TY - JOUR

T1 - Geometric properties of points on modular hyperbolas

AU - Ford, Kevin

AU - Khan, Mizan R.

AU - Shparlinski, Igor E.

N1 - 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.

PY - 2010/12

Y1 - 2010/12

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.

UR - http://www.scopus.com/inward/record.url?scp=78649857177&partnerID=8YFLogxK

U2 - 10.1090/S0002-9939-2010-10561-0

DO - 10.1090/S0002-9939-2010-10561-0

M3 - Article

AN - SCOPUS:78649857177

SN - 0002-9939

VL - 138

SP - 4177

EP - 4185

JO - Proceedings of the American Mathematical Society

JF - Proceedings of the American Mathematical Society

IS - 12

ER -