Geometric properties of points on modular hyperbolas

Kevin Ford; Mizan R. Khan; Igor E. Shparlinski

doi:10.1090/S0002-9939-2010-10561-0

Geometric properties of points on modular hyperbolas

Kevin Ford^*, Mizan R. Khan, Igor E. Shparlinski

^*Corresponding author for this work

School of Computing

Research output: Contribution to journal › Article › peer-review

6 Citations (Scopus)

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Abstract

Given an integer n ≥ 2, let H_n be the set H_n = {(a, b) : ab ≡1 (modn), 1 ≤ a, b ≤ n - 1} and let M(n) be the maximal difference of b-a for (a, b) ∈ H_n. We prove that for almost all n, n-M(n) = O (n^1/2+o(1)). We also improve some previously known upper and lower bounds on the number of vertices of the convex closure of H_n.

Original language	English
Pages (from-to)	4177-4185
Number of pages	9
Journal	Proceedings of the American Mathematical Society
Volume	138
Issue number	12
DOIs	https://doi.org/10.1090/S0002-9939-2010-10561-0
Publication status	Published - Dec 2010

Bibliographical note

Copyright [2010] American Mathematical Society. First published in Proceedings of the American Mathematical Society, 138:12, pp. 4177-4185, published by the American Mathematical Society. The original article can be found at http://dx.doi.org/10.1090/S0002-9939-2010-10561-0.

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title = "Geometric properties of points on modular hyperbolas",

abstract = "Given an integer n ≥ 2, let Hn be the set Hn = \{(a, b) : ab ≡1 (modn), 1 ≤ a, b ≤ n - 1\} and let M(n) be the maximal difference of b-a for (a, b) ∈ Hn. We prove that for almost all n, n-M(n) = O (n1/2+o(1)). We also improve some previously known upper and lower bounds on the number of vertices of the convex closure of Hn.",

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note = "Copyright [2010] American Mathematical Society. First published in Proceedings of the American Mathematical Society, 138:12, pp. 4177-4185, published by the American Mathematical Society. The original article can be found at http://dx.doi.org/10.1090/S0002-9939-2010-10561-0.",

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AU - Khan, Mizan R.

AU - Shparlinski, Igor E.

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N2 - Given an integer n ≥ 2, let Hn be the set Hn = {(a, b) : ab ≡1 (modn), 1 ≤ a, b ≤ n - 1} and let M(n) be the maximal difference of b-a for (a, b) ∈ Hn. We prove that for almost all n, n-M(n) = O (n1/2+o(1)). We also improve some previously known upper and lower bounds on the number of vertices of the convex closure of Hn.

AB - Given an integer n ≥ 2, let Hn be the set Hn = {(a, b) : ab ≡1 (modn), 1 ≤ a, b ≤ n - 1} and let M(n) be the maximal difference of b-a for (a, b) ∈ Hn. We prove that for almost all n, n-M(n) = O (n1/2+o(1)). We also improve some previously known upper and lower bounds on the number of vertices of the convex closure of Hn.

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JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 12

ER -

Geometric properties of points on modular hyperbolas

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