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

Abstract

Let $p$ be a prime and $n$ a positive integer such that $\sqrt{\frac{p}{2}}+1\le n\le \sqrt{p}$. 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.

22 pages, 1 figure

References in corpus (3)

• On the directions determined by Cartesian products and the clique number of generalized Paley graphs
• On The Solvability of Bilinear Equations in Finite Fields
• Character sums estimates and an application to a problem of Balog

Cited by in corpus (2)

• Exact values and improved bounds on the clique number of cyclotomic graphs
• Counting $2\times 2$ matrices with fixed determinant and bounded coefficients