On Product Sets in Fields of Prime Order and an Application of Burgess’ Inequality

This is the origin of paper ‘On a Question of Davenport and Lewis on Character Sums and Primitive Roots in Finite Fields’. There is still a little to be typed. Abstract Let A ⊂ Fp with |A| > p and |A + A| < C0|A|. We give explicit constants k = k(C0, ε) and κ = κ(C0, ε) such that |Ak| > κp. The tools we use are Garaev’s sum-product estimate, Freiman’s Theorem and a variant of Burgess’ method. As a by-product, we also obtain similar result for proper generalized progression in Fp.