M ar 2 00 7 A SLIGHT IMPROVEMENT TO GARAEV ’ S SUM PRODUCT ESTIMATE

Let A be a subset of F p , the field of p elements with p prime. We let A + A = {a + b : a ∈ A, b ∈ A}, and AA = {ab : a ∈ A, b ∈ A}. It is fun (and useful) to prove lower bounds on max(|A+A|, |AA|) (see e.g. [BKT],[BGK],[G]). Recently, Garaev [G] showed that when |A| < p 1 2 one has the estimate max(|A + A|, |AA|) |A| 15 14. By using Plunneke's inequality in a slightly more sophisticated way, we improve this exponent to 14 13. We believe that further improvements might be possible through aggressive use of Ruzsa covering. §1 Preliminaries Throughout this paper A will denote a fixed set in the field F p of p elements with p a prime. For B, any set, we will denote its cardinality by |B|. Whenever X and Y are quantities we will use X Y,