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 x1,...,xk ∈A exp(2πix1 . . . xkξ/p), where A ⊂ Fp.