Asymptotics for the number of directions determined by \([n] × [n]\) in \(\mathbb{F}_p^2\) - Martin, Greg; White, Ethan Patrick; Yip, Chi Hoi (arXiv.org, 2021)
Let \(p\) be a prime and \(n\) a positive integer such that \(\sqrt{\frac p2} + 1 ≤ n ≤ \sqrt{p}\). For any arithmetic progression \(A\) of length \(n\) in \(\mathbb{F}_p\), we establish an asymptotic formula for the number of directions determined by \(A × A \subset \mathbb{F}_p^2\). 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.