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Journal of the London Mathematical Society 2006 73(2):380-398; doi:10.1112/S0024610706022721
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© The London Mathematical Society

Estimates for the Number of Sums and Products and for Exponential Sums in Fields of Prime Order

J. Bourgain, A. A. Glibichuk and S. V. Konyagin

School of Mathematics, Institute for Advanced Study Olden Lane, Princeton, NJ 08540, USA bourgain{at}math.ias.edu
Department of Mechanics and Mathematics, Moscow State University Moscow, 119992, Russia aanatol{at}mail.ru
Department of Mechanics and Mathematics, Moscow State University Moscow, 119992, Russia konyagin{at}ok.ru

Received 28 July 2004. Revision received 23 August 2005.

Our first result is a ‘sum-product’ theorem for subsets A of the finite field Fp, p prime, providing a lower bound on max (|A + A|, |A · A|). The second and main result provides new bounds on exponential sums


Formula

where AsubFp.


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