On the number of directions formed by Cartesian products in $\mathbb{F}_{p^2}^2$

TL;DR

This paper establishes a lower bound on the number of directions determined by Cartesian products in the affine plane over _{p^2}, extending understanding of geometric configurations over finite fields.

Contribution

It introduces a new lower bound for directions in Cartesian products over _{p^2} and combines structural and algebraic methods to analyze these configurations.

Findings

Lower bound on directions for sets of size between p^{2/3} and p

Sets not contained in affine _p exhibit diverse directions

Explicit algebraic multiplicity parameters are derived for certain configurations

Abstract

We prove a lower bound on the number of directions determined by Cartesian products $A\times A$ in the affine plane over the finite field $\mathbb F_{p^2}$. Our lower bound holds for sets of size $p^{2/3}<|A|<p$, which are not contained in any affine copy of $\mathbb F_p$. The proof combines a structural result of Li and Roche-Newton on the set of directions formed by Cartesian products with a lower bound of Fancsali, Sziklai and Tak\'{a}ts. A key step shows that, unless the set of directions exhibits closure properties forcing subfield structure, one obtains a direction for which an algebraic multiplicity parameter in the latter theorem can be made explicit.