Asymptotics for the number of directions determined by $[n] \times [n]$ in $\mathbb{F}_p^2$

TL;DR

This paper derives an asymptotic formula for the number of directions determined by the Cartesian product of an arithmetic progression within a finite field, linking geometric combinatorics with solutions to a bilinear Diophantine equation.

Contribution

It introduces a novel asymptotic formula for counting directions in finite fields, reducing the problem to counting solutions of a specific bilinear Diophantine equation.

Findings

Asymptotic formula for directions in finite fields established

Reduction of geometric problem to Diophantine equation counting

Independent interest in solution count asymptotics

Abstract

Let $p$ be a prime and $n$ a positive integer such that $\sqrt{\frac p2} + 1 \leq n \leq \sqrt{p}$. For any arithmetic progression $A$ of length $n$ in $\mathbb{F}_p$, we establish an asymptotic formula for the number of directions determined by $A \times A \subset \mathbb{F}_p^2$. The key idea is to reduce the problem to counting the number of solutions to the bilinear Diophantine equation $ad+bc=p$ in variables $1\le a,b,c,d\le n$; our asymptotic formula for the number of solutions is of independent interest.