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-spacewhere 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 thatfor any A such that |A| < p 5 8 .for the sumset A + A of A ⊆ F , and similarly for the product set AA, alias A · A. Sometimes we write nA for multiple sumsets, e.g. A+A+A = 3A, as well as A −1 = {a −1 : a ∈ A\{0}}.

We prove a range of new sum-product type growth estimates over a general field F, in particular the special case F = F p . They are unified by the theme of "breaking the 3/2 threshold", epitomising the previous state of the art.This concerns two pivotal for the sum-product theory questions, which are lower bounds for the number of distinct cross-ratios determined by a finite subset of F, as well as the number of values of the symplectic form determined by a finite subset of F 2 .We establish the estimate |R[A]| |A| 8/5 for cardinality of the set R[A] of distinct cross-ratios, defined by triples of elements of a (sufficiently small if F has positive characteristic, similarly for the rest of the estimates) set A ⊂ F, pinned at infinity. The cross-ratio bound enables us to break the threshold in the second question: for a non-collinear point set P ⊂ F 2 , the number of distinct values of the symplectic form ω on pairs of distinct points u, u ′ of P is |ω(P )| |P | 2/3+c , with an explicit c. Symmetries of the cross-ratio underlie its local growth properties under both addition and multiplication, yielding an onset of growth of products of difference sets, which is another main result herein.Our proofs strongly use specially suited applications of new incidence bounds over F, which apply to higher moments of representation functions. The technical thrust of the paper is using additive combinatorics to relate and adapt these higher moment bounds to growth estimates. A particular instance of this is breaking the threshold in the few sums, many products question over any F, by showing that if A is sufficiently small and has additive doubling constant M , then * |AA| M −2 |A| 14/9 . This result has a second moment version, which allows for new upper bounds for the number of collinear point triples in the set A× A ⊂ F 2 , the quantity often arising in applications of geometric incidence estimates. (Expansion for large sets) IfF = F p , then |R[A]| ≫ min p, |A| 5/2 p 1/2 . In particular, if |A| ≥ p 3/5 , then |R[A]| ≫ p. 3. (Uniform expansion) If F = F p , then |R[A]| ≫ min{p, |A| 3 2 + 1 22 log − 4 9 |A|}.

Abstract. In this paper we further study the relationship between convexity and additive growth, building on the work of Schoen and Shkredov ([5]) to get some improvements to earlier results of Elekes, Nathanson and Ruzsa ([1]). In particular, we show that for any finite set A ⊂ R and any strictly convex or concave function f ,(log |A|) 2/19 and(log |A|) 2/11 . For the latter of these inequalities, we go on to consider the consequences for a sum-product type problem.

This paper considers various formulations of the sum-product problem. It is shown that, for a finite set A ⊂ R,giving a partial answer to a conjecture of Balog. In a similar spirit, it is established thata bound which is optimal up to constant and logarithmic factors. We also prove several new results concerning sum-product estimates and expanders, for example, showing thatholds for a typical element of A.

Given two points p, q in the real plane, the signed area of the rectangle with the diagonal [pq] equals the square of the Minkowski distance between the points p, q. We prove that N > 1 points in the Minkowski plane R 1,1 generate Ω( N log N ) distinct distances, or all the distances are zero. The proof follows the lines of the Elekes/Sharir/Guth/Katz approach to the Erdős distance problem, analysing the 3D incidence problem, arising by considering the action of the Minkowski isometry group ISO * (1, 1).The signature of the metric creates an obstacle to applying the Guth/Katz incidence theorem to the 3D problem at hand, since one may encounter a high count of congruent line intervals, lying on null lines, or "light cones", all these intervals having zero Minkowski length. In terms of the Guth/Katz theorem, its condition of the non-existence of "rich planes" generally gets violated. It turns out, however, that one can efficiently identify and discount incidences, corresponding to null intervals and devise a counting stratagem, where the rich planes condition happens to be just ample enough for the stratagem to succeed.As a corollary we establish the following near-optimal sum-product type estimate for finite sets A, B ⊂ R, with more than one element:|(A ± B) · (A ± B)| ≫ |A||B| log |A| + log |B| .