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Some remarks on Weyl sums. (English) Zbl 0545.10024
Topics in classical number theory, Colloq. Budapest 1981, Vol. II, Colloq. Math. Soc. János Bolyai 34, 1585-1602 (1984).
[For the entire collection see Zbl 0541.00002.]
This paper concerns sums $$f(\alpha)=\sum^{N}_{n=1}e(\alpha n^ k)$$ where $$e(z)=\exp(2\pi iz)$$. These sums occur in various contexts, but primarily in connection with Waring’s problem. If $$\alpha =a/q+\beta$$ with $$\beta$$ small, then $$f(\alpha)$$ is approximately $$q^{- 1}S(q,a)I(\beta),$$ where $$S(q,a)=\sum^{q}_{n=1}e((a/q)n^ k),\quad I(\beta)=\int^{N}_{0}e(\beta x^ k) dx.$$ Two new estimates are proved for the difference $$E=f(\alpha)-q^{-1}S(q,a)I(\beta).$$ Firstly, if $$\epsilon>0$$ and $$| \beta | \leq(2kqN^{k-1})^{-1},$$ then $$E\ll q^{{1\over2}+\epsilon}.$$ Secondly, if $$\epsilon>0$$, then $$E\ll q^{{1\over2}+\epsilon}(1+N^ k | \beta |)^{{1\over2}},$$ for all $$\beta$$. When $$k=3$$ the latter result yields a non-trivial bound for $$f(\alpha)$$ for all $$\alpha$$, and in particular reproduces the well-known minor arc estimate $$f(\alpha)\ll N^{3/4+\epsilon}.$$
The proofs depend on an improvement in the usual major arc analysis, so that non-trivial bounds for $$\sum_{n\leq N}e(\beta n^ k-(b/q)n)$$ can be used. The explicit nature of the formulae produced give one some hope that further progress might be possible. Thus, for example, it becomes an easy matter to bring Kloosterman’s device into play. The reviewer [Proc. Lond. Math. Soc., III. Ser. 47, 225-257 (1983; Zbl 0494.10012)] has used these ideas with cubic forms in many variables, in which context they are superior to the traditional Weyl approach.
Reviewer: D.R.Heath-Brown

##### MSC:
 11L40 Estimates on character sums
##### Keywords:
Weyl sums; Waring’s problem; major arc