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