Weyl’s inequality, Hua’s inequality and Waring’s problem. (English) Zbl 0619.10046

Heath-Brown, D. R.

J. Lond. Math. Soc., II. Ser. 38, No. 2, 216-230 (1988).

Let \(k\geq 3\) be a fixed integer, let \(\alpha\in {\mathbb{R}}\), and let \(S(\alpha)=\sum ^{P}_{n=1}e(\alpha n^ k)\). If \(| \alpha - (a/q)| \leq q^{-2}\) with \((a,q)=1\) then \(S(\alpha)\ll P^{1-2^{1- k}+\epsilon}\) providing that \(P\leq q\leq P^{k-1}\). This is Weyl’s inequality. The first result of the paper is the sharper bound \[ S(\alpha)\ll P^{1-(8/3)2^{-k}+\epsilon}, \] valid on the shorter range \(P^ 3\leq q\leq P^{k-3}\), for \(k\geq 6.\)
Hua’s inequality states that \(\int ^{1}_{0}| S(\alpha)| ^{2^ k} d\alpha \ll P^{2^ k-k+\epsilon}\). The second result of the paper is the better bound \[ \int ^{1}_{0}| S(\alpha)| ^{7.2^{k-3}} d\alpha \ll P^{7.2^{k-3}-k+\epsilon},\text{ for } k\geq 6. \] The third result, which is a simple corollary of the second, is that the Hardy-Littlewood asymptotic formula, for sums of s k-th powers, is valid for \(s\geq (7/8)2^{k-3}+1.\) The key idea in the proof is to estimate S(\(\alpha)\) by performing k-3 Weyl steps. This produces a large number of cubic sums, whose mean value is bounded using the integral \[ \int ^{1}_{0}\int ^{1}_{0}| \sum ^{P}_{1}e(\alpha n^ 3+\beta n)| ^ 6 d\alpha d\beta. \]

MSC:

11P05	Waring’s problem and variants
11L40	Estimates on character sums
11P55	Applications of the Hardy-Littlewood method

Keywords:

Weyl’s inequality; sharper bound; Hua’s inequality; Hardy-Littlewood asymptotic formula; cubic sums; small exponent; Waring problem