×

Diophantine methods for exponential sums, and exponential sums for Diophantine problems. (English) Zbl 1022.11051

Li, Ta Tsien (ed.) et al., Proceedings of the international congress of mathematicians, ICM 2002, Beijing, China, August 20-28, 2002. Vol. II: Invited lectures. Beijing: Higher Education Press. 207-217 (2002).
Summary: Recent developments in the theory and application of the Hardy-Littlewood method are discussed, concentrating on aspects associated with diagonal Diophantine problems. Recent efficient differencing methods for estimating mean values of exponential sums are described first, concentrating on developments involving smooth Weyl sums. Next, arithmetic variants of classical inequalities of Bessel and Cauchy-Schwarz are discussed. Finally, some emerging connections between the circle method and arithmetic geometry are mentioned.
For the entire collection see [Zbl 0993.00022].

MSC:

11P55 Applications of the Hardy-Littlewood method
11-02 Research exposition (monographs, survey articles) pertaining to number theory
11L07 Estimates on exponential sums
11P05 Waring’s problem and variants
11D72 Diophantine equations in many variables
14G05 Rational points
PDFBibTeX XMLCite
Full Text: arXiv