Growth Estimates in Positive Characteristic via Collisions

Abstract

Let $F$ be a field of characteristic $p>2$ and $A\subset F$ 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 $$|AA{|}^2|A+A{|}^3\gg |A{|}^6,|A(A+A)|\gg |A{|}^{3/2}.$$ We also prove several two-variable extractor estimates: $|A(A+1)|\gg |A{|}^{9/8},$ $$|A+A^2|\gg |A{|}^{11/10},|A+A^3|\gg |A{|}^{29/28},|A+1/A|\gg |A{|}^{31/30}.$$ Besides, we address questions of cardinalities $|A+A|$ vs $|f(A)+f(A)|$, for a polynomial $f$, where we establish the inequalities $$\max (|A+A|,|A^2+A^2|)\gg |A{|}^{8/7},\max (|A-A|,|A^3+A^3|)\gg |A{|}^{17/16}.$$ Szemerédi-Trotter type implications of the arithmetic estimates in question are that a Cartesian product point set $P=A\times B$ in $F^2$, of $n$ elements, with $|B|\le |A|<p^{2/3}$ makes $O(n^{3/4}{m}^{2/3}+m+n)$ incidences with any set of $m$ lines. In particular, when $|A|=|B|$, there are $\ll n^{9/4}$ collinear triples of points in $P$, $\gg n^{3/2}$ distinct lines between pairs of its points, in $\gg n^{3/4}$ distinct directions. Besides, $P=A\times A$ determines $\gg n^{9/16}$ 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.

24pp

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