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Geometry of the gauss map and lattice points in convex domains

Part of: Discrete geometry

Published online by Cambridge University Press:  26 February 2010

L. Brandolini ,
L. Colzani ,
A. Iosevich ,
A. Podkorytov  and
G. Travaglini
L. Brandolini
Affiliation:
Dipartimento di Ingegneria, Gestionale e de l'lnformazione, Universita degli Studi di Bergamo, Viale Marconi 5, 24044 Dalmine (BG), Italy E-mail: brandolini@unibg.it
L. Colzani
Affiliation:
Dipartimento di Matematica e Applicationi, Universita di Milano-Bicocca, Via Bicocca degli Arcimboldi 8, 20126 Milano, Italy E-mail: leonardo@matapp.unimib. it
A. Iosevich
Affiliation:
Department of Mathematics, University of Missouri, Columbia, MO 65211, U.S.A.http://www.math.missouri.edu/-iosevich/ E-mail: iosevich@math.missouri.edu.
A. Podkorytov
Affiliation:
Department of Mathematics, St. Petersburg State University, Bibliotecnaya pi. 2, Peterhof, St. Petersburg, 198904, Russia E-mail: pan@pan.usr.pu.ru.
G. Travaglini
Affiliation:
Dipartimento di Matematica e Applicazioni, Universita di Milano-Bicocca, Via Bicocca degli Arcimboldi 8, 20126 Milano, Italyhttp://www.matapp.unimib.it/~travaglini/ E-mail: travaglini@matapp.unimib.it.
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Abstract

Let Ω be a convex planar domain, with no curvature or regularity assumption on the boundary. Let Nθ(R) = card{RΩθ∩ℤ2}, where Ωθ denotes the rotation of Ω by θ. It is proved that, up to a small logarithmic transgression, Nθ(R) = |Ω|R2 + O(R2/3), for almost every rotation. A refined result based on the fractal structure of the image of the boundary of Ω under the Gauss map is also obtained.

MSC classification

Secondary: 52C05: Lattices and convex bodies in $2$ dimensions

Information

Type
Research Article
Information
Mathematika , Volume 48 , Issue 1-2 , December 2001 , pp. 107 - 117
Copyright
Copyright © University College London 2001

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References

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