Breaking classical convexity in Waring's problem: sums of cubes and quasi-diagonal behaviour
type: Item
instance of: scholarly article
title: Breaking classical convexity in Waring's problem: sums of cubes and quasi-diagonal behaviour
zbMATH Open document ID: 0851.11055
DOI: 10.1007/BF01231451
published in: Inventiones Mathematicae
publication date: 1996-06-24T00:00:00Z
full work available at URL: https://eudml.org/doc/144327
review text: The classical convexity referred to is the principle that once bounds are known for averages \(\int^1_0 |f (\alpha) |^s d \alpha\) of Weyl sums \(\sum_x e (\alpha x^k)\) for even integer exponents \(s\), as might be derived by differencing arguments, then bounds can be deduced for intermediate values of \(s\) by using Hölder's inequality. The author's main theorem improves on these bounds for averages of Weyl sums which (as in recent work on this topic) are taken over smooth numbers \(x\) (essentially those whose prime factors do not exceed \(x^\eta\), where \(\eta\) is sufficiently small). A number of applications, mostly to problems involving sums of cubes, are dealt with. A more direct treatment, giving a slightly stronger result, becomes possible for the asymptotic formula in the seven cubes problem previously treated by \textit{R. C. Vaughan} [J. Lond. Math. Soc., II. Ser. 39, 205--218 (1989; Zbl 0677.10034)]. Let \(N_k (X)\) denote the number of positive integers not exceeding \(X\) which are sums of three \(k\)th powers of positive integers. A small improvement is obtained in the lower bound for \(N_3 (X)\). For large \(k\) a substantial improvement, to \(\gg X^{3/k - \varepsilon_k}\) with \(\varepsilon_k = e^{- k/17}\), is made in the lower bound for \(N_k (X)\). The earlier result of \textit{R. C. Vaughan} [J. Lond. Math. Soc., II. Ser. 39, 219--230 (1989; Zbl 0677.10035)] had \(\varepsilon_k = A \exp (- C \log^2k)\).
reviewed by: George Greaves
Mathematics Subject Classification ID: 11P05
zbMATH DE Number: 871775
zbMATH Keywords: Waring's problem
zbMATH Keywords: averages of Weyl sums
zbMATH Keywords: sums of cubes
Wikidata QID: Q63230024
author: Trevor D. Wooley
MaRDI profile type: Publication
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OpenAlex ID: W1540761923
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description: scientific article; zbMATH DE number 871775