Abstract
Abstract Let F be a field with positive odd characteristic p. We prove a variety of new sumproduct type estimates over F. They are derived from the theorem that the number of incidences between m points and n planes in the projective threespace 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 stateoftheart sumproduct 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 .
Original language  English 

Pages (fromto)  589605 
Number of pages  17 
Journal  Advances in Mathematics 
Volume  293 
Early online date  1 Mar 2016 
DOIs  
Publication status  Published  30 Apr 2016 
Bibliographical note
This 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 artKeywords
 Sumproduct estimates
Fingerprint
Dive into the research topics of 'New sumproduct type estimates over finite fields'. Together they form a unique fingerprint.Profiles

Dr Misha Rudnev
 School of Mathematics  Associate Professor in Mathematics
 Number theory and combinatorics
 Pure Mathematics
Person: Academic , Member