Abstract 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 P G ( 3 , F ) , with m ≥ n = O ( p 2 ) , is O ( m n + k m ) , 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 | 1 + 1 5 ) , for any A such that | A | < p 5 8 .
Bibliographical noteThis is a revised version: Theorem 1 was incorrect as stated. We give its correct statement; this does not seriously affect the main arguments throughout the paper. Also added is a seres of remarks, placing the result in the context of the current state of the art
- Sum-product estimates