Abstract
Let F be a field of characteristic p > 2 and A ⊂ F have sufficiently small cardinality in terms of p. We improve the state of the art of a variety of sumproduct type inequalities. In particular, we prove thatAA2A + A3 ≫ A6, A(A + A) ≫ A3/2. We also prove several twovariable extractor estimates: A(A + 1) ≫ A9/8,A + A2 ≫ A11/10, A + A3 ≫ A29/28, A + 1/A ≫ A31/30. Besides, we address questions of cardinalities A+A vs f(A)+f(A), for a polynomial f, where we establish the inequalities max(A + A, A2 + A2) ≫ A8/7, max(A − A, A3 + A3) ≫ A17/16.Szemer´ediTrotter type implications of the arithmetic estimates in question are that a Cartesian product point set P = A×B in F2, of n elements, with B ≤ A < p2/3 makes O(n3/4m2/3+m+n) incidences with any set of m lines. In particular, when A = B, there are ≪ n9/4 collinear triples of points in P, ≫ n3/2 distinct lines between pairs of its points, in ≫ n3/4 distinct directions. Besides, P = A × A determines ≫ n9/16 distinct pairwise distances. These estimates are obtained on the basis of a new plane geometry interpretation of the incidence theorem between points and planes in three dimensions, which we call collisions of images.
Original language  English 

Article number  rnw206 
Pages (fromto)  71487189 
Number of pages  42 
Journal  International Mathematics Research Notices 
Volume  2017 
Issue number  23 
Early online date  28 Oct 2016 
DOIs  
Publication status  Published  Dec 2017 
Bibliographical note
24ppKeywords
 math.CO
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Dr Misha Rudnev
 School of Mathematics  Associate Professor in Mathematics
 Number theory and combinatorics
 Pure Mathematics
Person: Academic , Member