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ESTIMATES FOR THE NUMBER OF SUMS AND PRODUCTS AND FOR EXPONENTIAL SUMS IN FIELDS OF PRIME ORDER

  • J. BOURGAIN (a1), A. A. GLIBICHUK (a2) and S. V. KONYAGIN (a3)

Abstract

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

\[\sum_{x_1,\dots,x_k\in A} \exp(2\pi ix_1\dotsc x_k\xi/p),\]

where $A\subset{{\mathbb F}_p}$.

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