Breaking classical convexity in Waring's problem: sums of cubes and quasi-diagonal behaviour
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 \(\varepsilon_k = A \exp (- C \log^2k)\).
scientific article; zbMATH DE number 871775
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