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Mordell’s exponential sum estimate revisited

Author: J. Bourgain
Journal: J. Amer. Math. Soc. 18 (2005), 477-499
MSC (2000): Primary 11L07; Secondary 11T23
Published electronically: January 18, 2005
MathSciNet review: 2137982
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Abstract: The aim of this paper is to extend recent work of S. Konyagin and the author on Gauss sum estimates for large degree to the case of ‘sparse’ polynomials. In this context we do obtain a nearly optimal result, improving on the works of Mordell and of Cochrane and Pinner. The result is optimal in terms of providing some power gain under conditions on the exponents in the polynomial that are best possible if we allow arbitrary coefficients. As in earlier work referred to above, our main combinatorial tool is a sum-product theorem. Here we need a version for product spaces $\mathbb {F}_{p}\times \mathbb {F}_{p}$ for which the formulation is obviously not as simple as in the $\mathbb {F}_{p}$-case. Again, the method applies more generally to provide nontrivial bounds on (possibly incomplete) exponential sums involving exponential functions. At the end of the paper, some applications of these are given to issues of uniform distribution for power generators in cryptography.

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Additional Information

J. Bourgain
Affiliation: School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540
MR Author ID: 40280

Received by editor(s): July 16, 2004
Published electronically: January 18, 2005
Article copyright: © Copyright 2005 American Mathematical Society