Estimates for the number of sums and products and for exponential sums over subgroups in fields of prime order

Abstract

Our first result is a ‘sum–product’ theorem for subsets A of the finite field F p , p prime, providing a lower bound on max(| A+ A|,| A· A|). As corollary, the second and main result provides new bounds on exponential sums associated to subgroups of the multiplicative group F ∗ p . To cite this article: J. Bourgain, S.V. Konyagin, C. R. Acad. Sci. Paris, Ser. I 337 (2003).