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

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...

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Bibliographic Details
Published in:	arXiv.org
Main Authors:	Martin, Greg, White, Ethan Patrick, Yip, Chi Hoi
Format:	Paper
Language:	English
Published:	Ithaca Cornell University Library, arXiv.org 03.07.2021
Subjects:	Asymptotic properties Diophantine equation Mathematical analysis
ISSN:	2331-8422
Online Access:	Get full text
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Summary:	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.
Bibliography:	SourceType-Working Papers-1 ObjectType-Working Paper/Pre-Print-1 content type line 50
ISSN:	2331-8422
DOI:	10.48550/arxiv.2107.01311