-th powers (\(k\geq 2)\) was established by Hua Lookeng [Q. J. Math., Oxf. Ser. 9, 199-202 (1938; Zbl 0020.10504)] with \(s\geq 2^ k+1\). Vinogradov’s results are much stronger than this for large \(k\). However, for \(3\leq k\leq 10\), Hua’s result has been the best known result.
In this paper, the author shows that the asymptotic formula holds with \(s\geq 2^ k\). With \(P=N^{1/k}\), \(f(\alpha)=\sum_{1\leq x\leq P}e(\alpha x^ k)\), Hua’s inequality asserts that \[ \int^{1}_{0}| f(\alpha)|^{2^ k} d\alpha \ll P^{2^ k- k+\epsilon} \] (and more precisely, \(P^{\epsilon}\) can be replaced by \((\log P)^ C\) for some positive constant \(C\)). Thus, in using the Hardy-Littlewood method for establishing the asymptotic formula, Hua uses Weyl’s inequality for an additional \(k\)-th power in order to save more than \(P^{-k}\) over the minor arcs. With \(m\) denoting the minor arcs, the author shows that \[ \int_{m}| f(\alpha)|^{2^ k} d\alpha \ll P^{2^ k-k}\quad (\log P)^{-2+\epsilon}, \] and consequently, does not require the additional \(k\)-th power for the minor arcs (the major arcs presenting no problem).
Reviewer: K.Thanigasalam

MSC:

11P05 Waring’s problem and variants
11P55 Applications of the Hardy-Littlewood method
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References:

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