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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
DOI: https://doi.org/10.1090/S0002-9939-2010-10561-0
Published electronically: 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 $ 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 [Enhancements On Off] (What's this?)

  • 1. W. R. Alford, A. Granville and C. Pomerance, There are infinitely many Carmichael numbers, Annals of Math. (2) 139 (1994), 703-722. MR 1283874 (95k:11114)
  • 2. G. Andrews, A lower bound for the volume of strictly convex bodies with many boundary lattice points, Trans. Amer. Math. Soc. 106 (1963), 270-279. MR 0143105 (26:670)
  • 3. K. Ford, The distribution of integers with a divisor in a given interval, Annals. of Math. (2) 168 (2008), 367-433. MR 2434882 (2009m:11152)
  • 4. K. Ford, M. R. Khan, I. E. Shparlinski and C. L. Yankov, On the maximal difference between an element and its inverse in residue rings, Proc. Amer. Math. Soc. 133 (2005), 3463-3468. MR 2163580 (2006c:11108)
  • 5. É. Fouvry, Sur le problème des diviseurs de Titchmarsh, J. Reine Angew. Math. 357 (1985), 51-76. MR 783533 (87b:11090)
  • 6. G. Harman, On the number of Carmichael numbers up to $ x$, Bull. London Math. Soc. 37 (2005), 641-650. MR 2164825 (2006d:11106)
  • 7. D. R. Heath-Brown, Zero-free regions for Dirichlet $ L$-functions, and the least prime in an arithmetic progression, Proc. London Math. Soc. 64 (1992) 265-338. MR 1143227 (93a:11075)
  • 8. M. R. Khan and I. E. Shparlinski, On the maximal difference between an element and its inverse modulo $ n$, Periodica Math. Hungarica 47 (2003), 111-117. MR 2024977 (2004k:11006)
  • 9. M. R. Khan, I. E. Shparlinski and C. L. Yankov, On the convex closure of the graph of modular inversions, Experimental Math. 17 (2008), 91-104. MR 2410119 (2009e:11003)
  • 10. I. E. Shparlinski, On the distribution of points on multidimensional modular hyperbolas, Proc. Japan Acad. Sci., Ser. A 83 (2007), 5-9. MR 2303621 (2008g:11161)
  • 11. I. E. Shparlinski, Distribution of modular inverses and multiples of small integers and the Sato-Tate conjecture on average, Michigan Math. J. 56 (2008), 99-111. MR 2433659 (2009e:11154)
  • 12. T. Tao, Structure and Randomness, Amer. Math. Soc., Providence, RI, 2008. MR 2459552

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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: https://doi.org/10.1090/S0002-9939-2010-10561-0
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

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