New Sum-product Type Estimates over Finite Fields

Let F be a field with positive odd characteristic p. We prove a variety of new sum-product type estimates over F . They are derived from the theorem that the number of incidences between m points and n planes in the projective three-space PG(3, F ), with m ≥ n = O(p), is O(m √ n+ km), where k denotes the maximum number of collinear planes. The main result is a significant improvement of the state-of-the-art sum-product inequality over fields with positive characteristic, namely that (1) |A±A|+ |A ·A| = Ω ( |A| 5 ) , for any A such that |A| < p 5 8 .