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One and two dimensional exponential sums. (English) Zbl 0626.10034
Analytic number theory and diophantine problems, Proc. Conf., Stillwater/Okla. 1984, Prog. Math. 70, 205-222 (1987).
[For the entire collection see Zbl 0618.00005.]
In the theory of numbers one often encounters sums of the form \(\sum_{(n_ 1,...,n_ k)\in D}F(n_ 1,...,n_ k),\) where D is a bounded domain in \({\mathbb{R}}^ k\) and F is an exponential function \(F(n_ 1,...,n_ k)=e^{2\pi i f(n_ 1,...,n_ k)}.\) The object of the present paper is to give an exposition of van der Corput’s method for the estimation of such exponential sums, including the theory of exponent pairs (\(\ell,m)\in P\). In the one-dimensional case \(k=1\) the so-called A and B processes are discussed and a new algorithm is sketched. It yields a sequence of exponent pairs which provide approximations to the unknown infimum \(\inf_{(\ell,m)\in P}(\ell +m).\) The authors also give an outline of what is known and what is conjectured about the two- dimensional case \(k=2\).
Reviewer: J.Hinz

MSC:
11L03 Trigonometric and exponential sums, general
11L40 Estimates on character sums