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Close Lattice Points on Circles

Published online by Cambridge University Press:  20 November 2018

Javier Cilleruelo  and
Andrew Granville
Javier Cilleruelo
Affiliation:
Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM) and Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain email: franciscojavier.cilleruelo@uam.es
Andrew Granville
Affiliation:
Départment de Mathématiques et Statistique, Université de Montréal, CP 6128 succ Centre-Ville, Montréal, QC H3C 3J7 email: andrew@dms.umontreal.ca
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Abstract

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We classify the sets of four lattice points that all lie on a short arc of a circle that has its center at the origin; specifically on arcs of length $t{{R}^{1/3}}$ on a circle of radius $R$, for any given $t\,>\,0$. In particular we prove that any arc of length ${{\left( 40\,+\frac{40}{3}\sqrt{10} \right)}^{1/3}}\,{{R}^{1/3}}$ on a circle of radius $R$, with $R\,>\,\sqrt{65}$, contains at most three lattice points, whereas we give an explicit infinite family of 4-tuples of lattice points, $\left( {{v}_{1,n}},\,{{v}_{2,n}},\,{{v}_{3,n}},\,{{v}_{4,n}} \right)$, each of which lies on an arc of length ${{\left( 40+\frac{40}{3}\sqrt{10} \right)}^{1/3}}R_{n}^{1/3}\,+\,o\left( 1 \right)$ on a circle of radius ${{R}_{n}}$.

Keywords

11N36
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 61 , Issue 6 , 01 December 2009 , pp. 1214 - 1238
Copyright
Copyright © Canadian Mathematical Society 2009

References

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