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)PG(3,F), with m≥n=O(p2)m≥n=O(p2), isO(mn+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 thatequation(1)|A±A|+|A⋅A|=Ω(|A|1+15), for any A   such that |A|