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New estimates for Weyl sums. (English) Zbl 0855.11043

In this article the author sharpens further an upper bound for the classical Weyl sum obtained earlier [the author, Mathematika 39, 379-399 (1992; Zbl 0769.11036)]. Instead of stating the precise form of the main result, it is interesting to note the following consequence (Theorem 1):
Let \(\alpha_k\) be a real number such that there exist an integer \(a\) and a natural number \(q\) with \((a, q) =1\), \(|\alpha_k- aq^{-1} |\leq q^{-2}\) and \(P\leq q\leq P^{k-1}\). Then for large \(P\), the Weyl sum satisfies \[ \sum_{1\leq x\leq P} e(\alpha_k x^k+ \cdots+ \alpha_1 x)\;\ll_{\varepsilon, k} P^{1- \rho (k)+ \varepsilon}, \] where, when \(k\) is large, \(\rho (k)^{-1}= {3\over 2} k^2 (\log k+ O (\log \log k))\). The previous result has a factor ‘2’ in place of ‘3/2’ in the last expression for \(\rho (k)^{-1}\).
The main argument follows that in the author’s paper [J. Lond. Math. Soc., II. Ser. 51, 1-13 (1995; Zbl 0833.11041)]. The Weyl sum in question is first transformed, by using Weyl shifts, to another exponential sum based on well-factorable numbers. This is then further transformed and eventually bounded by applying the large sieve inequality. This yields a good bound for the Weyl sum when \(q\) is small. When \(q\) is large, a complimentary bound for Weyl sums is used to give the desired result.

MSC:

11L15 Weyl sums
11N36 Applications of sieve methods
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