Growth Estimates in Positive Characteristic via Collisions

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 sum-product type inequalities. In particular, we prove that|AA|2|A + A|3 ≫ |A|6, |A(A + A)| ≫ |A|3/2. We also prove several two-variable extractor estimates: |A(A + 1)| ≫ |A|9/8,|A + A2| ≫ |A|11/10, |A + A3| ≫ |A|29/28, |A + 1/A| ≫ |A|31/30. 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|) ≫ |A|8/7, max(|A − A|, |A3 + A3|) ≫ |A|17/16.Szemer´edi-Trotter 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 pair-wise 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.