Growth Estimates in Positive Characteristic via Collisions

Abstract

Let |$F$| be a field of characteristic |$p>2$| and |$A\subseteq F$| 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 show that |$|AA|^2|A+A|^3 \gg |A|^6$| and |$|A(A+A)|\gg |A|^{3/2}$|⁠. We also prove new two-variable extractor estimates, involving powers and reciprocals, namely |$|A+A^2|\gg |A|^{11/10}$|⁠, |$|A+A^3|\gg |A|^{29/28}$|⁠, and |$|A+1/A|\gg |A|^{31/30}$|⁠. More generally, we address questions of cardinalities |$|A+A|$| versus |$|f(A)+f(A)|$|⁠, for a polynomial |$f$|⁠, where we establish the inequalities |$ \max(|A+A|,\, |A^2+A^2|)\gg |A|^{8/7}$| and |$\max(|A-A|,\, |A^3+A^3|)\gg |A|^{17/16}$|⁠. Our results are obtained on the basis of a new plane geometry interpretation of the incidence theorem between points and planes in three dimensions, which we call collisions of images. They have strong implications in incidence geometry in two dimensions. We show that a Cartesian product point set |$P=A\times B$| in |$F^2$|⁠, of |$n$| elements, with |$|B|\leq |A|< p^{2/3}$| makes |$O(n^{3/4}m^{2/3} + m + n)$| incidences with any set of |$m$| lines. In particular, when |$|A|=|B|$|⁠, there are |$O(n^{9/4})$| collinear triples of points in |$P$|⁠, |$\Omega(n^{3/2})$| distinct lines between pairs of its points, in |$\Omega(n^{3/4})$| distinct directions. Further, |$P=A\times A$| determines |$\Omega(n^{9/16})$| distinct values of the polynomial, analogous to the Euclidean distance.

Locations

• International Mathematics Research Notices - View
• arXiv (Cornell University) - View - PDF
• Bristol Research (University of Bristol) - View - PDF