English

On distinct cross-ratios in positive characteristic

Combinatorics 2016-01-11 v6

Abstract

We prove that a finite subset AA of the projective line FP1FP^1 over a field FF of positive characteristic pp determines Ω(A2)\Omega(|A|^2) distinct cross-ratios, as long as A<p.|A|<\sqrt{p}.

Cite

@article{arxiv.1508.05142,
  title  = {On distinct cross-ratios in positive characteristic},
  author = {Misha Rudnev},
  journal= {arXiv preprint arXiv:1508.05142},
  year   = {2016}
}

Comments

This article had been earlier withdrawn by the author: the main proposition was false as stated.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

R2 v1 2026-06-22T10:38:28.624Z