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Geometric properties of points on modular hyperbolas

Author(s): Kevin Ford; Mizan R. Khan; Igor E. Shparlinski
Journal: Proc. Amer. Math. Soc. 138 (2010), 4177-4185.
MSC (2010): Primary 11A07; Secondary 11H06, 11N69
Posted: July 9, 2010
MathSciNet review: 2680044
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Abstract | References | Similar articles | Additional information

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

be the maximal difference of

for $(a,b) \in \mathcal{H}_n$ . We prove that for almost all

, $n-M(n)=O\left(n^{1/2+o(1)}\right).$ 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: 10.1090/S0002-9939-2010-10561-0
PII: S 0002-9939(2010)10561-0
Received by editor(s): February 11, 2010
Posted: 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
Copyright of article: Copyright 2010, American Mathematical Society

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