On distinct cross-ratios in positive characteristic

TL;DR

This paper proves that a finite set in the projective line over a field with positive characteristic determines a quadratic number of distinct cross-ratios, provided the set size is less than the square root of the characteristic.

Contribution

It establishes a lower bound on the number of distinct cross-ratios for sets in positive characteristic fields, extending understanding of geometric configurations in such fields.

Findings

Finite sets in positive characteristic fields determine many distinct cross-ratios.

The lower bound on cross-ratios is proportional to the square of the set size.

The result holds when the set size is less than the square root of the characteristic.

Abstract

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