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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Geometric properties of points on modular hyperbolas


Authors: Kevin Ford, Mizan R. Khan and Igor E. Shparlinski
Journal: Proc. Amer. Math. Soc. 138 (2010), 4177-4185
MSC (2010): Primary 11A07; Secondary 11H06, 11N69
Published electronically: July 9, 2010
MathSciNet review: 2680044
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Abstract: Given an integer $ n\ge 2$, let $ \mathcal{H}_n$ be the set

$\displaystyle \mathcal{H}_n= \{(a,b) : ab \equiv 1 \pmod n, 1\le a,b \le n-1\} $

and let $ M(n)$ be the maximal difference of $ b-a$ for $ (a,b) \in \mathcal{H}_n$. We prove that for almost all $ n$, $ n-M(n)=O\left(n^{1/2+o(1)}\right).$ We also improve some previously known upper and lower bounds on the number of vertices of the convex closure of $ \mathcal{H}_n$.


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Additional Information

Kevin Ford
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, Illinois 61801
Email: ford@math.uiuc.edu

Mizan R. Khan
Affiliation: Department of Mathematics and Computer Science, Eastern Connecticut State University, Willimantic, Connecticut 06226
Email: khanm@easternct.edu

Igor E. Shparlinski
Affiliation: Department of Computing, Macquarie University, Sydney, NSW 2109, Australia
Email: igor@ics.mq.edu.au

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10561-0
PII: S 0002-9939(2010)10561-0
Received by editor(s): February 11, 2010
Published electronically: July 9, 2010
Additional Notes: The research of the first author was supported in part by NSF grants DMS-0555367 and DMS-0901339.
The research of the third author was supported by ARC grants DP0556431 and DP1092835.
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2010 American Mathematical Society