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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