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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| Vydané v: | arXiv.org |
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| Hlavní autori: | , , |
| Médium: | Paper |
| Jazyk: | English |
| Vydavateľské údaje: |
Ithaca
Cornell University Library, arXiv.org
03.07.2021
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| Predmet: | |
| ISSN: | 2331-8422 |
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| Shrnutí: | 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. |
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| Bibliografia: | SourceType-Working Papers-1 ObjectType-Working Paper/Pre-Print-1 content type line 50 |
| ISSN: | 2331-8422 |
| DOI: | 10.48550/arxiv.2107.01311 |