On distinct cross-ratios in positive characteristic

Abstract

We prove that a finite subset $A$ of the projective line $F{P}^1$ over a field $F$ of positive characteristic $p$ determines $\Omega (|A{|}^2)$ distinct cross-ratios, as long as $|A|<\sqrt{p}.$

arXiv commentThis article had been earlier withdrawn by the author: the main proposition was false as stated. This update is to point out that the bound $\Omega (|A{|}^{3/2})$ for the number of distinct cross-ratios, and other incidence estimates have been proven in the preprint "Growth Estimates in Positive Characteristic via Collisions", with E. Aksoy, B. Murphy and I. Shkredov, arXiv:1512.06613