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

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 ∑ x 1 , … , x k ∈ A exp ( 2 π i x 1 … x k ξ / p ) where A⊂Fp.