Geometric properties of points on modular hyperbolas
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- by Kevin Ford, Mizan R. Khan and Igor E. Shparlinski PDF
- Proc. Amer. Math. Soc. 138 (2010), 4177-4185 Request permission
Abstract:
Given an integer $n\ge 2$, let $\mathcal {H}_n$ be the set \[ \mathcal {H}_n= \{(a,b) \ : \ ab \equiv 1 \pmod n,\ 1\le a,b \le n-1\} \] 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 $\mathcal {H}_n$.References
Additional Information
- Kevin Ford
- Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 West Green Street, Urbana, Illinois 61801
- MR Author ID: 325647
- ORCID: 0000-0001-9650-725X
- 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
- MR Author ID: 192194
- Email: igor@ics.mq.edu.au
- 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
- © Copyright 2010 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 138 (2010), 4177-4185
- MSC (2010): Primary 11A07; Secondary 11H06, 11N69
- DOI: https://doi.org/10.1090/S0002-9939-2010-10561-0
- MathSciNet review: 2680044